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+{"text":"\\section{Introduction}\n\nThe Dicke model describes the dynamics of $N$ identical two-level\natoms interacting with a quantized three-dimensional\nelectromagnetic (EM) field \\cite{Dicke}. Under certain conditions\nthe model predicts that the atoms interact with the quantized EM field collectively, giving rise to the widely studied phenomena of\nsuperradiance and subradiance \\cite{HarochePR,tanas02}. In free\nspace ideal superradiance and subradiance take place in the so\ncalled small sample limit, i.e., when the atoms are so close to\neach other that one can ignore any effect resulting from  their\ndifferent spatial positions. In this case the atoms are\nindistinguishable with respect to their emission and absorption\nproperties; hence, the presence of equivalent paths through which\nthe emission process may occur gives rise to fully constructive\n(superradiance) or destructive (subradiance) interference.\n\nIdeal superradiance or subradiance in free space is very difficult to\nobserve in the experiments since it requires that the atoms are\nplaced in a regular pattern within a sample smaller than the\nwavelength of the EM field they interact with (small sample case).\nThe requirement of a regular pattern is due to the presence of the\ndipole-dipole forces that would otherwise break the symmetry under\npermutation of any two atoms necessary to observe\nsuperradiant-subradiant behavior. Such a regularity can be\nachieved, e.g., with trapped-ion crystals \\cite{IonsRev} or atoms\nin optical lattices \\cite{Bloch08}. In these systems, however, the\nseparation between the particles is typically larger or on the same\norder of magnitude than the resonant wavelength (large sample\ncase). In the large sample case, cooperative effects still occur\nbut the subradiant state is not completely decoupled from the\ndynamics. Indeed, partial subradiance and superradiance have been \nobserved with trapped ions \\cite{DeVoe}.\n\nA way for relaxing the requirement for configuration regularity is\nto place the small sample in a cavity resonator. In this case,\nindeed, due to the Purcell effect, the cooperative atomic behavior\ncan be observed at much lower atomic density than in free space,\nmaking the van der Waals dephasing caused by the irregular atomic\nconfiguration negligible \\cite{HarochePR}. Experiments observing\nsuperradiance in the small sample case in a cavity have been\nperformed with Rydberg atoms \\cite{HarocheSuperR}, giving results\nin a very good agreement with the predictions of the single-mode\nsuperradiance theory. In this experiment, all of the atoms are\nequivalently coupled to the quantized mode of the EM field\n(homogeneous case).\n\nRecent advances in ion-cavity QED experiments make it possible to\nconfine arrays of ions inside an optical cavity in a regime in\nwhich the width of their wave packet in position space is smaller\nthan the wavelength of the cavity mode they interact with\n(Lamb--Dicke regime) \\cite{Walther,Piety}. Moreover, it is\npossible to accurately manipulate the position of the single ions\nwith respect to the intensity profile of the standing cavity mode, thereby allowing us to change the strength of the\ncoupling between each ion and the quantized EM field.\n\nIt has been  demonstrated theoretically that, when the atoms are\ncoupled with different strengths to the EM field, ideal\nsuperradiance or subradiance can still occur, depending on the\nparticular spatial distribution of the atoms\n\\cite{Benivegna,BenivegnaPL,BuzekZei}. However, no experiments\nhave up to now confirmed these predictions by the inhomogeneous Dicke model. Very recently, an important step in this direction has been achieved at the University of Aarhus, where a collective strong coupling between an ion crystal and a cavity mode was observed \\cite{Herskind2009}. In this paper, we\ninvestigate in detail how the inhomogeneous single-mode Dicke\nmodel (or Tavis-Cummings model \\cite{TCmodel}) can be realized in\nthe ion-cavity QED context and the conditions under which\nsubradiance and superradiance can be observed.\n\nBesides the importance in the study of fundamentals of quantum\ntheory, the realization of the Dicke model and the generation of\nthe subradiant state play a crucial role in quantum information\ntechnology and quantum communication. Indeed, arrays of ions are\nideal candidates for quantum registers and their controlled\ninteraction with photons allows us to realize atom-light quantum\ninterfaces \\cite{Kimble} and to distribute entanglement to\ndifferent nodes of quantum networks. The importance of the\nsubradiant states in this context stems from the fact that they\nare robust entangled atomic states since they are completely\ndecoupled from the EM field.\n\nThe aim of this work is to discuss a realistic setup that is able\nto show the collective behavior of trapped ions in a cavity. In\nparticular, since in the experiments performed so far the ions are\ncoupled to the EM mode via a Raman scheme in a $\\Lambda$-configuration, we will include the entire level structure, which\nis important in order to understand the decohering role of the spontaneous\nemission from the upper and essentially unpopulated level. We\nwill also include cavity losses in order to study in detail the\ndeviation from the ideal cooperative Dicke model and to identify\nthe parameter regions in which such deviations are as small as\npossible.\n\nIn fact, during the last two decades, several theoretical papers\nhave discussed issues such as entanglement generation, preparation\nof nonclassical states, or realization of quantum gates in the\nion-cavity QED context assuming that the conditions to realize an\nideal Tavis-Cummings model were met\n\\cite{Pellizzari,vanEnk,Plenio,Zheng,Pachos,Lougovski,Chimczak,Li,Li07,Chimczak08,Bina}.\nThus, either the spontaneous emission or the cavity losses (or both\nprocesses) are usually neglected \\cite{vanEnk,Zheng,Li,Li07}.\nConcerning spontaneous emission, for example, the assumption is\nmade that the emission rate is much smaller than the cavity\ncoupling constant\n\\cite{Pellizzari,Pachos,Lougovski,Chimczak,Chimczak08}. However,\nthis condition is not met in the ion-cavity QED experiments\n\\cite{Walther,Piety}. Furthermore, as we will demonstrate in this\npaper, if one deals with simplified atomic level structures\n\\cite{Plenio,Zheng,Bina,Natali2007}, it is not possible to single out those\nregions in parameter space for which the systems of trapped ions\nbehave collectively.\n\nIn this paper, we will take both the cavity losses and the spontaneous\nemissions into account and employ $\\Lambda$-type schemes to describe\nthe ions and to identify the experimental conditions under which\nthe coherent dynamics predicted by the single-mode Dicke model is\ndominant with respect to losses and decoherence. This will also\nallow us to present realistic protocols for entanglement\ngeneration and to discuss ways to optimize the generated\nentanglement using specific features of the trapped-ion system,\nsuch as the ability to manipulate in a controlled way the relative\ncoupling between the ions and the cavity field.\n\nThe structure of the paper is the following. In\nSec.~\\ref{sec:DickeModel} we review the properties of the\ninhomogeneous single-mode Dicke model. In\nSec.~\\ref{sec:effectiveModel} we present the Hamiltonian for two\nions in a cavity and we make the connection to the Dicke model by deriving an effective model describing the dynamics under realistic experimental conditions. Section~\\ref{sec:resonantRegime} is devoted to the description of\nthe experimental proposal to observe subradiance and verify the\ninhomogeneous Dicke model. Furthermore, in Sec.~\\ref{sec:dispersiveRegime}\nwe explore another way to optimize the entanglement generation by\nusing off-resonant transitions. Finally, a summary of the results\nand the conclusions are given in Sec.~\\ref{sec:summary}.\n\n\n\n\n\n\\section{\\label{sec:DickeModel}\nInhomogeneous single-mode Dicke model}\n\n\n\n\\subsection{Ideal cavity}\n\nThe single-mode Dicke model, or Tavis-Cummings model, is the\nsimplest quantum-mechanical model describing collective effects\nsuch as superradiance and subradiance in cavity. It describes the\nquasi-resonant interaction between $N$ identical two-level atoms\nand a single quantized cavity mode. The Tavis-Cummings\nHamiltonian is\n\\begin{align}\nH_\\textrm{D} =&\\,\\, \\omega_C \\left( a^{\\dag} a + \\frac{1}{2} \\right) + \\sum_{j=1}^N  \\omega_A \\sigma_+^{(j)} \\sigma_-^{(j)} \\nonumber \\\\\n& +\\sum_{j=1}^N \\left( \\alpha^{(j)} a^{\\dag} \\sigma_-^{(j)} + \\alpha^{(j)*} a \\sigma_+^{(j)}\\right),  \\label{eq:HDicke2}\n\\end{align}\nwhere $\\omega_C$ and $\\omega_A$ are the frequencies of the cavity\nmode and the atomic transition, respectively; $a$ and\n$a^{\\dag}$ are the annihilation and the creation operators for the\ncavity mode; and $ \\sigma_-^{(j)} =\\vert 0^{(j)} \\rangle \\langle\n1^{(j)} \\vert$ and $ \\sigma_+^{(j)}=(\\sigma_-^{(j)})^{\\dag}$ are\nthe lowering and the raising operators for the $j$th atom, $\\vert\n0^{(j)} \\rangle$ and $\\vert 1^{(j)} \\rangle$ being its ground and\nexcited states, respectively. Finally, $\\alpha^{(j)}$ is the\ncoupling strength of the $j$th atom with the cavity field.\nInhomogeneity of the coupling strengths originates from\ndifferent relative positions of the atoms with respect to the\nintensity profile of the standing EM mode supported by the cavity\nresonator.\n\nThis model assumes that the cavity is ideal, as photon escape is\nnot taken into account, and that atomic spontaneous emission from the\nexcited to the ground state is negligible. The model also neglects\nthe atomic motion as well as recoil effects due to the absorption\nand subsequent re-emission of a photon by the atoms. Moreover, the\ndipolar coupling of the atoms and the EM field is expressed within\na rotating wave approximation (RWA), thereby suppressing the\nnon-energy-conserving terms. Finally, it implicitly assumes that\nthe coupling between the atoms and the cavity mode does not\nchange, i.e., that the atoms are kept at fixed positions. While\nthe RWA has been proven to work extremely well in optical\nexperiments, all other assumptions need further consideration. In\nthe following sections we will examine them in detail for the\nion-cavity QED setup.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig01}\n\\end{center}\n\\caption{\\label{fig:dickeModel} (Color online) Two binary quantum\nobjects interacting through a quantized electromagnetic mode\nsupported by a cavity resonator. The dynamics of such an ideal system\nis described by the Dicke model. }\n\\end{figure}\n\nUsing a suitable canonical transformation it has been shown that,\nwhen only one excitation is present in the total system, the $N$\natoms interacting with the quantized field mode according to\nEq.~\\eqref{eq:HDicke2} cooperate in such a way that only one\ncollective atomic mode (superradiant state) is coupled to the\nfield  \\cite{BenivegnaPL}. Consequently, the energy exchange\nbetween the atoms and the field can be completely suppressed if\nthe only field-coupled collective mode is unexcited.\n\nFor simplicity, we will from now on focus on the $N=2$ case\nsketched in Fig.~\\ref{fig:dickeModel}, and  we will denote the\nenergy eigenstates for the free ions as $|a^{(1)} b^{(2)}\\rangle\n\\equiv |a^{(1)}\\rangle\\otimes |b^{(2)}\\rangle$ (with $a,b=0,1$) and the corresponding Fock states of the cavity mode as $|n^{(C)}\\rangle$, where $n= 0,1, \\ldots$ .\nThe time evolution generated by $H_D$ is easily obtained\nexplicitly. For a cavity initially prepared in the vacuum state,\nand in the presence of only one atomic excitation, the time\nevolution of the amplitudes $c_{10} (t)$ and $c_{01} (t)$ to find the ions\nin the states $\\vert 1^{(1)} 0^{(2)} \\rangle$ and $\\vert 0^{(1)}\n1^{(2)} \\rangle$, respectively, is given by\n\\begin{align}\n c_{10}(t) =& \\left[\\, |r^{(2)}|^2 + |r^{(1)}|^2 \\, {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& - r^{(1)*} r^{(2)} \\left[\\, 1- {\\cal E}(t)\\, \\right]c_{01}(0),\n\\label{eq:c1sing} \\\\\nc_{01}(t) =& - r^{(1)} r^{(2)*} \\left[\\, 1- {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& + \\left[\\, |r^{(1)}|^2 + |r^{(2)}|^2\\, {\\cal E}(t)\\,\n\\right]c_{01}(0).\n\\label{eq:c2sing}\n\\end{align}\nIn the equations above the relative coupling strengths are defined as $r^{(j)}=\\alpha^{(j)}\/|\\alpha_T |$, where $|\\alpha_T| =\n\\sqrt{ |\\alpha^{(1)}|^2 + |\\alpha^{(2)}|^2 }$ is  the total\ncoupling strength, and\n\\begin{equation}\n{\\cal E}(t) = e^{ i \\delta t \/2} \\Big[\n\\cos \\Big( \\frac{\\Omega_\\textrm{v} t}{2}\\Big) - i\\frac{\n\\delta}{\\Omega_\\textrm{v}} \\, \\sin \\Big( \\frac{\\Omega_\\textrm{v} t}{2}\\Big)\\Big],\n\\label{eq:Esing}\n\\end{equation}\nwhere $\\delta= \\omega_A-\\omega_C$ is the detuning and $\\Omega_\\textrm{v} = \\sqrt{ 4  |\\alpha_T|^2 + \\delta^2}$ is\nthe \\emph{vacuum Rabi frequency}. Note that $r^{(1)}$ and\n$r^{(2)}$ are not independent parameters, since\n$|r^{(2)}|=\\sqrt{1-|r^{(1)}|^2}$.\n\nThe subradiant $ \\ket{\\psi_-}$ and the superradiant $ \\ket{\\psi_+}$ states are\n\\begin{align}\n\\big|\\psi_- \\big\\rangle &= r^{(2)} \\big| 1^{(1)} 0^{(2)} \\big\\rangle  - r^{(1)} \\big| 0^{(1)} 1^{(2)} \\big\\rangle,\n\\label{eq:psim} \\\\\n\\big|\\psi_+ \\big\\rangle &= r^{(1)*} \\big| 1^{(1)} 0^{(2)} \\big\\rangle + r^{(2)*} \\big|0^{(1)} 1^{(2)} \\big\\rangle,\n\\label{eq:psip}\n\\end{align}\nand, in this case, they are position dependent through the\nrelative coupling strength parameters $r^{(1)}$ and $r^{(2)}$. As\none can see directly from Eq.~\\eqref{eq:HDicke2}, the  state\n$\\ket{\\psi_-} \\otimes \\ket{0^{(C)}}$ is an eigenstate of the\nTavis-Cummings Hamiltonian  with eigenvalue $\\frac{1}{2}\\omega_C + \\omega_A$.\nTherefore, when the atoms are prepared in this state, they are\ncompletely decoupled from the cavity field and the system does not\nevolve at all. In the case of equally strong couplings, i.e., for\n$|r^{(1)}|=|r^{(2)}|=1\/\\sqrt{2}$, the subradiant and the superradiant\nstates coincide with the maximally entangled Bell states. In\ngeneral, however, these states are not maximally entangled.\n\n\n\n\n\n\\subsection{\\label{sec:nonIdealCavity}\nNon-ideal cavity}\n\nWe now proceed to generalize Eq.~\\eqref{eq:HDicke2} to the case of\na lossy cavity. The imperfect reflectivity of the cavity mirrors\nand consequent leakage of photons causes a Lorentzian broadening\nof the spectral line corresponding to the mode supported by the ideal \ncavity. Accordingly, the microscopic atom-field interaction should\nnow take into account a continuum of modes described by a\nLorentzian distribution peaked at the central cavity frequency $\\omega_C$.\nFor the sake of simplicity, and in view of the discussion in the\nion-cavity QED context, we restrict our attention to a one-dimensional cavity\nmodel. Namely, we neglect the coupling with all the EM modes other\nthan the ones supported by the lossy cavity. In the rotating wave\napproximation, the Hamiltonian is given by\n\\begin{align}\nH =& \\sum_{k} \\omega_k \\Big( a_k^{\\dag}a_k + \\frac{1}{2}\\Big) + \\sum_{j=1}^N \\omega_A \\sigma_+^{(j)}\\sigma_-^{(j)} \\nonumber \\\\\n&+ \\sum_{k} \\sum_{j=1}^N \\Big[ i g_k \\sin \\Big(\\frac{\\omega_k}{c} x^{(j)} \\Big) a_k^\\dagger \\sigma_-^{(j)} + H.c. \\Big], \\label{eq:secondaeq}\n\\end{align}\nwhere $a_k$ and $a^{\\dag}_k$ are the annihilation and the creation\noperators of cavity photons of frequency $\\omega_k$, respectively.\nAbove, we have assumed that all the atoms have the same\nelectric dipole moment, which has been incorporated in the\ncoupling constants $g_k$, and we indicate with $x^{(j)}$ the\nposition of the atoms along the cavity axis. In the following we\nwill assume that each atom is kept at a fixed position inside the\ncavity and that they are all well localized, i.e., the spread of\ntheir wave function in position space is smaller than the\nwavelength of the central cavity field mode: $\\Delta x^{(j)} \\ll\nc\/\\omega_C$. Since all the significantly contributing modes are\nclose to the central mode (of frequency $\\omega_C$), we have\n\\begin{equation}\n\\sin \\left(\\frac{\\omega_k}{c} x^{(j)} \\right) \\simeq \\sin \\left(\\frac{\\omega_C}{c} x^{(j)} \\right),\n\\end{equation}\nand Eq.~\\eqref{eq:secondaeq} takes the form\n\\begin{align}\nH =& \\sum_{k} \\omega_k \\Big( a_k^{\\dag}a_k + \\frac{1}{2}\\Big) + \\sum_{j=1}^N \\omega_A \\sigma_+^{(j)}\\sigma_-^{(j)} \\nonumber \\\\\n&+ \\sum_{j=1}^N \\Big[ \\chi^{(j)} \\sigma_-^{(j)}  \\sum_{k} g_k   a_k^\\dagger + H.c. \\Big], \\label{eq:terzaeq}\n\\end{align}\nwith $\\chi^{(j)} = i \\sin (\\omega_C x^{(j)}\/c)$. In the continuum\nlimit the sum over the $k$-modes is replaced with an integral\n$$\n\\sum_k |g_k|^2 \\rightarrow \\int \\! d\\omega J(\\omega),\n$$\nwhere $J(\\omega)$ is the reservoir spectral density. As mentioned\nabove, we assume a Lorentzian distribution for the spectrum of the\nfield inside the cavity; therefore, we take a spectral density of\nthe form\n\\begin{equation}\nJ(\\omega) = \\frac{W^2}{2 \\pi} \\frac{\\kappa}{\\left( \\omega - \\omega_C \\right)^2 + (\\kappa \/ 2)^2},\n\\label{eq:J}\n\\end{equation}\nwhere the distribution is characterized by its full width at half\nmaximum value $\\kappa$ and by a normalization parameter $W^2 =\n\\int \\! d\\omega \\, J(\\omega)$. Hence,  $\\kappa$ describes the cavity\nlosses and $W$ describes the total coupling strength.\n\nWe focus again on the two-atom case, i.e., $N=2$, and we consider\nthe situation in which only one excitation is present in the total\natoms-field system. Starting from the Hamiltonian~\\eqref{eq:secondaeq} and using the\nLorentzian spectral density~\\eqref{eq:J}, it is possible to derive\nan effective master equation\n\\begin{equation}\n\\frac{d \\varrho}{dt} = - i \\left[ H_D, \\varrho \\right]\n-\\frac{\\kappa}{2} \\left[ a^{\\dag} a \\varrho + \\varrho a^{\\dag} a -\n2 a \\varrho a^{\\dag}\\right] \\label{eq:Dickeloss}\n\\end{equation}\nfor the dynamics of the atoms and the cavity mode of frequency\n$\\omega_C$ \\cite{ESDLaura}. Here, $a$ and $a^{\\dag}$ are the\nannihilation and the creation operators for the central cavity mode,\nwhich is damped at rate $\\kappa$, and the coherent dynamics is\ngenerated by $H_D$ in Eq.~\\eqref{eq:HDicke2}, where the\ncoupling constants are identified as $\\alpha^{(j)} = \\chi^{(j)}\nW$. From the exact solution of the effective master\nequation~\\eqref{eq:Dickeloss}, one can obtain the state of the\natomic system by tracing out the cavity degree of freedom: $\\rho\n(t) = \\textrm{tr}_C [\\varrho (t) ]$.\n\nAfter performing the trace, and for an initially empty cavity, the\nproblem can be solved exactly. In the ordered basis $\\left\\{ \\vert\n1^{(1)} 1^{(2)} \\rangle, \\vert 1^{(1)} 0^{(2)} \\rangle, \\vert\n0^{(1)} 1^{(2)} \\rangle, \\vert 0^{(1)} 0^{(2)} \\rangle \\right\\}$,\nthe atomic density matrix can be written in the form \\cite{Man08}\n\\begin{equation}\n\\rho(t) = \n\\begin{pmatrix}\n   0& 0 & 0 & 0 & \\\\ 0& |c_{10}|^2   &  c_{10} c_{01}^* & 0\\\\\n 0& c_{10}^* c_{01} & |c_{01}|^2  & 0\\\\\n   0&  0  & 0 & 1-|c_{10}|^2-|c_{01}|^2\n\\end{pmatrix}.\n\\label{eq:rhos}\n\\end{equation}\nThe dynamics of the two qubits is therefore completely characterized by the two amplitudes:\n\\begin{align}\nc_{10}(t) =& \\left[\\, |r^{(2)}|^2 + |r^{(1)}|^2 \\, {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& -r^{(1)*} r^{(2)} \\left[\\, 1- {\\cal E}(t)\\, \\right]c_{01}(0),\n\\label{eq:c1Sc1} \\\\\nc_{01}(t) =& - r^{(1)} r^{(2)*} \\left[\\, 1- {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& + \\left[\\, |r^{(1)}|^2 + |r^{(2)}|^2\\, {\\cal E}(t)\\, \\right]c_{01}(0),\n\\label{eq:c2Sc1}\n\\end{align}\nwith $r^{(j)} = \\chi^{(j)} \/ |\\chi_T |$, where $|\\chi_T| = \\sqrt{ |\\chi^{(1)}|^2 + |\\chi^{(2)}|^2 }$, and\n\\begin{eqnarray}\n{\\cal E}(t) = e^{-(\\kappa -i 2\\delta) t \/4} \\Big[\n\\cos \\Big( \\frac{\\Omega_g t}{2}\\Big) + \\frac{\\kappa- i 2\n\\delta}{2 \\Omega_g} \\sin \\Big( \\frac{\\Omega_g t}{2}\\Big) \\Big], \\nonumber \\\\\n\\label{eq:E}\n\\end{eqnarray}\nwhere $\\Omega_g = \\sqrt{4 |\\chi_T|^2 W^2 + \\delta^2 +i\n\\delta \\kappa -\\kappa^2 \/ 4}$ is the \\emph{generalized Rabi\nfrequency}. Note that Eqs.~\\eqref{eq:c1Sc1} and \\eqref{eq:c2Sc1} have\nexactly the same structure as\nEqs.~\\eqref{eq:c1sing} and \\eqref{eq:c2sing}, obtained for the\nsingle-mode Dicke model without losses. Formally, the cavity losses appear as an\nadditional imaginary part of the detuning $\\delta \\mapsto \\delta + i\\kappa\n\/2$. Accordingly, the effect of the cavity losses is described by\nthe modification of the time-dependent coefficient ${\\cal E}(t)$,\nwhich is now damped at rate $\\kappa \/4$, and by the\n$\\kappa$-dependent shift of the Rabi frequency. For $\\kappa\n\\rightarrow 0$, the Lorentzian spectral density~\\eqref{eq:J} tends\nto Dirac's delta distribution, $J(\\omega ) \\rightarrow W^2 \\delta\n(\\omega - \\omega_C )$, and Eq.~\\eqref{eq:E} reduces to\nEq.~\\eqref{eq:Esing}, with $\\alpha^{(j)} = \\chi^{(j)} W$.\n\nIt is worth noticing that, as one sees directly from\nEq.~\\eqref{eq:terzaeq}, the subradiant state $\\vert \\psi_-\n\\rangle$, given by Eq.~\\eqref{eq:psim}, is still decoupled from\nthe vacuum cavity field. Hence, if the atomic system is initially\nprepared in this state, no exchange of excitation with the cavity\nfield will take place.\n\n\n\n\n\n\\section{\\label{sec:effectiveModel}\nEffective model of ion-cavity interaction}\n\n\n\n\\subsection{Physical setup}\n\nIon-cavity QED experiments use calcium ions which are trapped in a linear\nPaul microtrap and interact with a quantized mode of a\nhigh-finesse optical cavity~\\cite{Walther,Piety}. In\nFig.~\\ref{fig:threeLevelModel} we show the relevant energy-level\nstructure, couplings, and decay channels for the compound system\nof two $^{40}$Ca$^+$ ions and a single cavity mode. The atomic\nground state $4^2 S_{1\/2}$ is coupled to the electronically\nexcited  state $4^2 P_{1\/2}$ by a (classical) pump laser injected\nfrom the side of the cavity. On the other hand, the excited state \n$4^2 P_{1\/2}$ is coupled to a metastable state $3^2 D_{3\/2}$ by the quantized cavity mode. The excited state\n$4^2 P_{1\/2}$ decays spontaneously to the states $4^2 S_{1\/2}$\nand $3^2 D_{3\/2}$ at rates $\\gamma_S$ and $\\gamma_D$,\nrespectively, and the cavity photon is damped at rate $\\kappa$,\nas explained in the previous section.\n\n\\begin{figure*}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig02}\n\\end{center}\n\\caption{\\label{fig:threeLevelModel} (Color online) The relevant\nelectronic states of two identical $^{40}$Ca$^+$ ions and\ncorresponding couplings provided by an external pump laser and a\nquantized cavity mode. The excited electronic state decays\nspontaneously to the ground and to the metastable states, and the\ncavity mode is damped as well.}\n\\end{figure*}\n\nA realistic theoretical description of the dynamics of a single\n$^{40}$Ca$^+$ ion coupled to the cavity mode has been given in\nRef. \\cite{Matthias}. The authors consider there also the effect\nof cavity losses and spontaneous emission, taking into account all\nthe Zeeman sublevels of the three relevant  electronic states. The\nmain consequence of the presence of the Zeeman sublevels is a\nreduction in the coupling driven by the cavity field by a factor of \n$\\sqrt{3}$ with respect to the simpler three-level model\nconsidered here. Therefore, we will use in the following a\nthree-level model scheme with such a reduced effective coupling to\naccount for the presence of the Zeeman sublevels. In the\nexperiments, the ions sit at the bottom of the trapping potential\nand are cooled down to the Lamb-Dicke regime. Under these\nconditions one can assume that the ions are kept at fixed\npositions and neglect recoil during the emission-absorption\nprocess.\n\nIn the following we will consider as initial atomic states those\nin which one of the two atoms is in its ground state and the other\none is in its metastable state, i.e., the states $\\ket{S^{(1)}\nD^{(2)}}$ and $\\ket{ D^{(1)} S^{(2)} }$. In order to prepare these\nstates, if the vibrational sidebands are not resolved, it is\nnecessary to use a selective laser  addressing of the individual\nions. This is routinely done in trapped-ion experiments with\n$^{40}$Ca$^+$ ions (see, e.g., \\cite{Blatt09}).\n\nThe two identical ions interact with the quantized cavity mode of\nfrequency $\\omega_C$ via laser-assisted two-photon processes, as\nshown in Fig.~\\ref{fig:threeLevelModel}. The ions are irradiated\nby a laser beam of frequency $\\omega_L+\\delta_L$. The laser beams\nand the cavity field are far detuned by $\\Delta$ from the\nelectronic level $\\ket{P^{(j)}}$, such that $\\omega_P - \\omega_D =\n\\omega_C + \\Delta$ and $\\omega_P - \\omega_S = \\omega_L + \\Delta$.\nTherefore, the setup provides each ion $j=1,2$ with a Raman coupling between the levels $\\ket{S^{(j)}}$ and $\\ket{D^{(j)}}$.\n\nThe time evolution of the composite system of the two ions and the cavity mode can be described by a master equation\n\\begin{align}\n\\frac{d \\varrho}{dt} =& -i \\big[ H(t),\\varrho \\big] - \\frac{\\kappa}{2} \\big(a^{\\dag}a \\varrho + \\varrho a^{\\dag}a-2a \\varrho a^{\\dag} \\big) \\nonumber \\\\\n&- \\frac{\\gamma_S}{2} \\sum_{j=1,2} \\Big( A_{PP}^{(j)} \\varrho + \\varrho A_{PP}^{(j)} - 2 A_{SP}^{(j)} \\varrho A_{PS}^{(j)} \\Big) \\nonumber \\\\\n& - \\frac{\\gamma_D}{2} \\sum_{j=1,2} \\Big( A_{PP}^{(j)} \\varrho + \\varrho A_{PP}^{(j)} - 2 A_{DP}^{(j)} \\varrho A_{PD}^{(j)} \\Big),\n\\label{eq:model2me}\n\\end{align}\nwhere we have included the cavity field damping at rate $\\kappa$, the\nspontaneous emission channels (two for each ion) at rates\n$\\gamma_S$ and $\\gamma_D$, and where the coherent dynamics is\ngenerated by a Hamiltonian\n\\begin{multline}\nH(t) = \\omega_C \\Big( a^{\\dag}a + \\frac{1}{2} \\Big) +  \\sum_{j=1,2} \\sum_{l=S,P,D}\\omega_l A_{ll}^{(j)}   \\\\\n+ \\sum_{j=1,2}\\Big( e^{-i (\\omega_L+\\delta_L)t} g^{(j)}_L A_{PS}^{(j)} + g_C^{(j)} a\\, A_{PD}^{(j)} + H.c. \\Big). \\label{eq:model3H}\n\\end{multline}\nThe atomic operators are defined as $A_{ll'}^{(j)}=\\vert l^{(j)}\n\\rangle \\langle l'^{(j)} \\vert$, with $l,l'=S,P,D$ and $j=1,2$.\nFinally, the coherent couplings provided by the laser and the\ncavity mode are, respectively,\n\\begin{align}\ng_L^{(j)} &=\\, \\Omega\\, e^{i k_L x^{(j)}}, \\label{eq:gl} \\\\\ng_C^{(j)} &=\\, g \\sin (k_C x^{(j)} ),  \\label{eq:gc}\n\\end{align}\nwith $k_L$ and $k_C$ being the wave numbers of the laser and the standing cavity mode.\n\n\n\n\n\n\\subsection{\\label{sec:effectiveProcesses}\nEffective two-level model}\n\nWhen the detuning $\\Delta$ is sufficiently large compared to the\ncouplings, $\\Delta \\gg g_L^{(j)}, g_C^{(j)}$, the excited\nelectronic states $\\ket{P^{(j)}}$ can be adiabatically eliminated\nfrom the dynamics, as described in detail in Appendix~\\ref{app:adiabaticElimination}. In this\ncase the system can be effectively described as composed of two\ntwo-level atoms interacting with a cavity mode. For this purpose,\nwe denote the ground and the metastable states of the $j$th atom\nas $\\ket{1^{(j)}} \\equiv \\ket{S^{(j)}}$ and $\\ket{0^{(j)}} \\equiv\n\\ket{D^{(j)}}$ (N.B., the true atomic ground state corresponds to \nthe excited state of the effective two-level system, since it is\nable to emit a cavity photon through the Raman transition).\n\nThe adiabatic elimination of the excited levels $\\{ |P^{(j)}\\rangle \\}$ is not at all\ntrivial due to the inclusion of the spontaneous emission\nprocesses \\cite{DiFidio}. We show in Appendix~\\ref{app:adiabaticElimination} that an effective\nTavis-Cummings Hamiltonian can be derived, describing an\nexcitation exchange between the ions and the cavity. However, one\nneeds to include (i) two Stark shift terms per ion (one, in\nparticular, being dependent on the state of the cavity mode) and (ii) an overall\nre-scaling  of both the free and the interaction energies by a\nfactor explicitly dependent on the emission rates.\n\nIt turns out that, in the interaction picture with respect to $H_0-\\Delta \\sum_{j}\nA_{PP}^{(j)}$, where $H_0$ is given by the first two terms on the\nright-hand side of Eq.~\\eqref{eq:model3H}, the coherent part of the evolution of the ion-cavity system is described by an effective Hamiltonian\n\\begin{align}\nH_{\\rm eff} = & -\\xi \\sum_{j=1,2} \\Big[ \\Big( e^{-i \\delta_L t} \\frac{\\beta^{(j)} g^* \\Omega}{\\Delta}\\, a^{\\dag} A_{01}^{(j)} + H.c. \\Big) \\nonumber \\\\\n& +\\frac{|\\beta^{(j)}g|^2}{\\Delta} \\, a^{\\dag} a A_{00}^{(j)} +\n\\frac{|\\Omega|^2}{\\Delta} A_{11}^{(j)} \\Big], \\label{eq:Heffion}\n\\end{align}\nwhere the position-dependent parameters $\\beta^{(j)}$ are defined\nas\n\\begin{equation}\n\\beta^{(j)} = e^{i k_L x^{(j)}} \\sin \\big( k_C x^{(j)} \\big),\n\\label{eq:coefficient}\n\\end{equation}\nand the dimensionless renormalizing prefactor is\n\\begin{equation}\n\\xi =  \\frac{\\Delta^2}{\\Delta^2 + (\\gamma_S+\\gamma_D)^2\/4}.\n\\label{eq:xi}\n\\end{equation}\nThis Hamiltonian resembles the Tavis-Cummings Hamiltonian~\\eqref{eq:HDicke2}, except for the photon-dependent Stark shift term. However, since the original microscopic model includes\ndissipative processes, the unitary evolution generated by $H_{\\rm\neff}$ needs to be supplemented by decohering terms that have a very peculiar structure. Indeed, the effective master equation that describes the time evolution of the ions and the cavity contains four (now both dissipative and non-dissipative) processes (described by jump operators) that take into account the effects of the spontaneous emission as seen in the\nrestricted atomic subspaces spanned by $\\{|0^{(j)}\\rangle, |1^{(j)}\\rangle \\}$. The cavity damping appears in the restricted subspace in the same form as in the original model. \n\nThe effective master equation reads\n\\begin{align}\n\\frac{d \\varrho}{dt} =& -i \\left[ H_{\\rm eff}, \\varrho\\right] - \\frac{\\kappa}{2} \\left( a^{\\dag}a \\varrho + \\varrho a^{\\dag}a -2a \\varrho a^{\\dag} \\right) \\nonumber \\\\\n& - \\sum_{\\substack{j=1,2\\\\m=S,D}} \\frac{\\Gamma_m^{(j)}}{2} \\Big[ C_m^{(j)\\dag}C_m^{(j)} \\varrho +\\varrho \\, C_m^{(j)\\dag}C_m^{(j)} \\nonumber \\\\\n& \\qquad\\qquad\\qquad\\, -2  C_m^{(j)} \\varrho \\, C_m^{(j)\\dag} \\Big],\n\\label{eq:meeffion}\n\\end{align}\nwhere the jump operators for each ion $j$ are\ngiven by\n\\begin{align}\nC_S^{(j)} &= e^{-i \\delta_L t} \\Omega \\, A_{11}^{(j)} + \\beta^{(j)*} g\\, a A_{10}^{(j)}, \\label{eq:jumpC1} \\\\\nC_D^{(j)} &= e^{-i \\delta_L t} \\Omega \\, A_{01}^{(j)} + \\beta^{(j)*} g\\, a A_{00}^{(j)}, \\label{eq:jumpC2}\n\\end{align}\nwhile the effective decay rates are $\\Gamma_m^{(j)} =\\xi \\gamma_m \/ \\Delta^2$, where $m=S,D$ and the prefactor $\\xi$ is given by Eq.~\\eqref{eq:xi}. The structure of these jump operators is easy to interpret once\nthe full level configurations of Fig.~\\ref{fig:threeLevelModel}\nare taken into account. Let us consider, for example, the operator\n$C_S^{(j)}$ of Eq.~(\\ref{eq:jumpC1}). It arises from the spontaneous emission process $4^2 P_{1\/2}\\rightarrow 4^2 S_{1\/2}$ of the $j$th atom, now being restricted to the two-dimensional subspace $\\{|0^{(j)}\\rangle,|1^{(j)}\\rangle\\}$. The jump operator $C_S^{(j)}$ has two contributions, both of\nthem describing non-dissipative decoherence by pure dephasing processes (as one understands from the fact that they do not produce any excitation loss). These two\ncontributions account for the interruption of the ion-cavity\nexcitation exchange (vacuum Rabi cycle) by the spontaneous\nemission. The first term is an unwanted repopulation of state  $|1^{(j)}\\rangle$ occurring after the laser has virtually brought the\nsystem to the intermediate level $|P^{(j)}\\rangle$ of the full Raman cycle. The\nsecond term is also due to decay into state $|1^{(j)}\\rangle$, but this time the virtual excitation of level $|P^{(j)}\\rangle$ is caused by the cavity field. In conclusion, both processes interrupt the\nvacuum Rabi cycle without the excitation being lost as, at the\nend, the two-level system is found in its excited state $|1^{(j)}\\rangle$. This\nimplies that the excitation exchange can restart, but with a different\nphase. Thus, $C_S^{(j)}$ describes a phase error.\n\nA similar interpretation scheme can be adopted for the two terms\nconstituting $C_D^{(j)}$ in Eq.~\\eqref{eq:jumpC2}. However, this time the involved process is the spontaneous emission $4^2 P_{1\/2}\\rightarrow 3^2 D_{3\/2}$. Whether it occurs after the\nvirtual excitation of level $|P^{(j)}\\rangle$ performed by the laser (first term) or by the\ncavity field (second term), the result is that at the end the two-level system is found in its ground state $|0^{(j)}\\rangle$ and that one excitation has been lost either from the atom or from the cavity mode. Therefore, this jump operator causes dissipative decoherence. We note that, at this stage, the four jump operators of\nEqs.~\\eqref{eq:jumpC1}-\\eqref{eq:jumpC2} are both explicitly time\ndependent and implicitly position dependent via the coefficients\n$\\beta^{(j)}$ [see Eq.~\\eqref{eq:coefficient}].\n\nA phase rotation within the restricted Hilbert space, spanned by\nthe states with at maximum one excitation, allows transforming\nthe effective Hamiltonian~\\eqref{eq:Heffion} into the Tavis-Cummings\nHamiltonian~\\eqref{eq:HDicke2} as well as removing simultaneously the\ntime dependence from the jump operators~\\eqref{eq:jumpC1} and \\eqref{eq:jumpC2}. This is described in Appendix~\\ref{app:phaseRotation}. Therefore,\nin a suitable rotating frame,  the following effective\nTavis-Cummings Hamiltonian is obtained:\n\\begin{align}\nH_D^\\textrm{eff} =&\\,\\, \\omega_C^\\textrm{eff} \\Big( a^{\\dag} a + \\frac{1}{2} \\Big) + \\sum_{j=1,2}  \\omega_A^\\textrm{eff} \\, \\sigma_+^{(j)} \\sigma_-^{(j)} \\nonumber \\\\\n& +\\sum_{j=1,2} \\Big( \\alpha_\\textrm{eff}^{(j)} \\, a^{\\dag} \\, \\sigma_-^{(j)} + \\alpha_\\textrm{eff}^{(j)*} \\, a \\, \\sigma_+^{(j)}\\Big), \\label{eq:Hdeff}\n\\end{align}\nwhere we have introduced again the spin inversion operators used in Sec.~\\ref{sec:DickeModel}. The effective Dicke model parameters are\n\\begin{align}\n\\omega_C^\\textrm{eff} & = -\\xi \\, \\frac{2 |\\beta_T g|^2}{3 \\Delta}, \\label{eq:omegaCEff}\\\\\n\\omega_A^\\textrm{eff} & = \\delta_L - \\xi \\Big( \\frac{|\\Omega|^2}{\\Delta} - \\frac{|\\beta_T g|^2}{3\\Delta} \\Big) , \\label{eq:omega0Eff}\\\\\n\\alpha_\\textrm{eff}^{(j)} & = -\\xi \\, \\frac{ \\beta^{(j)} g^* \\Omega }{\\Delta} \\equiv \\beta^{(j)} g_\\textrm{eff}, \\label{eq:alphaEff}\n\\end{align}\nwhere $|\\beta_T | = \\sqrt{|\\beta^{(1)} |^2 + |\\beta^{(2)} |^2 }$. The effective detuning is given by\n\\begin{equation}\n\\delta_\\textrm{eff} = \\omega_A^\\textrm{eff} - \\omega_C^\\textrm{eff} = \\delta_L -\\xi \\, \\frac{ |\\Omega|^2 - |\\beta_T g|^2 }{\\Delta }, \\label{eq:deltaEff}\n\\end{equation}\nand the relative effective coupling strengths $r^{(j)}$ [cf.~Eqs.~\\eqref{eq:c1sing} and\\eqref{eq:c2sing}] are directly given by the position-dependent parameters $\\beta^{(j)}$,\nsince now $r^{(j)} = \\alpha_\\textrm{eff}^{(j)} \/\n|\\alpha_{\\textrm{eff},T}| = \\beta^{(j)} \/ |\\beta_T|$.\n\nComparing Eqs.~\\eqref{eq:Hdeff} and \\eqref{eq:meeffion} with\nEqs.~\\eqref{eq:HDicke2} and \\eqref{eq:Dickeloss}, respectively, we\nsee that, when the effective atomic spontaneous emissions are\nnegligible, this system allows us to realize the Dicke model in the\nnon-ideal cavity case.\n\n\n\n\n\n\\subsection{\\label{sec:scaling}\nEffective spontaneous emission processes}\n\nAs mentioned before, we restrict our study to the case in which\nonly one or zero quanta are present in the composite system of the two ions and the cavity mode.\nTherefore, the compound state of the two atoms and the cavity\nphoton can be expressed in the basis  $\\{ \\ket{ 0^{(1)} 0^{(2)}\n0^{(C)} }$, $\\ket{ 0^{(1)} 0^{(2)} 1^{(C)} }$, $\\ket{ 0^{(1)}\n1^{(2)} 0^{(C)} }$, $\\ket{ 1^{(1)} 0^{(2)} 0^{(C)} } \\}$ (see\nAppendix~\\ref{app:phaseRotation}). Consequently, the jump operators~\\eqref{eq:jumpC1}-\\eqref{eq:jumpC2} can  be normalized with respect to the operator\nnorm $\\|A\\| = \\sup_{\\|\\phi\\|=1} \\|A\\ket{ \\phi } \\|$, where $\\ket{\n\\phi }$ belongs to the Hilbert space spanned by the basis defined\nabove. The introduction of the normalized jump operators allows to\ndefine the effective spontaneous emission decay rates\n$\\Gamma_m^{(j)}$ unambiguously.\n\nThe normalized jump operators are\n\\begin{align}\nC_S^{(1)} &= |1^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle \\Phi_1 |, \\label{eq:scaledC11}\\\\\nC_S^{(2)} &= |0^{(1)} 1^{(2)} 0^{(C)}\\rangle \\langle \\Phi_2 |, \\label{eq:scaledC12}\\\\\nC_D^{(1)} &= |0^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle \\Phi_1 |,\\\\\nC_D^{(2)} &= |0^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle \\Phi_2 |, \\label{eq:scaledD12}\n\\end{align}\nwhere the decaying states are\n\\begin{align}\n|\\Phi_1 \\rangle & = \\frac{ \\Omega^* \\, |1^{(1)} 0^{(2)} 0^{(C)}\\rangle  + \\beta^{(1)} g^*\\, |0^{(1)} 0^{(2)} 1^{(C)}\\rangle }{\\sqrt{|\\Omega |^2 + |\\beta^{(1)} g|^2 } }, \\label{eq:decayingState1} \\\\\n|\\Phi_2 \\rangle & = \\frac{ \\Omega^* \\, |0^{(1)} 1^{(2)} 0^{(C)}\\rangle + \\beta^{(2)} g^*\\, |0^{(1)} 0^{(2)} 1^{(C)}\\rangle }{\\sqrt{|\\Omega |^2 + |\\beta^{(2)} g|^2 } }. \\label{eq:decayingState2}\n\\end{align}\nThe corresponding rescaled decay rates are given by\n\\begin{align}\n\\Gamma_S^{(j)} &= \\xi \\, \\big( |\\Omega|^2 + |\\beta^{(j)} g|^2 \\big) \\frac{ \\gamma_S }{ \\Delta^2 }, \\label{eq:scaledDecayRateS}\\\\\n\\Gamma_D^{(j)} &= \\xi \\, \\big( |\\Omega|^2 + |\\beta^{(j)} g|^2 \\big) \\frac{ \\gamma_D }{ \\Delta^2 }. \\label{eq:scaledDecayRateD}\n\\end{align}\nThe cavity photon annihilation operator $a = |0^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle 0^{(1)} 0^{(2)} 1^{(C)} |$ is already normalized in our restricted basis.\n\nThe spontaneous emission decay rates for the considered states of\na calcium atom are $\\gamma_S \/ 2 \\pi = 22.3$ MHz and\n$\\gamma_D \/ 2 \\pi = 1.7$ MHz. Therefore, $\\Gamma_S^{(j)} \\gg\n\\Gamma_D^{(j)}$ and the dominant effective spontaneous emission\njump processes are described by the operators\n$C_S^{(j)}$. Consequently, according to the discussion above, the\nmain decoherence sources are the non-dissipative dephasing processes that conserve the energy of the ion--cavity system.\n\nThe character of the decaying state, and hence the corresponding\njump operator is defined by the balance between the strengths of the laser pumping $\\Omega$ and the cavity coupling $\\beta^{(j)} g$. In the \\emph{strong laser pumping} case ($|\\Omega |\n\\gg |\\beta^{(j)} g|$) the non-unitary dynamics of the atomic\nreduced system is dominated by phase diffusion processes described\nby the operators $A_{11}^{(j)}$. In the \\emph{weak laser pumping}\ncase ($|\\Omega | \\ll |\\beta^{(j)} g|$), on the contrary, the\nprocesses described by the operators $a A_{10}^{(j)}$\ndominate. Moreover, as one can see from\nEqs.~\\eqref{eq:decayingState1} and \\eqref{eq:decayingState2}, one can\nfurther modify the character of the specific atomic jump operators\nby changing the relative position of the ions with respect to the\ncavity field through the $\\beta^{(j)}$ parameters.\n\nThe significance of the spontaneous emissions can be estimated by the ratio\n\\begin{equation}\n\\bigg| \\frac{ \\Gamma_S^{(j)} }{ \\alpha_\\textrm{eff}^{(j)} } \\bigg| = \\frac{ 1 + |\\beta^{(j)} g \/ \\Omega |^2 }{ | \\beta^{(j)} g \/ \\Omega | } \\, \\frac{ \\gamma_S }{ \\Delta }.\n\\label{eq:decayVersusCoupling}\n\\end{equation}\nFor a fixed detuning $\\Delta$ this ratio has its minimum value $2\n\\gamma_S \/ \\Delta $ when $|\\beta^{(j)} g \/  \\Omega| = 1$, i.e.,\nwhen the couplings provided by the laser and the cavity field are\nequally strong. On the other hand, for fixed coupling strengths,\nthe ratio is inversely proportional to the detuning $\\Delta$. This\ncan be exploited in order to minimize the role of the effective spontaneous decay. The\ncavity damping $\\kappa$ is neither affected by the detuning nor the couplings.\n\nFinally, we note that for large detunings, $\\Delta \\gg \\gamma_S, \\gamma_D $, the dimensionless prefactor $\\xi \\sim 1$ and the effective decay rates as well as the effective coupling terms have simplified expressions. The effective couplings are then given by $\\alpha_\\textrm{eff}^{(j)} \\sim -\\beta^{(j)} g^* \\Omega \/ \\Delta$, while in the limit of strong and weak laser pumpings the dominating decay rates are $\\Gamma_S^{(j)} \\sim |\\Omega|^2 \\gamma_S \/ \\Delta^2$ and $\\Gamma_S^{(j)} \\sim |\\beta^{(j)} g|^2 \\gamma_S \/ \\Delta^2$, respectively. \n\n\n\n\n\n\\section{\\label{sec:resonantRegime}\nEnvironment-induced entanglement: Resonant regime}\n\nIn this section we study, analytically and numerically, the\ndynamics of the entanglement between the electronic degrees of\nfreedom of the two atoms. The generation of entanglement between\nthe ions and its persistence at long times are, indeed, a clear\nmanifestation of the collective (subradiant) behavior. In\nparticular, entanglement generation is mediated by the interaction\nwith the quantized cavity field which is initially prepared in the\nvacuum state. If the atomic spontaneous emission processes are negligible\nand we face the bare Dicke model, the dynamics can be described\nexactly. We compare these exact analytical results to numerical\nsimulations including the spontaneous emission effects. The\nsimulations were implemented by using the Monte Carlo wave\nfunction (MCWF) method \\cite{MCWF,ZollerCarmichael}. We begin by\nconsidering the resonant case, where the effective detuning\n$\\delta_\\textrm{eff} = 0$, with $\\delta_\\textrm{eff} $ given by\nEq.~\\eqref{eq:deltaEff}.\n\n\n\n\n\n\\subsection{\\label{sec:analytical}\nAnalytical solution neglecting spontaneous emission}\n\nThe effective model describing the dynamics when spontaneous emissions are negligible is given by the master equation~\\eqref{eq:Dickeloss} with the effective Tavis--Cummings Hamiltonian~\\eqref{eq:Hdeff}, as described in Sec.~\\ref{sec:nonIdealCavity}. The analytical solution for the atomic density matrix is given by Eqs.~\\eqref{eq:rhos} and \\eqref{eq:E}, with $\\chi^{(j)} W = \\alpha_\\textrm{eff}^{(j)} = \\beta^{(j)} g_\\textrm{eff}$.\n\nWe are interested in the collective dynamics when initially one excitation is present in the atomic system and the cavity is in its vacuum state. Any initial atomic state containing one excitation can be written in terms of the superradiant and subradiant states~\\eqref{eq:psim}-\\eqref{eq:psip} as\n\\begin{equation}\n\\ket{\\psi(0) }= \\beta_+  \\ket{\\psi_+}  + \\beta_- \\ket{\\psi_-}. \\label{eq:nonloso}\n\\end{equation}\nAs time passes, the collective atomic state decays via the evolution of the superradiant component,\n\\begin{equation}\n\\langle \\psi_+ \\vert \\psi (t) \\rangle = {\\cal E}(t) \\, \\beta_+,\n\\end{equation}\nwith ${\\cal E}(t)$ given by Eq.~\\eqref{eq:E}. The subradiant component $\\langle \\psi_- \\vert \\psi (t) \\rangle = \\beta_-$, however, remains unchanged. Consequently, for times, such that $\\kappa t \\gg 1$, the atomic state will be in general a statistical mixture of the collective ground state $| 0^{(1)} 0^{(2)} \\rangle$ and the subradiant state $\\ket{\\psi_-}$ with weights dependent on $\\beta_-$, which in turn depends on the relative coupling strengths $r^{(j)}$.\n\nIn the following we focus on the dynamics of entanglement between the atoms. In order to quantify the stationary asymptotic entanglement of the final state we use Wootters's concurrence \\cite{wootte} which, for a density matrix of the form of Eq.~\\eqref{eq:rhos}, is given by\n\\begin{equation}\nC(t) = 2 \\left| c_{10}(t) c_{01}^*(t)\\right|, \\label{eq:concurrdef}\n\\end{equation}\nwith $c_{10}(t)$ and $c_{01}(t)$ given by Eqs.~\\eqref{eq:c1Sc1} and \\eqref{eq:c2Sc1}. In general, the concurrence is zero for factorized states and unity for maximally entangled states. For $\\kappa t \\gg 1$ we obtain a stationary concurrence value\n\\begin{equation}\nC_\\textrm{stat}=2 |r^{(1)}r^{(2)}| \\left| \\beta_- \\right|^2.\n\\end{equation}\nAs expected, the value of the stationary concurrence is directly\nrelated to the subradiant component of the initial state. If both\natoms are coupled to the EM field, the stationary value of the\nconcurrence, for any initial state with $\\beta_- \\ne 0$, will be\nnonzero. When the atoms are initially prepared in the\nsuperradiant state, i.e., $\\beta_- =0$, the system approaches\nasymptotically the pure factorized state $| 0^{(1)} 0^{(2)}\n\\rangle$.\n\nFor the initially factorized states $\\ket{1^{(1)} 0^{(2)}}$ and\n$\\ket{0^{(1)} 1^{(2)}}$, the interaction with the environment\ngenerates entanglement in the atomic system. For these initial\nstates the stationary concurrence takes the values $C_\\textrm{stat} =\n2 |r^{(1)}|(1- |r^{(1)}|^2)^{3\/2} $ and $C_\\textrm{stat} = 2\n|r^{(1)}|^3 \\sqrt{1- |r^{(1)}|^2}$, respectively. As we have\nnoticed in Ref. \\cite{Man08}, the factorized states are those that\nmaximize the stationary concurrence for certain values of\n$r^{(1)}$. The maximum value of stationary concurrence, for both\nthe two factorized initial states considered here, is\n$C_\\textrm{stat}^\\textrm{max} = \\max_{\\,|r^{(1)}| \\in [0,1]}\nC_\\textrm{stat} \\simeq 0.65$. This value is obtained with\n$|r^{(1)}|= 0.5$ and $|r^{(1)}| \\simeq 0.87$ (i.e., $|r^{(2)}|=\n0.5$) for initial states $\\ket{1^{(1)} 0^{(2)}}$ and $\\ket{0^{(1)}\n1^{(2)}}$, respectively.\n\nWe note in passing that when only one of the two atoms is coupled\nto the EM field, i.e., $r^{(1)}=0$ or $r^{(2)}=0$, the stationary\nconcurrence is zero. In this case, indeed, the subradiant and the\nsuperradiant states coincide with states $\\ket{1^{(1)} 0^{(2)}}$\nand $\\ket{0^{(1)} 1^{(2)}}$ as one can see from\ndefinitions~\\eqref{eq:psim} and \\eqref{eq:psip}.\n\nFrom the definition of the generalized Rabi frequency given by\nEq.~\\eqref{eq:E}, which in the resonant case reads as\n$\\Omega_g = \\sqrt{4 |\\beta_T g_\\textrm{eff}|^2 - \\kappa^2\n\/ 4}$, two extreme regimes can be defined. In the \\textsl{weak ion-cavity\ncoupling regime}, defined by  $4 |\\beta_T g_\\textrm{eff} |\n\\ll \\kappa$,  the generalized Rabi frequency is purely imaginary.\nTherefore, according to Eq.~\\eqref{eq:E}, the Dicke model predicts\na solution given by monotonic hyperbolic sine and cosine\nfunctions. The opposite limit is the \\textsl{strong ion-cavity coupling regime}, defined by $4 |\\beta_T g_\\textrm{eff} | \\gg\n\\kappa $. In this case the generalized Rabi frequency is real and\nthe Dicke model predicts damped oscillatory dynamics.\n\n\n\n\n\n\\subsection{MCWF simulations in the presence of spontaneous emission}\n\nIn this section, we focus on the effect of the spontaneous emissions on\nthe subradiant-state-based entanglement generation described in\nthe previous section. We consider again as initial atomic\nstate $|\\psi (0) \\rangle = |1^{(1)} 0^{(2)}\\rangle$ with the cavity\nin the vacuum state $|0^{(C)}\\rangle$. For a given value of $r^{(1)} \\in [0,1]$, we choose $\\beta^{(1)}$ and $\\beta^{(2)}$ to be positive real numbers such that the larger of the two is always unity and the smaller one is $\\min \\{ r^{(1)}\/ \\sqrt{ 1 - r^{(1)2}}, \\sqrt{1 - r^{(1)2}}\/ r^{(1)} \\}$ [cf.~definition~\\eqref{eq:coefficient}]. Now $|\\beta_T|^2 = |\\beta^{(1)}|^2 + |\\beta^{(2)}|^2 = \\textrm{min} \\{ 1\/r^{(1)2},1\/ ( 1-r^{(1)2}) \\} \\in [1,2]$. The physical parameters\nhave been chosen in accordance to the experiments of Ref.\n\\cite{Walther} and  are summarized in\nTable~\\ref{tab:physicalValues}.  The size of the ensemble in the\nMCWF simulations is $N=1000$. We are using the variant of MCWF\nmethod described in \\cite{ZollerCarmichael}.\n\n\\begin{table}[tb]\n\\caption{ \\label{tab:physicalValues} Values of physical quantities\nused in the simulations. Note that the cavity coupling is here\nexplicitly scaled by the Clebsch--Gordan factor $1\/\\sqrt{3}$ and,\nin the text, also by the position-dependent parameters\n$\\beta^{(j)}$.}\n\\begin{center}\n\\begin{tabular}{l@{\\quad}c@{\\quad}r@{}l}\n\\hline\n\\hline\nQuantity & Symbol & \\multicolumn{2}{c}{Value (2$\\pi$ MHz)} \\\\ \\hline\nLaser coupling & $\\Omega$ & \\qquad\\quad 9&.0 \\\\\nCavity coupling & $g$ & 6&.5 $\/ \\sqrt{3}$\\\\\nDecay rate $4^2 P_{1\/2}\\rightarrow 4^2 S_{1\/2}$ & $\\gamma_S$ & 22&.3 \\\\\nDecay rate $4^2 P_{1\/2}\\rightarrow 3^2 D_{3\/2}$ & $\\gamma_D$ & 1&.7 \\\\\nDetuning & $\\Delta_0$ & 20&.0 \\\\\nCavity damping & $\\kappa_0$ & 1&.2 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe value of the cavity coupling constant $g$ in\nTable~\\ref{tab:physicalValues} refers to  the new miniature trap\nrecently realized at the University of Sussex \\cite{W}. The\nreference value $\\kappa_0$ for the cavity damping can nowadays be\nimproved by at least one order of magnitude. Finally, the detuning\n$\\Delta$ can be easily increased in the experiments, with respect\nto the reference value $\\Delta_0$.\n\nWith the experimental parameters of Table~\\ref{tab:physicalValues}, the coupling strengths $\\Omega$ and $g$ are of the same order. Therefore, neither the strong nor the weak laser pumping regimes, introduced\nin Sec.~\\ref{sec:scaling}, are reached and, consequently, all the effective\ndecay processes caused by the spontaneous emission are\ncombinations of two different physical operations, as interpreted in Sec.~\\ref{sec:effectiveProcesses}. \n\nLet us denote the atomic density-matrix components as\n$\\rho_{ab,cd} \\equiv \\langle a^{(1)} b^{(2)} | \\rho | c^{(1)}\nd^{(2)} \\rangle$, where $a,b,c,d=0,1$. The density matrix remains\nstill in the same block form of Eq.~\\eqref{eq:rhos} even in the\npresence of spontaneous emissions. The concurrence is therefore\ngiven by $C(t) = 2 |\\rho_{01,10}(t)|$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig03}\n\\end{center}\n\\caption{\\label{fig:scaling} (Color online) Scaling of the effective ion-cavity coupling $|g_\\textrm{eff}|$ (middle line for large $\\Delta$) and effective spontaneous emission decay rates $\\Gamma_S^{(j)}$ (lowest line for large $\\Delta$; with $\\beta^{(j)}=1$) as a function of the detuning $\\Delta$. The isocurves  $4 |\\beta_T g_\\textrm{eff}|\/\\kappa =$ const (thin lines) are parallel to the $|g_\\textrm{eff}|$ curve, so that the weak ion-cavity coupling regime is in the upper right corner and the strong ion-cavity coupling regime in the lower left one. The cavity decay rate $\\kappa$ (horizontal line) does not depend on the detuning. The effective spontaneous emission events are suppressed for large detunings.}\n\\end{figure}\n\nIn the following we will examine the effect of the spontaneous\nemissions by comparing the concurrence as a function of time for\nfixed values of $4 |\\beta_T g_\\textrm{eff}|\/\\kappa = 4 |\\beta_T\n\\xi g \\Omega  \/ \\kappa \\Delta|$. We study large detunings ($\\Delta \\gg \\gamma_S, \\gamma_D$), so the prefactor $\\xi \\sim 1$. In the examples we change $\\kappa$ and $\\Delta$, such that $\\kappa \/\\kappa_0 = 0.1, 0.01$ and $\\Delta \/ \\Delta_0 = 10, 100, 1000$, while keeping the product $\\kappa\n\\Delta$ constant. Physically, this corresponds to using different cavity qualities and detunings which, furthermore, influences the effective dynamical parameters. Larger\ndetunings, indeed, suppress the effective spontaneous emissions in\nfavor of the coherent dynamics, as explained in\nSec.~\\ref{sec:scaling}. The situation is clarified in\nFig.~\\ref{fig:scaling} which shows the scaling of the effective\ncoupling strength $g_\\textrm{eff}$ and the dominant spontaneous\nemission decay rate $\\Gamma_S^{(j)}$ [cf.\nEqs.~\\eqref{eq:alphaEff}, \\eqref{eq:scaledDecayRateS}, and \\eqref{eq:scaledDecayRateD}] as functions of detuning $\\Delta$. The cavity damping rate $\\kappa$ is not\naffected by the detuning. The relative values of the three key\nparameters $g_\\textrm{eff}$, $\\Gamma_S^{(j)}$, and $\\kappa$\ncharacterize the dynamical regime: (i) the ratio\n$|g_\\textrm{eff}|\/\\kappa$ defines the strong and the weak\nion-cavity coupling regimes and (ii) the magnitude of $\\Gamma_S^{(j)}$ compared to\n$|g_\\textrm{eff}|$ and $\\kappa$, in turn, describes the significance of the spontaneous emission processes and tells us whether the dynamics is well described by the Dicke model or not.\n\n\n\n\n\n\\subsubsection{\\label{sec:subsub}Weak ion-cavity coupling regime}\n\nIn this regime, the oscillatory dynamics stemming from the\ncoherent coupling between the atoms and the cavity is heavily\ndamped. In Fig.~\\ref{fig:weakCoupling} we plot the concurrence as\na function of both time and the relative coupling strength $r^{(1)}$\nfor $\\Delta \/ \\Delta_0 = 100$ and $\\kappa \/ \\kappa_0 = 0.1$, giving $|g_\\textrm{eff}|\/ 2\\pi = \\xi g \\Omega\/2\\pi\\Delta = 17$~kHz. All the other parameters are chosen as in\nTable~\\ref{tab:physicalValues}. We recall that initially the atomic state $|\\psi (0) \\rangle = |1^{(1)} 0^{(2)}\\rangle$ is factorized. The initial dynamics of the concurrence\nshows a monotonic increase, as the superradiant component [see\nEq.~\\eqref{eq:nonloso}] rapidly fades away while the subradiant\ncomponent remains intact. However, because of the presence of\nspontaneous emission, the subradiant state is not anymore\nperfectly decoupled from the dynamics and, consequently, the concurrence\nwill not reach a steady-state value.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig04}\n\\end{center}\n\\caption{\\label{fig:weakCoupling} (Color online) Concurrence as a\nfunction of time and the relative coupling strength $r^{(1)}$\nin the weak ion-cavity coupling regime. The dynamics is initially\nmonotonic since the existing superradiant component decays rapidly\ncompared to other dynamical time scales. The subradiant state\ncomponent decays eventually because of the atomic spontaneous\nemissions. The best entanglement production occurs with asymmetric couplings ($r^{(1)} \\neq 1\/\\sqrt{2}$). Parameters: $\\Delta \/ \\Delta_0 = 100, \\kappa \/ \\kappa_0 = 0.1$}\n\\end{figure}\n\nWe note that the best peak value of the concurrence $C\\simeq0.6$\nis achieved for $r^{(1)}\\simeq0.55$, i.e., as expected, for an\nasymmetric configuration ($r^{(1)} \\neq 1\/ \\sqrt{2}$) of the ions with respect to the cavity\nfield. However, this value of  $r^{(1)}$ is now slightly different\nthan the one obtained in Sec.~\\ref{sec:analytical} where\nspontaneous emissions were neglected ($r^{(1)}=0.5$). We will\nfurther discuss this point when considering the position\ndependence of the jumps statistics at the end of this subsection.\n\nIn Fig.~\\ref{fig:weakCouplingSlides} we further study the effect\nof the spontaneous emissions in the weak ion-cavity coupling case. In this\nfigure, we compare the predictions of the Dicke model, described\nin Sec.~\\ref{sec:nonIdealCavity}, with the dynamics of the\nion-cavity system in the presence of the spontaneous emissions for $\\Delta \/ \\Delta_0\n= 100, \\kappa \/ \\kappa_0 = 0.1$ and  $\\Delta \/ \\Delta_0 = 1000, \\kappa \/ \\kappa_0 = 0.01$. The dynamics\nof the concurrence clearly shows that, in the first case ($\n\\kappa \/ \\kappa_0 = 0.1$), the system approximates the\nDicke model well while $|\\Omega_g| t \/ 2\\pi < 2.5 $, where\nthe generalized Rabi frequency $|\\Omega_g|$ is given by Eq.~\\eqref{eq:E}. For\na better cavity ($ \\kappa \/ \\kappa_0 = 0.01$), the concurrence\napproaches its quasi-stationary value and the system approximates\nthe ideal Dicke dynamics for longer times, $|\\Omega_g| t \/ 2\\pi < 20$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig05}\n\\end{center}\n\\caption{\\label{fig:weakCouplingSlides} (Color online) Dynamics of the concurrence in the weak ion-cavity coupling regime for $r^{(1)}=0.55$. In the Dicke model with cavity losses (highest line) a stationary value of the concurrence is reached as the superradiant component is over-damped. Parameters: $\\Delta \/ \\Delta_0 = 100, \\kappa \/ \\kappa_0 = 0.1$ (lowest line, $2\\pi\/|\\Omega_g|=23$ $\\mu$s); $\\Delta \/ \\Delta_0 = 1000, \\kappa \/ \\kappa_0 = 0.01$ (middle line, $2\\pi\/|\\Omega_g|=230$ $\\mu$s).}\n\\end{figure}\n\nWe finally look at the statistics of the quantum jumps, described by the\njump operators $C_m^{(j)}$ in Eqs.~\\eqref{eq:scaledC11}-\\eqref{eq:scaledD12}.\nFirst of all, we note that the source states $|\\Phi_j\\rangle$ [see Eqs.~\\eqref{eq:decayingState1} and \\eqref{eq:decayingState2}] of the jump operators $C_S^{(j)}$ and $C_D^{(j)}$ are identical for a given atom $j=1,2$. Therefore, the jump statistics of the two corresponding decay channels will\nalso be the same with a branching ratio given by  $\\Gamma_S^{(j)} \/ \\Gamma_D^{(j)} = \\gamma_S \/ \\gamma_D \\simeq 13$. Our MCWF simulations confirm that the\ndominant jump processes are those corresponding to the effective spontaneous emission operators\n$C_S^{(j)}$ and the cavity photon annihilation operator $a$. In Fig.~\\ref{fig:weakCouplingJumps} we plot\nthe average cumulative number of quantum jumps per ensemble member for the jump operators $C_S^{(1)}$, $C_S^{(2)}$, and $a$.\n\nLooking at the statistics helps us to understand how the reservoir-mediated entanglement generation\nprocess depends on $r^{(1)}$. We notice that the jump statistics of processes originating from the spontaneous\nemissions of atoms 1 and 2 are different. This is of course due to the asymmetry in the initial condition. Since initially the excitation is present in\natom 1, the average cumulative number of jumps  per ensemble\nmember is typically greater for  $C_S^{(1)}$ than for $C_S^{(2)}$.\nThe peak in the cumulative number of jumps, for the three\ndifferent jump operators considered in\nFig.~\\ref{fig:weakCouplingJumps}, moreover, is reached in\ncorrespondence of different values of $r^{(1)}$. This indicates\nthat the value $r^{(1)}\\simeq 0.55$, which optimizes the concurrence generation (see Fig.~\\ref{fig:weakCoupling}), corresponds to a compromise\nbetween the different  $r^{(1)}$-dependent jump statistics. In\nparticular, the deviation from the optimal value in the absence of spontaneous emission ($r^{(1)}=0.5$) might be due to the fact that the number of $C_S^{(1)}$-jumps increases for decreasing values of\n$r^{(1)}$. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig06}\n\\end{center}\n\\caption{\\label{fig:weakCouplingJumps} (Color online) Average\ncumulative number of quantum jumps per ensemble member for each\ndecay channel in the weak ion-cavity coupling regime. From above:\n$C_S^{(1)}$, $C_S^{(2)}$, and $a$. Parameters: $\\Delta \/ \\Delta_0 = 100, \\kappa \/ \\kappa_0 = 0.1$.}\n\\end{figure}\n\n\n\n\n\n\\subsubsection{Strong ion-cavity coupling regime}\n\nIn the strong ion-cavity coupling regime, the cavity damping is slow\ncompared to the coherent dynamics. Therefore, a slowly damped\noscillatory behavior of the concurrence is expected. In\nFig.~\\ref{fig:strongCoupling} we plot the concurrence as a\nfunction of both time and the relative coupling strength $r^{(1)}$\nfor $\\Delta \/ \\Delta_0 = 10$ and $\\kappa \/ \\kappa_0 = 0.1$, giving $|g_\\textrm{eff}|\/ 2\\pi = \\xi g \\Omega \/ 2\\pi\\Delta = 170$~kHz. All the other parameters are chosen as in\nTable~\\ref{tab:physicalValues}. Note that the ratio\n$|g_\\textrm{eff}|\/\\kappa$  is now one order of magnitude bigger\nthan in Sec.~\\ref{sec:subsub}. The dynamics has an oscillatory\ncharacter, since the superradiant component survives much longer\nthan in the weak ion-cavity coupling regime. However, due to the presence of the spontaneous emissions the concurrence does not reach a steady-state value in this regime either.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig07}\n\\end{center}\n\\caption{\\label{fig:strongCoupling} (Color online) Concurrence as\na function of time for different values of the relative coupling\nstrength $r^{(1)}$ in the strong ion-cavity coupling regime.\nOscillations appear because of a relative phase evolution between the\nsuperradiant and subradiant states. Parameters: $\\Delta \/ \\Delta_0 = 10$ and $\\kappa \/ \\kappa_0 = 0.1$.}\n\\end{figure}\n\nThe best peak value of the concurrence, $C\\simeq0.6$, is\nnow obtained for $r^{(1)}\\simeq0.46$. In\nFig.~\\ref{fig:strongCouplingSlides} we choose this value of\n$r^{(1)}$ and we compare the dynamics of the single-mode Dicke\nmodel with cavity losses to the dynamics of the ion-cavity system\nin the presence of effective spontaneous emissions for the cases of\n$\\Delta \/ \\Delta_0 = 10$ with $\\kappa \/ \\kappa_0 = 0.1$,\nand $\\Delta \/ \\Delta_0 = 100$ with $\\kappa \/ \\kappa_0 = 0.01$.\nIn the second case, i.e., for a better quality factor, the system\napproximates the Dicke model for longer time scales, as one\nwould expect. In this case one can clearly observe the damped Rabi\noscillation at the generalized Rabi frequency given by\nEq.~\\eqref{eq:E}.\n\nIt is worth noticing that, in the strong ion-cavity coupling regime, the\nlaser-mediated interaction with the cavity vacuum allows us to\ngenerate a highly entangled state of the two ions, as one can see\nin  Fig.~\\ref{fig:strongCouplingSlides}. In particular, for\n$\\Delta \/ \\Delta_0 = 100$ with $\\kappa \/ \\kappa_0 = 0.01$,\nusing a laser pulse of duration $t \\simeq 2\\pi \/ |\\Omega_g|$, the generated state is close to a maximally entangled Bell state.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig08}\n\\end{center}\n\\caption{\\label{fig:strongCouplingSlides} (Color online) Time\nevolution of the concurrence in the strong ion-cavity coupling regime\nwith relative coupling strength $r^{(1)}=0.46$. For the Dicke\nmodel with cavity losses (highest line), the concurrence approaches\na constant stationary value after strong oscillations caused by\nthe slowly decaying superradiant component. Parameters: $\\Delta \/ \\Delta_0 =\n10, \\kappa \/ \\kappa_0 = 0.1$ (lowest line, $2\\pi\/|\\Omega_g|=2.7$ $\\mu$s); $\\Delta \/ \\Delta_0 = 100, \\kappa \/ \\kappa_0 = 0.01$ (middle line, $2\\pi\/|\\Omega_g|=27$ $\\mu$s).}\n\\end{figure}\n\n\n\n\n\n\\section{\\label{sec:dispersiveRegime}\nEnvironment-induced entanglement: Dispersive regime}\n\nIn the previous section we have seen that by placing the ions properly, i.e., by adjusting the relative coupling strength $r^{(1)}$, it is possible to optimize the reservoir-mediated entanglement generation. The\nexamples discussed above deal with the resonant effective model, which is defined by the condition $\\delta_\\textrm{eff} = 0$, which in turn corresponds to a physical detuning $\\delta_L = \\xi [ \\Omega^2 - ( \\beta_T g )^2 ]\/ \\Delta$ [cf.~Eq.~\\eqref{eq:deltaEff}]. We have seen that the highest value of\nthe concurrence is obtained in the strong ion-cavity coupling regime.\n\nIn Ref.~\\cite{Francica}, however, the single-mode Dicke model with\ncavity losses is studied in the dispersive regime, showing that a\nhigh degree of entanglement can be obtained also in the weak ion-cavity coupling regime. For this reason we now look at the off-resonant\nentanglement generation process in the ion-cavity QED, i.e., we\nconsider the case in which $\\delta_\\textrm{eff} \\neq 0$. In the\ndispersive regime, the relative position of the ions does not play\nan essential role and in fact one shows that the optimal value of\n$r^{(1)}$  is obtained for equal coupling of the two ions, i.e.,\n$r^{(1)} = r^{(2)} = 1\/\\sqrt{2}$~\\cite{Francica}.\n\nWe consider once more the initial atomic state $|\\psi (0) \\rangle =\n|1^{(1)} 0^{(2)}\\rangle$ combined with the cavity in the vacuum state\n$|0^{(C)}\\rangle$. We set $r^{(1)} = r^{(2)} = 1\/\\sqrt{2}$ (by choosing maximally strong cavity-driven couplings $\\beta^{(1)} = \\beta^{(2)} = 1$), $\\Delta \/ \\Delta_0 = 10$, and $\\kappa \/ \\kappa_0 = 0.1$,\ncorresponding to the weak ion-cavity coupling regime of\nSec.~\\ref{sec:subsub}. We now look at the time evolution of the\nconcurrence for different values of the laser detuning $\\delta_L$.\nFigure~\\ref{fig:dispersiveRegime} shows the concurrence as a\nfunction of both time and detuning $\\delta_L$. One can see clearly that the\nStark shift terms appearing in the effective Hamiltonian of\nEq.~\\eqref{eq:Heffion} relocate the resonance condition from\nthe origin to  $\\delta_L \/2\\pi = \\xi [ \\Omega^2 - ( \\beta_T g )^2 ]\/ 2\\pi\\Delta = 120$~kHz. Figure~\\ref{fig:dispersiveRegime}\nalso shows that selecting the detuning $\\delta_L$ further away from the resonance produces higher values of concurrence. In particular, with the chosen parameters the maximum value of concurrence $C \\simeq 0.62$ is obtained with $\\delta_L \/ 2\\pi \\simeq 600$~kHz.\n\nAs demonstrated in Ref.~\\cite{Francica}, increasing the detuning $|\\delta_\\textrm{eff}|$ correspondingly increases the time it takes for the concurrence to\nreach its peak value. The longer is the entanglement generation\ntime, however, the stronger is the effect of the spontaneous\nemissions. In other words, the achieved gain in the entanglement\ngeneration obtained by increasing the effective detuning is quickly suppressed due to the\nspontaneous decay, as the overall time of the entanglement\ngeneration process increases. The maximum value of entanglement\nachievable in the dispersive regime is therefore determined by the\ninterplay between these two effects.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig09}\n\\end{center}\n\\caption{\\label{fig:dispersiveRegime} (Color online) Concurrence\nas a function of time for different values of the detuning\n$\\delta_L$ for a homogeneously coupled case ($r^{(1)}=r^{(2)} = 1\/\\sqrt{2}$). The entanglement generation slows down when passing into the dispersive regime. The resonance is dislocated from the origin because of the Stark shifts. Parameters: $\\Delta \/\\Delta_0 = 10$ and $\\kappa \/ \\kappa_0 = 0.1$. }\n\\end{figure}\n\nIt is worth noticing that going from the resonant into the\ndispersive regime changes the character of the generated\nentangled state as well. To illustrate this point, we plot in\nFigs.~\\ref{fig:dispersiveRegimePopulations}\nand~\\ref{fig:dispersiveRegimeCoherences}  the populations and\ncoherences, respectively, of the reduced atomic density matrix versus\ntime and detuning $\\delta_L$. These plots confirm the increase in the\nentanglement generation time when going deeper and deeper into the dispersive regime ($|\\delta_\\textrm{eff}| > 0$). If we then focus on the dynamics of the coherences and, in particular, on the real and imaginary parts of the only nonzero off-diagonal element $\\rho_{01,10}$, we see that on resonance the imaginary part vanishes in accordance with the predictions of Sec.~\\ref{sec:resonantRegime}. Therefore, in the resonant regime the generated entangled state approximates the subradiant state. On the other hand, in the dispersive regime Re$[\\rho_{01,10}] \\simeq 0$ and Im$[\\rho_{01,10}]\\neq 0$. Indeed, in the absence of the spontaneous emissions, the generated state in the dispersive regime would be $\\left( \\big| 1^{(1)} 0^{(2)} \\big\\rangle  \\pm i \\big| 0^{(1)} 1^{(2)} \\big\\rangle\n\\right)\/\\sqrt{2}$ (positive sign for negative $\\delta_\\textrm{eff}$ and vice versa).\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig10}\n\\end{center}\n\\caption{\\label{fig:dispersiveRegimePopulations} (Color online)\nPopulations of the atomic states $\\rho_{00,00}$, $\\rho_{01,01}$,\nand $\\rho_{10,10}$ (from above) as a function of time for\ndifferent values of detuning $\\delta_L$. Parameters are as in Fig.~\\ref{fig:dispersiveRegime}. }\n\\end{figure}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig11}\n\\end{center}\n\\caption{\\label{fig:dispersiveRegimeCoherences} (Color online)\nDynamics of $\\textrm{Re}[\\rho_{01,10}]$,\n$\\textrm{Im}[\\rho_{01,10}]$, and $|\\rho_{01,10}|$ (from above) as\na function of time for different values of detuning $\\delta_L$\n(observed concurrence is given by $C=2|\\rho_{01,10}|$). Parameters are as in Fig.~\\ref{fig:dispersiveRegime}.}\n\\end{figure}\n\n\n\n\n\n\\section{\\label{sec:summary}\nSummary and Conclusions}\n\nIn this paper we have investigated how the single-mode Dicke model\ncan be realized under experimentally feasible conditions using two\ntrapped $^{40}$Ca$^+$ ions inside a high-finesse optical cavity.\nWe have taken into account the spontaneous emissions of the ions\nas well as the damping of the electromagnetic field inside the\ncavity. In particular, we have derived an effective two-level\ndescription of the three-level ions interacting with the cavity\nmode.\n\nWe have shown that under suitable conditions the two ions indeed behave\ncollectively, with a coherent dynamical evolution well\ndescribed by the Dicke model: two effective two-level systems exchanging an excitation with an effective one-dimensional cavity mode. The presence of decohering processes, such as the atomic spontaneous emission or the cavity field damping, modifies this ideal picture. However, in the effective\nmodel, the spontaneous emission decay rates are proportional to $1\/\\Delta^2$ whereas the ion-cavity couplings scale as $1 \/ \\Delta$, where\n$\\Delta$ is the detuning of the physical cavity frequency from the electronic\ntransition that it is driving. This difference in the scaling can\nbe exploited in order to partly suppress the destructive effect of\nthe atomic spontaneous emissions.\n\nWe have identified the generation of entanglement as a fingerprint\nof the cooperative atomic behavior and analyzed this process in\ndetail. In particular, we have proven that it is possible to enhance the entanglement generation process by positioning the ions appropriately at different locations with respect to the standing mode of the electromagnetic field inside the cavity. In the\nresonant case, where the two-level systems and the cavity mode\nhave the same frequency, we have shown that asymmetric coupling\nwith the cavity mode produces the highest degree of entanglement,\neven in presence of spontaneous emissions. We have studied both the\nweak and the strong ion-cavity coupling regimes, defined by the strength of \nthe ion-cavity excitation exchange compared to the cavity field damping rate, and found out the optimal conditions for entanglement generation in both cases.\n\nAnother possibility to optimize the entanglement generation is to\ngo to the dispersive regime in the ion-cavity coupling by using an off-resonant Raman\ntransition. The maximum degree of entanglement in the dispersive\nand in the resonant regimes, for realistic values of the\nparameters, is similar. Our results indicate, however, that the\ncharacter of the generated entangled state in the dispersive\nregime changes compared to the resonant case.\n\nOur experimental proposal is based on existing technology used in\nthe context of ion-cavity QED experiments\n\\cite{Walther,Piety,Matthias}.  In order to detect the generated\nentanglement, the state tomography of the atomic systems is needed. In\nrecent years, this has been routinely performed in similar trapped-ion systems, e.g., in the context of quantum computation and measuring the quality of quantum gates \\cite{Blatt09}. Therefore,\nwe expect our proposal to be within the reach of the experimental\ncommunity.\n\n\n\n\n\n\\acknowledgments\n\nThe authors thank K.-A. Suominen for useful discussions. S.M. thanks B. Garraway, M. Keller, and W. Lange for discussions on the experimental implementation of the ion-cavity QED setup and for the kind hospitality at the University of Sussex. This work was supported by the National Graduate School of Modern Optics and Photonics and the Magnus Ehrnrooth Foundation (K.H.), the Academy of Finland (Projects\nNo.~108699, No.~115682, No.~115982, and No.~8125004), the V\\\"ais\\\"al\\\"a Foundation, and the Turku Collegium of Science and Medicine (S.M.). \n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nThe Dicke model describes the dynamics of $N$ identical two-level\natoms interacting with a quantized three-dimensional\nelectromagnetic (EM) field \\cite{Dicke}. Under certain conditions\nthe model predicts that the atoms interact with the quantized EM field collectively, giving rise to the widely studied phenomena of\nsuperradiance and subradiance \\cite{HarochePR,tanas02}. In free\nspace ideal superradiance and subradiance take place in the so\ncalled small sample limit, i.e., when the atoms are so close to\neach other that one can ignore any effect resulting from  their\ndifferent spatial positions. In this case the atoms are\nindistinguishable with respect to their emission and absorption\nproperties; hence, the presence of equivalent paths through which\nthe emission process may occur gives rise to fully constructive\n(superradiance) or destructive (subradiance) interference.\n\nIdeal superradiance\/subradiance in free space is very difficult to\nobserve in the experiments since it requires that the atoms are\nplaced in a regular pattern within a sample smaller than the\nwavelength of the EM field they interact with (small sample case).\nThe requirement of a regular pattern is due to the presence of the\ndipole-dipole forces that would otherwise break the symmetry under\npermutation of any two atoms necessary to observe\nsuperradiant\/subradiant behavior. Such a regularity can be\nachieved, e.g., with trapped ion crystals \\cite{IonsRev} or atoms\nin optical lattices \\cite{Bloch08}. In these systems, however, the\nseparation between the particles is typically larger or of the same\norder of magnitude than the resonant wavelength (large sample\ncase). In the large sample case, cooperative effects still occur\nbut the subradiant state is not completely decoupled from the\ndynamics. Indeed, partial subradiance and superradiance have been \nobserved with trapped ions \\cite{DeVoe}.\n\nA way for relaxing the requirement for configuration regularity is\nto place the small sample in a cavity resonator. In this case,\nindeed, due to the Purcell effect, the cooperative atomic behavior\ncan be observed at much lower atomic density than in free space,\nmaking the van der Waals dephasing caused by the irregular atomic\nconfiguration negligible \\cite{HarochePR}. Experiments observing\nsuperradiance in the small sample case in a cavity have been\nperformed with Rydberg atoms \\cite{HarocheSuperR}, giving results\nin a very good agreement with the predictions of the single-mode\nsuperradiance theory. In this experiment, all of the atoms are\nequivalently coupled to the quantized mode of the EM field\n(homogeneous case).\n\nRecent advances in ion--cavity QED experiments make it possible to\nconfine arrays of ions inside an optical cavity in a regime in\nwhich the width of their wavepacket in position space is smaller\nthan the wavelength of the cavity mode they interact with\n(Lamb--Dicke regime) \\cite{Walther,Piety}. Moreover, it is\npossible to accurately manipulate the position of the single ions\nwith respect to the intensity profile of the standing cavity mode, thereby allowing to change the strength of the\ncoupling between each ion and the quantized EM field.\n\nIt has been  demonstrated theoretically that, when the atoms are\ncoupled with different strengths to the EM field, ideal\nsuperradiance or subradiance can still occur, depending on the\nparticular spatial distribution of the atoms\n\\cite{Benivegna,BenivegnaPL,BuzekZei}. However, no experiments\nhave up to now confirmed these predictions by the inhomogeneous Dicke model. Very recently, an important step in this direction has been achieved at the University of Aarhus, where a collective strong coupling between an ion crystal and a cavity mode was observed \\cite{Herskind2009}. In this paper, we\ninvestigate in detail how the inhomogeneous single-mode Dicke\nmodel (or Tavis--Cummings model \\cite{TCmodel}) can be realized in\nthe ion--cavity QED context and the conditions under which\nsubradiance and superradiance can be observed.\n\nBesides the importance in the study of fundamentals of quantum\ntheory, the realization of the Dicke model and the generation of\nthe subradiant state play a crucial role in quantum information\ntechnology and quantum communication. Indeed, arrays of ions are\nideal candidates for quantum registers and their controlled\ninteraction with photons allows to realize atom--light quantum\ninterfaces \\cite{Kimble} and to distribute entanglement to\ndifferent nodes of quantum networks. The importance of the\nsubradiant states in this context stems from the fact that they\nare robust entangled atomic states since they are completely\ndecoupled from the EM field.\n\nThe aim of this work is to discuss a realistic setup that is able\nto show the collective behavior of trapped ions in a cavity. In\nparticular, since in the experiments performed so far the ions are\ncoupled to the EM mode via a Raman scheme in a $\\Lambda$-configuration, we will include the entire level structure, which\nis important in order to understand the decohering role of the spontaneous\nemission from the upper and essentially unpopulated level. We\nwill also include cavity losses in order to study in detail the\ndeviation from the ideal cooperative Dicke model and to identify\nthe parameter regions in which such deviations are as small as\npossible.\n\nIn fact, during the last two decades, several theoretical papers\nhave discussed issues such as entanglement generation, preparation\nof nonclassical states, or realization of quantum gates in the\nion--cavity QED context assuming that the conditions to realize an\nideal Tavis--Cummings model were met\n\\cite{Pellizzari,vanEnk,Plenio,Zheng,Pachos,Lougovski,Chimczak,Li,Li07,Chimczak08,Bina}.\nThus, either the spontaneous emission or cavity losses (or both\nprocesses) are usually neglected \\cite{vanEnk,Zheng,Li,Li07}.\nConcerning spontaneous emission, for example, the assumption is\nmade that the emission rate is much smaller than the cavity\ncoupling constant\n\\cite{Pellizzari,Pachos,Lougovski,Chimczak,Chimczak08}. However,\nthis condition is not met in the ion--cavity QED experiments\n\\cite{Walther,Piety}. Furthermore, as we will demonstrate in this\npaper, if one deals with simplified atomic level structures\n\\cite{Plenio,Zheng,Bina} it is not possible to single out those\nregions in parameter space for which the systems of trapped ions\nbehave collectively.\n\nIn this paper, we will take both the cavity losses and the spontaneous\nemissions into account and employ $\\Lambda$-type schemes to describe\nthe ions and to identify the experimental conditions under which\nthe coherent dynamics predicted by the single-mode Dicke model is\ndominant with respect to losses and decoherence. This will also\nallow us to present realistic protocols for entanglement\ngeneration and to discuss ways to optimize the generated\nentanglement using specific features of the trapped-ion system,\nsuch as the ability to manipulate in a controlled way the relative\ncoupling between the ions and the cavity field.\n\nThe structure of the paper is the following. In\nSec.~\\ref{sec:DickeModel} we review the properties of the\ninhomogeneous single-mode Dicke model. In\nSec.~\\ref{sec:effectiveModel} we present the Hamiltonian for two\nions in a cavity and we make the connection to the Dicke model by deriving an effective model describing the dynamics under realistic experimental conditions. Section~\\ref{sec:resonantRegime} is devoted to the description of\nthe experimental proposal to observe subradiance and verify the\ninhomogeneous Dicke model. Furthermore, in Sec.~\\ref{sec:dispersiveRegime}\nwe explore another way to optimize the entanglement generation by\nusing off-resonant transitions. Finally, a summary of the results\nand the conclusions are given in Sec.~\\ref{sec:summary}.\n\n\n\n\n\n\\section{\\label{sec:DickeModel}\nInhomogeneous single-mode Dicke model}\n\n\n\n\\subsection{Ideal cavity}\n\nThe single-mode Dicke model, or Tavis--Cummings model, is the\nsimplest quantum mechanical model describing collective effects\nsuch as superradiance and subradiance in cavity. It describes the\nquasi-resonant interaction between $N$ identical two-level atoms\nand a single quantized cavity mode. The Tavis--Cummings\nHamiltonian is\n\\begin{align}\nH_\\textrm{D} =&\\,\\, \\omega_C \\left( a^{\\dag} a + \\frac{1}{2} \\right) + \\sum_{j=1}^N  \\omega_A \\sigma_+^{(j)} \\sigma_-^{(j)} \\nonumber \\\\\n& +\\sum_{j=1}^N \\left( \\alpha^{(j)} a^{\\dag} \\sigma_-^{(j)} + \\alpha^{(j)*} a \\sigma_+^{(j)}\\right),  \\label{eq:HDicke2}\n\\end{align}\nwhere $\\omega_C$ and $\\omega_A$ are the frequencies of the cavity\nmode and the atomic transition, respectively, $a$ and\n$a^{\\dag}$ are the annihilation  and creation operators for the\ncavity mode, and $ \\sigma_-^{(j)} =\\vert 0^{(j)} \\rangle \\langle\n1^{(j)} \\vert$ and $ \\sigma_+^{(j)}=(\\sigma_-^{(j)})^{\\dag}$ are\nthe lowering and raising operators for the $j$th atom, $\\vert\n0^{(j)} \\rangle$ and $\\vert 1^{(j)} \\rangle$ being its ground and\nexcited states, respectively. Finally, $\\alpha^{(j)}$ is the\ncoupling strength of the $j$th atom with the cavity field.\nInhomogeneity of the coupling strengths originates from\ndifferent relative positions of the atoms with respect to the\nintensity profile of the standing EM mode supported by the cavity\nresonator.\n\nThis model assumes that the cavity is ideal, as photon escape is\nnot taken into account, and that atomic spontaneous emission from the\nexcited to the ground state is negligible. The model also neglects\nthe atomic motion as well as recoil effects due to the absorption\nand subsequent re-emission of a photon by the atoms. Moreover, the\ndipolar coupling of the atoms and the EM field is expressed within\na rotating wave approximation (RWA), thereby suppressing the\nnon-energy-conserving terms. Finally, it implicitly assumes that\nthe coupling between the atoms and the cavity mode does not\nchange, i.e., that the atoms are kept at fixed positions. While\nthe RWA has been proven to work extremely well in optical\nexperiments, all other assumptions need further consideration. In\nthe following sections we will examine them in detail for the\nion--cavity QED setup.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig01}\n\\end{center}\n\\caption{\\label{fig:dickeModel} (Color online) Two binary quantum\nobjects interacting through a quantized electromagnetic mode\nsupported by a cavity resonator. The dynamics of such ideal system\nis described by the Dicke model. }\n\\end{figure}\n\nUsing a suitable canonical transformation it has been shown that,\nwhen only one excitation is present in the total system, the $N$\natoms interacting with the quantized field mode according to\nEq.~\\eqref{eq:HDicke2} cooperate in such a way that only one\ncollective atomic mode (superradiant state) is coupled to the\nfield  \\cite{BenivegnaPL}. Consequently, the energy exchange\nbetween the atoms and the field can be completely suppressed if\nthe only field-coupled collective mode is unexcited.\n\nFor simplicity, we will from now on focus on the $N=2$ case\nsketched in Fig.~\\ref{fig:dickeModel}, and  we will denote the\nenergy eigenstates for the free ions as $|a^{(1)} b^{(2)}\\rangle\n\\equiv |a^{(1)}\\rangle\\otimes |b^{(2)}\\rangle$, with $a,b=0,1$, and the corresponding Fock states of the cavity mode as $|n^{(C)}\\rangle$, where $n= 0,1, \\ldots$ .\nThe time evolution generated by $H_\\textrm{D}$ is easily obtained\nexplicitly. For a cavity initially prepared in the vacuum state,\nand in the presence of only one atomic excitation, the time\nevolution of the amplitudes $c_{10} (t)$ and $c_{01} (t)$ to find the ions\nin the states $\\vert 1^{(1)} 0^{(2)} \\rangle$ and $\\vert 0^{(1)}\n1^{(2)} \\rangle$, respectively, is given by\n\\begin{align}\n c_{10}(t) =& \\left[\\, |r^{(2)}|^2 + |r^{(1)}|^2 \\, {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& - r^{(1)*} r^{(2)} \\left[\\, 1- {\\cal E}(t)\\, \\right]c_{01}(0),\n\\label{eq:c1sing} \\\\\nc_{01}(t) =& - r^{(1)} r^{(2)*} \\left[\\, 1- {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& + \\left[\\, |r^{(1)}|^2 + |r^{(2)}|^2\\, {\\cal E}(t)\\,\n\\right]c_{01}(0).\n\\label{eq:c2sing}\n\\end{align}\nIn the equations above the relative coupling strengths are defined as $r^{(j)}=\\alpha^{(j)}\/|\\alpha_T |$, where $|\\alpha_T| =\n\\sqrt{ |\\alpha^{(1)}|^2 + |\\alpha^{(2)}|^2 }$ is  the total\ncoupling strength, and\n\\begin{equation}\n{\\cal E}(t) = e^{ i \\delta t \/2} \\Big[\n\\cos \\Big( \\frac{\\Omega_\\textrm{v} t}{2}\\Big) - i\\frac{\n\\delta}{\\Omega_\\textrm{v}} \\, \\sin \\Big( \\frac{\\Omega_\\textrm{v} t}{2}\\Big)\\Big],\n\\label{eq:Esing}\n\\end{equation}\nwhere $\\delta= \\omega_A-\\omega_C$ is the detuning and $\\Omega_\\textrm{v} = \\sqrt{ 4  |\\alpha_T|^2 + \\delta^2}$ is\nthe \\emph{vacuum Rabi frequency}. Note that $r^{(1)}$ and\n$r^{(2)}$ are not independent parameters, since\n$|r^{(2)}|=\\sqrt{1-|r^{(1)}|^2}$.\n\nThe subradiant $ \\ket{\\psi_-}$ and superradiant $ \\ket{\\psi_+}$ states are\n\\begin{align}\n\\big|\\psi_- \\big\\rangle &= r^{(2)} \\big| 1^{(1)} 0^{(2)} \\big\\rangle  - r^{(1)} \\big| 0^{(1)} 1^{(2)} \\big\\rangle,\n\\label{eq:psim} \\\\\n\\big|\\psi_+ \\big\\rangle &= r^{(1)*} \\big| 1^{(1)} 0^{(2)} \\big\\rangle + r^{(2)*} \\big|0^{(1)} 1^{(2)} \\big\\rangle,\n\\label{eq:psip}\n\\end{align}\nand, in this case, they are position dependent through the\nrelative coupling strength parameters $r^{(1)}$ and $r^{(2)}$. As\none can see directly from Eq.~\\eqref{eq:HDicke2}, the  state\n$\\ket{\\psi_-} \\otimes \\ket{0^{(C)}}$ is an eigenstate of the\nTavis--Cummings Hamiltonian  with eigenvalue $\\frac{1}{2}\\omega_C + \\omega_A$.\nTherefore, when the atoms are prepared in this state, they are\ncompletely decoupled from the cavity field and the system does not\nevolve at all. In the case of equally strong couplings, i.e., for\n$|r^{(1)}|=|r^{(2)}|=1\/\\sqrt{2}$, the subradiant and superradiant\nstates coincide with the maximally entangled Bell states. In\ngeneral, however, these states are not maximally entangled.\n\n\n\n\n\n\\subsection{\\label{sec:nonIdealCavity}\nNon-ideal cavity}\n\nWe now proceed to generalize Eq.~\\eqref{eq:HDicke2} to the case of\na lossy cavity. The imperfect reflectivity of the cavity mirrors\nand consequent leakage of photons causes a Lorentzian broadening\nof the spectral line corresponding to the mode supported by the ideal \ncavity. Accordingly, the microscopic atom--field interaction should\nnow take into account a continuum of modes described by a\nLorentzian distribution peaked at the central cavity frequency $\\omega_C$.\nFor the sake of simplicity, and in view of the discussion in the\nion--cavity QED context, we restrict our attention to a 1D cavity\nmodel. Namely, we neglect the coupling with all the EM modes other\nthan the ones supported by the lossy cavity. In the rotating wave\napproximation, the Hamiltonian is given by\n\\begin{align}\nH =& \\sum_{k} \\omega_k \\Big( a_k^{\\dag}a_k + \\frac{1}{2}\\Big) + \\sum_{j=1}^N \\omega_A \\sigma_+^{(j)}\\sigma_-^{(j)} \\nonumber \\\\\n&+ \\sum_{k} \\sum_{j=1}^N \\Big[ i g_k \\sin \\Big(\\frac{\\omega_k}{c} x^{(j)} \\Big) a_k^\\dagger \\sigma_-^{(j)} + h.c. \\Big], \\label{eq:secondaeq}\n\\end{align}\nwhere $a_k$ and $a^{\\dag}_k$ are the annihilation and creation\noperators of cavity photons of frequency $\\omega_k$, respectively.\nAbove, we have assumed that all the atoms have the same\nelectric dipole moment, which has been incorporated in the\ncoupling constants $g_k$, and we indicate with $x^{(j)}$ the\nposition of the atoms along the cavity axis. In the following we\nwill assume that each atom is kept at a fixed position inside the\ncavity and that they are all well localized, i.e., the spread of\ntheir wave function in position space is smaller than the\nwavelength of the central cavity field mode: $\\Delta x^{(j)} \\ll\nc\/\\omega_C$. Since all the significantly contributing modes are\nclose to the central mode (of frequency $\\omega_C$), we have\n\\begin{equation}\n\\sin \\left(\\frac{\\omega_k}{c} x^{(j)} \\right) \\simeq \\sin \\left(\\frac{\\omega_C}{c} x^{(j)} \\right),\n\\end{equation}\nand Eq.~\\eqref{eq:secondaeq} takes the form\n\\begin{align}\nH =& \\sum_{k} \\omega_k \\Big( a_k^{\\dag}a_k + \\frac{1}{2}\\Big) + \\sum_{j=1}^N \\omega_A \\sigma_+^{(j)}\\sigma_-^{(j)} \\nonumber \\\\\n&+ \\sum_{j=1}^N \\Big[ \\chi^{(j)} \\sigma_-^{(j)}  \\sum_{k} g_k   a_k^\\dagger + h.c. \\Big], \\label{eq:terzaeq}\n\\end{align}\nwith $\\chi^{(j)} = i \\sin (\\omega_C x^{(j)}\/c)$. In the continuum\nlimit the sum over the $k$-modes is replaced by an integral\n$$\n\\sum_k |g_k|^2 \\rightarrow \\int \\! d\\omega J(\\omega),\n$$\nwhere $J(\\omega)$ is the reservoir spectral density. As mentioned\nabove, we assume a Lorentzian distribution for the spectrum of the\nfield inside the cavity; therefore, we take a spectral density of\nthe form\n\\begin{equation}\nJ(\\omega) = \\frac{W^2}{2 \\pi} \\frac{\\kappa}{\\left( \\omega - \\omega_C \\right)^2 + (\\kappa \/ 2)^2},\n\\label{eq:J}\n\\end{equation}\nwhere the distribution is characterized by its full width at half\nmaximum (FWHM) value $\\kappa$ and by a normalization parameter $W^2 =\n\\int \\! d\\omega J(\\omega)$. Hence,  $\\kappa$ describes the cavity\nlosses and $W$ the total coupling strength.\n\nWe focus again on the two-atom case, i.e., $N=2$, and we consider\nthe situation in which only one excitation is present in the total\natoms--field system.\n\nStarting from the Hamiltonian~\\eqref{eq:secondaeq} and using the\nLorentzian spectral density~\\eqref{eq:J}, it is possible to derive\nan effective master equation\n\\begin{equation}\n\\frac{d \\varrho}{dt} = - i \\left[ H_\\textrm{D}, \\varrho \\right]\n-\\frac{\\kappa}{2} \\left[ a^{\\dag} a \\varrho + \\varrho a^{\\dag} a -\n2 a \\varrho a^{\\dag}\\right] \\label{eq:Dickeloss}\n\\end{equation}\nfor the dynamics of the atoms and the cavity mode of frequency\n$\\omega_C$ \\cite{ESDLaura}. Here $a$ and $a^{\\dag}$ are the\nannihilation and creation operators for the central cavity mode,\nwhich is damped at rate $\\kappa$, and the coherent dynamics is\ngenerated by $H_\\textrm{D}$ in Eq.~\\eqref{eq:HDicke2}, where the\ncoupling constants are identified as $\\alpha^{(j)} = \\chi^{(j)}\nW$. From the exact solution of the effective master\nequation~\\eqref{eq:Dickeloss} one can obtain the state of the\natomic system by tracing out the cavity degree of freedom: $\\rho\n(t) = \\textrm{tr}_C [\\varrho (t) ]$.\n\nAfter performing the trace, and for an initially empty cavity, the\nproblem can be solved exactly. In the ordered basis $\\left\\{ \\vert\n1^{(1)} 1^{(2)} \\rangle, \\vert 1^{(1)} 0^{(2)} \\rangle, \\vert\n0^{(1)} 1^{(2)} \\rangle, \\vert 0^{(1)} 0^{(2)} \\rangle \\right\\}$\nthe atomic density matrix can be written in the form \\cite{Man08}\n\\begin{equation}\n\\rho(t) = \n\\begin{pmatrix}\n   0& 0 & 0 & 0 & \\\\ 0& |c_{10}(t)|^2   &  c_{10}(t)c_{01}^*(t)  & 0\\\\\n 0& c_{10}^*(t)c_{01}(t)      & |c_{01}(t)|^2  & 0\\\\\n   0&  0  & 0 & 1-|c_{10}|^2-|c_{01}|^2\n\\end{pmatrix}.\n\\label{eq:rhos}\n\\end{equation}\nThe dynamics of the two qubits is therefore completely characterized by the two amplitudes\n\\begin{align}\nc_{10}(t) =& \\left[\\, |r^{(2)}|^2 + |r^{(1)}|^2 \\, {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& -r^{(1)*} r^{(2)} \\left[\\, 1- {\\cal E}(t)\\, \\right]c_{01}(0),\n\\label{eq:c1Sc1} \\\\\nc_{01}(t) =& - r^{(1)} r^{(2)*} \\left[\\, 1- {\\cal E}(t)\\,\n\\right]c_{10}(0) \\nonumber \\\\\n& + \\left[\\, |r^{(1)}|^2 + |r^{(2)}|^2\\, {\\cal E}(t)\\, \\right]c_{01}(0),\n\\label{eq:c2Sc1}\n\\end{align}\nwith $r^{(j)} = \\chi^{(j)} \/ |\\chi_T |$, where $|\\chi_T| = \\sqrt{ |\\chi^{(1)}|^2 + |\\chi^{(2)}|^2 }$, and\n\\begin{eqnarray}\n{\\cal E}(t) = e^{-(\\kappa -i 2\\delta) t \/4} \\Big[\n\\cos \\Big( \\frac{\\Omega_\\textrm{g} t}{2}\\Big) + \\frac{\\kappa- i 2\n\\delta}{2 \\Omega_\\textrm{g}} \\sin \\Big( \\frac{\\Omega_\\textrm{g} t}{2}\\Big) \\Big], \\nonumber \\\\\n\\label{eq:E}\n\\end{eqnarray}\nwhere $\\Omega_\\textrm{g}= \\sqrt{4 |\\chi_T|^2 W^2 + \\delta^2 +i\n\\delta \\kappa -\\kappa^2 \/ 4}$ is the \\emph{generalized Rabi\nfrequency}. Note that Eqs.~\\eqref{eq:c1Sc1}-\\eqref{eq:c2Sc1} have\nexactly the same structure as\nEqs.~\\eqref{eq:c1sing}-\\eqref{eq:c2sing}, obtained for the\nsingle-mode Dicke model without losses. Formally, the cavity losses appear as an\nadditional imaginary part of the detuning $\\delta \\mapsto \\delta + i\\kappa\n\/2$. Accordingly, the effect of the cavity losses is described by\nthe modification of the time-dependent coefficient ${\\cal E}(t)$,\nwhich is now damped at rate $\\kappa \/4$, and by the\n$\\kappa$-dependent shift of the Rabi frequency. For $\\kappa\n\\rightarrow 0$, the Lorentzian spectral density~\\eqref{eq:J} tends\nto Dirac's delta distribution, $J(\\omega ) \\rightarrow W^2 \\delta\n(\\omega - \\omega_C )$, and Eq.~\\eqref{eq:E} reduces to\nEq.~\\eqref{eq:Esing}, with $\\alpha^{(j)} = \\chi^{(j)} W$.\n\nIt is worth noticing that, as one sees directly from\nEq.~\\eqref{eq:terzaeq}, the subradiant state $\\vert \\psi_-\n\\rangle$, given by Eq.~\\eqref{eq:psim}, is still decoupled from\nthe vacuum cavity field. Hence, if the atomic system is initially\nprepared in this state, no exchange of excitation with the cavity\nfield will take place.\n\n\n\n\n\n\\section{\\label{sec:effectiveModel}\nEffective model of ion--cavity interaction}\n\n\n\n\\subsection{Physical setup}\n\nIon-cavity QED experiments use calcium ions which are trapped in a linear\nPaul microtrap and interact with a quantized mode of a\nhigh-finesse optical cavity~\\cite{Walther,Piety}. In\nFig.~\\ref{fig:threeLevelModel} we show the relevant energy level\nstructure, couplings, and decay channels for the compound system\nof two $^{40}$Ca$^+$ ions and a single cavity mode. The atomic\nground state $4^2$S$_{1\/2}$ is coupled to the electronically\nexcited  state $4^2$P$_{1\/2}$ by a (classical) pump laser injected\nfrom the side of the cavity. On the other hand, the excited state \n$4^2$P$_{1\/2}$ is coupled to a metastable state $3^2$D$_{3\/2}$ by the quantized cavity mode. The excited state\n$4^2$P$_{1\/2}$ decays spontaneously to the states $4^2$S$_{1\/2}$\nand $3^2$D$_{3\/2}$ at rates $\\gamma_S$ and $\\gamma_D$,\nrespectively, and the cavity photon is damped at rate $\\kappa$,\nas explained in the previous section.\n\n\\begin{figure*}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig02}\n\\end{center}\n\\caption{\\label{fig:threeLevelModel} (Color online) The relevant\nelectronic states of two identical $^{40}$Ca$^+$ ions and\ncorresponding couplings provided by an external pump laser and a\nquantized cavity mode. The excited electronic state decays\nspontaneously to the ground and to the metastable states, and the\ncavity mode is damped as well.}\n\\end{figure*}\n\nA realistic theoretical description of the dynamics of a single\n$^{40}$Ca$^+$ ion coupled to the cavity mode has been given in\nRef. \\cite{Matthias}. The authors consider there also the effect\nof cavity losses and spontaneous emission, taking into account all\nthe Zeeman sublevels of the three relevant  electronic states. The\nmain consequence of the presence of the Zeeman sublevels is a\nreduction of the coupling driven by the cavity field by a factor of \n$\\sqrt{3}$ with respect to the simpler three-level model\nconsidered here. Therefore, we will use in the following a\nthree-level model scheme with such a reduced effective coupling to\naccount for the presence of the Zeeman sublevels. In the\nexperiments, the ions sit at the bottom of the trapping potential\nand are cooled down to the Lamb-Dicke regime. Under these\nconditions one can assume that the ions are kept at fixed\npositions and neglect recoil during the emission-absorption\nprocess.\n\nIn the following we will consider as initial atomic states those\nin which one of the two atoms is in its ground state and the other\none is in its metastable state, i.e., the states $\\ket{S^{(1)}\nD^{(2)}}$ and $\\ket{ D^{(1)} S^{(2)} }$. In order to prepare these\nstates, if the vibrational sidebands are not resolved, it is\nnecessary to use a selective laser  addressing of the individual\nions. This is routinely done in trapped-ion experiments with\n$^{40}$Ca$^+$ ions (see, e.g., \\cite{Blatt09}).\n\nThe two identical ions interact with the quantized cavity mode of\nfrequency $\\omega_C$ via laser-assisted two-photon processes, as\nshown in Fig.~\\ref{fig:threeLevelModel}. The ions are irradiated\nby a laser beam of frequency $\\omega_L+\\delta_L$. The laser beams\nand the cavity field are far detuned by $\\Delta$ from the\nelectronic level $\\ket{P^{(j)}}$, such that $\\omega_P - \\omega_D =\n\\omega_C + \\Delta$ and $\\omega_P - \\omega_S = \\omega_L + \\Delta$.\nTherefore, the setup provides each ion $j=1,2$ with a Raman coupling between the levels $\\ket{S^{(j)}}$ and $\\ket{D^{(j)}}$.\n\nThe time evolution of the composite system of the two ions and the cavity mode can be described by a master equation\n\\begin{align}\n\\frac{d \\varrho}{dt} =& -i \\big[ H(t),\\varrho \\big] - \\frac{\\kappa}{2} \\big(a^{\\dag}a \\varrho + \\varrho a^{\\dag}a-2a \\varrho a^{\\dag} \\big) \\nonumber \\\\\n&- \\frac{\\gamma_S}{2} \\sum_{j=1,2} \\Big( A_{PP}^{(j)} \\varrho + \\varrho A_{PP}^{(j)} - 2 A_{SP}^{(j)} \\varrho A_{PS}^{(j)} \\Big) \\nonumber \\\\\n& - \\frac{\\gamma_D}{2} \\sum_{j=1,2} \\Big( A_{PP}^{(j)} \\varrho + \\varrho A_{PP}^{(j)} - 2 A_{DP}^{(j)} \\varrho A_{PD}^{(j)} \\Big),\n\\label{eq:model2me}\n\\end{align}\nwhere we have included the cavity field damping at rate $\\kappa$, the\nspontaneous emission channels (two for each ion) at rates\n$\\gamma_S$ and $\\gamma_D$, and where the coherent dynamics is\ngenerated by a Hamiltonian\n\\begin{multline}\nH(t) = \\omega_C \\Big( a^{\\dag}a + \\frac{1}{2} \\Big) +  \\sum_{j=1,2} \\sum_{l=S,P,D}\\omega_l A_{ll}^{(j)}   \\\\\n+ \\sum_{j=1,2}\\Big( e^{-i (\\omega_L+\\delta_L)t} g^{(j)}_L A_{PS}^{(j)} + g_C^{(j)} a\\, A_{PD}^{(j)} + h.c. \\Big). \\label{eq:model3H}\n\\end{multline}\nThe atomic operators are defined as $A_{ll'}^{(j)}=\\vert l^{(j)}\n\\rangle \\langle l'^{(j)} \\vert$, with $l,l'=S,P,D$ and $j=1,2$.\nFinally, the coherent couplings provided by the laser and the\ncavity mode are, respectively,\n\\begin{align}\ng_L^{(j)} &=\\, \\Omega\\, e^{i k_L x^{(j)}}, \\label{eq:gl} \\\\\ng_C^{(j)} &=\\, g \\sin (k_C x^{(j)} ),  \\label{eq:gc}\n\\end{align}\nwith $k_L$ and $k_C$ being the wavenumbers of the laser and the standing cavity mode.\n\n\n\n\n\n\\subsection{\\label{sec:effectiveProcesses}\nEffective two-level model}\n\nWhen the detuning $\\Delta$ is sufficiently large compared to the\ncouplings, $\\Delta \\gg g_L^{(j)}, g_C^{(j)}$, the excited\nelectronic states $\\ket{P^{(j)}}$ can be adiabatically eliminated\nfrom the dynamics, as described in detail in Appendix~\\ref{app:adiabaticElimination}. In this\ncase the system can be effectively described as composed of two\ntwo-level atoms interacting with a cavity mode. For this purpose,\nwe denote the ground and the metastable states of the $j$th atom\nas $\\ket{1^{(j)}} \\equiv \\ket{S^{(j)}}$ and $\\ket{0^{(j)}} \\equiv\n\\ket{D^{(j)}}$ (N.B. the true atomic ground state corresponds to \nthe excited state of the effective two-level system, since it is\nable to emit a cavity photon through the Raman transition).\n\nThe adiabatic elimination of the excited levels $\\{ |P^{(j)}\\rangle \\}$ is not at all\ntrivial due to the inclusion of the spontaneous emission\nprocesses \\cite{DiFidio}. We show in Appendix~\\ref{app:adiabaticElimination} that an effective\nTavis--Cummings Hamiltonian can be derived, describing an\nexcitation exchange between the ions and the cavity. However, one\nneeds to include (i) two Stark shift terms per ion (one, in\nparticular, being dependent on the state of the cavity mode) and (ii) an overall\nre-scaling  of both the free and the interaction energies by a\nfactor explicitly dependent on the emission rates.\n\nIt turns out that in the interaction picture with respect to $H_0-\\Delta \\sum_{j}\nA_{PP}^{(j)}$, where $H_0$ is given by the first two terms on the\nright-hand side of Eq.~\\eqref{eq:model3H}, the coherent part of the evolution of the ion--cavity system is described by an effective Hamiltonian\n\\begin{align}\nH_{\\rm eff} = & -\\xi \\sum_{j=1,2} \\Big[ \\Big( e^{-i \\delta_L t} \\frac{\\beta^{(j)} g^* \\Omega}{\\Delta}\\, a^{\\dag} A_{01}^{(j)} + h.c. \\Big) \\nonumber \\\\\n& +\\frac{|\\beta^{(j)}g|^2}{\\Delta} \\, a^{\\dag} a A_{00}^{(j)} +\n\\frac{|\\Omega|^2}{\\Delta} A_{11}^{(j)} \\Big], \\label{eq:Heffion}\n\\end{align}\nwhere the position-dependent parameters $\\beta^{(j)}$ are defined\nas\n\\begin{equation}\n\\beta^{(j)} = e^{i k_L x^{(j)}} \\sin \\big( k_C x^{(j)} \\big),\n\\label{eq:coefficient}\n\\end{equation}\nand the dimensionless renormalizing prefactor is\n\\begin{equation}\n\\xi =  \\frac{\\Delta^2}{\\Delta^2 + (\\gamma_S+\\gamma_D)^2\/4}.\n\\label{eq:xi}\n\\end{equation}\nThis Hamiltonian resembles the Tavis--Cummings Hamiltonian~\\eqref{eq:HDicke2}, except for the photon-dependent Stark shift term. However, since the original microscopic model includes\ndissipative processes, the unitary evolution generated by $H_{\\rm\neff}$ needs to be supplemented by decohering terms that have a very peculiar structure. Indeed, the effective master equation that describes the time evolution of the ions and the cavity contains four (now both dissipative and non-dissipative) processes (described by jump operators) that take into account the effects of the spontaneous emission as seen in the\nrestricted atomic subspaces spanned by $\\{|0^{(j)}\\rangle, |1^{(j)}\\rangle \\}$. The cavity damping appears in the restricted subspace in the same form as in the original model. \n\nThe effective master equation reads\n\\begin{align}\n\\frac{d \\varrho}{dt} =& -i \\left[ H_{\\rm eff}, \\varrho\\right] - \\frac{\\kappa}{2} \\left( a^{\\dag}a \\varrho + \\varrho a^{\\dag}a -2a \\varrho a^{\\dag} \\right) \\nonumber \\\\\n& - \\sum_{\\substack{j=1,2\\\\m=S,D}} \\frac{\\Gamma_m^{(j)}}{2} \\Big[ C_m^{(j)\\dag}C_m^{(j)} \\varrho +\\varrho \\, C_m^{(j)\\dag}C_m^{(j)} \\nonumber \\\\\n& \\qquad\\qquad\\qquad\\, -2  C_m^{(j)} \\varrho \\, C_m^{(j)\\dag} \\Big],\n\\label{eq:meeffion}\n\\end{align}\nwhere the jump operators for each ion $j$ are\ngiven by\n\\begin{align}\nC_S^{(j)} &= e^{-i \\delta_L t} \\Omega \\, A_{11}^{(j)} + \\beta^{(j)*} g\\, a A_{10}^{(j)}, \\label{eq:jumpC1} \\\\\nC_D^{(j)} &= e^{-i \\delta_L t} \\Omega \\, A_{01}^{(j)} + \\beta^{(j)*} g\\, a A_{00}^{(j)}, \\label{eq:jumpC2}\n\\end{align}\nwhile the effective decay rates are $\\Gamma_m^{(j)} =\\xi \\gamma_m \/ \\Delta^2$, where $m=S,D$ and the prefactor $\\xi$ is given by Eq.~\\eqref{eq:xi}. The structure of these jump operators is easy to interpret once\nthe full level configurations of Fig.~\\ref{fig:threeLevelModel}\nis taken into account. Let us consider, for example, the operator\n$C_S^{(j)}$ of Eq.~(\\ref{eq:jumpC1}). It arises from the spontaneous emission process $4^2$P$_{1\/2}\\rightarrow 4^2$S$_{1\/2}$ of the $j$th atom,  being now restricted to the two-dimensional subspace $\\{|0^{(j)}\\rangle,|1^{(j)}\\rangle\\}$. The jump operator $C_S^{(j)}$ has two contributions, both of\nthem describing non-dissipative decoherence by pure dephasing processes (as one understands from the fact that they do not produce any excitation loss). These two\ncontributions account for the interruption of the ion--cavity\nexcitation exchange (vacuum Rabi cycle) by the spontaneous\nemission. The first term is an unwanted repopulation of state  $|1^{(j)}\\rangle$ occurring after the laser has virtually brought the\nsystem to the intermediate level $|P^{(j)}\\rangle$ of the full Raman cycle. The\nsecond term is also due to decay into state $|1^{(j)}\\rangle$, but this time the virtual excitation of level $|P^{(j)}\\rangle$ is caused by the cavity field. In conclusion, both processes interrupt the\nvacuum Rabi cycle without the excitation being lost as, at the\nend, the two-level system is found in its excited state $|1^{(j)}\\rangle$. This\nimplies that the excitation exchange can restart, but with a different\nphase. Thus, $C_S^{(j)}$ describes a phase error.\n\nA similar interpretation scheme can be adopted for the two terms\nconstituting $C_D^{(j)}$ in Eq.~\\eqref{eq:jumpC2}. However, this time the involved process is the\nspontaneous emission $4^2$P$_{1\/2}\\rightarrow 3^2$D$_{3\/2}$. Whether it occurs after the\nvirtual excitation of level $|P^{(j)}\\rangle$ performed by the laser (first term) or by the\ncavity field (second term), the result is that at the end the two-level system is found in its ground state $|0^{(j)}\\rangle$ and that one excitation has been lost either from the atom or from the cavity mode. Therefore, this jump operator causes dissipative decoherence.\n\nWe note that, at this stage, the four jump operators of\nEqs.~\\eqref{eq:jumpC1}-\\eqref{eq:jumpC2} are both explicitly time\ndependent and implicitly position dependent via the coefficients\n$\\beta^{(j)}$ [see Eq.~\\eqref{eq:coefficient}].\n\nA phase rotation within the restricted Hilbert space, spanned by\nthe states with at maximum one excitation, allows transforming\nthe effective Hamiltonian~\\eqref{eq:Heffion} into the Tavis--Cummings\nHamiltonian~\\eqref{eq:HDicke2} as well as removing simultaneously the\ntime dependence from the jump operators~\\eqref{eq:jumpC1}-\\eqref{eq:jumpC2}. This is described in Appendix~\\ref{app:phaseRotation}. Therefore,\nin a suitable rotating frame,  the following effective\nTavis--Cummings Hamiltonian is obtained\n\\begin{align}\nH_\\textrm{D}^\\textrm{eff} =&\\,\\, \\omega_C^\\textrm{eff} \\Big( a^{\\dag} a + \\frac{1}{2} \\Big) + \\sum_{j=1,2}  \\omega_A^\\textrm{eff} \\, \\sigma_+^{(j)} \\sigma_-^{(j)} \\nonumber \\\\\n& +\\sum_{j=1,2} \\Big( \\alpha_\\textrm{eff}^{(j)} \\, a^{\\dag} \\, \\sigma_-^{(j)} + \\alpha_\\textrm{eff}^{(j)*} \\, a \\, \\sigma_+^{(j)}\\Big), \\label{eq:Hdeff}\n\\end{align}\nwhere we have introduced again the spin inversion operators used in Sec.~\\ref{sec:DickeModel}. The effective Dicke model parameters are\n\\begin{align}\n\\omega_C^\\textrm{eff} & = -\\xi \\, \\frac{2 |\\beta_T g|^2}{3 \\Delta}, \\label{eq:omegaCEff}\\\\\n\\omega_A^\\textrm{eff} & = \\delta_L - \\xi \\Big( \\frac{|\\Omega|^2}{\\Delta} - \\frac{|\\beta_T g|^2}{3\\Delta} \\Big) , \\label{eq:omega0Eff}\\\\\n\\alpha_\\textrm{eff}^{(j)} & = -\\xi \\, \\frac{ \\beta^{(j)} g^* \\Omega }{\\Delta} \\equiv \\beta^{(j)} g_\\textrm{eff}, \\label{eq:alphaEff}\n\\end{align}\nwhere $|\\beta_T | = \\sqrt{|\\beta^{(1)} |^2 + |\\beta^{(2)} |^2 }$. The effective detuning is given by\n\\begin{equation}\n\\delta_\\textrm{eff} = \\omega_A^\\textrm{eff} - \\omega_C^\\textrm{eff} = \\delta_L -\\xi \\, \\frac{ |\\Omega|^2 - |\\beta_T g|^2 }{\\Delta }, \\label{eq:deltaEff}\n\\end{equation}\nand the relative effective coupling strengths $r^{(j)}$ [cf.~Eqs.~\\eqref{eq:c1sing}-\\eqref{eq:c2sing}] are directly given by the position-dependent parameters $\\beta^{(j)}$,\nsince now $r^{(j)} = \\alpha_\\textrm{eff}^{(j)} \/\n|\\alpha_{\\textrm{eff},T}| = \\beta^{(j)} \/ |\\beta_T|$.\n\nComparing Eqs.~\\eqref{eq:Hdeff} and \\eqref{eq:meeffion} with\nEqs.~\\eqref{eq:HDicke2} and \\eqref{eq:Dickeloss}, respectively, we\nsee that, when the effective atomic spontaneous emissions are\nnegligible, this system allows to realize the Dicke model in the\nnon-ideal cavity case.\n\n\n\n\n\n\\subsection{\\label{sec:scaling}\nEffective spontaneous emission processes}\n\nAs mentioned before, we restrict our study to the case in which\nonly one or zero quanta are present in the composite system of the two ions and the cavity mode.\nTherefore, the compound state of the two atoms and the cavity\nphoton can be expressed in the basis  $\\{ \\ket{ 0^{(1)} 0^{(2)}\n0^{(C)} }$, $\\ket{ 0^{(1)} 0^{(2)} 1^{(C)} }$, $\\ket{ 0^{(1)}\n1^{(2)} 0^{(C)} }$, $\\ket{ 1^{(1)} 0^{(2)} 0^{(C)} } \\}$ (see\nAppendix~\\ref{app:phaseRotation}). Consequently, the jump operators~\\eqref{eq:jumpC1}-\\eqref{eq:jumpC2} can  be normalized with respect to the operator\nnorm $\\|A\\| = \\sup_{\\|\\phi\\|=1} \\|A\\ket{ \\phi } \\|$, where $\\ket{\n\\phi }$ belongs to the Hilbert space spanned by the basis defined\nabove. The introduction of the normalized jump operators allows to\ndefine the effective spontaneous emission decay rates\n$\\Gamma_m^{(j)}$ unambiguously.\n\nThe normalized jump operators are\n\\begin{align}\nC_S^{(1)} &= |1^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle \\Phi_1 |, \\label{eq:scaledC11}\\\\\nC_S^{(2)} &= |0^{(1)} 1^{(2)} 0^{(C)}\\rangle \\langle \\Phi_2 |, \\label{eq:scaledC12}\\\\\nC_D^{(1)} &= |0^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle \\Phi_1 |,\\\\\nC_D^{(2)} &= |0^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle \\Phi_2 |, \\label{eq:scaledD12}\n\\end{align}\nwhere the decaying states are\n\\begin{align}\n|\\Phi_1 \\rangle & = \\frac{ \\Omega^* \\, |1^{(1)} 0^{(2)} 0^{(C)}\\rangle  + \\beta^{(1)} g^*\\, |0^{(1)} 0^{(2)} 1^{(C)}\\rangle }{\\sqrt{|\\Omega |^2 + |\\beta^{(1)} g|^2 } }, \\label{eq:decayingState1} \\\\\n|\\Phi_2 \\rangle & = \\frac{ \\Omega^* \\, |0^{(1)} 1^{(2)} 0^{(C)}\\rangle + \\beta^{(2)} g^*\\, |0^{(1)} 0^{(2)} 1^{(C)}\\rangle }{\\sqrt{|\\Omega |^2 + |\\beta^{(2)} g|^2 } }. \\label{eq:decayingState2}\n\\end{align}\nThe corresponding rescaled decay rates are given by\n\\begin{align}\n\\Gamma_S^{(j)} &= \\xi \\, \\big( |\\Omega|^2 + |\\beta^{(j)} g|^2 \\big) \\frac{ \\gamma_S }{ \\Delta^2 }, \\label{eq:scaledDecayRateS}\\\\\n\\Gamma_D^{(j)} &= \\xi \\, \\big( |\\Omega|^2 + |\\beta^{(j)} g|^2 \\big) \\frac{ \\gamma_D }{ \\Delta^2 }. \\label{eq:scaledDecayRateD}\n\\end{align}\nThe cavity photon annihilation operator $a = |0^{(1)} 0^{(2)} 0^{(C)}\\rangle \\langle 0^{(1)} 0^{(2)} 1^{(C)} |$ is already normalized in our restricted basis.\n\nThe spontaneous emission decay rates for the considered states of\na calcium atom are $\\gamma_S = 2 \\pi \\times 22.3$ MHz and\n$\\gamma_D = 2 \\pi \\times 1.7$ MHz. Therefore, $\\Gamma_S^{(j)} \\gg\n\\Gamma_D^{(j)}$ and the dominant effective spontaneous emission\njump processes are described by the operators\n$C_S^{(j)}$. Consequently, according to the discussion above, the\nmain decoherence sources are the non-dissipative dephasing processes that conserve the energy of the ion--cavity system.\n\nThe character of the decaying state, and hence the corresponding\njump operator is defined by the balance between the strengths of the laser pumping $\\Omega$ and the cavity coupling $\\beta^{(j)} g$. In the \\emph{strong laser pumping} case ($|\\Omega |\n\\gg |\\beta^{(j)} g|$) the non-unitary dynamics of the atomic\nreduced system is dominated by phase diffusion processes described\nby the operators $A_{11}^{(j)}$. In the \\emph{weak laser pumping}\ncase ($|\\Omega | \\ll |\\beta^{(j)} g|$), on the contrary, the\nprocesses described by the operators $a A_{10}^{(j)}$\ndominate. Moreover, as one can see from\nEqs.~\\eqref{eq:decayingState1}-\\eqref{eq:decayingState2}, one can\nfurther modify the character of the specific atomic jump operators\nby changing the relative position of the ions with respect to the\ncavity field through the $\\beta^{(j)}$ parameters.\n\nThe significance of the spontaneous emissions can be estimated by the ratio\n\\begin{equation}\n\\bigg| \\frac{ \\Gamma_S^{(j)} }{ \\alpha_\\textrm{eff}^{(j)} } \\bigg| = \\frac{ 1 + |\\beta^{(j)} g \/ \\Omega |^2 }{ | \\beta^{(j)} g \/ \\Omega | } \\, \\frac{ \\gamma_S }{ \\Delta }.\n\\label{eq:decayVersusCoupling}\n\\end{equation}\nFor a fixed detuning $\\Delta$ this ratio has its minimum value $2\n\\gamma_S \/ \\Delta $ when $|\\beta^{(j)} g \/  \\Omega| = 1$, i.e.,\nwhen the couplings provided by the laser and the cavity field are\nequally strong. On the other hand, for fixed coupling strengths,\nthe ratio is inversely proportional to the detuning $\\Delta$. This\ncan be exploited in order to minimize the role of the effective spontaneous decay. The\ncavity damping $\\kappa$ is neither affected by the detuning nor the couplings.\n\nFinally, we note that for large detunings, $\\Delta \\gg \\gamma_S, \\gamma_D $, the dimensionless prefactor $\\xi \\sim 1$ and the effective decay rates as well as the effective coupling terms have simplified expressions. The effective couplings are then given by $\\alpha_\\textrm{eff}^{(j)} \\sim -\\beta^{(j)} g^* \\Omega \/ \\Delta$, while in the limit of strong and weak laser pumping the dominating decay rates are $\\Gamma_S^{(j)} \\sim |\\Omega|^2 \\gamma_S \/ \\Delta^2$ and $\\Gamma_S^{(j)} \\sim |\\beta^{(j)} g|^2 \\gamma_S \/ \\Delta^2$, respectively. \n\n\n\n\n\n\\section{\\label{sec:resonantRegime}\nEnvironment-induced entanglement: Resonant regime}\n\nIn this section we study, analytically and numerically, the\ndynamics of the entanglement between the electronic degrees of\nfreedom of the two atoms. The generation of entanglement between\nthe ions and its persistence at long times are, indeed, a clear\nmanifestation of the collective (subradiant) behavior. In\nparticular, entanglement generation is mediated by the interaction\nwith the quantized cavity field which is initially prepared in the\nvacuum state. If the atomic spontaneous emission processes are negligible\nand we face the bare Dicke model, the dynamics can be described\nexactly. We compare these exact analytic results to numerical\nsimulations including the spontaneous emission effects. The\nsimulations were implemented by using the Monte Carlo wave\nfunction (MCWF) method \\cite{MCWF,ZollerCarmichael}. We begin by\nconsidering the resonant case, where the effective detuning\n$\\delta_\\textrm{eff} = 0$, with $\\delta_\\textrm{eff} $ given by\nEq.~\\eqref{eq:deltaEff}.\n\n\n\n\n\n\\subsection{\\label{sec:analytical}\nAnalytical solution neglecting spontaneous emission}\n\nThe effective model describing the dynamics when spontaneous emissions are negligible is given by the master equation~\\eqref{eq:Dickeloss} with the effective Tavis--Cummings Hamiltonian~\\eqref{eq:Hdeff}, as described in Sec.~\\ref{sec:nonIdealCavity}. The analytical solution for the atomic density matrix is given by Eqs.~\\eqref{eq:rhos}-\\eqref{eq:E}, with $\\chi^{(j)} W = \\alpha_\\textrm{eff}^{(j)} = \\beta^{(j)} g_\\textrm{eff}$.\n\nWe are interested in the collective dynamics when initially one excitation is present in the atomic system and the cavity is in its vacuum state. Any initial atomic state containing one excitation can be written in terms of the superradiant and subradiant states~\\eqref{eq:psim}-\\eqref{eq:psip} as\n\\begin{equation}\n\\ket{\\psi(0) }= \\beta_+  \\ket{\\psi_+}  + \\beta_- \\ket{\\psi_-}. \\label{eq:nonloso}\n\\end{equation}\nAs time passes, the collective atomic state decays via the evolution of the superradiant component\n\\begin{equation}\n\\langle \\psi_+ \\vert \\psi (t) \\rangle = {\\cal E}(t) \\, \\beta_+,\n\\end{equation}\nwith ${\\cal E}(t)$ given by Eq.~\\eqref{eq:E}. The subradiant component $\\langle \\psi_- \\vert \\psi (t) \\rangle = \\beta_-$, however, remains unchanged. Consequently, for times, such that $\\kappa t \\gg 1$, the atomic state will be in general a statistical mixture of the collective ground state $| 0^{(1)} 0^{(2)} \\rangle$ and the subradiant state $\\ket{\\psi_-}$ with weights dependent on $\\beta_-$, which in turn depends on the relative coupling strengths $r^{(j)}$.\n\nIn the following we focus on the dynamics of entanglement between the atoms. In order to quantify the stationary asymptotic entanglement of the final state we use Wootters's concurrence \\cite{wootte} which, for a density matrix of the form of Eq.~\\eqref{eq:rhos}, is given by\n\\begin{equation}\nC(t) = 2 \\left| c_{10}(t) c_{01}^*(t)\\right|, \\label{eq:concurrdef}\n\\end{equation}\nwith $c_{10}(t)$ and $c_{01}(t)$ given by Eqs.~\\eqref{eq:c1Sc1}-\\eqref{eq:c2Sc1}. In general, the concurrence is zero for factorized states and unity for maximally entangled states. For $\\kappa t \\gg 1$ we obtain a stationary concurrence value\n\\begin{equation}\nC_\\textrm{stat}=2 |r^{(1)}r^{(2)}| \\left| \\beta_- \\right|^2.\n\\end{equation}\nAs expected, the value of the stationary concurrence is directly\nrelated to the subradiant component of the initial state. If both\natoms are coupled to the EM field, the stationary value of the\nconcurrence, for any initial state with $\\beta_- \\ne 0$ will be\nnonzero. When the atoms are initially prepared in the\nsuperradiant state, i.e., $\\beta_- =0$, the system approaches\nasymptotically the pure factorized state $| 0^{(1)} 0^{(2)}\n\\rangle$.\n\nFor the initially factorized states $\\ket{1^{(1)} 0^{(2)}}$ and\n$\\ket{0^{(1)} 1^{(2)}}$ the interaction with the environment\ngenerates entanglement in the atomic system. For these initial\nstates the stationary concurrence takes the values $C_\\textrm{stat} =\n2 |r^{(1)}|(1- |r^{(1)}|^2)^{3\/2} $ and $C_\\textrm{stat} = 2\n|r^{(1)}|^3 \\sqrt{1- |r^{(1)}|^2}$, respectively. As we have\nnoticed in Ref. \\cite{Man08}, the factorized states are those that\nmaximize the stationary concurrence for certain values of\n$r^{(1)}$. The maximum value of stationary concurrence, for both\nthe two factorized initial states considered here, is\n$C_\\textrm{stat}^\\textrm{max} = \\max_{\\,|r^{(1)}| \\in [0,1]}\nC_\\textrm{stat} \\simeq 0.65$. This value is obtained with\n$|r^{(1)}|= 0.5$ and $|r^{(1)}| \\simeq 0.87$ (i.e., $|r^{(2)}|=\n0.5$) for initial states $\\ket{1^{(1)} 0^{(2)}}$ and $\\ket{0^{(1)}\n1^{(2)}}$, respectively.\n\nWe note in passing that when only one of the two atoms is coupled\nto the EM field, i.e., $r^{(1)}=0$ or $r^{(2)}=0$, the stationary\nconcurrence is zero. In this case, indeed, the subradiant and\nsuperradiant states coincide with states $\\ket{1^{(1)} 0^{(2)}}$\nand $\\ket{0^{(1)} 1^{(2)}}$ as one can see from\nthe definitions~\\eqref{eq:psim}-\\eqref{eq:psip}.\n\nFrom the definition of the generalized Rabi frequency given by\nEq.~\\eqref{eq:E}, which in the resonant case reads as\n$\\Omega_\\textrm{g} = \\sqrt{4 |\\beta_T g_\\textrm{eff}|^2 - \\kappa^2\n\/ 4}$, two extreme regimes can be defined. In the \\textsl{weak ion--cavity\ncoupling regime}, defined by  $4 |\\beta_T g_\\textrm{eff} |\n\\ll \\kappa$,  the generalized Rabi frequency is purely imaginary.\nTherefore, according to Eq.~\\eqref{eq:E}, the Dicke model predicts\na solution given by monotonic hyperbolic sine and cosine\nfunctions. The opposite limit is the \\textsl{strong ion--cavity coupling regime}, defined by $4 |\\beta_T g_\\textrm{eff} | \\gg\n\\kappa $. In this case the generalized Rabi frequency is real and\nthe Dicke model predicts damped oscillatory dynamics.\n\n\n\n\n\n\\subsection{MCWF simulations in presence of spontaneous emission}\n\nIn this section, we focus on the effect of the spontaneous emissions on\nthe subradiant-state-based entanglement generation described in\nthe previous section. We consider again as initial atomic\nstate $|\\psi (0) \\rangle = |1^{(1)} 0^{(2)}\\rangle$ with the cavity\nin the vacuum state $|0^{(C)}\\rangle$. For a given value of $r^{(1)} \\in [0,1]$, we choose $\\beta^{(1)}$ and $\\beta^{(2)}$ to be positive real numbers such that the larger of the two is always unity and the smaller one is $\\min \\{ r^{(1)}\/ \\sqrt{ 1 - r^{(1)2}}, \\sqrt{1 - r^{(1)2}}\/ r^{(1)} \\}$ [cf.~definition~\\eqref{eq:coefficient}]. Now $|\\beta_T|^2 = |\\beta^{(1)}|^2 + |\\beta^{(2)}|^2 = \\textrm{min} \\{ 1\/r^{(1)2},1\/ ( 1-r^{(1)2}) \\} \\in [1,2]$. The physical parameters\nhave been chosen in accordance to the experiments of Ref.\n\\cite{Walther} and  are summarized in\nTab.~\\ref{tab:physicalValues}.  The size of the ensemble in the\nMCWF simulations is $N=1000$. We are using the variant of MCWF\nmethod described in \\cite{ZollerCarmichael}.\n\n\\begin{table}[tb]\n\\caption{ \\label{tab:physicalValues} Values of physical quantities\nused in the simulations. Note that the cavity coupling is here\nexplicitly scaled by the Clebsch--Gordan factor $1\/\\sqrt{3}$ and,\nin the text, also by the position-dependent parameters\n$\\beta^{(j)}$.}\n\\begin{center}\n\\begin{tabular}{l@{\\quad}c@{\\quad}r@{}l}\n\\hline\n\\hline\nQuantity & Symbol & \\multicolumn{2}{c}{Value [2$\\pi \\times$MHz]} \\\\ \\hline\nLaser coupling & $\\Omega$ & \\qquad\\quad 9&.0 \\\\\nCavity coupling & $g$ & 6&.5 $\/ \\sqrt{3}$\\\\\nDecay rate $4^2$P$_{1\/2}\\rightarrow 4^2$S$_{1\/2}$ & $\\gamma_S$ & 22&.3 \\\\\nDecay rate $4^2$P$_{1\/2}\\rightarrow 3^2$D$_{3\/2}$ & $\\gamma_D$ & 1&.7 \\\\\nDetuning & $\\Delta_0$ & 20&.0 \\\\\nCavity damping & $\\kappa_0$ & 1&.2 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe value of the cavity coupling constant $g$ in\nTab.~\\ref{tab:physicalValues} refers to  the new miniature trap\nrecently realized at the University of Sussex \\cite{W}. The\nreference value $\\kappa_0$ for the cavity damping can nowadays be\nimproved by at least one order of magnitude. Finally, the detuning\n$\\Delta$ can be easily increased in the experiments, with respect\nto the reference value $\\Delta_0$.\n\nWith the experimental parameters of Tab.~\\ref{tab:physicalValues}, the coupling strengths $\\Omega$ and $g$ are of the same order. Therefore, neither the strong nor the weak laser pumping regimes, introduced\nin Sec.~\\ref{sec:scaling}, are reached and, consequently, all the effective\ndecay processes caused by the spontaneous emission are\ncombinations of two different physical operations, as interpreted in Sec.~\\ref{sec:effectiveProcesses}. \n\nLet us denote the atomic density matrix components as\n$\\rho_{ab,cd} \\equiv \\langle a^{(1)} b^{(2)} | \\rho | c^{(1)}\nd^{(2)} \\rangle$, where $a,b,c,d=0,1$. The density matrix remains\nstill in the same block form of Eq.~\\eqref{eq:rhos} even in the\npresence of spontaneous emissions. The concurrence is therefore\ngiven by $C(t) = 2 |\\rho_{01,10}(t)|$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig03}\n\\end{center}\n\\caption{\\label{fig:scaling} (Color online) Scaling of the\neffective coupling $|g_\\textrm{eff}|$ and the effective\nspontaneous emission decay rates $\\Gamma_S^{(j)}$ (with\n$\\beta^{(j)}=1$) as a function of the detuning $\\Delta$. The\nisocurves  $4 |\\beta_T g_\\textrm{eff}|\/\\kappa = const.$ (thin lines) are parallel to the $|g_\\textrm{eff}|$ curve so that the\nweak ion--cavity coupling regime is in the upper right corner and the\nstrong ion--cavity coupling regime in the lower left one. The cavity\ndecay rate $\\kappa$ does not depend on the detuning. The effective\nspontaneous emission events are suppressed for large detunings.}\n\\end{figure}\n\nIn the following we will examine the effect of the spontaneous\nemissions by comparing the concurrence as a function of time for\nfixed values of $4 |\\beta_T g_\\textrm{eff}|\/\\kappa = 4 |\\beta_T\n\\xi g \\Omega  \/ \\kappa \\Delta|$. We study large detunings ($\\Delta \\gg \\gamma_S, \\gamma_D$), so the prefactor $\\xi \\sim 1$. In the examples we change $\\kappa$ and $\\Delta$, such that $\\kappa \/\\kappa_0 = 0.1, 0.01$ and $\\Delta \/ \\Delta_0 = 10, 100, 1000$, while keeping the product $\\kappa\n\\Delta$ constant. Physically, this corresponds to using different cavity qualities and detunings which, furthermore, influences the effective dynamical parameters. Larger\ndetunings, indeed, suppress the effective spontaneous emissions in\nfavor of the coherent dynamics, as explained in\nSec.~\\ref{sec:scaling}. The situation is clarified in\nFig.~\\ref{fig:scaling} which shows the scaling of the effective\ncoupling strength $g_\\textrm{eff}$ and the dominant spontaneous\nemission decay rate $\\Gamma_S^{(j)}$ [cf.\nEqs.~\\eqref{eq:alphaEff} and\n\\eqref{eq:scaledDecayRateS}-\\eqref{eq:scaledDecayRateD}] as functions of detuning $\\Delta$. The cavity damping rate $\\kappa$ is not\naffected by the detuning. The relative values of the three key\nparameters $g_\\textrm{eff}$, $\\Gamma_S^{(j)}$ and $\\kappa$\ncharacterize the dynamical regime: (i) the ratio\n$|g_\\textrm{eff}|\/\\kappa$ defines the strong and weak\nion--cavity coupling regimes; (ii) the magnitude of $\\Gamma_S^{(j)}$ compared to\n$|g_\\textrm{eff}|$ and $\\kappa$, in turn, describes the significance of the spontaneous emission processes and tells us whether the dynamics is well described by the Dicke model or not.\n\n\n\n\n\n\\subsubsection{Weak ion--cavity coupling regime}\\label{sec:subsub}\n\nIn this regime, the oscillatory dynamics stemming from the\ncoherent coupling between the atoms and the cavity is heavily\ndamped. In Fig.~\\ref{fig:weakCoupling} we plot the concurrence as\na function of both time and the relative coupling strength $r^{(1)}$\nfor $\\Delta = 100 \\times \\Delta_0$ and $\\kappa=0.1 \\times\n\\kappa_0$, giving $|g_\\textrm{eff}| = \\xi g \\Omega\/\\Delta = 2\\pi\n\\times 17$~kHz. All the other parameters are chosen as in\nTab.~\\ref{tab:physicalValues}. We recall that initially the atomic state $|\\psi (0) \\rangle = |1^{(1)} 0^{(2)}\\rangle$ is factorized. The initial dynamics of the concurrence\nshows a monotonic increase, as the superradiant component [see\nEq.~\\eqref{eq:nonloso}] rapidly fades away while the subradiant\ncomponent remains intact. However, because of the presence of\nspontaneous emission, the subradiant state is not anymore\nperfectly decoupled from the dynamics and, consequently, the concurrence\nwill not reach a steady state value.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig04}\n\\end{center}\n\\caption{\\label{fig:weakCoupling} (Color online) Concurrence as a\nfunction of time and the relative coupling strength $r^{(1)}$\nin the weak ion--cavity coupling regime. The dynamics is initially\nmonotonic since the existing superradiant component decays rapidly\ncompared to other dynamical time scales. The subradiant state\ncomponent decays eventually because of the atomic spontaneous\nemissions. The best entanglement production occurs with asymmetric couplings ($r^{(1)} \\neq 1\/\\sqrt{2}$). Parameters: $\\Delta = 100 \\times \\Delta_0, \\kappa=0.1\n\\times \\kappa_0$}\n\\end{figure}\n\nWe note that the best peak value of the concurrence $C\\simeq0.6$\nis achieved for $r^{(1)}\\simeq0.55$, i.e., as expected, for an\nasymmetric configuration ($r^{(1)} \\neq 1\/ \\sqrt{2}$) of the ions with respect to the cavity\nfield. However, this value of  $r^{(1)}$ is now slightly different\nthan the one obtained in Sec.~\\ref{sec:analytical} where\nspontaneous emissions were neglected ($r^{(1)}=0.5$). We will\nfurther discuss this point when considering the position\ndependence of the jumps statistics at the end of this subsection.\n\nIn Fig.~\\ref{fig:weakCouplingSlides} we further study the effect\nof the spontaneous emissions in the weak ion--cavity coupling case. In this\nfigure, we compare the predictions of the Dicke model, described\nin Sec.~\\ref{sec:nonIdealCavity}, with the dynamics of the\nion--cavity system in presence of the spontaneous emissions for $\\Delta\n= 100 \\times \\Delta_0, \\kappa=0.1 \\times \\kappa_0$ and  $\\Delta =\n1000 \\times \\Delta_0, \\kappa=0.01 \\times \\kappa_0$. The dynamics\nof the concurrence clearly shows that in the first case ($\n\\kappa=0.1 \\times \\kappa_0$), the system approximates well the\nDicke model for times $t < 2.5 \\times 2\\pi \/ |\\Omega_\\textrm{g}|$, where\nthe generalized Rabi frequency $|\\Omega_\\textrm{g}|$ is given by Eq.~\\eqref{eq:E}. For\na better cavity ($ \\kappa=0.01 \\times \\kappa_0$), the concurrence\napproaches its quasi-stationary value and the system approximates\nthe ideal Dicke dynamics for longer times, $t < 20 \\times 2\\pi \/ |\\Omega_\\textrm{g}|$.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig05}\n\\end{center}\n\\caption{\\label{fig:weakCouplingSlides} (Color online) Dynamics of\nthe concurrence in the weak ion--cavity coupling regime for\n$r^{(1)}=0.55$. In the Dicke model with cavity losses (highest line)\na stationary value of the concurrence is reached as the\nsuperradiant component is over-damped. Parameters: $\\Delta = 100\n\\times \\Delta_0, \\kappa=0.1 \\times \\kappa_0$ (lowest line,\n$2\\pi\/|\\Omega_\\textrm{g}|=23$ $\\mu$s); $\\Delta = 1000 \\times\n\\Delta_0, \\kappa=0.01 \\times \\kappa_0$ (middle line,\n$2\\pi\/|\\Omega_\\textrm{g}|=230$ $\\mu$s).}\n\\end{figure}\n\nWe finally look at the statistics of the quantum jumps, described by the\njump operators $C_m^{(j)}$ in Eqs.~\\eqref{eq:scaledC11}-\\eqref{eq:scaledD12}.\nFirst of all, we note that the source states $|\\Phi_j\\rangle$ (see Eqs.~\\eqref{eq:decayingState1}-\\eqref{eq:decayingState2}) of the jump operators $C_S^{(j)}$ and $C_D^{(j)}$ are identical for a given atom $j=1,2$. Therefore, the jump statistics of the two corresponding decay channels will\nalso be the same with a branching ratio given by  $\\Gamma_S^{(j)} \/ \\Gamma_D^{(j)} = \\gamma_S \/ \\gamma_D \\simeq 13$. Our MCWF simulations confirm that the\ndominant jump processes are those corresponding to the effective spontaneous emission operators\n$C_S^{(j)}$ and the cavity photon annihilation operator $a$. In Fig.~\\ref{fig:weakCouplingJumps} we plot\nthe average cumulative number of quantum jumps per ensemble member for the jump operators $C_S^{(1)}$, $C_S^{(2)}$, and $a$.\n\nLooking at the statistics helps to understand how the reservoir-mediated entanglement generation\nprocess the depends on $r^{(1)}$. We notice that the jump statistics of processes originating from the spontaneous\nemissions of atom 1 and atom 2 are different. This is of course due to the asymmetry in the initial condition. Since initially the excitation is present in the\natom 1, the average cumulative number of jumps  per ensemble\nmember is typically greater for  $C_S^{(1)}$ than for $C_S^{(2)}$.\nThe peak in the cumulative number of jumps, for the three\ndifferent jump operators considered in\nFig.~\\ref{fig:weakCouplingJumps}, moreover, is reached in\ncorrespondence of different values of $r^{(1)}$. This indicates\nthat the value $r^{(1)}\\simeq 0.55$, which optimizes the concurrence generation (see Fig.~\\ref{fig:weakCoupling}), corresponds to a compromise\nbetween the different  $r^{(1)}$-dependent jump statistics. In\nparticular, the deviation from the optimal value in the absence of spontaneous emission ($r^{(1)}=0.5$) might be due to the fact that the number of $C_S^{(1)}$-jumps increases for decreasing values of\n$r^{(1)}$. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig06}\n\\end{center}\n\\caption{\\label{fig:weakCouplingJumps} (Color online) Average\ncumulative number of quantum jumps per ensemble member for each\ndecay channel in the weak ion--cavity coupling regime. From above:\n$C_S^{(1)}$, $C_S^{(2)}$, and $a$. Parameters: $\\Delta = 100\n\\times \\Delta_0, \\kappa=0.1 \\times \\kappa_0$.}\n\\end{figure}\n\n\n\n\n\n\\subsubsection{Strong ion--cavity coupling regime}\n\nIn the strong ion--cavity coupling regime, the cavity damping is slow\ncompared to the coherent dynamics. Therefore, a slowly damped\noscillatory behavior of the concurrence is expected. In\nFig.~\\ref{fig:strongCoupling} we plot the concurrence as a\nfunction of both time and the relative coupling strength $r^{(1)}$\nfor $\\Delta = 10 \\times \\Delta_0$ and $\\kappa=0.1 \\times\n\\kappa_0$, giving $|g_\\textrm{eff}| = \\xi g \\Omega \/ \\Delta = 2\\pi\n\\times 170$~kHz. All the other parameters are chosen as in\nTab.~\\ref{tab:physicalValues}. Note that the ratio\n$|g_\\textrm{eff}|\/\\kappa$  is now one order of magnitude bigger\nthan in Sec.~\\ref{sec:subsub}. The dynamics has an oscillatory\ncharacter, since the superradiant component survives much longer\nthan in the weak ion--cavity coupling regime. However, due to the presence of the spontaneous emissions the concurrence does not reach a steady state value in this regime either.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig07}\n\\end{center}\n\\caption{\\label{fig:strongCoupling} (Color online) Concurrence as\na function of time for different values of the relative coupling\nstrength $r^{(1)}$ in the strong ion--cavity coupling regime.\nOscillations appear because of a relative phase evolution between the\nsuperradiant and subradiant states. Parameters: $\\Delta = 10 \\times \\Delta_0$ and $\\kappa=0.1 \\times \\kappa_0$.}\n\\end{figure}\n\nThe best peak value of the concurrence, $C\\simeq0.6$, is\nnow obtained for $r^{(1)}\\simeq0.46$. In\nFig.~\\ref{fig:strongCouplingSlides} we choose this value of\n$r^{(1)}$ and we compare the dynamics of the single-mode Dicke\nmodel with cavity losses to the dynamics of the ion--cavity system\nin presence of effective spontaneous emissions for the cases of\n$\\Delta = 10 \\times \\Delta_0$ with $ \\kappa=0.1 \\times \\kappa_0$,\nand $\\Delta=100\\times \\Delta_0$ with $\\kappa=0.01\\times \\kappa_0$.\nIn the second case, i.e., for a better quality factor, the system\napproximates the Dicke model for longer time scales, as one\nwould expect. In this case one can clearly observe the damped Rabi\noscillation at the generalized Rabi frequency, given by\nEq.~\\eqref{eq:E}.\n\nIt is worth noticing that, in the strong ion--cavity coupling regime, the\nlaser-mediated interaction with the cavity vacuum allows to\ngenerate a highly entangled state of the two ions, as one can see\nin  Fig.~\\ref{fig:strongCouplingSlides}. In particular, for\n$\\Delta=100\\times \\Delta_0$ with $\\kappa=0.01\\times \\kappa_0$,\nusing a laser pulse of duration $t \\simeq 2\\pi \/ |\\Omega_\\textrm{g}|$, the generated state is close to a maximally entangled Bell state.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig08}\n\\end{center}\n\\caption{\\label{fig:strongCouplingSlides} (Color online) Time\nevolution of the concurrence in the strong ion--cavity coupling regime\nwith relative coupling strength $r^{(1)}=0.46$. For the Dicke\nmodel with cavity losses (highest line), the concurrence approaches\na constant stationary value after strong oscillations caused by\nthe slowly decaying superradiant component. Parameters: $\\Delta =\n10 \\times \\Delta_0, \\kappa=0.1 \\times \\kappa_0$ (lowest line,\n$2\\pi\/|\\Omega_\\textrm{g}|=2.7$ $\\mu$s); $\\Delta = 100 \\times\n\\Delta_0, \\kappa=0.01 \\times \\kappa_0$ (middle line,\n$2\\pi\/|\\Omega_\\textrm{g}|=27$ $\\mu$s).}\n\\end{figure}\n\n\n\n\n\n\\section{\\label{sec:dispersiveRegime}\nEnvironment-induced entanglement: Dispersive regime}\n\nIn the previous section we have seen that by placing the ions properly, i.e., by adjusting the relative coupling strength $r^{(1)}$, it is possible to optimize the reservoir-mediated entanglement generation. The\nexamples discussed above deal with the resonant effective model, which is defined by the condition $\\delta_\\textrm{eff} = 0$, which in turn corresponds to a physical detuning $\\delta_L = \\xi [ \\Omega^2 - ( \\beta_T g )^2 ]\/ \\Delta$ [cf.~Eq.~\\eqref{eq:deltaEff}]. We have seen that the highest value of\nthe concurrence is obtained in the strong ion--cavity coupling regime.\n\nIn Ref.~\\cite{Francica}, however, the single-mode Dicke model with\ncavity losses is studied in the dispersive regime, showing that a\nhigh degree of entanglement can be obtained also in the weak ion--cavity coupling regime. For this reason we now look at the off-resonant\nentanglement generation process in the ion--cavity QED, i.e., we\nconsider the case in which $\\delta_\\textrm{eff} \\neq 0$. In the\ndispersive regime, the relative position of the ions does not play\nan essential role and in fact one shows that the optimal value of\n$r^{(1)}$  is obtained for equal coupling of the two ions, i.e.,\n$r^{(1)} = r^{(2)} = 1\/\\sqrt{2}$~\\cite{Francica}.\n\nWe consider once more the initial atomic state $|\\psi (0) \\rangle =\n|1^{(1)} 0^{(2)}\\rangle$ combined with the cavity in the vacuum state\n$|0^{(C)}\\rangle$. We set $r^{(1)} = r^{(2)} = 1\/\\sqrt{2}$ (by choosing maximally strong cavity-driven couplings $\\beta^{(1)} = \\beta^{(2)} = 1$), $\\Delta = 10 \\times \\Delta_0$ and $\\kappa=0.1 \\times \\kappa_0$,\ncorresponding to the weak ion--cavity coupling regime of\nSec.~\\ref{sec:subsub}. We now look at the time evolution of the\nconcurrence for different values of the laser detuning $\\delta_L$.\nFigure~\\ref{fig:dispersiveRegime} shows the concurrence as a\nfunction of both time and detuning $\\delta_L$. One can see clearly that the\nStark shift terms appearing in the effective Hamiltonian of\nEq.~\\eqref{eq:Heffion} relocate the resonance condition from\nthe origin to  $\\delta_L = \\xi [ \\Omega^2 - ( \\beta_T g )^2 ]\/ \\Delta = 2\\pi \\times 120$~kHz. Figure~\\ref{fig:dispersiveRegime}\nalso shows that selecting the detuning $\\delta_L$ further away from the resonance produces higher values of concurrence. In particular, with the chosen parameters the maximum value of concurrence $C \\simeq 0.62$ is obtained with $\\delta_L \\simeq 2 \\pi \\times 600$~kHz.\n\nAs demonstrated in Ref.~\\cite{Francica}, increasing the detuning $|\\delta_\\textrm{eff}|$ correspondingly increases the time it takes for the concurrence to\nreach its peak value. The longer is the entanglement generation\ntime, however, the stronger is the effect of the spontaneous\nemissions. In other words, the achieved gain in the entanglement\ngeneration obtained by increasing the effective detuning is quickly suppressed due to the\nspontaneous decay, as the overall time of the entanglement\ngeneration process increases. The maximum value of entanglement\nachievable in the dispersive regime is therefore determined by the\ninterplay between these two effects.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig09}\n\\end{center}\n\\caption{\\label{fig:dispersiveRegime} (Color online) Concurrence\nas a function of time for different values of the detuning\n$\\delta_L$ for an homogeneously coupled case ($r^{(1)}=r^{(2)} = 1\/\\sqrt{2}$). The entanglement generation slows down when passing into the dispersive regime. The resonance is dislocated from the origin by $2\\pi \\times 120$~kHz  because of the Stark shifts. Parameters: $\\Delta = 10 \\times \\Delta_0$ and $\\kappa=0.1 \\times \\kappa_0$. }\n\\end{figure}\n\nIt is worth noticing that going from the resonant into the\ndispersive regime changes the character of the generated\nentangled state as well. To illustrate this point, we plot in\nFigs.~\\ref{fig:dispersiveRegimePopulations}\nand~\\ref{fig:dispersiveRegimeCoherences}  the populations and\ncoherences, respectively, of the reduced atomic density matrix versus\ntime and detuning $\\delta_L$. These plots confirm the increase in the\nentanglement generation time when going deeper and deeper into the dispersive regime ($|\\delta_\\textrm{eff}| > 0$). If we then focus on the dynamics of the coherences, and in particular on the real and imaginary parts of the only nonzero off-diagonal element $\\rho_{01,10}$, we see that on resonance the imaginary part vanishes in accordance with the predictions of Sec.~\\ref{sec:resonantRegime}. Therefore, in the resonant regime the generated entangled state approximates the subradiant state. On the other hand, in the dispersive regime Re$[\\rho_{01,10}] \\simeq 0$ and Im$[\\rho_{01,10}]\\neq 0$. Indeed, in the absence of the spontaneous emissions, the generated state in the dispersive regime would be $\\left( \\big| 1^{(1)} 0^{(2)} \\big\\rangle  \\pm i \\big| 0^{(1)} 1^{(2)} \\big\\rangle\n\\right)\/\\sqrt{2}$ (positive sign for negative $\\delta_\\textrm{eff}$ and vice versa).\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig10}\n\\end{center}\n\\caption{\\label{fig:dispersiveRegimePopulations} (Color online)\nPopulations of the atomic states $\\rho_{00,00}$, $\\rho_{01,01}$,\nand $\\rho_{10,10}$ (from above) as a function of time for\ndifferent values of detuning $\\delta_L$. Parameters are as in Fig.~\\ref{fig:dispersiveRegime}. }\n\\end{figure}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics{Harkonen_DickeModel_fig11}\n\\end{center}\n\\caption{\\label{fig:dispersiveRegimeCoherences} (Color online)\nDynamics of $\\textrm{Re}[\\rho_{01,10}]$,\n$\\textrm{Im}[\\rho_{01,10}]$, and $|\\rho_{01,10}|$ (from above) as\na function of time for different values of detuning $\\delta_L$\n(observe concurrence is given by $C=2|\\rho_{01,10}|$). Parameters are as in Fig.~\\ref{fig:dispersiveRegime}.}\n\\end{figure}\n\n\n\n\n\n\\section{\\label{sec:summary}\nSummary and Conclusions}\n\nIn this paper we have investigated how the single-mode Dicke model\ncan be realized under experimentally feasible conditions using two\ntrapped $^{40}$Ca$^+$ ions inside a high-finesse optical cavity.\nWe have taken into account the spontaneous emissions of the ions\nas well as the damping of the electromagnetic field inside the\ncavity. In particular, we have derived an effective two-level\ndescription of the three-level ions interacting with the cavity\nmode.\n\nWe have shown that under suitable conditions the two ions indeed behave\ncollectively, with a coherent dynamical evolution well\ndescribed by the Dicke model: two effective two-level systems exchanging an excitation with an effective one-dimensional cavity mode. The presence of decohering processes, such as the atomic spontaneous emission or the cavity field damping, modifies this ideal picture. However, in the effective\nmodel, the spontaneous emission decay rates are proportional to $1\/\\Delta^2$ whereas the ion--cavity couplings scale as $1 \/ \\Delta$, where\n$\\Delta$ is the detuning of the physical cavity frequency from the electronic\ntransition that it is driving. This difference in the scaling can\nbe exploited in order to partly suppress the destructive effect of\nthe atomic spontaneous emissions.\n\nWe have identified the generation of entanglement as a fingerprint\nof the cooperative atomic behavior and analyzed this process in\ndetail. In particular, we have proven that it is possible to enhance the entanglement generation process by positioning the ions appropriately at different locations with respect to the standing mode of the electromagnetic field inside the cavity. In the\nresonant case, where the two-level systems and the cavity mode\nhave the same frequency, we have shown that asymmetric coupling\nwith the cavity mode produces the highest degree of entanglement,\neven in presence of spontaneous emissions. We have studied both\nweak and strong ion--cavity coupling regimes, defined by the strength of \nthe ion--cavity excitation exchange compared to the cavity field damping rate, and found out the optimal conditions for entanglement generation in both cases.\n\nAnother possibility to optimize the entanglement generation is to\ngo to the dispersive regime in the ion--cavity coupling by using an off-resonant Raman\ntransition. The maximum degree of entanglement in the dispersive\nand in the resonant regimes, for realistic values of the\nparameters, is similar. Our results indicate, however, that the\ncharacter of the generated entangled state in the dispersive\nregime changes compared to the resonant case.\n\nOur experimental proposal is based on existing technology used in\nthe context of ion--cavity QED experiments\n\\cite{Walther,Piety,Matthias}.  In order to detect the generated\nentanglement, state tomography of the atomic systems is needed. In\nrecent years, this has been routinely performed in similar trapped-ion systems, e.g., in the context of quantum computation and measuring the quality of quantum gates \\cite{Blatt09}. Therefore,\nwe expect our proposal to be within the reach of the experimental\ncommunity.\n\n\n\n\n\n\\acknowledgments\n\nThe authors thank K.-A. Suominen for useful discussions. S.M. thanks B. Garraway, M. Keller, and W. Lange for discussions on the experimental implementation of the ion--cavity QED setup and for the kind hospitality at the University of Sussex. This work has been supported by the National Graduate School of Modern Optics and Photonics and the Magnus Ehrnrooth Foundation (K.H.), and the Academy of Finland (Projects\nNo.~108699, No.~115682, No.~115982, and No.~8125004), the V\\\"ais\\\"al\\\"a Foundation and the Turku Collegium of Science and Medicine (S.M.). \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Supplementary Materials}\n\n\\subsection{2-D generalization of nonlinear L\\'{e}vy walk model}\n\nIn 2-D we consider the random motion of an individual that runs in the\ndirection $\\mathbf{\\theta }=(\\cos \\varphi ,\\sin \\varphi )$ with the constant\nvelocity $v$ \\ during the running time $\\tau $, and changes the direction at\n$(\\mathbf{x},t)$ to $\\mathbf{\\theta }^{\\prime }=(\\cos \\varphi ^{\\prime\n},\\sin \\varphi ^{\\prime })$. The turning rate $\\mathbb{T}_{\\rho }\\left(\n\\mathbf{x},t,\\tau ,\\varphi ,\\varphi ^{\\prime }\\right) $ from $\\varphi $ to \n\\varphi ^{\\prime }$ at $(\\mathbf{x},t)$ depends on the running time $\\tau $\nand the non-local interactions with neighboring conspecifics. We define the\nmean structural density of individuals, $n(\\mathbf{x},t,\\tau ,\\varphi ),$ at\npoint $\\mathbf{x}$ and time $t$ moving in the direction $\\mathbf{\\theta }\n\\textbf{\\ }and having started the move a time $\\tau $ ago\\textbf{.} The\ngoverning equation for $n(\\mathbf{x},t,\\tau ,\\varphi )$ takes the form \\cit\n{Alt}\n\\begin{equation}\n\\frac{\\partial n}{\\partial t}+v\\mathbf{\\theta \\cdot \\nabla }n+\\frac{\\partial\nn}{\\partial \\tau }=-\\gamma _{\\rho }\\left( \\mathbf{x},t,\\tau ,\\varphi \\right)\nn,  \\label{mainbal}\n\\end{equation\nwhere $\\gamma _{\\rho }$ can be defined in terms of the turning rate $\\mathbb\nT}_{\\rho }$ as follows\n\\begin{equation}\n\\gamma _{\\rho }\\left( \\mathbf{x},t,\\tau ,\\varphi \\right) =\\int_{-\\pi }^{\\pi \n\\mathbb{T}_{\\rho }\\left( \\mathbf{x},t,\\tau ,\\varphi ,\\varphi ^{\\prime\n}\\right) d\\varphi ^{\\prime }.  \\label{gamma}\n\\end{equation\nThe function $\\gamma _{\\rho }\\left( \\mathbf{x},t,\\tau ,\\varphi \\right) $\ndescribes the rate at which the individual changes the direction at\\ $\n\\mathbf{x},t)$ from $\\varphi $ to other directions due to interactions with\nneighboring conspecifics. We assume that at the initial time $t=0$ all\nindividuals have zero running time\n\\begin{equation}\nn(\\mathbf{x},0,\\tau ,\\varphi )=\\rho (\\mathbf{x},0,\\varphi )\\delta (\\tau ).\n\\label{initial}\n\\end{equation\nThe total population density density is\n\\begin{equation}\n\\rho (\\mathbf{x},t,\\varphi )=\\int_{0}^{t}n(\\mathbf{x},t,\\tau ,\\varphi )d\\tau\n.  \\label{den}\n\\end{equation\nWe set up the boundary condition at zero running time $\\tau =0$:\n\\begin{equation}\nn(\\mathbf{x},t,0,\\varphi )=\\int_{0}^{t}\\int_{-\\pi }^{\\pi }\\mathbb{T}_{\\rho\n}\\left( \\mathbf{x},t,\\tau ,\\varphi ^{\\prime },\\varphi \\right) n(\\mathbf{x\n,t,\\tau ,\\varphi ^{\\prime })d\\varphi ^{\\prime }d\\tau .  \\label{in0}\n\\end{equation\nFrom the Markovian equation (\\ref{mainbal}) together with (\\ref{initial})\nand (\\ref{in0}) one can\\textbf{\\ }obtain the equation for $\\rho (\\mathbf{x\n,t,\\varphi )$ \\cite{Jake}\n\\begin{equation}\n\\frac{\\partial \\rho }{\\partial t}+v\\mathbf{\\theta \\cdot \\nabla }\\rho =-i\n\\mathbf{x},t,\\varphi )+n(\\mathbf{x},t,0,\\varphi ),  \\label{basic}\n\\end{equation\nwher\n\\begin{equation}\ni(\\mathbf{x},t,\\varphi )=\\int_{0}^{t}\\gamma _{\\rho }\\left( \\mathbf{x},t,\\tau\n,\\varphi \\right) n(\\mathbf{x},t,\\tau ,\\varphi )d\\tau .\n\\end{equation\nIn the linear case without interactions, one can find $i(\\mathbf{x\n,t,\\varphi )$ in terms of the total density $\\rho (\\mathbf{x},t,\\varphi )\n\\cite{Jake}\n\\begin{equation}\ni(\\mathbf{x},t,\\varphi )=\\int_{0}^{t}K(t-\\tau )\\rho (\\mathbf{x}-v\\mathbf\n\\theta }(t-\\tau ),\\tau ,\\varphi )d\\tau ,  \\label{i}\n\\end{equation\nwhere $K(t)$ is the standard memory kernal \\cite{Hel}.\n\nNon-local interactions involving alignment rate $\\mathbb{T}_{al}$ and\nrepulsion\/collision rate $\\mathbb{T}_{r}$ can be modelled as follows\n\\begin{equation}\n\\mathbb{T}_{\\rho }=\\mathbb{T}_{al}+\\mathbb{T}_{r},\n\\end{equation\nwhere\n\\begin{eqnarray}\n\\mathbb{T}_{al} &=&\\frac{\\mu }{\\tau _{0}+\\tau }\\int_{-\\pi }^{\\pi }\\int_\n\\mathbb{R}^{2}}K_{al}^{d}(\\mathbf{x-y})K_{al}^{o}\\left( \\chi ,\\varphi\n^{\\prime }\\right) \\times  \\notag \\\\\n&&\\omega _{al}\\left( \\varphi ^{\\prime }-\\varphi ,\\varphi ^{\\prime }-\\chi\n\\right) \\rho (\\mathbf{y},t,\\chi )d\\mathbf{y}d\\chi ,\n\\end{eqnarray\n\\begin{eqnarray}\n\\mathbb{T}_{r} &=&r\\left( \\tau \\right) \\int_{-\\pi }^{\\pi }\\int_{\\mathbb{R\n^{2}}K_{r}^{d}(\\mathbf{x-y})K_{r}^{o}\\left( \\mathbf{x,y},\\varphi ^{\\prime\n}\\right) \\times  \\notag \\\\\n&&\\omega _{r}\\left( \\varphi ^{\\prime }-\\varphi ,\\varphi ^{\\prime }-\\psi\n\\right) \\rho (\\mathbf{y},t,\\chi )d\\mathbf{y}d\\chi .\n\\end{eqnarray\nThe explicit expressions for the functions $K_{al,r}^{d}$, $K_{al,r}^{o}$\nand $\\omega _{al,r}$ can be found in \\cite{Fete,Car}. Detailed study of 2-D\nmodel will follow.\n\n\\subsection{Equations for the unstructured densities $\\protect\\rho _{\\pm\n}(x,t)$}\n\nBalance equations for the unstructured densities can be found by\ndifferentiating\n\\begin{equation}\n\\rho _{\\pm }(x,t)=\\int_{0}^{t}n_{\\pm }(x,t,\\tau )d\\tau  \\label{ii3}\n\\end{equation\nwith respect to time $t.$ Because of the initial condition\n\\begin{equation}\nn_{\\pm }(x,0,\\tau )=\\frac{\\rho (x,0)}{2}\\delta (\\tau ),  \\label{iniini}\n\\end{equation\n\\ the running time $\\tau $ varies from $0$ to $t$. We obtain for $\\rho _{\\pm\n}$ the following equation\n\\begin{eqnarray*}\n\\frac{\\partial \\rho _{_{\\pm }}}{\\partial t} &=&n_{_{\\pm }}(x,t,t)\\mp\nv\\int_{0}^{t}\\frac{\\partial n_{_{\\pm }}}{\\partial x}d\\tau -\\int_{0}^{t}\\frac\n\\partial n_{_{\\pm }}}{\\partial \\tau }d\\tau \\\\\n&&-\\int_{0}^{t}\\mathbb{T}_{\\pm }\\left( \\tau ,\\rho _{+},\\rho _{-}\\right)\nn_{\\pm }d\\tau .\n\\end{eqnarray*\nSince a zero running time condition ($\\tau =0)$ involves the proliferation\nof the individuals with the proliferation rate $k\\left( \\rho \\right) :$\n\\begin{equation}\nn_{\\pm }(x,t,0)=\\int_{0}^{t}\\left[ \\mathbb{T}_{\\mp }\\left( \\tau ,\\rho\n_{+},\\rho _{-}\\right) n_{\\mp }+k\\left( \\rho \\right) n_{\\pm }\\right] d\\tau ,\n\\end{equation\nwe rewrite the equation for $\\frac{\\partial \\rho _{_{\\pm }}}{\\partial t}$ as\nfollows\n\\begin{equation}\n\\frac{\\partial \\rho _{_{\\pm }}}{\\partial t}\\pm v\\frac{\\partial \\rho _{\\pm }}\n\\partial x}=i_{_{\\mp }}(x,t)-i_{_{\\pm }}(x,t)+k\\left( \\rho \\right) \\rho\n_{\\pm },\n\\end{equation\nwhere\n\\begin{equation}\ni_{\\pm }(x,t)=\\int_{0}^{t}\\mathbb{T}_{\\pm }\\left( \\tau ,\\rho _{+},\\rho\n_{-}\\right) n_{\\pm }(x,t,\\tau )d\\tau .  \\label{ii1}\n\\end{equation}\n\n\\subsection{Nonlinear transition from superdiffusion to diffusion.}\n\nNow let us find the switching rate $i_{\\pm }(x,t)$ in terms of the density \n\\rho _{\\pm }(x,t)$ for the rat\n\\begin{equation}\n\\mathbb{T}_{\\pm }\\left( \\tau ,\\rho _{+},\\rho _{-}\\right) =\\frac{\\mu }{\\tau\n_{0}+\\tau }+\\gamma _{\\pm }\\left( \\rho _{\\mp }\\right) .  \\label{rrrr}\n\\end{equation\nThe purpose is show that $\\gamma _{\\pm }$ plays the role of nonlinear\ntempering. By using the method of characteristics we solve the equation\n\\begin{equation}\n\\frac{\\partial n_{\\pm }}{\\partial t}\\pm v\\frac{\\partial n_{\\pm }}{\\partial x\n+\\frac{\\partial n_{\\pm }}{\\partial \\tau }=-\\mathbb{T}_{\\pm }\\left( \\tau\n,\\rho _{+},\\rho _{-}\\right) n_{\\pm }.\n\\end{equation\nWe find for $\\tau <t$\n\\begin{eqnarray}\nn_{\\pm }(x,t,\\tau ) &=&n_{\\pm }(x\\mp v\\tau ,t-\\tau ,0)\\times\n\\label{solution} \\\\\n&&\\Psi (\\tau )e^{-\\int_{t-\\tau }^{t}\\gamma _{\\pm }(\\rho _{\\mp }(x\\mp\nv(t-u),u))du}.  \\notag\n\\end{eqnarray\nwhere the survival function $\\Psi (\\tau )$ is\n\\begin{equation}\n\\Psi (\\tau )=\\left( \\frac{\\tau _{0}}{\\tau _{0}+\\tau }\\right) ^{\\mu }.\n\\label{ss}\n\\end{equation\n\\ The formula (\\ref{solution}) can be rewritten as\n\\begin{equation}\nn_{_{\\pm }}(x,t,\\tau )=n_{_{\\pm }}(x\\mp v\\tau ,t-\\tau ,0)\\Psi (\\tau )\\frac\nF_{\\rho }(x,t)}{F_{\\rho }(x,\\tau )}  \\label{new5}\n\\end{equation\nwhere\n\\begin{equation}\nF_{\\rho }(x,t)=e^{-\\int_{0}^{t}\\gamma _{\\pm }(\\rho _{\\mp }(x\\mp\nv(t-u),u))du}.\n\\end{equation\nTaking into account the initial condition (\\ref{iniini}) and substituting \n\\ref{new5}) into we obtain\n\\begin{eqnarray}\ni_{\\pm }(x,t) &=&\\int_{0}^{t}n_{_{\\pm }}(x\\mp v\\tau ,t-\\tau ,0)\\psi (\\tau \n\\frac{F_{\\rho }(x,t)}{F_{\\rho }(x,\\tau )}d\\tau  \\notag \\\\\n&&+\\frac{1}{2}\\rho (x\\mp vt,0)\\psi (t)F_{\\rho }(x,t)  \\label{jjj}\n\\end{eqnarray\nan\n\\begin{eqnarray}\n\\rho _{\\pm }(x,t) &=&\\int_{0}^{t}n_{_{\\pm }}(x\\mp v\\tau ,t-\\tau ,0)\\Psi\n(\\tau )\\frac{F_{\\rho }(x,t)}{F_{\\rho }(x,\\tau )}d\\tau  \\notag \\\\\n&&+\\frac{1}{2}\\Psi (t)\\rho (x\\mp vt,0)F_{\\rho }(x,t).  \\label{rr}\n\\end{eqnarray\nwhere $\\psi (\\tau )=-d\\Psi (\\tau )\/d\\tau .$ By using the Laplace transforms\none can eliminate $n_{\\pm }(x,t,0)$ from the above equations and obtain\nexplicit expressions for the total turning rates $i_{\\pm }$ in terms of the\ndensity of walkers $\\rho _{\\pm }:$\n\\begin{eqnarray}\ni_{\\pm }(x,t) &=&\\int_{0}^{t}K(t-\\tau )\\rho _{\\pm }(x\\mp v(t-\\tau ),\\tau\n)\\times  \\notag \\\\\n&&e^{-\\int_{\\tau }^{t}\\gamma _{\\pm }(\\rho _{\\mp }(x\\mp v(t-u),u))du}d\\tau .\n\\label{rrr}\n\\end{eqnarray\nThis term involves the exponential factor in which the rate $\\gamma _{\\pm }$\nplays the role of a tempering parameter. This rate leads to the shift of the\nsuperdiffusive L\\'{e}vy walk towards Brownian motion as the density $\\rho\n_{\\pm }$ increases. The main feature of the rate $i_{\\pm }(x,t)$ is that\nalthough the rates $\\frac{\\mu }{\\tau _{0}+\\tau }$ and $\\gamma _{\\pm }\\left(\n\\rho _{\\mp }\\right) $\\ are additive (see (\\ref{rrrr})), the corresponding\nterms in the rate (\\ref{rrr}) are not additive. This is clearly a\nnon-Markovian tempering effect.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nThe goal of these lectures is to discuss the theory of relativistic real fluids under the excuse of its application to the description of relativistic heavy ion collisions \\cite{Roma09,Ruus86}. More concretely, our goal is to discuss, in the simplest possible terms, why the ``Einstein elevator'' concept is not useful in the development of this theory. Namely, we cannot build a theory of relativistic real fluids by just asking that it reduces to its nonrelativistic counterpart, namely the Navier - Stokes equations, in an inertial frame where the fluid is at rest. For this reason we shall begin with a ``derivation'' of the nonrelativistic theory of ideal fluids (those which  flow with no entropy production) from thermodynamics \\cite{ChaLub95}. For ideal fluids, the relativistic theory may be found as the covariant extension of the nonrelativistic theory enforced in the rest frame. We shall then show that already in this simple framework we may obtain a working picture of how a relativistic heavy ion collision works, yielding remarkably accurate predictions regarding several concrete observables. \n\nAmong these observables there is the so-called elliptic flow, namely the anisotropy of the particle yield in the transverse plane to the beam direction. If the matter deposited in the collision region behaves as a nearly ideal fluid, then elliptic flow arises as the simple transduction of pressure gradients in the original configuration into velocities in the asymptotic expanding state. This is an obvious prediction of the Euler equations which is surprisingly hard to reproduce in non-hydrodynamic models. The success in predicting that there must be an elliptic flow is one of the most compelling reasons to believe that matter in the collision region, from some very early time after the actual collision up to the break up in the individual hadrons which are eventually detected, behaves as a fluid. However, ideal hydrodynamics overestimates the amount of elliptic flow \\cite{Roma09}. This suggests there must be a mechanism that tends to isotropize flow, and indeed viscosity acts in precisely that direction. Therefore, the next natural step is to consider hydrodynamic models based on real fluid hydrodynamics.\n\nAt this point we return to the formal theory, and generalize our previous ``thermodynamic'' derivation to the case of real fluids. The resulting non relativistic theory is given by the Navier - Stokes equations, which is satisfactory, but the relativistic theory emerging from the ``Einstein elevator'' is essentially flawed - if covariance is enforced, then the theory has no stable solutions \\cite{HisLin83}. How to get out of this cul de sac is the subject of the rest of the lectures.\n\nSince the ``thermodynamic'' approach led us nowhere, we shall begin anew from a more fundamental point of view, that provided by relativistic kinetic theory. In this case, there is an agreed upon relativistic formulation to build on \\cite{Isr72,Isr88}. The problem is how to reduce it to the hydrodynamic level. In the nonrelativistic case, the method of choice is the so-called Chapman - Enskog expansion, which leads to the Navier - Stokes equations \\cite{CHACO39}. In the relativistic case, it works likewise, and so it must be rejected. However, a second well known technique, the Grad expansion, is successful in building a relativistic hydrodynamics where linear perturbations of an equilibrium state evolve causally. \n\nHowever, the application to relativistic heavy ion collisions forces us to face the relationship between kinetic and hydro descriptions a second time, because what is actually measured are individual hadrons. This means that there must be some space time ``break up'' or ``freeze out'' surface (or region) where the fluid nucleates into individual particles that fly away to the detectors (a process that is by no means simple, but does not belong to these lectures). We must be able to read the one particle distribution function of these particles out of the hydrodynamic state of the fluid just before break up. If we simple invert the Grad expansion, we are led to a one particle distribution function which is not nonnegative throughout phase space, a defect that contaminates the predictions of the theory regarding observables which are sensitive to high momenta.\n\nThis drawback of the Grad approach, and also the need to include nonlinear corrections to the linearized equations of motion without spoiling stability and covariance, prompted us to suggest that the Grad approach may be just the linear approximation to a more comprehensive view of the kinetic - hydro relationship \\cite{CalPR10}. The basic insight is that any realistic kinetic theory displays a whole hierarchy of relaxation times, from the relaxation times characteristic of large scale inhomogeneities to the much shorter relaxation times of hard modes. Fluctuations in the hydrodynamic modes relax on the longer time scales, and are perceived by the harder modes as externally imposed thermodynamic forces which prevent their relaxation to true equilibrium. In other words, the truly kinetic modes relax not to equilibrium but to a nonequilibrium steady state constrained by the instantaneous configuration of the hydrodynamic modes. It has been known from long ago that such steady states are the solutions to variational problems \\cite{Ono61,Jay80}. Prigogine among others has proposed that they are the extrema of the entropy \\emph{production}, as opposed to the extrema of the entropy itself, which are the true equilibria \\cite{Pri55}. Known proofs of the so-called ``Prigogine theorem'' are restricted to linear irreversible thermodynamics \\cite{Lan75,Bru06}; we shall appeal to it on a heuristic rather than formal basis - the idea is that most kinetic modes are in the linear regime most of the time anyway, so a theory which is good in the linear regime is good enough to compute global observables such as stress tensor components, but we shall  not attempt to formulate this insight in any rigorous way \\cite{Kli91,Kli95}.\n\nWe shall therefore conclude these lectures with a brief presentation of this ``entropy production variational method'', its relationship to positivity of the one particle distribution function, and its nonlinear generalization.\n\nThe lectures are organized as follows. In next section we present the theory of relativistic ideal fluids, as derived from thermodynamics, and its application to the description of relativistic heavy ion collisions (RHICs). The main success of the theory, namely the prediction of elliptic flow, is also its fatal drawback, because experimental data show that ideal hydrodynamics overestimates the flow anisotropy \\cite{Roma09}. We are therefore motivated to extend the theory to include viscosity, whereby we hit on the stability problem \\cite{HisLin83,Ols90,OlsHis90}. We present the theory of real fluids in Section III; since the thermodynamic derivation fails, we base our discussion on relativistic kinetic theory. We must confront the \\emph{closure} problem, that is, to relate the one particle distribution function in the kinetic description to the hydrodynamic degrees of freedom. We present the Chapman-Enskog solution (which reproduces the results from the thermodynamic analysis, and therefore is unsuitable) and the Grad solution, and finally identify the Grad solution of the closure problem as the linearized approximation to an approach based on the entropy production variational method (EPVM). We conclude with some brief final remarks.\n\nIf relativistic hydrodynamics may be founded on kinetic theory, kinetic theory itself comes from quantum field theory, the main subject matter of this school. We show the main lines of the derivation of kinetic theory from quantum field theory in the Appendix, which relies heavily on ref. \\cite{CalHu08}.\n\n\\section{From Thermodynamics to Hydrodynamics}\n\\subsection{Nonrelativistic Ideal fluids}\nLet us begin by reviewing non relativistic hydrodynamics, as ``derived'' from thermodynamics.\nThe basic tenets of thermodynamics we need to keep in mind are the following: we\nhave a (simple) system described by some intensive parameters (temperature $T$,\nchemical potential $\\mu$, pressure $p$, etc.) whose meaning we take for granted (e.g.\nwe already know everything about the zeroth law) and extensive parameters,\nsuch as energy $U$, entropy $S$, volume $V$ , and particle number $N$ (we use particle\nnumber for concreteness, but any, or several, conserved charge(s) would serve\njust as well). In equilibrium, all these quantities are position independent. Their\nfirst deviations from equilibrium are related by the first law\n\n\\begin{equation}\nTdS = dU + pdV -\\mu dN \n\\label{(12.1)}\n\\end{equation}\nExtensive quantities are homogeneous functions of each other, so we must have\n\n\\begin{equation}\nTS = U + pV - \\mu N \n\\label{(12.2)}\n\\end{equation}\nFrom the differential of this second identity we obtain the Gibbs-Duhem relation\n\n\\begin{equation}\ndp = sdT + nd\\mu \n\\label{(12.3)}\n\\end{equation}\nwhere $s = S\/V$ and $n = N\/V$ are the entropy and particle number densities,\nrespectively. This means\n\n\\begin{eqnarray}\n\\left.\\frac{\\partial p}{\\partial T}\\right|_{\\mu}\n&=& s = \\frac{\\rho + p - \\mu n}T\\nonumber\\\\\n\\left.\\frac{\\partial p}{\\partial \\mu}\\right|_T\n&=& n \n\\label{(12.4)}\n\\end{eqnarray}\nwhere $\\rho = U\/V$ is the energy density.\n\nLet us now consider a non equilibrium, inhomogeneous configuration. The above relations still hold for the densities of the conserved quantities energy, momentum (which we must include) and particle number. The dynamics is given by the conservation laws\n\n\\begin{eqnarray}\n\\frac{\\partial \\rho}{\\partial t}&=&-\\nabla_j J^ j_u\\nonumber\\\\\n\\frac{\\partial p^ i}{\\partial t}&=&-\\nabla_j T^{ij}\\nonumber\\\\\n\\frac{\\partial n}{\\partial t}&=&-\\nabla_j J^j\n\\end{eqnarray} \nHere $\\rho$, $p^i$ and $n$ are the energy, momentum and charge densities, $J^j_u$, $T^{ij}$ and $J^j$ the respective currents. $T^{ij}$ must be symmetric \\cite{Bat67}. The entropy density\n\n\\begin{equation} \ns=\\frac 1T\\left[ \\rho+p-v_ip^i-\\mu n\\right] \n\\end{equation} \nobeys the first Law\n\n\\begin{equation} \nTds=d\\rho-v_idp^i-\\mu dn\n\\end{equation} \nwhich implies the Gibbs - Duhem relation\n\n\\begin{equation} \ndp=sdT+p^idv_i+nd\\mu \n\\end{equation} \nIn these formulae we have introduced the velocity $v_i$ and the chemical potential $\\mu $. We therefore have\n\n\\begin{eqnarray} \n\\frac{\\partial s}{\\partial t}&=& \\frac 1T\\left\\lbrace \\frac{\\partial \\rho}{\\partial t}-v_i\\frac{\\partial p^ i}{\\partial t}-\\mu\\frac{\\partial n}{\\partial t}\\right\\rbrace \\nonumber\\\\\n&=& \\frac {-1}T\\left\\lbrace \\nabla_j J^ j_u-v_i\\nabla_j T^{ij} -\\mu\\nabla_j J^j\\right\\rbrace \n\\end{eqnarray} \nWe then have\n\n\\begin{equation} \n\\frac{\\partial s}{\\partial t}=-\\nabla_j\\left\\lbrace\\frac 1T\\left[J^ j_u-v_iT^{ij}-\\mu J^j \\right]  \\right\\rbrace +\\left[J^ j_u-v_iT^{ij}-\\mu J^j \\right]\\nabla_j\\left(\\frac 1T \\right) -\\frac 1TT^{ij}\\nabla_jv_i-\\frac {1}TJ^j\\nabla_j\\mu \n\\end{equation}\nTo proceed we write $J^j=nv^j+\\chi^j$. In the terms involving the velocity we eliminate the  gradient of the chemical potential by using the Gibbs - Duhem relation \n\n\\begin{eqnarray} \n\\frac{\\partial s}{\\partial t}&=&-\\nabla_j\\left\\lbrace\\frac 1T\\left[J^ j_u-v_iT^{ij}-\\mu J^j \\right]  \\right\\rbrace +\\left[J^ j_u-v_iT^{ij}-\\mu J^j \\right]\\nabla_j\\left(\\frac 1T \\right) -\\frac 1TT^{ij}\\nabla_jv_i\\nonumber\\\\\n&-&\\frac 1T{v}^j\\left[\\nabla_jp-s\\nabla_jT-p^i\\nabla_jv_i \\right] - \\frac {1}T\\chi^j\\nabla_j\\mu  \n\\end{eqnarray}\nso far\n\n\\begin{eqnarray} \n\\frac{\\partial s}{\\partial t}&=&-\\nabla_j\\left\\lbrace\\frac 1T\\left[J^ j_u+p{v}^j-v_iT^{ij}-\\mu J^j \\right]  \\right\\rbrace \\nonumber\\\\\n&+&\\left[J^ j_u+p{v}^j-v_iT^{ij}-\\mu J^j-Ts{v}^j \\right]\\nabla_j\\left(\\frac 1T \\right) \\nonumber\\\\\n&-&\\frac 1T\\left[ T^{ij}-p^i{v}^j-p\\delta^{ij}\\right] \\nabla_jv_i- \\frac {1}T\\chi^j\\nabla_j\\mu\n\\end{eqnarray}\nUsing again the second law, we may write\n\n\\begin{equation}\nJ^ j_u+p{v}^j-v_iT^{ij}-\\mu J^j-Ts{v}^j =J^ j_u-\\left(\\rho+p\\right)v^j-v_i\\left(T^{ij}-p^iv^j-p\\delta^{ij}\\right)-\\mu\\chi^j \n\\end{equation}\nand so\n\n\\begin{eqnarray} \n\\frac{\\partial s}{\\partial t}&=&-\\nabla_j\\left\\lbrace\\frac 1T\\left[J^ j_u+p{v}^j-v_iT^{ij}-\\mu J^j \\right]  \\right\\rbrace \\nonumber\\\\\n&+&\\left[J^ j_u-\\left(\\rho+p\\right)v^j-v_i\\left(T^{ij}-p^iv^j-p\\delta^{ij}\\right)\\right]\\nabla_j\\left(\\frac 1T \\right) \\nonumber\\\\\n&-&\\frac 1T\\left[ T^{ij}-p^i{v}^j-p\\delta^{ij}\\right] \\nabla_jv_i-\\chi^j\\nabla_j\\alpha\n\\label{dsdt}\n\\end{eqnarray}\nWhere $\\alpha=\\mu \/T$. Since we have allowed for the possibility of the energy and particle number flux not being collinear, we must adopt a convention about what is the fluid velocity. We shall adopt the so-called Landau-Lifshitz prescription, namely, that $J_u^j$ and $p^j$ vanish when $v^j=0$ \\cite{LL6}.\n\nWe define an ideal fluid as one that flows with no entropy production. Thus for an ideal fluid\n\n\\begin{eqnarray}\nT^{ij}&=&p^i{v}^j+p\\delta^{ij}\\nonumber\\\\\nJ^ j_u&=&\\left(\\rho+p\\right)v^j\\nonumber\\\\\n\\chi^j&=&0\n\\end{eqnarray}\nThe symmetry of $T^{ij}$ requires $p^i$ to be proportional to $v^i$. The equations for an ideal fluid are thus\n\n\\begin{eqnarray}\n\\frac{\\partial n}{\\partial t}+\\nabla_j nv^j&=&0\\nonumber\\\\\n\\frac{\\partial \\rho}{\\partial t}+\\nabla_j \\left(\\rho+p\\right)v^j&=&0\\nonumber\\\\\n\\frac{\\partial p^ i}{\\partial t}+\\nabla_j p^iv^j+\\delta^{ij}\\nabla_jp&=&0\n\\end{eqnarray} \n\n\\subsection{Relativistic Ideal Fluids}\nWe now generalize the above framework of thermodynamics to a relativistic fluid\nevolving in a spacetime with an arbitrary metric $g_{\\mu\\nu}$. All derivatives shall be covariant derivatives\nwith respect to the Levi-Civita connection, so that $g_{\\mu\\nu;\\rho}=0$. We write the Minkowsky metric as $\\eta^{00}=-1$, $\\eta^{0i}=0$ and $\\eta^{ij}=\\delta^{ij}$, where a $0$ refers to coordinate $x^0=ct$\n\nTo simplify matters we will describe the construction of a covariant theory in\nterms of a set of rules \\cite{Isr88}:\n\n(a) Intensive quantities ($T$, $p$, $\\mu$) are associated with scalars, which represent the\nvalue of the quantity at a given event, as measured by an observer at rest\nwith respect to the fluid.\n\n(b) Extensive quantities ($S$, $V$, $N$) are associated with vector currents $S^{\\mu}$, $u^{\\mu}$, $N^{\\mu}$. If a given observer measures a density $x$ and a flux $J_x^i$ for the quantity $X$, then $X ^{\\mu}=\\left(x,J_x^i\/c\\right)$ The quantity $u^{\\mu}$ associated with volume is the fluid\n4-velocity; in Minkowsky space\n\n\\begin{equation}\nu^{\\mu}=\\frac1{\\sqrt{1-\\frac{v^2}{c^2}}}\\left(1,\\frac{v^i}c\\right)\n\\end{equation}\n$u^{\\mu}$ obeys the additional constraint $u^2 = -1$. We call density tout\ncourt the density measured by an observer comoving with the fluid, namely\n$x = -u_{\\mu}X^{\\mu}$.\n\nIf the quantity $X$ is conserved,\nthen $X^{\\mu}_{;\\mu} = 0$. If we consider the variation of the entropy content within some spatial region \nas a function of time, the second law demands that the increase in entropy should\nbe higher than the entropy flow through the boundary . Thus the covariant\nstatement of the second law is that entropy production must be positive, i.e.\n$S^{\\mu}_{;\\mu}\\ge 0$.\n\n\n(c) Energy and momentum are combined into a single extensive quantity\ndescribed by an energy-momentum tensor $T^{\\mu\\nu}$ which is symmetric. Therefore any given observer will identify the energy density as $T^{00}$, the energy flux as $cT^{0i}$, the momentum density as $T^{i0}\/c$ and the momentum flux as $T^{ij}$. The symmetry of the energy momentum tensor implies that the momentum density is $c^{-2}$ times the energy flux \\cite{LL6}. We define the rest frame of the fluid as the frame where the energy flux vanishes. As before, the energy density $\\rho$ is defined as the energy density in the rest frame. Both the fluid velocity and energy density may be found from the eigenvalue equation $T^{\\mu\\nu}u_{\\nu}=-\\rho u^{\\mu}$\n\nLet us now describe a relativistic ideal fluid, namely one that flows with no entropy production. We shall proceed by correspondence between the relativistic and non relativistic theories. To this end, we assume that in the rest frame we have the decomposition\n\n\\begin{equation}\nN^{\\mu}=\\left(n,0\\right)\n\\end{equation}\n\n\\begin{equation}\nT^{\\mu\\nu}=\\left(\n\\begin{array}{cc}\n\\rho&0\\\\\n0&p\\delta^{ij}\n\\end{array}\\right)\n\\end{equation}\nSo in an arbitrary frame we must have\n\n\\begin{eqnarray}\nN^{\\mu} &=& nu^{\\mu}\\nonumber\\\\\nT^{\\mu\\nu}&=& \\rho u^{\\mu}u^{\\nu}+ p\\Delta^{\\mu\\nu}\n\\label{(12.11)}\n\\end{eqnarray}\nwhere $\\Delta^{\\mu\\nu}=g^{\\mu\\nu}+u^{\\mu}u^{\\nu}$. The entropy\ncurrent $S^{\\mu}$ in the rest frame is given by $S^{\\mu}=\\left(s,0\\right)$, where $Ts = p + \\rho - \\mu n$. In an arbitrary frame $S^{\\mu}=su^{\\mu}$ or\n\n\\begin{equation}\nS^{\\mu}= -T^{\\mu\\nu}\\beta_{\\nu} + pu^{\\mu} - {\\alpha}N^{\\mu}\n\\end{equation}\nWhere the inverse temperature vector $\\beta^{\\mu} = T^{-1}u^{\\mu}$ and $\\alpha=\\mu \/T$. Observe that $T^{-2} = -\\beta_{\\mu}\\beta^{\\mu}$. Then \n\n\\begin{equation}\nS^{\\mu}_{;\\mu} = -\\beta_{\\nu}T^{\\mu\\nu}_{;\\mu}- \\alpha N^{\\mu}_{;\\mu} \n\\label{ (12.15)}\n\\end{equation}\nThis means that entropy production vanishes for an ideal fluid, provided the conservation\nlaws of energy-momentum and particle number hold. \n\n\n\\subsection{Ideal Hydrodynamic models of heavy ion collisions}\n\n\nThe theory of relativistic real fluids has been brought to the limelight by the mounting evidence that one such system has been seen in actual experiments, namely, relativistic heavy ion collisions (RHICs). Indeed, one can get a working understanding of RHICs from the theory of ideal fluids we have just described. Nevertheless, precisely the demands of a better fit between theory and experiment will prompt us to develop a theory or real fluids, namely, to add viscosity to the flow equations. In this section we shall briefly describe RHICs, and discuss what can and cannot be reproduced by models based on ideal hydrodynamics.\n\nRHICs have been produced in several experiments, namely the Super Proton Synchrotron (SPS) at CERN, the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven and\nthe Large Hadron Collider (LHC) once again at CERN. The\nRHIC experiments in particular are described in detail in the so-called `white\npapers', which are possibly the most reliable source on the subject \\cite{BRAHMS05,PHOBOS05,STAR05,PHENIX05}. \nOne of the goals of the RHIC program is to probe into possible new\nphases of nuclear matter at high energies.  In such a high energy phase, matter is expected to form a plasma of gluons and (massless) quarks (quark-gluon\nplasma, QGP). \n\n\nVirtually all theoretical analysis of RHICs assume a\nspace-time picture of the collision provided by the Bjorken\nmodel \\cite{BJOR83}. The colliding nuclei are\nseen as slabs of quark and gluon matter. In the center of mass frame,\nboth slabs approach each other at near light speed. Upon collision,\nthe two slabs of matter will mostly go through each other, leaving\nbehind a wake of hot plasma. We may then distinguish three different\nregions: the two fragmentation regions\ncorresponding to the receding slabs, and the central\nregion corresponding to the plasma in between.\nWe are interested in phenomena in the central region.\n\nAt the time of crossing a number of hard scattering processes will\noccur, whose products will reach directly the detectors. These hard\nprocesses are unrelated to the nonequilibrium dynamics of the plasma;\nand may presumably be predicted on perturbative QCD grounds. In what\nfollows, we will assume this hard component has been isolated despite\ngreat difficulty to achieve this in reality.\n\nThe hot plasma will expand and cool, and eventually fragment into\nordinary particles in flight intercepted by the detectors. We wish to\npredict the number of particles of each species to be detected, as a\nfunction of the angle $\\theta $ between the direction of flight and\nthe direction $z$ of the beam. It is remarkable that with this simple\npicture we can state a first observable prediction already.\n\nIndeed, because of Lorentz contraction, we may think of the\napproaching slabs as infinitely thin in the direction of motion $z$,\nand in a first approach to the problem, as infinite and homogeneous\nin the transverse directions $x$ and $y$. This picture is invariant\nunder boost in the $z$ direction, and so is the final\ndistribution of particles. So if we parametrize the momentum of an\nout-going particle as $p^{0}=E$, $p^{3}=p$ and $\\left(\np^{1},p^{2}\\right) =p_{\\perp }$, then the distribution of\nparticles may depend only upon the transverse momentum and $E^{2}-p^{2}=m^{2}+p_{\\perp }^{2}$. In particular, it must be independent of $\\theta ,$ since $\\cos \\theta \\sim p\/E$ is not invariant. It is\nconventional to plot the yield of the collision in terms of the\n\\textit{rapidity} $Y$, defined by $p\/E\\equiv \\tanh\nY$, or rather the pseudorapidity $\\eta =-\\ln\n\\tan \\left[ \\theta \/2\\right] ,$ $\\tanh \\eta =p\/\\left|\n\\mathbf{p}\\right| .$ Rapidity and pseudo-rapidity agree at momenta\nwhich are large compared to the mass of the particle. Then the\nprediction in this picture is that there is a \\textit{plateau} in the\n(pseudo) rapidity distribution, at least for\nsmall rapidity ($\\left| \\eta \\right| \\rightarrow \\infty $ corresponds\nto the fragmentation rather than the central region). Although this\nprediction is not quantitatively borne out by the RHIC data \\cite{PHOBOS05}, the experimental curve flattens enough at low rapidity that it may be accepted as a working rough approximation.\n\nWe may elaborate the Bjorken picture further. Let us assume that the\nplasma is formed on the plane $z=0$ at the time $t=0$ of the\ncollision, and then expands along the $z$ direction. A given plasma\nelement will cool according to its own proper time $\\tau $. \nEventually, at some given constant $\\tau $ surface, the plasma will\nbe cold enough (and\/or dilute enough) to break up into hadrons.\nAssuming that the product hadrons are thermally distributed,\nmassless and at zero chemical potential, the Bose-Einstein\ndistribution predicts that the energy per particle is $\\epsilon \/n=2.7$\n$T$. Since temperature is constant on the break up surface, this\nmeans that in all collisions particles should have the same average\nenergy. Indeed, it is observed that the energy per particle is about\n$0.8$ \\textrm{GeV}, regardless of center-of-mass energy and impact\nparameter.\n\nTo obtain a more quantitative description of the process, we \ndescribe the plasma as a relativistic ideal fluid.  To\nclose the hydrodynamic system of equations we must provide the\nequation of state. The central feature of this is the\n``softening'' near the critical point, meaning that the speed of sound $%\nc_{s}^{2}=\\partial p\/\\partial \\epsilon \\rightarrow 0$ as we approach the\ntransition point. The softening of the equation of state affects the\nevolution of the fireball, which then becomes a signal of whether the\ntransition point has been reached or not.\n\nSince perfect fluids conserve entropy, the total entropy within the\nfireball remains constant, and $T$ scales as $V^{-1\/3}.$ So, if the\nexpansion is one-dimensional, and we consider the volume enclosed\nbetween two fixed rapidities, then $T\\sim \\tau ^{-1\/3},$ where $\\tau\n$ is the proper time. In particular, the energy density scales as\n$\\tau ^{-4\/3}$.\n\n\nWe now consider more closely the phenomenon of\nbreak-up \\cite{Roma09,Ruus86}. Assume this occurs on a\n3-dimensional surface $\\Sigma$ defined by some equation $\\Sigma\n\\left( x^{\\mu }\\right) =0.$ If $x_{0}$ is a solution, then the normal\nvector at $x_{0}$ is $n^{\\mu }=\\left( \\alpha \\right) \\Sigma^{,\\mu\n}$, $\\alpha =\\left( -\\Sigma _{,\\mu }\\Sigma ^{,\\mu }\\right) ^{-1\/2}.$\nWe shall assume that $n^{\\mu }$ is timelike. The invariant measure on $\\Sigma $ is given by\n$d^{3}\\sigma =d^{4}x\\;\\delta \\left( \\Sigma \\right) \\alpha ^{-1}.$\n\n\nLet us assume that right up to break-up we can describe\nmatter as a perfect relativistic fluid, and as a noninteracting relativistic gas thereafter. Let $K_{a}=\\partial \/\\partial\nx^{a}$ be the four Killing vectors of Minkowski space. Then Gauss'\ntheorem shows that the quantities $n_{\\mu }K_{a\\nu }T^{\\mu \\nu }$ and\n$n_{\\mu }N^{\\mu }$ are continuous across the break-up surface (we\nshall consider only one conserved current, corresponding to, say, the\nbaryon number). These conditions plus the equation of state of the\nhadronic phase define the energy density, pressure, baryon number\ndensity (or equivalently, the temperature and chemical potential) and\nthe four-velocity of the hadrons at break-up. The detailed spectrum\nis found by assuming that the hadrons are thermally distributed.\n\nThe total number of emitted particles is\n\n\\begin{equation}\n\\int d^{3}{\\mathbf {x}}\\;K_{0\\mu }N_{had}^{\\mu }\n\\end{equation}\nwhere the integral is over some $t=$\\textrm{cons}$\\tan $\\textrm{t}\nsurface well to the future of the collision. Because of Gauss\ntheorem, we may replace the integral by an integral over the\nbreak-up surface (we may have to complete this surface to get a\nCauchy surface, but the particle density flux will vanish on these\nadditions anyway). But then we may use the matching conditions to\nexpress this integral in terms of the particle current before\nbreak-up. We obtain the total number of emitted particles as\n\n\\begin{equation}\n\\int d^{4}x\\;\\delta \\left( \\Sigma \\right) \\Sigma _{,\\mu }N_{hydro}^{\\mu }\n\\end{equation}\nIn practice, we may wish to smear a little the position of the\nbreak-up surface, thus writing the total number of emitted particles\nas\n\n\\begin{equation}\n\\int d^{4}x\\;\\left[ \\frac{e^{-\\Sigma ^{2}\/2\\left( \\Delta \\Sigma \\right) ^{2}}%\n}{\\sqrt{2\\pi }\\left( \\Delta \\Sigma \\right) }\\right] \\Sigma _{,\\mu\n}N_{hydro}^{\\mu }\n\\end{equation}\n\nThe total number of particles of species $i$ with momentum $p^{\\mu }$ is\n\n\\begin{equation}\ng_{i}\\int {d^{4}x}\\;\\left[ \\frac{e^{-\\Sigma\n^{2}\/2\\left( \\Delta \\Sigma \\right) ^{2}}}{\\sqrt{2\\pi }\\left( \\Delta \\Sigma\n\\right) }\\right] \\delta \\left(\np_{i}^{2}-m_{i}^{2}\\right) \\frac{\\Sigma _{,\\mu }p_{i}^{\\mu }}{\\left[\n\\exp \\left( -\\beta _{\\nu }p_{i}^{\\nu }-\\mu_{i}\\right) -\\varepsilon _{i}%\n\\right] }  \\label{spec1}\n\\end{equation}\nwhere the temperature four vector and chemical potentials are read out from the (ideal) energy momentum tensor and charged currents of the fluid just before break up.\n\nThe two basic observables are the total number of particles with\ntransverse (respect to the beam axis) momentum $p_{\\perp }$, which is\nusually given in terms of the transverse mass\n$m_{\\perp }^{2}=m^{2}+p_{\\perp }^{2},$ and the elliptic\nflow coefficient $v_{2},$ which results from\nfitting the particle spectrum in the transverse plane to a second\nharmonic $\\left( 1+2v_{2}\\left( p_{\\perp }\\right) \\cos 2\\phi \\right)\n,$ where $\\phi $ is the angle measured from the reaction plane. This\nis equivalent to considering an elliptic fireball, in which case\n$v_{2}$ measures the eccentricity of the ellipse. The first harmonic\nis called directed flow, and would represent a\nshifted spherical fireball in the transverse plane.\n\n\nThe agreement of predictions from hydrodynamical simulations with\nexperimental data is good, provided the simulation is started very\nearly (earlier than $1$ \\textrm{fm}\/$c$ after the collision). If one\nbelieves that the validity of hydrodynamics demands (local)\nequilibration, this very short time is somewhat of a puzzle. \n\n\\subsection{Elliptic flow}\n\nOf the several predictions of hydrodynamic models, elliptic flow is one of the most compelling, because it is hard to match by alternative models. Let us show how elliptic flow arises in hydro models. \n \nLet the velocity of the fluid (with respect to laboratory time) in the longitudinal direction be $dz\/dt=v^z$. We may define a longitudinal rapidity\n\n\\begin{equation} \ny_L=\\frac12\\ln\\frac{1+v^z}{1-v^z}\n\\end{equation} \nLet us take a boost invariant flow in the beam direction as background solution. Then $v^z=z\/t$ and the longitudinal rapidity becomes identical with the space time rapidity\n\n\\begin{equation} \ny_L=\\eta=\\frac12\\ln\\frac{t+z}{t-z}\n\\end{equation} \nWe introduce Milne time $\\tau=\\sqrt{t^2-z^2}$ and Milne coordinates $\\xi^{\\alpha}=\\left( \\tau,\\eta,x,y\\right) $. Define\n${dx^{\\mu}}\/{d\\tau}= \\left( t\/\\tau,z\/\\tau,\\vec{v}_{\\perp}\\right) $. We find \n\n\\begin{equation} \n-\\eta_{\\mu\\nu}\\frac{dx^{\\mu}}{d\\tau}\\frac{dx^{\\nu}}{d\\tau}=1-{v}_{\\perp}^2\n\\end{equation} \nIntroducing the transverse rapidity \n\n\\begin{equation}\ny_{\\perp}=\\frac12\\ln\\frac{1+{v}_{\\perp}}{1-{v}_{\\perp}}\n\\end{equation} \nthen $v_{\\perp}=\\tanh y_{\\perp}$ and the four velocity becomes\n\n\\begin{equation} \nu^{\\mu}=\\cosh y_{\\perp}\\;\\left( \\cosh\\eta,\\sinh\\eta,\\vec{v}_{\\perp}\\right)\n\\end{equation} \nor else in Milne coordinates\n\n\\begin{equation} \nu^{\\alpha}=\\cosh y_{\\perp}\\;\\left( 1,0,\\vec{v}_{\\perp}\\right)\n\\end{equation} \nThe interval element in Milne coordinates is $ds^2=-d\\tau^2+\\tau^2d\\eta^2+dx^2+dy^2$, and so there are nontrivial Christoffel symbols\n\n\\begin{equation} \n\\Gamma^{\\tau}_{\\eta\\eta}=\\tau\n\\end{equation} \nand\n\n\\begin{equation} \n\\Gamma^{\\eta}_{\\tau\\eta}=\\frac 1{\\tau}\n\\end{equation} \nIn the strictly boost invariant case $y_{\\perp}=0$ we have $u_{\\eta ;\\eta}=\\tau$ and $u^{\\alpha}_{;\\alpha}=1\/\\tau$, although $u^{\\beta}u_{\\alpha ;\\beta}=\\Gamma^{\\tau}_{\\alpha\\tau}=0$. The covariant Euler equation is identically satisfied, $n\\propto \\tau^{-1}$, while the energy density $\\rho\\propto \\tau^{-4\/3}$ for conformal matter. \n\nObviously boost invariant expansion cannot create elliptic flow. The next step is to consider a nontrivial dynamics in the transverse plane. \nIt is convenient to parametrize the transverse plane in terms of Lagrangian coordinates $\\vec{q}$. Then the full coordinates are $\\zeta^{\\rho}=\\left( \\tau,\\eta,q^ 1,q^ 2\\right) $; the transformation back to Euler coordinates is given by the system\n\n\\begin{equation} \n \\frac{\\partial x^i }{\\partial \\tau}\\vert_{q^ a}= v^i\\left( \\tau, x^i\\right) \n\\end{equation} \nwith initial condition $x^i\\left( \\tau =0\\right) =q^ i$. \nNow write\n\n\\begin{eqnarray}\nds^2&=&-d\\tau^2+\\tau^2d\\xi^2+\\delta_{ij}dx^idx^j\\nonumber\\\\\n&=&-d\\tau^2+\\tau^2d\\xi^2+\\delta_{ij}\\left( \\frac{\\partial x^i}{\\partial q^ a}dq^a+v^id\\tau\\right) \\left( \\frac{\\partial x^j}{\\partial q^ b}dq^b+v^jd\\tau\\right)\n\\end{eqnarray}\nIn the new coordinates the metric reads\n\n\\begin{equation}\n{g_{\\rho\\sigma}}=\\left(\n\\begin{array}{ccc}\n-\\left( 1-g^{ab}N_aN_b\\right)  & 0  & N_b\n\\\\\n0 & \\tau^{2} & 0\n\\\\\nN_a & 0 & g_{ab}\n\\end{array}\n\\right)\n\\end{equation}\nwhere\n\n\\begin{equation} \ng_{ab}=\\delta_{ij} \\frac{\\partial x^i}{\\partial q^ a}\\frac{\\partial x^j}{\\partial q^ b}\n\\end{equation} \n\n\\begin{equation} \nN_a=v_i\\frac{\\partial x^i}{\\partial q^ a}\n\\end{equation} \nObserve that $g^{ab}N_aN_b=v_iv^i=\\tanh^2y_{\\perp}$, so $g_{00}=-1\/\\cosh^2y_{\\perp}$. \nThe lapse function is then $N=1$ and the four metric determinant is $g=- \\tau^{2} g^{\\left( 2\\right) }$, where $g^{\\left( 2\\right) }=\\mathrm{det}\\;g^{ab}$. In Milne-Lagrange coordinates the four velocity is $u^{\\alpha}=\\left( \\cosh y_{\\perp},\\vec{0}\\right) $, so \n\n\\begin{equation} \nu^{\\nu}_{;\\nu}=\\frac 1{\\tau\\sqrt{g^{\\left( 2\\right) }}}\\frac{\\partial}{\\partial \\tau}\\tau\\sqrt{g^{\\left( 2\\right) }}\\cosh y_{\\perp}\n\\end{equation}\nand the ideal hydrodynamic equations\n\n\\begin{eqnarray}\n\\frac{\\dot{T}}T+\\frac 13u^{\\nu}_{,\\nu}&=&0\\nonumber\\\\\n\\dot{u}^{\\mu}+\\Delta^{\\mu\\nu}\\frac{T_{,\\nu}}T&=&0\n\\end{eqnarray}\nGive\n\n\\begin{equation} \nT=\\frac{T_0\\left(\\vec{q} \\right) }{\\left[ \\tau\\sqrt{g^{\\left( 2\\right) }}\\cosh y_{\\perp}\\right] ^{1\/3}}\n\\end{equation} \nand\n\n\\begin{equation} \n\\cosh^2y_{\\perp}\\Gamma^a_{00}+\\frac 1T\\left[ N^aT_{,\\tau}+g^{ab}T_{,b}-N^aN^bT_{,b}\\right] =0\n\\end{equation}\nWe shall consider only linear deviations from homogeneity, where\n\n\\begin{equation} \nx^a=q^a+\\xi^a\\left(\\vec{q},\\tau \\right) \n\\end{equation}\n$\\xi^a\\left(\\vec{q},0 \\right) =0$\n\\begin{equation} \nN^a=v^a=\\frac{\\partial}{\\partial \\tau}\\xi^a\\left(\\vec{q},\\tau \\right) \n\\end{equation}\n\n\\begin{equation}\n\\sqrt{g^{\\left( 2\\right) }}=1+\\xi^a_{,a}\n\\end{equation}\n\n\\begin{equation} \n\\Gamma^a_{00}=\\frac{\\partial^2}{\\partial \\tau^2}\\xi^a\\left(\\vec{q},\\tau \\right) \n\\end{equation}\nSo, writing $T_0\\left(\\vec{q} \\right) =T_0\\left(1+\\delta \\left(\\vec{q} \\right)\\right)$\n\n\\begin{equation} \n\\frac{\\partial^2\\xi_a}{\\partial \\tau^2} -\\frac 1{3\\tau}\\frac{\\partial\\xi_a}{\\partial \\tau}-\\frac 1{3}\\xi^b_{,ba} =-\\delta_{,a}\n\\end{equation}\nGiven our initial conditions, there are no transverse velocity components. Writing $\\xi_a=\\phi_{,a}$ we get\n\n\\begin{equation} \n\\frac{\\partial^2\\phi}{\\partial \\tau^2} -\\frac 1{3\\tau}\\frac{\\partial\\phi}{\\partial \\tau}-\\frac 1{3}\\bf{\\Delta}\\phi=-\\delta\n\\end{equation}\nWrite\n\n\\begin{equation}\n\\delta=\\int\\frac{d^2k}{\\left(2\\pi\\right)^2}\\;e^{ikq}\\delta_k\n\\end{equation}\nThen\n\n\\begin{equation}\n\\phi=\\int\\frac{d^2k}{\\left(2\\pi\\right)^2}\\;e^{ikq}\\phi_k\n\\end{equation}\n\n\\begin{equation} \n\\frac{\\partial^2\\phi_k}{\\partial \\tau^2} -\\frac 1{3\\tau}\\frac{\\partial\\phi_k}{\\partial \\tau}+\\frac {k^2}{3}\\phi_k=-\\delta_k\n\\end{equation}\nWrite $\\dot{\\phi_k}=\\varphi_k$\n\n\\begin{equation} \n\\frac{\\partial^2\\varphi_k}{\\partial \\tau^2} -\\frac 1{3\\tau}\\frac{\\partial\\varphi_k}{\\partial \\tau}+\\frac13 {\\left[k^2+\\frac 1{\\tau^2}\\right]}\\varphi_k=0\n\\end{equation}\nThe solution which is regular at $\\tau=0$ is $\\varphi_k=c_k\\tau^{2\/3}J_{1\/3}\\left(k\\tau\/\\sqrt{3}\\right)$. As $\\tau\\to 0$ we get $\\varphi_k\\approx c_k\\tau\\left(k\/\\sqrt{3}\\right)^{1\/3}$, $\\phi_k\\approx c_k\\tau^2\\left(k\/\\sqrt{3}\\right)^{1\/3}\/2$, so\n\n\\begin{equation}\nc_k=\\frac{-3}{2}\\left(\\frac k{\\sqrt{3}}\\right)^{-1\/3}\\delta_k\n\\end{equation}\nWe may now find the velocities\n\n\\begin{equation}\nv_a=\\frac{\\partial}{\\partial \\tau}\\phi_{,a}=\\varphi_{,a}=\\left(\\frac{-3^{3\/2}}{2}\\right)\\int\\frac{d^2k}{\\left(2\\pi\\right)^2}\\;e^{ikq}\\frac{ik_a}k\\left(\\frac {k\\tau}{\\sqrt{3}}\\right)^{2\/3}\\delta_kJ_{1\/3}\\left(k\\tau\/\\sqrt{3}\\right)\n\\end{equation}\nAt early times $v\\propto -\\tau\\nabla\\delta$, and thus the flow transforms an initial density (or pressure) gradient into a flow anisotropy. This simple effect is remarkably hard to reproduce in parton models, under any realistic initial condition.\n\n\n\n\nThe important conclusion is that even in this simple model we get a qualitative picture of a relativistic heavy ion collision that serves as a framework for further analysis. In reality the flow is not really boost invariant, and it is certainly not homogeneous in the transverse plane. The hydrodynamic models offer a natural way to couple flow to pressure gradients in the transverse plane, and in this respect are superior to all other known alternatives. On the other hand, ideal hydrodynamic models overestimate the amount of anisotropy in the transverse flow \\cite{Roma09}; this underlines the need for a relativistic theory of real fluids to really model relativistic heavy ion collisions.\n\n\n\\section{Real Fluids}\n\n\\subsection{Relativistic Real Fluids from Thermodynamics}\nLet us return to our analysis of non relativistic fluids. Our starting point is eq. (\\ref{dsdt}), where \nnow we allow for entropy production, which must be positive. This suggests modifying the constitutive relations for an ideal fluid. We must take into account that $T^{ij}$must be symmetric, and the relativistic constraint $p^i=J_u^i\/c^2$. This suggests writing\n\n\\begin{equation}\nT_{ij}=\\frac12\\left(p^jv^i+v^jp^i\\right)+p\\delta^{ij}+\\tau_{ij}\n\\end{equation}\nwhere\n\n\\begin{equation}\n\\tau_{ij}=-\\eta\\sigma_{ij}-\\xi\\delta_{ij}\\nabla_kv^k\n\\end{equation} \n\n\\begin{equation} \n\\sigma_{ij}=\\nabla_iv_j+\\nabla_jv_i-\\frac 23\\delta_{ij}\\nabla_kv^k\n\\end{equation} \nand\n\n\\begin{equation}\nJ^ j_u=\\left(\\rho+p\\right)v^j+\\frac12\\left(p^jv^2-v^j\\left(v_ip^i\\right)\\right)+v_i\\tau^{ij}\n\\end{equation}\n\n\\begin{equation} \n\\chi_j=-\\kappa\\nabla_j\\alpha\n\\end{equation} \nIn what follows we assume $\\xi=0$. With these constitutive relations we find\n\n\\begin{equation} \n\\frac{\\partial s}{\\partial t}=-\\nabla_jS^j+{\\kappa}\\left(\\nabla_j\\alpha\\right)^2 +\\frac{\\eta}T\\sigma^{ij}\\sigma_{ij}+\\frac{\\xi}T\\left(\\nabla_jv^j\\right)^2 +\\frac 1T\\left(\\vec{p}\\times\\vec{v}\\right)\\nabla\\times\\vec{v}\n\\end{equation}\nThe last term vanishes in the rest frame and will not affect our argument.\n\nTo find the relativistic generalization of this theory, we observe that in the rest frame\n\n\\begin{equation}\nT^{\\mu\\nu}=\\left(\n\\begin{array}{cc}\n\\rho&0\\\\\n0&p\\delta^{ij}+\\tau^{ij}\n\\end{array}\\right)\n\\end{equation}\nThus in any frame we may write\n\n\\begin{equation}\nT^{\\mu\\nu}= T^{\\mu\\nu}_0+\\Pi^{\\mu\\nu}\n\\end{equation}\nwhere $T^{\\mu\\nu}_0=\\rho u^{\\mu}u^{\\nu}+ p\\Delta^{\\mu\\nu}$ and\n\\begin{equation}\n\\Pi^{\\mu\\nu}=-\\eta\\sigma^{\\mu\\nu}\n\\end{equation}\nwhere $\\sigma$ is the shear tensor\n\n\\begin{equation}\n\\sigma_{\\mu\\nu}=\\Delta^{\\lambda}_{\\mu}\\Delta^{\\rho}_{\\nu}\\left[u_{\\rho,\\lambda}+u_{\\lambda,\\rho}-\\frac23\\Delta_{\\lambda\\rho}u^{\\alpha}_{,\\alpha}\\right]\n\\end{equation}\nThis is the so-called Chapman-Enskog prescription, and although it leads to a covariant generalization of the Navier-Stokes equations, it is untenable.\n\nThe conservation equations read\n\\begin{equation}\nT^{\\mu\\nu}_{0,\\nu}+\\Pi^{\\mu\\nu}_{,\\nu}=0\n\\end{equation}\nNow\n\n\\begin{equation}\nT^{\\mu\\nu}_{0,\\nu}=\\rho_{,\\nu}u^{\\mu}u^{\\nu}+\\left(\\rho+p\\right)u^{\\mu}_{,\\nu}u^{\\nu}+\\left(\\rho+p\\right)u^{\\mu}u^{\\nu}_{,\\nu}+\\Delta^{\\mu\\nu}p_{,\\nu}=0\n\\end{equation}\nTherefore\n\n\\begin{eqnarray}\n\\dot{\\rho}+\\left(\\rho+p\\right)u^{\\nu}_{,\\nu}-u_{\\mu}\\Pi^{\\mu\\nu}_{,\\nu}&=&0\\nonumber\\\\\n\\dot{u}^{\\mu}+\\Delta^{\\mu\\nu}\\frac{p_{,\\nu}+\\Pi^{\\lambda}_{\\nu,\\lambda}}{\\left(\\rho+p\\right)}&=&0\n\\end{eqnarray}\nMoreover, since $\\Pi^{\\mu\\nu}$ is both traceless and transverse, we may write the first equation as\n\n\\begin{equation}\n\\dot{\\rho}+\\left(\\rho+p\\right)u^{\\nu}_{,\\nu}+\\frac12\\Pi^{\\mu\\nu}\\sigma_{\\mu\\nu}=0\n\\end{equation}\n\n\\subsection{The causality problem}\nSince in the Chapman-Enskog approach the viscous energy - momentum tensor (VEMT) is slaved to the four velocity, the only dynamic degrees of freedom are the components of the four velocity themselves. Let us consider the linear perturbations around a solution with $\\beta^{\\mu}=$ constant. Let us  write $u^ i=un^ i$, $n^ in_i=1$ and $u^0=\\sqrt{u^ 2+1}$. We seek solutions that depend on a single variable $k_{\\mu}x^{\\mu}=k_ix^ i-\\omega t$. We further decompose $k^i=Kn^i+k^ i_{\\perp}$, $x^ i=xn^ i+x^i_{\\perp}$. Since the theory is covariant, we may always perform a Lorentz transformation to the rest frame, namely we introduce new time and space variables\n\n\\begin{eqnarray} \nt'&=&u^0t-ux\\nonumber\\\\\nx'&=&u^ 0x-ut\n\\end{eqnarray}\nand new frequency and wave number\n\n\\begin{eqnarray} \n\\omega'&=&u^0\\omega-uK\\nonumber\\\\\nK'&=&u^ 0K-u\\omega\n\\end{eqnarray}\nwith $x'^i_{\\perp}=x^i_{\\perp}$ and $k'^i_{\\perp}=k^i_{\\perp}$. We seek solutions where $k^ i$ is real and $\\mathrm{Im}\\left[\\omega\\right]\\le 0$.\n\nIn this frame $u'^ i=0$, $u'^ 0=1$ and $T'=T$. The linearized perturbations must have $\\delta u'^0=0$. For the other variables we seek a solution \n\n\\begin{eqnarray} \n\\delta' T'=T\\delta' e^{-i\\omega' t'+ik'_jx'^j} \\nonumber\\\\\n\\delta' u'^i=\\delta'^ i e^{-i\\omega' t'+ik'_jx'^j}\n\\end{eqnarray}\nThe equations of motion become\n\n\\begin{eqnarray}\n-i\\omega'\\delta'+ic^2k'_j\\delta'^{j}&=&0\\nonumber\\\\\n-i\\omega'\\delta'^{j}+ik'^j\\delta'+{\\gamma}\\left(\\left( K'^2+k_{\\perp}^2\\right)\\delta'^{j}+\\frac 13k'^jk'_k\\delta'^{k}\\right) &=&0\n\\end{eqnarray}\nwhere $\\gamma =\\eta\/\\rho +p$. For a transverse perturbation $k'_k\\delta'^{k}=0$, $\\delta'=0$ we must have $\\omega'=-i\\gamma \\left( K'^2+k_{\\perp}^2\\right)$. We observe that because $u^0>0$, the imaginary parts of $\\omega$ and $\\omega'$ have the same sign, so we can check stability by checking the imaginary part of $\\omega'$.\nUnder a Galilean transformation $K'=K$ and $\\omega'=\\omega-uK$, so the theory is stable in all frames. However, under a relativistic transformation $K'=\\left(  K-u\\omega'\\right) \/u^0$. If $u\\not= 0$ we get a quadratic equation for $\\omega'$. For $k_{\\perp}=0$ this is\n\n\\begin{equation} \n\\omega'^2-2\\left(\\frac Ku+\\frac{iu^{02}}{2\\gamma u^2} \\right)\\omega' +\\frac{K^2}{u^2}=0\n\\end{equation} \nwith solution\n\n\\begin{equation} \n\\omega'=\\frac Ku\\left[ 1+\\frac{iu^{02}}{2\\gamma Ku}\\pm\\sqrt{\\left(1+\\frac{iu^{02}}{2\\gamma Ku} \\right)^2-1 }\\right] \n\\end{equation}\nWhen $u\\to 0$, one solution is the expected one $\\omega'=-i\\gamma K^ 2$ but the second solution becomes $\\omega'=iu^{02}\/\\gamma u^2$ and is unstable. For a longitudinal perturbation $\\delta'^j=k'^j\\delta''$ and the dispersion relation is\n\n\\begin{equation} \n\\omega'^2-c^2\\left( K'^2+k_{\\perp}^2\\right)+\\frac 43i\\gamma \\omega'\\left( K'^2+k_{\\perp}^2\\right)=0\n\\end{equation}\nThe same analysis as before leads to a cubic equation. When $u\\to 0$ two roots correspond to damped sound waves; the third root $\\omega'\\approx 3i\/4\\gamma u^2$ is unstable.\n\n\\subsection{Relativistic kinetic theory}\nIn the kinetic theory description \\cite{Isr72} the transport equation reads \n\n\\begin{equation}\np^{\\mu}\\partial_{\\mu}f=\\frac{-1}{\\tau}\\mathrm{sign}\\left(p^0\\right)I_{col}\n\\end{equation}\n$\\tau$ is the so-called relaxation time. The current is\n\n\\begin{equation}\nJ^{\\mu}=e\\int\\;Dp\\;p^{\\mu}f\n\\end{equation}\nand the EMT is\n\n\\begin{equation}\nT^{\\mu\\nu}=\\int\\;Dp\\;p^{\\mu}p^{\\nu}f\n\\end{equation}\nwhere \n\n\\begin{equation}\nDp =\\frac{2d^4p\\delta\\left(p^2\\right)}{\\left(2\\pi\\right)^3}=\\frac{d^4p}{\\left(2\\pi\\right)^3 p}\\left(\\delta\\left(p^0-p \\right) + \\delta\\left(p^0+p \\right)\\right) \n\\end{equation}\nFor simplicity we assume massless particles. We do not assume particle number conservation. To enforce energy-momentum conservation  we require\n\n\\begin{equation}\n\\int\\;Dp\\;p^{\\mu}\\mathrm{sign}\\left(p^0\\right)I_{col}=0\n\\end{equation}\nassuming for simplicity Maxwell - Boltzmann statistics, the equilibria are of the form\n\n\\begin{equation}\nf_0=\\exp\\left\\{-\\left|\\beta_{\\mu}p^{\\mu}\\right|\\right\\}\n\\end{equation}\nFor a given $T^{\\mu\\nu}$ we can always find a  local equilibrium distribution  $f_0$ such that the ideal fluid energy momentum tensor built from it\n\n\\begin{equation} \nT_0^{\\mu\\nu}=\\int\\;Dp\\;p^{\\mu}p^{\\nu}f_0=\\rho u^{\\mu}u^{\\nu}+p\\Delta^{\\mu\\nu}\n\\label{EMT0}\n\\end{equation} \nobeys\n\n\\begin{equation} \nT_0^{\\mu\\nu}u_{\\nu}=-\\rho u^{\\mu}\n\\label{LL0}\n\\end{equation}\nwith the same energy density and four velocity as from the Landau-Lifshitz prescription. We then define a temperature from\n\n\\begin{equation}\n\\rho=\\sigma T^4\n\\end{equation}\nwhere $\\sigma=\\pi^2\/15\\approx 0.66$. Using the Maxwell - Juttner distribution rather than the Bose-Einstein one is equivalent to the approximation\n\n\\begin{equation} \n\\sum_{n=1}n^{-4}=\\frac{\\pi^4}{90}\\approx 1\n\\end{equation}\nwhereby $\\sigma$ becomes $6\/\\pi^2\\approx 0.61$. We make this approximation from now on.\n\nIt follows that the viscous energy momentum tensor\n\n\\begin{equation} \n\\Pi^{\\mu\\nu}=T^{\\mu\\nu}-T_0^{\\mu\\nu}\n\\label{VEMT}\n\\end{equation} \nis traceless and transverse\n\n\\begin{equation} \n\\Pi^{\\mu\\nu}u_{\\nu}=0\n\\label{TTC}\n\\end{equation} \nWe parametrize\n\n\\begin{equation} \nf=f_0\\left[1+Z\\right]\n\\end{equation} \nwith $Z=0$ at equilibrium. The transversality condition becomes\n\n\\begin{equation}\n\\int\\;Dp\\;p^{\\mu}\\left(-u_{\\nu}p^{\\nu}\\right)f_0Z=0\n\\label{trans}\n\\end{equation}\nWe assume a simple Boltzmann type entropy flux \\cite{Sag13}\n\n\\begin{equation}\nS^{\\mu}=-\\int\\;Dp\\;\\left(\\mathrm{sign}\\left(p^0 \\right)\\right)p^{\\mu}f\\left[\\ln \\frac{f}{f_0}-1\\right]\n\\end{equation}\nwe get the entropy production\n\n\\begin{equation}\nS^{\\mu}_{,\\mu}=\\frac{1}{\\tau}\\int\\;Dp\\;I_{col}\\ln\\left[1+Z\\right] \n\\end{equation}\n\n\\subsection{Chapman-Enskog and Grad}\n\nThe Chapman-Enskog procedure seeks a formal expansion for $Z$ in powers of $\\tau$. To this end, one parametrizes\n\n\\begin{equation}\nZ=\\tau Z_1+\\tau^2Z_2+\\ldots\n\\end{equation}\nThe ``spatial'' derivatives $\\Delta^{\\mu\\nu}T_{,\\nu}$ and $\\Delta^{\\mu\\nu}u^{\\lambda}_{,\\nu}$ are regarded as zeroth order quantities, while the ``time'' derivatives $\\dot{T}=u^{\\mu}T_{,\\mu}$ and\n$\\dot{u}^{\\lambda}=u^{\\mu}u^{\\lambda}_{,\\mu}$ are derived from energy-momentum conservation, which is also a necessary consistency condition. \n\nTo find $Z_1$ we only need the linearized collision integral. This must be a symmetric operator, it must obey the energy momentum conservation constraint, must lead to non negative entropy production and must admit thermal distributions as the only homogeneous distributions. To satisfy these requirements, we write \n\n\\begin{equation} \nI_{col}\\left( p\\right) =F\\left[p\\right]  f_0\\left( p\\right) \\left[Z\\left(p \\right) -\\int\\;Dp'\\;K\\left[ p,p'\\right] F\\left[p'\\right]  f_0\\left( p'\\right) Z\\left( p'\\right) \\right] \n\\end{equation}\nThe second term is there to enforce the constraints, namely\n\n\\begin{equation}\n\\int\\;Dp\\;\\mathrm{sign}\\left(p^0\\right)p^{\\mu}F  f_0Z=\\int\\;DpDp'\\;\\mathrm{sign}\\left(p^0\\right)p^{\\mu}Ff_0K\\left[ p,p'\\right] F' f'_0Z' \n\\end{equation}\nSince the kernel $K$ is symmetric and these hold for any $Z$ we must have\n\n\\begin{equation}\n\\int\\;Dp'\\;K\\left[ p,p'\\right] \\mathrm{sign}\\left(p'^0\\right)p'^{\\mu}F' f'_0=p^{\\mu}\\mathrm{sign}\\left(p^0\\right)\n\\end{equation}\nIt also makes $I_{col}$ to vanish when $Z$ is just a variation in $\\beta_{\\mu}$.\n\nWe adopt the Anderson - Witting prescription $F=\\left|-u_{\\mu}p^{\\mu}\\right|$ \\cite{AndWit74,TakInu10}. Write\n\n\\begin{equation} \nK \\left[ p,p'\\right] =K_{\\rho\\sigma}p^{\\rho}\\mathrm{sign}\\left(p^0\\right)p'^{\\sigma}\\mathrm{sign}\\left(p'^0\\right)\n\\end{equation}\nThen\n\n\\begin{equation}\n K_{\\rho\\sigma}I^{\\sigma\\mu}=\\delta^{\\mu}_{\\rho}\n\\end{equation}\nwhere\n\n\\begin{equation}\nI^{\\sigma\\mu}=u_{\\lambda}\\int\\;Dp\\;\\mathrm{sign}\\left(p^0\\right)p^{\\sigma}p^{\\mu}p^{\\lambda} f_0=A_3\\left[u^{\\sigma}u^{\\mu}+\\frac13\\Delta^{\\sigma\\mu}\\right]\n\\end{equation}\n\n\\begin{equation}\nA_3=\\int\\;Dp\\;\\left|-u_{\\mu}p^{\\mu}\\right|^3 f_0=\\frac{24}{\\pi^2}T^5\n\\end{equation}\nThis means that\n\n\\begin{equation}\nK_{\\rho\\sigma}=A_3^{-1}\\left[u_{\\rho}u_{\\sigma}+3\\Delta_{\\rho\\sigma}\\right]\n\\end{equation}\nFinally the collision integral is\n\n\\begin{equation} \nI_{col}\\left( p\\right) =\\left| -u_{\\mu}p^{\\mu}\\right|   f_0\\left( p\\right) \\left[Z\\left(p \\right) -K_{\\rho\\sigma}p^{\\rho}\\mathrm{sign}\\left(p^0\\right)\\int\\;Dp'\\;p'^{\\sigma}\\left(-u_{\\mu}p'^{\\mu}\\right)  f_0\\left( p'\\right) Z\\left( p'\\right) \\right] \n\\end{equation}\nIf $Z$ satisfies the constraint eq. (\\ref{trans}), then the second term is zero and the entropy production is positive, provided $Z\\ge -1$.\n\nTo first order\n\n\\begin{equation}\nZ_1=-\\frac{p^{\\mu}p^{\\nu}}{\\left|-p^{\\alpha}u_{\\alpha}\\right|}\\beta_{\\nu,\\mu}\n\\label{foc}\n\\end{equation}\nNow in general\n\n\\begin{eqnarray}\n\\beta_{\\nu,\\mu}&=&\\frac1T\\left[u_{\\nu,\\mu}-u_{\\nu}\\frac{T_{,\\mu}}{T}\\right]\\nonumber\\\\\n&=&\\frac1T\\left[u_{\\mu}u_{\\nu}\\frac{\\dot{T}}{T}+\\frac12\\left[\\sigma_{\\mu\\nu}+\\frac23\\Delta_{\\mu\\nu}u^{\\lambda}_{,\\lambda}\\right]-\\frac12\\left[u_{\\nu}\\Delta^{\\lambda}_{\\mu}+u_{\\mu}\\Delta^{\\lambda}_{\\nu}\\right]\\left(\\frac{T_{,\\lambda}}{T}+\\dot{u}_{,\\lambda}\\right)\\right.\\nonumber\\\\\n&-&\\left.\\frac12\\left[u_{\\nu}\\Delta^{\\lambda}_{\\mu}-u_{\\mu}\\Delta^{\\lambda}_{\\nu}\\right]\\left(\\frac{T_{,\\lambda}}{T}-\\dot{u}_{,\\lambda}\\right)+\\frac12\\Delta^{\\lambda}_{\\nu}\\Delta^{\\rho}_{\\mu}\\left[u_{\\lambda,\\rho}-u_{\\rho,\\lambda}\\right]\\right]\n\\end{eqnarray}\nso, using the zeroth order time derivatives and $c^2=1\/3$\n\n\\begin{equation}\nZ_1=\\frac{-1}{2T\\left|p^{\\alpha}u_{\\alpha}\\right|}\\sigma_{\\mu\\nu}{p^{\\mu}p^{\\nu}}\n\\label{trans2}\n\\end{equation}\nThe viscous energy momentum tensor is\n\n\\begin{equation} \n\\Pi_1^{\\mu\\nu}=\\frac{-\\tau}{2T}\\int\\;Dp\\;f_0\\frac{p^{\\mu}p^{\\nu}p^{\\rho}p^{\\lambda}}{\\left|p^{\\alpha}u_{\\alpha}\\right|}\\sigma_{\\rho\\lambda}=-\\eta\\sigma^{\\mu\\nu}\n\\label{VEMTCE1}\n\\end{equation} \nwhere\n\n\\begin{equation}\n\\eta=\\frac{\\tau}{15T}\\int\\;Dp\\;f_0\\left|-p^{\\alpha}u_{\\alpha}\\right|^3=\\frac{8}{5\\pi^2}\\tau T^4\n\\end{equation}\nThe Chapman-Enskog procedure cannot generate a true dynamical equation for the viscous EMT, and thus does not solve the causality\/stability problems. The Grad approach takes the different strategy of keeping the form eq. (\\ref{trans2}), but replacing $\\sigma_{\\mu\\nu}$ by a new tensor $C_{\\mu\\nu}$ regarded as a new independent variable.\n\n\\begin{equation}\nZ_G=\\frac{-1}{2T\\left|-p^{\\alpha}u_{\\alpha}\\right|}C_{\\mu\\nu}{p^{\\mu}p^{\\nu}}\n\\label{Gradansatz}\n\\end{equation}\n $C_{\\mu\\nu}$ is defined up to  a multiple of $g_{\\mu\\nu}$, so we may assume it is traceless, and because of the constraint eq. (\\ref{trans}) it must be transverse. \nThe viscous energy momentum tensor becomes\n\n\\begin{equation} \n\\Pi_G^{\\mu\\nu}=\\frac{-1}{2T}\\int\\;Dp\\;f_0\\frac{p^{\\mu}p^{\\nu}p^{\\rho}p^{\\lambda}}{\\left|-p^{\\alpha}u_{\\alpha}\\right|}C_{\\rho\\lambda}=-\\frac{\\eta}{\\tau}C^{\\mu\\nu}\n\\label{VEMTG}\n\\end{equation}\nThe next step is to substitute this into the Boltzmann equation. The nonlocal term vanishes and we get \n\n\\begin{equation}\n\\frac{1}{\\tau}\\left(p^{\\nu}u_{\\nu}\\right)f_0Z=\\frac{\\partial}{\\partial x^{\\mu}}\\left[p^{\\mu}f_0\\left(1+Z \\right) \\right]\n\\label{BE2}\n\\end{equation}\nWe wish to extract from there an equation for $C^{\\mu\\nu}$. The time-honored procedure is to consider the moments of this equation. Since $Z$ is even the zeroth moment vanishes, and the first moment gives back energy - momentum conservation.\n\n\\begin{equation}\n\\frac{\\partial}{\\partial x^{\\lambda}}\\int\\;Dp\\;p^{\\rho}p^{\\lambda}f_0\\left(1+Z \\right) =0\n\\end{equation}\nTherefore the first non trivial choice is to use the second moments\n\n\\begin{equation}\n\\frac{\\partial}{\\partial x^{\\mu}}\\int\\;Dp\\;\\mathrm{sign}\\left(p^0\\right)p^{\\rho}p^{\\lambda}p^{\\mu}f_0\\left(1+Z \\right) =\\frac{-1}{\\tau}\\int\\;Dp\\;p^{\\rho}p^{\\lambda}\\left|-p^{\\nu}u_{\\nu}\\right|f_0Z\n\\end{equation}\nTo evaluate these equations, we use the following identities\n\n\\begin{eqnarray} \n\\int\\;Dp\\;p^{\\rho}p^{\\lambda}f_0&=&{A_2}\\left[u^{\\rho}u^{\\lambda}+\\frac 1{3}\\Delta^{\\rho\\lambda}\\right]\\nonumber\\\\\n\\int\\;Dp\\;\\mathrm{sign}\\left(p^0\\right)p^{\\rho}p^{\\lambda}p^{\\mu}f_0&=&{A_3}\\left[u^{\\rho}u^{\\lambda}u^{\\mu}\n+\\frac 1{3}\\left( \\Delta^{\\rho\\lambda}u^{\\mu}+\\Delta^{\\mu\\lambda}u^{\\rho}+\\Delta^{\\mu\\rho}u^{\\lambda}\\right) \\right]\\nonumber\\\\\n\\int\\;Dp\\;p^{\\rho}p^{\\lambda}p^{\\mu}p^{\\nu}f_0&=&{A_4}\\left\\lbrace u^{\\rho}u^{\\lambda}u^{\\mu}u^{\\nu}+\\frac 1{3}\\left( \\Delta^{\\rho\\lambda}u^{\\mu}u^{\\nu}+...\\right) +\\frac 1{15}\\left( \\Delta^{\\rho\\lambda}\\Delta^{\\mu\\nu}+...\\right)\\right\\rbrace\\nonumber\\\\\n\\int\\;Dp\\;\\mathrm{sign}\\left(p^0\\right)p^{\\rho}p^{\\lambda}p^{\\mu}p^{\\nu}p^{\\theta}f_0&=&  {A_5}\\left\\lbrace u^{\\rho}u^{\\lambda}u^{\\mu}u^{\\nu}u^{\\theta}+\\frac 1{3}\\left( \\Delta^{\\rho\\lambda}u^{\\mu}u^{\\nu}u^{\\theta}+...\\right) +\\frac 1{15}\\left( \\Delta^{\\rho\\lambda}\\Delta^{\\mu\\nu}u^{\\theta}+...\\right)\\right\\rbrace \\nonumber\\\\\n\\int\\;Dk\\;k^{\\rho}k^{\\lambda}k^{\\mu}k^{\\nu}k^{\\theta}k^{\\phi}F\\left[\\left(k^{\\alpha}u_{\\alpha}\\right) \\right]&=&  {A_6}\\left\\lbrace u^{\\rho}u^{\\lambda}u^{\\mu}u^{\\nu}u^{\\theta}u^{\\phi}+\\frac13\\left( \\Delta^{\\rho\\lambda}u^{\\mu}u^{\\nu}u^{\\theta}u^{\\phi}+...\\right)\\right. \\nonumber\\\\\n &+&\\left. \\frac1{15}\\left( \\Delta^{\\rho\\lambda}\\Delta^{\\mu\\nu}u^{\\theta}u^{\\phi}+...\\right)+\\frac1{105}\\left( \\Delta^{\\rho\\lambda}\\Delta^{\\mu\\nu}\\Delta^{\\theta\\phi}+...\\right)\\right\\rbrace \n\\end{eqnarray} \nIn the following, we shall adopt\n\\begin{equation} \nA_a=\\int\\;Dp\\;\\left|-p^{\\alpha}u_{\\alpha}\\right|^a f_0=\\left(a+1 \\right) !\\frac{T^{a+2}}{ \\pi^2 }\n\\end{equation} \nThe equation becomes\n\n\\begin{equation} \n\\frac{\\partial}{\\partial x^{\\mu}}{T^5}C^{\\rho\\lambda\\mu}=\\frac{T^5}{\\tau}C^{\\rho\\lambda}\n\\end{equation}\nwhere\n\n\\begin{equation} \nC^{\\rho\\lambda\\mu}= \\left\\{3u^{\\rho}u^{\\lambda}u^{\\mu}\n+ \\Delta^{\\rho\\lambda}u^{\\mu}+\\Delta^{\\mu\\lambda}u^{\\rho}+\\Delta^{\\mu\\rho}u^{\\lambda}-u^{\\lambda}C^{\\rho\\mu}- u^{\\rho}C^{\\mu\\lambda}-u^{\\mu}C^{\\rho\\lambda} \\right\\}\n\\end{equation}\nIt is not really possible to kill all second moments simultaneously. We shall be happy to kill the transverse traceless contribution, namely\n\n\\begin{equation} \n\\left[\\Delta_{\\tau\\rho}\\Delta_{\\sigma\\lambda}-\\frac13\\Delta_{\\tau\\sigma}\\Delta_{\\rho\\lambda}\\right]\\frac{\\partial}{\\partial x^{\\mu}}T^{5}C^{\\rho\\lambda\\mu}=\\frac{T^5}{\\tau}C_{\\tau\\sigma}\n\\end{equation} \nwhich reduces to\n\n\\begin{eqnarray} \n\\frac {-1}{T^5}\\frac{\\partial}{\\partial x^{\\mu}}{T^{5}}u^{\\mu}C_{\\tau\\sigma}&-&\\Delta_{\\sigma\\lambda}C^{\\rho\\lambda\\mu}\\frac{\\partial u_{\\tau}u_{\\rho}}{\\partial x^{\\mu}}\n-\\Delta_{\\tau\\rho}C^{\\rho\\lambda\\mu}\\frac{\\partial u_{\\sigma}u_{\\lambda}}{\\partial x^{\\mu}}\\nonumber\\\\\n&+&\\frac13\\Delta_{\\tau\\sigma}C^{\\rho\\lambda\\mu}\\frac{\\partial u_{\\rho}u_{\\lambda}}{\\partial x^{\\mu}}+\\frac13\\Delta_{\\rho\\lambda}C^{\\rho\\lambda\\mu}\\frac{\\partial u_{\\sigma}u_{\\tau}}{\\partial x^{\\mu}}=\\frac{1}{\\tau}C_{\\tau\\sigma}\n\\label{GE}\n\\end{eqnarray} \nKeeping only linear terms, this is an equation of Maxwell - Cattaneo type for $C_{\\mu\\nu}$ \\cite{Max67,JosPre89}\n\n\\begin{equation}\n\\dot{C}_{\\tau\\sigma}+\\frac{1}{\\tau}C_{\\tau\\sigma}-\\sigma_{\\tau\\sigma}+\\mathrm{ho}=0\n\\end{equation} \nActually, we may get an expression for the time derivative of $Z$ directly from the transport equation and then use it to find an equation for the viscous energy momentum tensor \\cite{DMNR12}. \nHowever, beyond the leading order this equation involves integrals which cannot be simple expressed in terms of the known moments of the distribution function. Therefore to obtain a definite equation we must provide a \\emph{closure}, i. e., an expression for $Z$ allowing us to compute the integrals. For example, if we use the Grad ansatz $Z_G$ we get an equation with the same structure as eq. (\\ref{GE}), though the coefficients may be different.\n\nWe wish to check that the Grad approach is consistent with causality and stability \\cite{HisLin83,Ols90,OlsHis90}. Introducing a new perturbation  for $C^{ij}$ we get the equations\n\n\\begin{eqnarray}\n-i\\omega'\\delta'+ic^2k'_j\\delta'^j&=&0\\nonumber\\\\\n-i\\omega'\\delta'^i+ik'^i\\delta' -i\\frac{\\gamma}{\\tau}k'_jC'^{ij}&=&0\\nonumber\\\\\n-i\\omega'C'^{ij}+\\frac{1}{\\tau}C'^{ij}-i\\left( k'^i\\delta'^j+k'^j\\delta'^i-\\frac 23\\delta^{ij}k'_k\\delta'^k\\right) &=&0\n\\end{eqnarray}\nWe revert to the Chapman - Enskog equations with the replacement $\\gamma\\to\\gamma \/1-i\\omega'\\tau$. For transverse perturbations we now get a quadratic equation\n\n\\begin{equation} \n\\left(1-i\\omega'\\tau\\right)\\omega'+i\\gamma K'^2=0\n\\end{equation} \nIt is easy to see that both roots are stable when $u=0$. For general $u$ stability obtains if $\\tau\\ge\\gamma$. It is interesting to observe that with the expressions above $\\tau\\approx 5\\gamma$.\n\nWe now consider longitudinal waves with $k_{\\perp}=0$. The dispersion relation is\n\n\\begin{equation} \n\\left(1-i\\omega'\\tau\\right)\\left[\\omega'^2-\\frac{c^2}{u^2_0}\\left(  K-u\\omega'\\right)^2\\right]+\\frac 4{3u^2_0}i\\gamma \\omega'\\left(  K-u\\omega'\\right)^2=0\n\\end{equation}\nWe rearrange this as\n\n\\begin{equation}\n\\omega'\\left[\\omega'^2-C^2\\left(  K-u\\omega'\\right)^2\\right]\n+\\frac{i}{\\tau}\\left[\\omega'^2-\\frac{c^2}{u^2_0}\\left(  K-u\\omega'\\right)^2\\right]=0\n\\end{equation}\nwhere \n\n\\begin{equation}\nC^2=\\frac{c^2}{u^2_0}\\left(1+\\frac {4\\gamma}{3c^2\\tau}\\right)\n\\end{equation}\nThis is\n\n\\begin{equation}\n\\omega'\\left(\\omega'-\\frac{CK}{\\left(1+Cu\\right)}\\right)\\left(\\omega'+\\frac{CK}{\\left(1-Cu\\right)}\\right)\n+\\frac{i}{\\left(1-C^2u^2\\right)\\tau}\\left[\\omega'^2-\\frac{c^2}{u^2_0}\\left(  K-u\\omega'\\right)^2\\right]=0\n\\end{equation}\nIn the formal limit $\\tau\\to\\infty$ we find three real roots $\\omega'=0,\\pm\\omega_{\\pm}$, where $\\omega_{\\pm}= CK\/1\\pm Cu$. At large but finite $\\tau$, the solution that goes to zero behaves as\n\n\\begin{equation}\n\\omega_0=\\frac{-i}{\\tau}\\frac{c^2}{u^2_0C^2}\n\\end{equation}\nand it is stable. The solutions which converge to $\\pm\\omega_{\\pm}$ behave as\n\n\\begin{equation}\n2CK\\omega_{\\pm}\\left(\\omega'-\\omega_{\\pm}\\right)+\\frac {4i\\gamma}{3u^2_0\\tau^2}\\left(  K\\mp u\\omega_{\\pm}\\right)^2=0\n\\end{equation}\nso they are stable too. Of course, we also have solutions with $\\delta=\\delta^i=k'_jC^{ij}=0$ and $\\omega'=-i\/\\tau$, which are obviously stable. \n\n\\subsection{Entropy production variational method}\n\nThe Grad approach as we have presented it still has the problems that the Grad ansatz eq. (\\ref{Gradansatz}) does not lead to a non negative one particle distribution function (quite the opposite, since $C_{\\mu\\nu}{p^{\\mu}p^{\\nu}}$ must be negative in some direction in momentum space) and that it is unclear how to introduce nonlinear terms. To overcome these problems we need a better motivated closure for $Z$. We shall try to find it by seeking the value of $Z$ which minimizes entropy production for a given VEMT, and satisfies the constraint eq. (\\ref{TTC}) \\cite{CalPR10}. Adding Lagrange multipliers $\\zeta_{\\mu\\nu}$  we get the equation\n\n\\begin{equation}\n\\left\\{I_{col}\\left[\\ln\\left[1+Z\\right]\\right] +\\frac{I_{col}\\left[Z\\right] }{1+Z}\\right\\}= \n-\\tau\\zeta_{\\mu\\nu} p^{\\mu}p^{\\nu}f_0\n\\label{EPVM}\n\\end{equation}\nThe point is that this equation has bounded solutions for any value of the right hand side. We may also regard it as a means to obtain a formal series solution for $Z$ in powers of $\\tau$, where, if we keep only the first order term, we recover the Grad ansatz. \n\nIt is convenient to introduce a new unknown $\\chi=\\ln\\left[1+Z\\right]$, with inverse transformation $Z=e^{\\chi}-1$. In terms of the new unknown, the equation reads \n\n\\begin{equation}\nI_{col}\\left[\\chi\\right]= \n-\\frac{\\tau}2\\zeta_{\\mu\\nu} p^{\\mu}p^{\\nu}f_0-\\frac12\\left[e^{-\\chi}I_{col}\\left[e^{\\chi}-1\\right]-I_{col}\\left[\\chi\\right]\\right]\n\\label{EPVM3}\n\\end{equation}\nTo lowest order we may neglect the second term in the right hand side. Observe that in any case $\\zeta_{\\mu\\nu}$ is defined up to a multiple of $g_{\\mu\\nu}$. We use this freedom to require $\\zeta_{\\mu\\nu}u^{\\mu}u^{\\nu}=0$. Transversality requires\n\n\\begin{equation}\n0=\\int\\;Dp\\;\\mathrm{sign}\\left(p^0\\right)p^{\\lambda}f_0\\zeta_{\\mu\\nu} p^{\\mu}p^{\\nu}=\\zeta_{\\mu\\nu}A_3\\left[u^{\\lambda}u^{\\mu}u^{\\nu}+\\frac13\\left(u^{\\lambda}\\Delta^{\\mu\\nu}+u^{\\mu}\\Delta^{\\lambda\\nu}+u^{\\nu}\\Delta^{\\lambda\\mu}\\right)\\right]\n\\end{equation}\nTherefore $\\zeta_{\\mu\\nu}$ must be traceless and transverse.  To lowest order \n\n\\begin{equation}\n\\chi=\\frac{-1}2\\frac{\\zeta^{\\left(0\\right)}_{\\mu\\nu} p^{\\mu}p^{\\nu}}{\\left| -u_{\\rho}p^{\\rho}\\right|}\n\\end{equation} \nLet us further expand the exponential. In the rest frame  the energy momentum tensor reads\n\n\\begin{eqnarray}\nT^{00}&=&\\sigma T^4\\equiv \\rho\\nonumber\\\\\nT^{0i}&=&0\\nonumber\\\\\nT^{ij}&=&\\frac13 \\sigma T^4 \\delta^{ij}-\\frac{A_3}{15}\\tau\\zeta^{ij}\n\\label{tfromz}\n\\end{eqnarray}\nThe kinetic equation to lowest order in $\\tau$ reads\n\n\\begin{equation}\np^{\\lambda}\\partial_{\\lambda}f_0\\left[1-\\tau\\frac{1}2\\frac{\\zeta_{\\mu\\nu} p^{\\mu}p^{\\nu}}{\\left| -u_{\\rho}p^{\\rho}\\right|}\\right]=\\frac{1}2\\left(  -u_{\\mu}p^{\\mu}\\right)   f_0 \\frac{\\zeta_{\\mu\\nu} p^{\\mu}p^{\\nu}}{\\left| -u_{\\rho}p^{\\rho}\\right|}  \n\\end{equation}\nThe meaning of this equation is as a generating equation for its moments. The first order moments give energy momentum conservation. To find an equation for $\\zeta_{\\mu\\nu}$ we go to the rest frame and multiply both sides by $\\mathrm{sign}\\left(p^0\\right)p^ip^j$. Integrating and discarding the trace part, we get\n\n\\begin{equation} \n\\frac{A_4}{15 T}\\sigma_{ij}-\\frac{\\tau A_5 \\dot{T}}{15 T^2}\\zeta_{ij}-\\frac{\\tau A_5}{105 T}\\left[ u_{k,k}\\zeta_{ij}+\\zeta_{ik}\\sigma ^k_j+\\sigma_{i}^k\\zeta_{kj}-\\frac 23\\Delta_{ij}\\zeta^{\\left(0\\right)kl}\\sigma_{kl}\\right] =\\frac{A_4}{15 } \\left[\\zeta_{ij}+\\tau\\zeta_{ij,0}\\right] \n\\end{equation}\n\n\\subsection{Non linear corrections to boost invariant flow}\nWe now turn to consider nonlinear corrections to this equations.\n\nThe strategy we are following consists on improving the stability of the theory by adding new variables obeying dynamical equations of their own. In many approaches, such as the so-called  Israel - Stewart theory \\cite{is1,is2} or Extended Thermodynamics \\cite{extended} the extra variables are the components of the viscous energy-momentum tensor $\\Pi_{\\mu\\nu}$ itself. The dynamical equations for $\\Pi_{\\mu\\nu}$ have been derived in a number of ways, such as carefully taking moments of the kinetic equation \\cite{DMNR12}, a systematic gradient expansion of the kinetic theory \\cite{BRW06}, from AdS-CFT correspondence \\cite{cft} or simply writing down all terms consistent with the symmetries of the theory up to a certain order \\cite{BRSSS}. We shall call these theories ``second order fluid dynamics'' (SOFD) for short, and take the presentation in \\cite{EXG10} as a suitable representative.\n\nIn our approach (EPVM, or entropy production variational method), on the other hand, the new variables are the Lagrange multipliers enforcing the constraints on entropy production; $\\Pi_{\\mu\\nu}$ itself may depend nonlinearly on the Lagrange multipliers. Our approach is therefore closer to the Geroch - Linblom Divergence type theories \\cite{dtt}, although we shall not demand that the resulting theory conforms to the dissipative type framework.\n\nTo obtain further insight on the meaning of these theories, we shall apply them to the case of boost invariant flow. Adopting Milne coordinates $T^{\\mu}_{\\nu}$ is diagonal; we write $T^{\\tau}_{\\tau}=-\\rho$, $T^{\\eta}_{\\eta}= \\rho \/3-\\Pi$, $T^x_x=T^y_y=\\rho\/3+\\Pi\/2$. Energy-momentum conservation yields\n\n\\begin{equation} \n\\dot{\\rho}+\\frac1{\\tau}\\left( \\frac 43\\rho-\\Pi\\right) =0\n\\end{equation}\nWe also write $\\zeta^{\\left(0\\right)\\eta}_{\\eta}=\\zeta$, $\\zeta^{\\left(0\\right)x}_{x}=\\zeta^{\\left(0\\right)y}_{y}=-\\zeta\/2$. Recall that $\\sigma^{\\eta}_{\\eta}=4\/3\\tau$, $\\sigma^{x}_{x}=\\sigma^{y}_{y}=-2\/3\\tau$. We get, discarding a derivative of the temperature,\n\n\\begin{equation} \n\\tau_R\\dot{\\zeta}+\\zeta =\\frac 4{3T\\tau}-a_1 \\tau_R\\frac{\\zeta}{\\tau} \n\\label{foeq}\n\\end{equation} \nwhere we have written $\\tau_R$ for the relaxation time to avoid confusion with Milne time, and \n\n\\begin{equation}\na_1=\\frac{ A_5}{3A_4T}\n\\end{equation}\nThe linearized collision term we are using is too simplistic to allow for a reliable derivation of nonlinear terms; however, in this case we may exploit the symmetries of the problem, which indicate that there is a single true degree of freedom $\\zeta$ underlying the viscous energy momentum tensor. Therefore we generalize eq. (\\ref{tfromz}) to \n\n\\begin{equation} \n\\Pi=\\frac{\\tau_RA_3}{15}\\zeta + h_1 \\left( \\frac{\\tau_RA_3}{15}\\zeta\\right) ^2\n\\end{equation} \nand eq. (\\ref{foeq}) to \n\n\\begin{equation} \n\\tau_R\\dot{\\zeta}+\\zeta =\\frac 4{3T\\tau}-a_1\\tau_R \\frac{\\zeta}{\\tau} - h_2\\frac{\\tau_RA_3}{15}\\zeta^ 2\n\\label{soeq}\n\\end{equation}  \n\nWe determine the new transport coefficients $h_{1,2}$ by asking that, in a stationary situation, the relation between $\\Pi$ and the shear $4\/3\\tau$, to second order in $\\tau_R$, agrees with a systematic expansion for a Boltzmann gas \\cite{EXG10}. Indeed, to second order we may write\n\n\\begin{equation} \n\\zeta = \\frac{15}{\\tau_RA_3}\\left[ \\Pi - h_1\\Pi^2\\right] \n\\end{equation} \nand so, neglecting time derivatives\n\n\\begin{equation} \n\\Pi =\\frac{\\tau_R A_3}{15T}\\frac 4{3\\tau}-a_1\\tau_R \\frac{\\Pi}{\\tau}- \\left( h_2-h_1\\right) \\Pi^2\n\\end{equation} \nMatching against SOFD yields \\cite{EXG10}\n\n\\begin{eqnarray} \n\\tau_R &=&\\frac{6\\eta}{sT}\\nonumber\\\\\n\\frac{A_3}{15T}&=&\\frac{sT}{6}\\nonumber\\\\\na_1&=&\\frac 43\\nonumber\\\\\nh_2-h_1&=&\\frac1{4\\pi T\\eta}\n\\end{eqnarray}\nwhere $\\eta$ is the shear viscosity and $s$ the entropy density, both extracted from a realistic equation of state. An analysis of the possible origins of the $h_{1,2}$ coefficients shows that both are of the same order of magnitude, $\\approx\\tau_R^{-1}$. When this holds, the behavior of the model is relatively insensitive to the actual values of $h_1$ and $h_2$.\n\nTo break the degeneracy between $h_1$ and $h_2$, let us consider a situation where the shear $\\tau^{-1}$ terms are negligible and the transport coefficients are constant. The SOFD equation admits a stationary solution with $\\Pi=\\Pi_{0SOFD}=-1\/\\left( h_2-h_1\\right) $; $\\Pi$ is otherwise unbounded. The EPVM equations on the other hand imply a lower bound $\\Pi\\ge -1\/\\left( 4h_1\\right) $, which is realized when $\\zeta=-1\/\\left( 2h_1\\eta T\\right) $. There is a steady solution when $\\zeta=-1\/\\left( h_2\\eta T\\right) $, which implies $\\Pi=\\Pi_{0EPVM}=(-1\/h_2)\\left[1-\\left( h_1\/h_2\\right) \\right]$. In both cases, the steady solutions are unstable. For SOFD, this implies the existence of runaway solutions, namely, solutions which begin below the fixed point (if it is negative) run away to minus infinity. In the EPVM, on the other hand, we may eliminate the runaway solutions by demanding that the steady solution coincides with the lower bound, and adopting, for each allowed value of $\\Pi$, the value of $\\zeta$ above the fixed point. This requires $h_1=h_2\/2$, \nand therefore $h_2=2h_1=1\/\\left( 2\\pi T\\eta\\right) $. Introducing the transport coefficient \n\n\\begin{equation} \n\\lambda =\\frac{\\eta}{2\\pi T}\n\\end{equation} \nThe SOFD equation reads \\cite{EXG10}\n\n\\begin{equation} \n\\tau_R\\dot{\\Pi}+\\Pi =\\frac {4\\eta}{3\\tau}-\\frac 43\\tau_R \\frac{\\Pi}{\\tau}- \\frac{\\lambda}{2\\eta^2} \\Pi^2\n\\label{sofd}\n\\end{equation} \nwhile the extended EPVM yields the system \\cite{CalPR10}\n\n\\begin{eqnarray} \n\\tau_R\\dot{\\zeta}+\\zeta &=&\\frac 4{3T\\tau}-\\frac 43\\tau_R \\frac{\\zeta}{\\tau} - \\frac{\\zeta^ 2}{2\\pi}\\nonumber\\\\\n\\Pi &=&\\eta T\\left[ \\zeta + \\frac{\\zeta^2}{4\\pi}\\right] \n\\end{eqnarray}\nOne way to visualize the difference between these models is to consider the free decay of $\\Pi$. Suppose $\\Pi$ is observed to have the value $\\Pi_0$ at some time $\\tau_0$ late enough that the $\\tau^ {-1}$ terms in the equations may be neglected, and the transport coefficients regarded as constant. Then, according to SOFD, the further decay of the VEMT is given by\n\n\\begin{equation} \n\\frac{\\Pi}{\\Pi_0}=\\frac{1}{\\left( 1+\\frac{x_0}{4}\\right)e^{t}-\\frac{x_0}{4} }\n\\end{equation} \nwhere $t=\\tau-\\tau_0\/\\tau_R$ and $x=\\Pi\/\\pi\\eta T$, while from EPVM we get\n\n\\begin{equation} \n\\zeta=\\frac{\\zeta_0}{\\left( 1+\\frac{\\zeta_0}{2\\pi}\\right)e^{t}-\\frac{\\zeta_0}{2\\pi} }\n\\end{equation} \nwhere\n\n\\begin{equation} \n\\frac{\\zeta_0}{2\\pi}=\\sqrt{1+x_0}-1\n\\end{equation}\nBesides the absence of runaway solutions already noted, the EPVM provides for a faster decay of large fluctuations (see fig. (\\ref{decay})). \n\n\\begin{center}\n\n\\begin{figure}[htb]\n\n\\scalebox{0.47}{\\includegraphics{decay.pdf}}\n\n\\caption{(Color online) The evolution of the viscous energy momentum tensor, as predicted by SOFD (dashes and dots) and EPVM (full line) starting form $x_0=5$. We see that EPVM predicts a faster approach towards $\\Pi=0$. }\n\n\\label{decay}\n\n\\end{figure}\n\n \\end{center}\n\n\n\\section{Final remarks}\n\nThe theory of relativistic real fluids has a curious history because while the stability problems we have discussed have been known for a long time \\cite{JosPre89} , yet they do not seem to have elicited any strong response until fairly recently. There were known ways to improve the theory (foremost the Israel - Stewart \\cite{is1,is2} and extended thermodynamics theories \\cite{extended}), and also a family of theories which were known to be free of such problems on a rigorous basis (the Geroch-Lindblom dissipative type theories \\cite{dtt}). However, the former were presented as successive approximations to a yet unknown theory, and the physical foundations of the latter remained elusive \\cite{dtt2}. It was only the realization that relativistic real fluids might be produced in RHICs that triggered an all-out attack on the problem, to the extent that it would be impossible to describe all this activity in a short review such as this. We have therefore aimed to present just the fundamental ideas behind the theory, what the main problems are, and which lines of thought seem to us likely to be fruitful, and this almost entirely from the formal side, leaving phenomenology to more knowledgeable authors.\n\nConcerning the present state of the theory, it seems fair to say that we have a reliable understanding of evolution during the hydrodynamic era, and the beginnings of a theory of the freeze out transition. The early times of the collision are relatively much more poorly understood. In particular, we do not know how hydrodynamic behavior may arise on such short time scales as demanded by theory, though the work on non-abelian instabilities \\cite{inst} and QGP turbulence \\cite{turb} , on one hand, and on AdS-CFT correspondence on the other \\cite{cfttherm}, makes those scales look not so unrealistic as they used to. \n\nWe can only conclude that we are only witnessing  the early childhood of the relativistic real fluids - RHICs connection, and this is what makes this such an exciting field to work on. \n\n\\section*{Acknowledgment}\nThis work has been developed in collaboration with Jer\\'onimo Peralta Ramos. It is supported in part by Universidad de Buenos Aires, CONICET and ANPCYT (Argentina)\n\n\n\\section*{ Appendix: Kadanoff-Baym equations and quantum kinetic field theory}\n\nIn this appendix we shall discuss the derivation of kinetic theory from quantum field theory. The presentation follows \\cite{CalHu08}.\n\nThe CTP generating functional depends on two external sources\n\n\\begin{equation}\ne^{iW\\left[J^{1},J^{2}\\right]}=\\int D\\Phi^{1}D\\Phi^{2}\\:e^{i\\left\\{S\\left[\\Phi^{1}\\right]-S\\left[\\Phi^{2}\\right]+\\int \\left(J^{1}\\Phi^{1}-J^{2}\\Phi^{2}\\right)\\right\\}}\n\\label{five}\n\\end{equation}\n\nIt defines two background fields through\n\n\\begin{equation}\n\\phi^{1}\\left[x\\right]=\\frac{\\delta W\\left[J^{1},J^{2}\\right]}{\\delta J^{1}\\left[x\\right]};\\ \\ \\phi^{2}\\left[x\\right]=-\\frac{\\delta W\\left[J^{1},J^{2}\\right]}{\\delta J^{2}\\left[x\\right]}\n\\label{six}\n\\end{equation}\n\n\n\n\nThe CTPEA is the full Legendre transform \n\n\\begin{equation}\n\\Gamma\\left[\\phi^1,\\phi^2\\right]=W\\left[J^{1},J^{2}\\right]-\\int \\left(J^{1}\\phi^{1}-J^{2}\\phi^{2}\\right)\n\\label{seven}\n\\end{equation}\n\nIt generates the equations of motion \n\n\\begin{equation}\n\\frac{\\delta \\Gamma\\left[\\phi^1,\\phi^2\\right]}{\\delta\\phi^1\\left[x\\right]}=-J^1\\left[x\\right];\\ \\ \\frac{\\delta \\Gamma\\left[\\phi^1,\\phi^2\\right]}{\\delta \\phi^2\\left[x\\right]}=J^2\\left[x\\right]\n\\label{eight}\n\\end{equation}\n\nThe equation of motion for the mean field is obtained when $J^1=J^2$ by setting $\\phi^1=\\phi^2$ in the equations (\\ref{eight}) \\emph{after} computing the variational derivatives.\n\n\n\nTo compute the CTPEA, observe that\n\n\\begin{equation}\n\\Gamma\\left[\\phi^1,\\phi^2\\right]=S\\left[\\phi^1\\right]-S\\left[\\phi^2\\right]+\\mathrm{quantum\\: corrections}\n\\label{nine}\n\\end{equation}\n\nThe quantum corrections are the sum of all the one-particle irreducible (1PI) graphs in the theory.\nThe linearized one-particle irreducible ($%\n1PI$) effective action has the structure\n\n\\begin{eqnarray}\n\\Gamma _{1PI} &=&\\int d^{d}xd^{d}y\\;\\left\\{ \\varphi_- \\left( x\\right) \\left[D\\left(\nx,y\\right) +\\mathbf{D}\\left( x,y\\right) \\right] \\varphi_+\\left( y\\right) \\right.  \\nonumber \\\\\n&&\\left. +\\frac{i}{2}\\varphi_- \\left( x\\right) \\mathbf{N}%\n\\left( x,y\\right) \\varphi_-\\left( y\\right)\n\\right\\}\n\\end{eqnarray}\n$\\varphi_-=\\left[ \\varphi\n^{1}-\\varphi ^{2}\\right]$, $\\varphi_+=\\left[ \\varphi\n^{1}+\\varphi ^{2}\\right]\/2$.\n\n\\begin{equation}\nD(x,y)=\\left[ \\partial _{x}^{2}-m_{b}^{2}\\right] \\delta (x-y)\n\\end{equation}\n$\\mathbf{D}$ is causal and $\\mathbf{N}$ is even, and both are real. \nA good deal of our discussion will revolve around the different properties\nof the propagators of the theory, that is, the expectation values of binary\nproducts of field operators with respect to the initial state. Since field\noperators at different locations do not generally commute, we have several\ndifferent propagators according to the ordering of the field operators\nwithin the expectation value. \n\nThe equations of motion for the propagators are derived from the\nidentity\n\n\\begin{equation}\n\\frac{{\\cal D} ^{2}\\Gamma _{1PI}}{{\\cal D} \\varphi ^{a}{\\cal D} \\varphi ^{b}}G^{bc}%\n=i\\hbar \\delta^c_{a}\n\\end{equation}\n\n\nThe $G^{ab}$ above denote the four basic propagators\n\\medskip\n\nFeynman $G_{F}\\equiv <T\\left( \\Phi \\left( x\\right) \\Phi \\left( x^{\\prime\n}\\right) \\right) >=G^{11}$,\n\\medskip\n\n\nDyson $G_{D}\\equiv <\\tilde{T}\\left( \\Phi \\left(\nx\\right) \\Phi \\left( x^{\\prime }\\right) \\right) >=G^{22}$,\n\\medskip\n\nPositive frequency $G^{+}\\equiv <\\Phi \\left( x\\right) \\Phi \\left( x^{\\prime }\\right) >=G^{21}$,\n\\medskip\n\nNegative frequency $G^{-}\\equiv <\\Phi \\left( x^{\\prime }\\right) \\Phi \\left(\nx\\right) >=G^{12}$,\n\\medskip\n\nwhere $T$ stands for time ordering and $\\tilde{T}$ stands for\nanti-time ordering.\n\nExplicitly\n\n\\begin{eqnarray}\n\\left[D+\\mathbf{D}_{even}+i\\mathbf{N}\\right]G^{11}+\\left[\\mathbf{D}_{odd}-i\\mathbf{N}\\right]G^{21}&=&i\\mathbf{1}\\nonumber\\\\\n\\left[\\mathbf{D}_{odd}+i\\mathbf{N}\\right]G^{11}+\\left[D+\\mathbf{D}_{even}-i\\mathbf{N}\\right]G^{21}&=&0\\nonumber\\\\\n\\left[D+\\mathbf{D}_{even}+i\\mathbf{N}\\right]G^{12}+\\left[\\mathbf{D}_{odd}-i\\mathbf{N}\\right]G^{22}&=&0\\nonumber\\\\\n\\left[\\mathbf{D}_{odd}+i\\mathbf{N}\\right]G^{12}+\\left[D+\\mathbf{D}_{even}-i\\mathbf{N}\\right]G^{22}&=&-i\\mathbf{1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\mathbf{D}_{even}\\left( x,y\\right)=\\frac12\\left[\\mathbf{D}\\left( x,y\\right)+\\mathbf{D}\\left( y,x\\right)\\right]\\nonumber\\\\\n\\mathbf{D}_{odd}\\left( x,y\\right)=\\frac12\\left[\\mathbf{D}\\left( x,y\\right)-\\mathbf{D}\\left( y,x\\right)\\right]\n\\end{eqnarray}\nWe obtain a more efficient representation of the dynamics by introducing new propagators\n\nThe Hadamard propagator\n\n\\begin{equation}\nG_{1}=G^{21}+G^{12}\\equiv <\\left\\{ \\Phi \\left(\nx\\right) ,\\Phi \\left( x^{\\prime }\\right) \\right\\} >\\label{dd19r}\n\\end{equation}\nis real and even. The\nJordan propagator\n\n\\begin{equation}\nG=G^{21}-G^{12}\\equiv <\\left[ \\Phi \\left( x\\right) ,\\Phi\n\\left( x^{\\prime }\\right) \\right] >\\label{ddd19r}\n\\end{equation}\nis imaginary and odd\n\n\nThe advanced and retarded propagators are the fundamental solutions for the\nequations of motion for linear fluctuations in the field.\n\n\\begin{eqnarray}\nG_{ret}\\left( x,x^{\\prime }\\right) &=&i\\left[G^{11}-G^{12}\\right]=\niG\\left( x,x^{\\prime }\\right) \\theta \\left( t-t^{\\prime\n}\\right)\\nonumber\\\\\nG_{adv}\\left( x,x^{\\prime }\\right) &=&-i\\left[G^{21}-G^{11}\\right]=-iG\\left( x,x^{\\prime\n}\\right) \\theta \\left( t^{\\prime }-t\\right) \n\\end{eqnarray}\n\nAll propagators may be expressed in terms of the Jordan and Hadamard ones\n\\begin{equation}\nG^{\\pm }\\left( x,x^{\\prime }\\right) =\\frac 12\\left[ G_1\\left( x,x^{\\prime\n}\\right) \\pm G\\left( x,x^{\\prime }\\right) \\right]\n\\end{equation}\n\n\\begin{equation}\nG_{F,D}\\left( x,x^{\\prime }\\right) =\\frac{1}{2}\\left[ G_{1}\\left(\nx,x^{\\prime }\\right) \\pm G\\left( x,x^{\\prime }\\right) \\mathrm{sign}\\left(\nt-t^{\\prime }\\right) \\right] \n\\end{equation}\nIn terms of the new propagators the equations of motion are\n\n\\begin{eqnarray}\n\\left[D+\\mathbf{D}_{even}+\\mathbf{D}_{odd}\\right]G_{ret}&=&-\\mathbf{1}\\nonumber\\\\\n\\left[D+\\mathbf{D}_{even}+\\mathbf{D}_{odd}\\right]G_1&=&\\mathbf{N}G_{adv}\n\\end{eqnarray}\n\n\n\nWe can now begin the discussion of our subject matter. Our goal is to recast the equations for the propagators in a way suitable to discuss the last stages of the equilibration process. We shall reduce the equations to the form of a kinetic equation, the so-called Kadanoff-Baym equation. Further approximations reduce this to the Boltzmann equation. To succeed, we need a way to identify the important terms and discard the irrelevant ones. This will be provided by the so-called adiabatic expansion, under the assumption that the propagators are almost translation invariant close enough to equilibrium.\n\nWe say that a $G^{ab}\\left(x.x'\\right)$ is almost\ntranslation-invariant\nif, when partially Fourier transformed with respect to $u=x-x'$, the\nFourier transform is weakly dependent on the ``centroid\"\nvariable $X=\\left(x+x'\\right)\/2$, i.e.,\n\n\\begin{equation}\nG^{ab}\\left( x,x^{\\prime }\\right) = \\int \\frac{d^{d}k}{\\left( 2\\pi \\right) ^{d}}%\n\\;e^{iku}G^{ab}\\left( X,k\\right)\n\\label{fast-slowb}\n\\end{equation}\nObserve that\n\\begin{eqnarray}\n\\int\\;dy\\;A\\left(x,y\\right)B\\left(y,x'\\right)=\\int \\frac{d^{d}k}{\\left( 2\\pi \\right) ^{d}}%\n\\;e^{iku}\\nonumber\\\\\n\\left\\{A\\left(X,k\\right)B\\left(X,k\\right)-\\frac i2\\left\\{A,B\\right\\}+\\ldots\\right\\}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\left\\{A,B\\right\\}=\\frac{\\partial A}{\\partial k}\\frac{\\partial B}{\\partial X}-\\frac{\\partial A}{\\partial X}\\frac{\\partial B}{\\partial k}\n\\end{equation}\n\nExpressions involving $G^{ab}\\left(\nX,k\\right) $ may be classified according to their \\emph{adiabatic order}, namely, the number of $X$ derivatives\nappearing in the expression. We call this the \\emph{adiabatic\nexpansion}. When almost\ntranslation-invariance is verified, we may further reject all terms\nabove a given adiabatic order. We call such a truncation of an\nadiabatic expansion an \\emph{adiabatic approximation}. In other words, the adiabatic order is used as a tag\nto bunch together certain terms in the equations of motion in\naccordance to their derivative orders and the adiabatic approximation\ndetermines how many of those terms are kept.\n\n\n\nLet\n\n\\begin{equation}\ni\\Gamma\\left( X,k\\right)=i\\pi\\gamma\\left( X,k\\right)\\mathrm{sign}\\left(\\omega\\right)=\\mathbf{D}_{odd}\\left( X,k\\right)\n\\end{equation}\n$\\omega=k^0$\n\n\\begin{equation}\nR\\left( X,k\\right) =\\left( k^{2}+m_{b}^{2}\\right) -\n\\mathbf{D}_{even}\\left( X,k\\right)\n\\end{equation}\n\n\nThen\n\n\\begin{equation} \nG_{ret}\\left( X,k\\right)=\\frac 1{R-i\\Gamma}\n\\end{equation}\n\n\\begin{equation} \nG_{adv}\\left( X,k\\right)=G_{ret}\\left( X,-k\\right)=\\frac 1{R+i\\Gamma}\n\\end{equation}\n\nThe relationship $G_{ret}=\niG\\theta \\left( t-t^{\\prime\n}\\right)$ implies\n\n\\begin{equation} \nG_{ret}\\left(X,\\left(\\omega,\\vec{k}\\right)\\right)=-\\int\\;\\frac{d\\omega'}{2\\pi}\\frac{G\\left(X,\\left(\\omega',\\vec{k}\\right)\\right)}{\\omega-\\omega'+i\\epsilon}\n\\end{equation}\nor else\n\\begin{eqnarray}\n\\mathrm{Re}G_{ret}\\left(X,\\left(\\omega,\\vec{k}\\right)\\right)&=&-PV\\int\\;\\frac{d\\omega'}{2\\pi}\\frac{G\\left(X,\\left(\\omega',\\vec{k}\\right)\\right)}{\\omega-\\omega'}\\nonumber\\\\\n\\mathrm{Im}G_{ret}\\left( X,k\\right)&=&\\frac12G\\left( X,k\\right)\n\\end{eqnarray}\nThus\n\n\\begin{equation}\nG\\left( X,k\\right)=\\frac{2\\Gamma}{R^2+\\Gamma^2}\n\\end{equation}\n\n\n\nUnder equilibrium conditions the Kubo - Martin - Schwinger theorem implies that\n\n\\begin{equation}\nG_{1eq}\\left( X,k\\right)=\\mathrm{sign}\\left(\\omega\\right)G_{eq}\\left( X,k\\right)\\left[1+2f_{BE}\\right]\n\\end{equation}\nwhere $f_{BE}$ is the Bose - Einstein distribution. We generalize this to a nonequilibrium situation\nby defining the \\emph{density of states} ${\\cal D} \\left( X,k\\right) $ out of\nthe Fourier transform of the Jordan propagator \n\n\\begin{equation}\n{\\cal D} \\left( X,k\\right)\\equiv \\frac 1{2\\pi}G\\left( X,k\\right)   \\;\\mathrm{sign}%\n\\left( \\omega\\right)=\\frac{\\gamma}{R^2+\\Gamma^2}\\label{cald_ch11}\n\\end{equation}\n\n\n\nWe now define the distribution function $f\\left( X,k\\right) $\nthrough the partial Fourier transform of the Hadamard propagator\n\n\\begin{equation}\nG_{1}\\left( X,k\\right) \\equiv 2\\pi {\\cal D} \\left( X,k\\right) \\;F_{1}\\left(\nX,k\\right)  \\label{hadamard}\n\\end{equation}\n\n\\begin{equation}\nF_{1}\\left( X,k\\right) =1+2f\\left( X,k\\right)\n\\end{equation}\n\n\n\nTo obtain the dynamics of the distribution function $f$, we make use\nof the equation involving the noise kernel. Let us call $F^{21}=\\theta\\left(\\omega\\right)+f$,  $F^{12}=\\theta\\left(-\\omega\\right)+f$,\n$\\Sigma_{12}=i\\left(\\mathbf{N}-\\Gamma\\right)$ and $\\Sigma_{21}=i\\left(\\mathbf{N}+\\Gamma\\right)$. Then to first adiabatic order we get\n\\begin{equation}\nA\\left\\{ R,F_{1}\\right\\} -B\\left\\{ \\mathbf{\\Gamma ,}F_{1}\\right\\}\n=I_{col}\\;\\mathrm{sign}\\left( k^{0}\\right)   \\label{ke1}\n\\end{equation}\nwhere\n\\begin{equation}\nA=\\frac{\\mathbf{\\Gamma }^{2}}{R^{2}+\\mathbf{\\Gamma }^{2}}\n\\end{equation}\n\\begin{equation}\nB=\\frac{R\\mathbf{\\Gamma }}{R^{2}+\\mathbf{\\Gamma }^{2}}\n\\end{equation}\nand  $I_{col}$ is the \\emph{collision integral}\n\n\\begin{equation}\nI_{col} =-i\\left[ \\Sigma_{12}F^{21}-\\Sigma_{21}F^{12}\\right]\n\\end{equation}\n\nFor weakly coupled theories, a series of approximations allow us to reduce\nthe  off-shell kinetic equation  to the more familiar Boltzmann kinetic equation.\nWe observe that in terms of the coupling constant $\\lambda $ we have,\nfor a generic momentum $p$, $R\\sim O\\left( 1\\right) $ while\n$\\mathbf{\\Gamma }\\sim O\\left( \\lambda ^{2}\\right) $.\n\n\nA second observation is that in general $\\mathbf{\\Gamma}$, which\ninvolves the coupling constants, will be much smaller than $R$ for a\ngeneric choice of $p$. When the coupling constants go to zero\n$\\mathbf{\\Gamma}\\to 0$, but the retarded propagator has a\nwell-defined asymptotic value, and the density of states becomes\n$ \\mathcal{D}=\\delta (R)$\n\n In this limit the propagators are\ninsensitive to the behavior of the distribution function ``off shell\"\n(i. e., when $R\\neq 0$), because the distribution function is always\nmultiplied by the density of states, and this is very small there.\nTherefore, only ``on shell'' modes (i. e., those for which $R=0$)\nreally contribute to the field correlation functions. If our only\nconcern is to follow the evolution of the distribution function on\nshell, we are allowed to replace the $A$ and $B$ coefficients in\n(\\ref{ke1}) by their ``on shell'' values, namely $A=1$ and $B=0.$ We\nthus obtain the Kadanoff-Baym equations \\cite{kad62}\n\n\\begin{equation}\n\\left\\{ R,F_{1}\\right\\} =-i\\;\\mathrm{sign}\\left( k^{0}\\right) \\;\\left[\n\\Sigma_{12}F^{21}-\\Sigma_{21}F^{12}\\right]   \\label{ke2}\n\\end{equation}\nThe\nnontrivial content of the Kadanoff-Baym equations is given by the form of the collision\nintegral, namely, which Feynman graphs contribute to the self energies. We\nrecognize the structure of the collision term as the difference between a\ngain and a loss term for particles moving in or out of a phase space cell\naround the point $\\left( X,k\\right) $ per unit time. Taking $\\omega >0$ for\nsimplicity, we see that $ \\Sigma_{12}F^{21}$ is the gain\nterm, with $F^{21}=1+f$ accounting for stimulated emission of particles\ninto the cell, while the other term is the loss term, which is proportional\nto the number of particles $F^{12}=f$ already there.\n\nIf we only keep the first term in the expansion, which for a $\\lambda\\phi^4$ theory is the setting-sun graph, we recover the Boltzmann's collision term \\cite{CH88}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\\subsection{The Tisserand Parameter}\n\\label{section:tisserandparam}\n\nThe Tisserand parameter, $T_P$, or Tisserand invariant, of a small solar system body under the influence of gravity from the Sun and a major planetary perturber is defined by\n\\begin{equation}\nT_P = {a_P\\over a} + 2 \\cos i \\left[{\\left(1-e^2\\right){a\\over a_P}}\\right]^{1\/2}\n\\end{equation}\nwhere $a_P$ is the semimajor axis of the planetary perturber, and $a$, $e$, and $i$ are the semimajor axis, eccentricity, and inclination of the small body in question.  Derived from Jacobi's integral, the long-term value of this quantity is largely conserved in the restricted three-body problem \\citep{tis96,vag73}, even in the event of close encounters with the planetary perturber \\citep{car95}.\n\nIn the study of small solar system body dynamics, the Tisserand parameter with respect to Jupiter, $T_J$, is frequently employed as a discriminant between asteroids and comets.  Main-belt asteroids typically have $T_J$$\\,>\\,$3, and comets typically have $T_J$$\\,<\\,$3 \\citep{kre72}.\nHowever, despite the appealing simplicity of a clear-cut boundary between asteroids and comets at $T_J$$\\,=\\,$3, $T_J$ is well-known to be an inexact means of dynamically classifying real solar system objects.\nThe expression for $T_P$ is derived using an idealized physical approximation in which the orbit of the planetary perturber is assumed to be circular ($e$$\\,=\\,$0) and non-inclined ($i$$\\,=\\,$0$^{\\circ}$), but Jupiter's actual orbit has both non-zero $e$ and non-zero $i$ ($e_J$$\\,=\\,$0.0489; $i_J$$\\,=\\,$1.304$^{\\circ}$).\nFurthermore, while Jupiter is the dominant planetary perturber in the solar system, the other outer planets as well as the terrestrial planets can also affect cometary orbits \\citep[e.g.,][]{mor99,lev06,gal14}.  Lastly, non-gravitational forces such as the Yarkovsky effect \\citep[cf.][]{rub95} and cometary outgassing \\citep[cf.][]{mar73,yeo04} can potentially play a significant role in the dynamical evolution of solar system objects, such as in the case of 2P\/Encke \\citep[e.g.,][]{ste96,jfer02,pit04}, but are unaccounted for in the formulation of $T_P$.\n\n\\subsection{$T_J$ as an Asteroid-Comet Discriminant}\n\n\\citet{fer01,fer05} demonstrated that near-Earth objects (NEOs) with $T_J$$\\,<\\,$3 (sometimes referred to as asteroids in cometary orbits, or ACOs, in the literature) showed significantly lower albedos than NEOs with $T_J$$\\,>\\,$3, consistent with the low-$T_J$ objects being dormant or extinct comet nuclei.  This finding led \\citet{bin04} and \\citet{dem08} to use a combination of low albedos and low $T_J$ values to identify extinct comet candidates in other NEO surveys.  \\citet{bin04} reported, however, that dynamical models indicated that $\\sim\\,$35\\% of low-albedo $T_J$$\\,\\leq\\,$3 NEOs were likely to originate from the outer asteroid belt, not the outer solar system as their $T_J$ values might normally suggest.  \\citet{zif05} also found that near-infrared spectra of two asteroids with $T_J$$\\,<\\,$3 showed that they had more in common with X-type asteroids than cometary nuclei.  In a study of asteroids with $T_J$$\\,<\\,$3, \\citet{lic06} similarly found a reflectivity gradient distribution more consistent with outer main-belt asteroids than with cometary nuclei, though cautioned that their results were preliminary.\n\n\\begin{figure*}\n \\centering{\\includegraphics[width=4.0in]{fig_asteroids_comets_tiss_all_aei_2panels.pdf}}\n\\caption{\\small Plots of $a$ vs.\\ $e$ (top half of each panel) and $i$ (bottom half of each panel) for the first 50\\,000 numbered asteroids (pale blue dots) and all comets catalogued by the Minor Planet Center as of 2014 April 1 (pale red dots), where asteroids and comets with $T_J$ values of (a) $T_J$$\\,<\\,$3.00, and (b) $T_J$$\\,>\\,$3.00 are highlighted with dark blue and dark red dots, respectively.  Solid vertical lines mark $a$ for Mars and Jupiter ($a_M$ and $a_J$), while the 4:1, 3:1, 5:2, 7:3, and 2:1 MMRs with Jupiter are marked with dashed vertical lines.  The loci of Mars-crossing orbits (where $q$$\\,=\\,$$Q_{M}$) and Jupiter-crossing orbits (where $Q$$\\,=\\,$$q_{J}$) are marked with light green and dark green curved solid lines, respectively, on each $a$-$e$ plot, while the loci of orbits for which objects can potentially come within 1.5 Hill radii of Jupiter ($Q$$\\,=\\,$$q_{J}$$\\,-\\,$$1.5R_H$) are marked with dark green dashed lines.\n}\n\\label{figure:asteroids_comets_tiss}\n\\end{figure*}\n\nWhile physical studies indicate that $T_J$ is probably a reasonable first-order indication of an object's probable dynamical origin, the aforementioned caveats mean that it should not be regarded as an absolute criterion.  Plots of the orbital elements of the first 50\\,000 numbered asteroids and all comets catalogued by the Minor Planet Center as of 2014 April 1 (Figure~\\ref{figure:asteroids_comets_tiss})\nshows that most asteroids have $T_J$$\\,>\\,$3, and most comets have $T_J$$\\,<\\,$3, though there are some asteroids with $T_J$$\\,<\\,$3 and a handful of comets with $T_J$$\\,>\\,$3.\nIn particular, Jovian Trojan asteroids ($a$$\\,\\sim\\,$5.2~AU) and Hilda asteroids ($a$$\\,\\sim\\,$3.9$\\,-\\,$4.0~AU) comprise a large portion of the $T_J$$\\,<\\,$3 asteroid population.  There are also some $T_J$$\\,<\\,$3 asteroids within the $a$ bounds of the main asteroid belt (between\nthe 4:1 and 2:1 mean-motion resonances, or MMRs, with Jupiter), where these objects have larger $e$, larger $i$, or both, relative to the rest of the main-belt asteroid population.  Notably, we see that the vast majority of comets in Figure~\\ref{figure:asteroids_comets_tiss} have perihelion distances, $Q$, within 1.5 Hill radii (1.5$R_H$) of Jupiter's perihelion, $q_J$, or beyond, indicating the possibility of very close encounters with the planet and therefore a strong degree of dynamical coupling.  Meanwhile, the vast majority of main-belt asteroids (Hilda and Jovian Trojan asteroids aside) do not have $Q$ closer than 1.5$R_H$ from $q_J$, indicating a low degree of dynamical coupling.  We therefore find that the locus of orbits in $a$-$e$ space with $Q$$\\,=\\,$$q_J$$\\,-\\,$$1.5R_H$ forms a reasonably effective alternative dynamical dividing line separating main-belt asteroids and Jupiter-family comets, consistent with the findings of \\citet{tan14}.\n\nContinuing to study Figure~\\ref{figure:asteroids_comets_tiss}, we see that there are very few comets with $T_J$$\\,>\\,$3. Most of these have $a$ placing them outside the main asteroid belt (i.e., beyond the 2:1 MMR with Jupiter at 3.277~AU).  A few other comets have $a$ that actually place them within the main asteroid belt, but also have $e$ values larger than those commonly associated with main-belt asteroids.  In almost all of these cases, the orbits of these comets meet the $Q$$\\,>\\,$$q_J$$\\,-\\,$$1.5R_H$ criterion for cometary orbits discussed above, with the notable exception of main-belt comets (described below).\n\n\n\n\\subsection{Main-Belt Comets}\n\nAside from the aforementioned handful of comets with $T_J$$\\,>\\,$3 that have $a$ or $e$ placing them beyond the commonly recognized bounds of the main asteroid belt, there exists a newly identified class of comets known as main-belt comets \\citep[MBCs;][]{hsi06} that exhibit cometary activity indicative of the sublimation of volatile ices, yet have $T_J$$\\,>\\,$3, have semimajor axes and eccentricities completely consistent with main-belt asteroids, and do not have close encounters with Jupiter.\nMBCs constitute a subset of the group of small solar system bodies known as active asteroids \\citep{jew12,jew15}, which also includes disrupted asteroids, which are objects that exhibit comet-like activity that is produced by non-sublimation-driven effects such as impacts or rotational destabilization \\citep[cf.][]{hsi12a}.  MBCs are particularly interesting from a dynamical perspective though, since the implication that they are icy bodies raises natural questions about whether they may have originated in the outer solar system like other comets, or whether they were formed in situ as their largely stable main-belt orbits appear to suggest \\citep[cf.][]{hsi14}.\n\nAttempts have been made in the past to find plausible dynamical pathways by which Jupiter-family comets (JFCs) could possibly have evolved onto MBC-like orbits, given the unexpectedness of objects on apparently dynamically stable main-belt orbits currently exhibiting active sublimation, but no such pathways were found \\citep[e.g.,][]{jfer02}.  The results of numerical integrations attesting to the long-term dynamical stability of individual MBCs \\citep{hag09,jew09,hsi12b,hsi12c} appears to indicate that those objects have resided in their current locations in the asteroid belt for some time, and may have even originated there.  There are however a few MBCs which have been found to be unstable on timescales of $\\lesssim30$~Myr at their present locations, suggesting that they cannot have resided there for long and must have originated elsewhere \\citep[e.g., 238P and 259P;][]{hag09,jew09}.\n\n\nIn this work, we are interested in understanding to what extent $T_J$ and other dynamical characteristics can be used to infer information about an object's possible dynamical origin based on current orbital elements.\nThe presumed in-situ formation of MBCs is the foundation on which efforts to use them as tracers of primordial ice in the inner solar system are based.  As such, it is very important to determine whether objects currently on main-belt-like orbits can in fact be assumed to be native to the main asteroid belt, or if non-native objects (e.g., from the outer solar system) may be able to occasionally assume main-belt-like orbits, and thus effectively masquerade (at least temporarily) as members of the local native population.  If the latter is the case, identifying the dynamical characteristics of such interlopers would then be very useful for improving our ability to exclude such objects when attempting to infer the distribution and abundance of inner solar system ice from the observed distribution of MBCs.  This issue is of particular interest in astrobiology given that MBCs represent a potential means for constraining solar system formation models that posit that icy objects from the main asteroid belt may be a significant primordial source of Earth's present-day water content.\n\n\n\n\\section{EXPERIMENTAL DESIGN\\label{section:expdesign}}\n\n\nFor this study, we seek to explore the range of dynamical paths that could be followed by small solar system objects in a designated region of interest in orbital element space (i.e., near the canonical $T_J$$\\,=\\,$3 boundary between asteroids and comets), with the ultimate objective of determining the degree to which an object's osculating orbital elements (or a parameter derived from those elements, e.g., $T_J$) at some arbitrary point in its dynamical evolution can be relied upon to infer its dynamical history.  To accomplish this, we conducted relatively short-duration integrations of a large number of test particles with starting orbital elements meeting our specified criteria.\nSpecifically, we generated a sample of 10\\,000 test particles spanning a range of $a$, $e$, and $i$ values required to produce starting $T_J$ values ($T_{J,s}$) of 2.80$\\,<\\,$$T_{J,s}$$\\,<\\,$3.20, with the expectation that particles close to the canonical dividing line between asteroids and comets should be the most likely to cross that boundary, and thus represent the most interesting cases for study.  To create these test particles, we randomly selected starting $T_{J,s}$, $a_s$, and $e_s$ values from within pre-defined ranges (2.8$\\,<\\,$$T_{J,s}$$\\,<\\,$3.2, $a_s$$\\,<\\,$$a_J$, and 0$\\,<\\,$$e_s$$\\,<\\,$0.99, where $a_J$$\\,=\\,$5.204~AU), and then computed the corresponding $i_s$ value needed to produce the selected $T_{J,s}$ value for each test particle.  Sets of $a_s$ and $e_s$ for which no value of $i_s$ could produce the target $T_{J,s}$ value were discarded and regenerated.  Finally, random values between $0^{\\circ}$ to $360^{\\circ}$ were selected as arguments of perihelion, longitudes of ascending nodes, and mean anomalies.  The starting orbital elements of all of our test particles generated in this way, separated into individual $T_{J,s}$ bins for added clarity, are plotted in Figure~\\ref{figure:testparticles_distribution}.\n\nOf course, by generating test particles that span the entire region of orbital element space where 2.80$\\,<\\,$$T_{J,s}$$\\,<\\,$3.20, we sample portions of orbital element space that are only sparsely populated, if at all, by real solar system objects, as can be seen by comparing Figures~\\ref{figure:asteroids_comets_tiss} and \\ref{figure:testparticles_distribution}.  In particular, our initial test particle set includes objects on polar orbits (i.e., $i_s$$\\,\\sim\\,$90$^{\\circ}$) and retrograde orbits (i.e., with $i_s$$\\,>\\,$90$^{\\circ}$).  Our goal in performing this general study, though, is to fully explore the available parameter space to search for possible dynamical pathways for particles defined by a specific dynamical criterion (i.e., $T_{J}$), and investigate what additional dynamical criteria, if any, are able to specify or exclude particular pathways of interest.\n\n\n\\begin{figure*}\n  \\includegraphics[width=6.4in]{fig_tj_all_stability.pdf}\n\\caption{\\small Plots of $a$ vs.\\ $e$ (top half of each panel) and $i$ (bottom half of each panel) for all test particles integrated as part of our study, where test particles with dynamical lifetimes of $t_{\\rm dyn}<2$~Myr and $t_{\\rm dyn}>2$~Myr are marked with orange and purple dots, respectively, and test particles are separated into individual $T_{J,s}$ bins, as labeled.  Solid vertical lines mark $a_M$ and $a_J$, and the 4:1, 3:1, 5:2, 7:3, and 2:1 MMRs with Jupiter are marked with dashed vertical lines.  The loci of Mars-crossing orbits (where $q=Q_{M}$) and Jupiter-crossing orbits (where $Q=q_{J}$) are marked with light green and dark green curved solid lines, respectively, on each $a$-$e$ plot, while the loci of orbits for which objects can potentially come within 1.5 Hill radii of Jupiter ($Q=q_{J}-1.5R_H$) are marked with dark green dashed lines.\n}\n\\label{figure:testparticles_distribution}\n\\end{figure*}\n\nWe conducted ``snapshot'' integrations of all test particles by integrating each of their orbits forward (using 10-day timesteps) for 2~Myr using the Bulirsch-St\\\"oer integrator in the Mercury numerical integration software package \\citep{cha99}.  The length of our snapshot integrations was chosen so that our study would not require an unmanageably large expenditure of computing resources, while still producing physically meaningful results.\n\\citet{lev94} found a median dynamical lifetime of $4.5\\times10^5$~years for short-period comets before ejection from the solar system or collision with the Sun, and so our integration period of 2~Myr should extend past the dynamical lifetimes of most of the comet-like particles in our integrations.\n\n\\begin{figure*}\n  \\includegraphics[width=6.4in]{fig_hist_comets_time_ejec_all_a_imo.pdf}\n\\caption{\\small Histograms of dynamical lifetimes for test particles representing clones of known comets with $a\\leq5.204$~AU with $T_{J,s}$ values of (a) $T_{J,s}<3.00$, (b) $3.00<T_{J,s}<3.05$, (c) $3.05<T_{J,s}<3.10$, and (d) $T_{J,s}>3.10$, where light grey bars indicate the fraction of comets with $2.064~{\\rm AU} < a < 3.277$~AU that are lost due to ejection or planetary\/solar impact within a particular time interval, and dark grey bars indicate the fraction of comets with $a>3.277$~AU that are lost due to ejection or planetary\/solar impact within a particular time interval.\n}\n\\label{figure:timeejec_comets}\n\\end{figure*}\n\nOf course, our chosen integration length means that our results are not appropriate for characterizing the evolution of test particles over much longer timescales, such as the age of the solar system, but this does not mean they are not physically meaningful.  Since we are considering the dynamical evolution of fictitious test particles distributed throughout our orbital element space region of interest in this study, our test particles can be thought of as both representing different individual objects, and also different stages of the long-term dynamical evolution of single objects, hence our characterization of these integrations as snapshot integrations.  Considering the situation from another perspective, longer integrations would certainly more directly characterize the long-term dynamical behavior of our test particles, but in no cases would they be expected to exclude or prevent any dynamical behavior observed during shorter integration periods.  As such, our snapshot integrations can be considered entirely applicable to the study of both the short-term and long-term dynamical evolution of our test particles.\n\n\nFor these integrations, we treated the Sun and the eight major planets as massive bodies (where the mass of Mercury was added to that of the Sun) and all test particles as massless bodies.  Non-gravitational effects were not included.\n\nFor reference, we used the same experimental design as we used for our test particles to study the dynamical evolution of known comets.  Since our interest is in comets in the inner solar system, we restrict our sample to those comets with $a<a_{J}$.  We created a set of test particles with the orbital elements of the 639 comets catalogued by the MPC as of 2014 January 1 that met our orbital criteria.  We also created four additional dynamical clones per comet to account for possible chaotic effects due to orbital element uncertainties, giving a total of 3195 cometary test particles and dynamical clones.  For the latter task, we utilized code that generates an arbitrary number of clones of an object that are Gaussian-distributed in orbital element space, centered on the object's osculating orbital elements, and have a distribution characterized by the 1-$\\sigma$ uncertainties of the orbital elements of those objects \\citep[previously used by][]{hsi12a,hsi12b,hsi12c,hsi13}.\n\nWe plot histograms of the dynamical lifetimes of the comets and their clones in each $T_{J,s}$ bin (Figure~\\ref{figure:timeejec_comets}), where the dynamical lifetime of an object is the amount of time it spends in the integration before reaching $a=100$~AU and is considered to have been ejected from the solar system, reaching $e>1$, or colliding with a planet or the Sun.  One caveat is that current orbital elements are not available for all comets. Since our integrations were all run from a single starting epoch, it was therefore necessary to integrate objects with non-updated orbital elements up to the present epoch, and in some cases, objects were eliminated from the integrations before even reaching the current epoch.  For the purposes of this analysis, the dynamical lifetimes for these objects were recorded as being zero years.\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=5.0in]{fig_semimaj_tiss_tji_all.pdf}}\n\\caption{\\small Plots of $a$ vs.\\ $T_{J,i}$ (small grey dots) for test particles with (a) $2.80<T_{J,s}<3.00$, (b) $3.00<T_{J,s}<3.05$, (c) $3.05<T_{J,s}<3.10$, and (d) $3.10<T_{J,s}<3.20$.  SOEs are plotted with purple dots, while main-belt-like IOEs and FOEs are plotted with light blue and dark blue dots, respectively.  Solid vertical lines mark $a_M$ and $a_J$ in each panel, while the 4:1, 3:1, 5:2, and 2:1 MMRs with Jupiter (from left to right) are marked with dashed vertical lines.  The $T_J=3.05$ boundary is marked by horizontal dashed lines in each panel.\n}\n\\label{figure:semimaj_tiss_tji_all}\n\\end{figure*}\n\nOverall, we find that $>$95\\% of our comet test particles are lost prior to the end of our 2~Myr integrations (consistent with the results of integrations of JFCs by \\citet{jfer02} over an identical integration period), indicating that our integrations are indeed longer than the dynamical lifetimes of most comets in the inner solar system.  We also immediately see that the dynamical behavior of comets with $3.00<T_{J,s}<3.05$ is extremely similar to those with $T_{J,s}<3.00$, indicating that $T_J=3$ may not in fact be an appropriate dividing line between comet-like and asteroid-like orbits. A shift towards greater stability is seen for comet test particles with $3.05<T_{J,s}<3.10$, while all comets with $T_{J,s}>3.10$ are found to be stable over the entirety of our integrations.  No appreciable differences in dynamical lifetimes were seen for comets with semimajor axes interior to and exterior to the 2:1 MMR with Jupiter (i.e., the outer boundary of the main asteroid belt).\nThe short ($<1$~Myr) dynamical lifetime of a typical comet is a key dynamical characteristic indicating a likely recent insertion onto an inner-solar-system-crossing orbit given the low likelihood of it residing on that orbit for significantly longer than its calculated dynamical lifetime.  As such, hereafter, we will consider $T_J=3.05$ to be, in practice, a more appropriate approximate upper bound on ``comet-like'' orbits.  This is consistent with the modified $T_J$ criterion for differentiating asteroids and comets used by \\citet{tan14}, as well as our previous discussion of how the physical simplifications used to derive $T_J$ are inexact (Section~\\ref{section:tisserandparam}).\n\n\n\\section{RESULTS \\& ANALYSIS\\label{results}}\n\n\\subsection{Reliability of $T_J$ as a Dynamical Discriminant\\label{section:tisserand_reliability}}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=3in]{fig_histogram_tji_all.pdf}}\n\\caption{\\small Histograms of $T_{J,i}$ values (grey bars) for test particles with (a) $2.80<T_{J,s}<3.00$, (b) $3.00<T_{J,s}<3.05$, (c) $3.05<T_{J,s}<3.10$, and (d) $3.10<T_{J,s}<3.20$.  The ranges of $T_{J,s}$ values are marked with shaded purple regions, while the $T_J$ distribution of main-belt-like IOEs are over-plotted as light blue bars.  The $T_J=3.05$ boundary is marked by vertical dashed lines in each panel.\n}\n\\label{figure:histograms_tji_all}\n\\end{figure}\n\nIn order to investigate the degree to which $T_J$ remains a reliable dynamical parameter for identifying the origins of individual test particles, even over relatively short periods of time, we compare the starting orbital elements (SOEs), intermediate orbital elements (IOEs) in 10\\,000-year intervals over our entire 2~Myr integration period or until particles are lost due to ejection or a planetary or solar impact, and final orbital elements (FOEs) of our test particles.  We plot starting, intermediate, and final $a$ and $T_J$ values for test particles in different $T_{J,s}$ bins in Figure~\\ref{figure:semimaj_tiss_tji_all}.  We immediately see from these plots that even $T_J$$\\,=\\,$3.05 is not a particularly impenetrable dynamical boundary, with a significant fraction of particles with starting $T_J$ values, $T_{J,s}$, of 2.80$\\,<\\,$$T_{J,s}$$\\,<\\,$3.00 reaching intermediate $T_J$ values ($T_{J,i}$) of $T_{J,i}$$\\,>\\,$3.05 over the course of the integration, and particles with 3.10$\\,<\\,$$T_{J,s}$$\\,<\\,$3.20 also reaching $T_{J,i}$$\\,<\\,$3.05.  Notably, some particles with 2.80$\\,<\\,$$T_{J,s}$$\\,<\\,$3.00 even attain main-belt-like IOEs\\footnote{For the purposes of the analyses presented here, we define ``main-belt-like'' orbits as those having $T_J$$\\,>\\,$3.05 (i.e., dynamically decoupled from Jupiter), $2.064~{\\rm AU}<a<3.277~{\\rm AU}$ (i.e., $a$ between the 4:1 and 2:1 MMRs with Jupiter, which bound the canonical main asteroid belt region), and $q>(Q_{M}+1.5R_{H,{M}})$ and $Q<(q_{J}-1.5R_{H,{J}})$ (i.e., confined within the orbits of Mars and Jupiter and prevented from approaching within 1.5$R_H$ of either planet), where $(Q_{M}+1.5R_{H,{M}})=1.65$~AU and $(q_{J}-1.5R_{H,{J}})=4.50$~AU.  ``Comet-like'' orbits are defined as those having $T_J$$\\,<\\,$3.05 and $Q>(q_{J}-1.5R_{H,{J}})$ (i.e., dynamically coupled to Jupiter).  In all cases, descriptions of orbits as ``main-belt-like'' or ``comet-like'' are intended only to refer to an object's or test particle's orbital elements at a particular moment in time, and are not meant to imply anything further about the object's dynamical history, long-term stability, or evolutionary fate.} during the integration period, and some even have main-belt-like FOEs at the end of the integrations.\n\n\\begin{table*}[ht]\n\\normalsize\n\\caption{\\normalsize Distribution of $T_{J,i}$ Values}\n\\smallskip\n\\begin{tabular}{ccccc}\n\\hline\\hline\n \\multicolumn{1}{c}{$T_{J,s}$ Bin} &\n \\multicolumn{1}{c}{$T_{J,i}<3.00^a$} &\n \\multicolumn{1}{c}{$T_{J,i}<3.05^b$} &\n \\multicolumn{1}{c}{$T_{J,i}>3.05^c$} &\n \\multicolumn{1}{c}{$T_{J,i}>3.10^d$} \\\\\n\\hline\n$T_{J,s}<3.00$      & 0.87 & 0.93 & 0.07 & 0.04 \\\\\n$3.00<T_{J,s}<3.05$ & 0.34 & 0.71 & 0.29 & 0.12 \\\\\n$3.05<T_{J,s}<3.10$ & 0.16 & 0.35 & 0.65 & 0.27 \\\\\n$T_{J,s}>3.10$      & 0.07 & 0.12 & 0.88 & 0.74 \\\\\n\\hline\\hline\n\\end{tabular}\n\\newline{$^a$ Fraction of IOEs where $T_{J,i}<3.00$.}\n\\newline{$^b$ Fraction of IOEs where $T_{J,i}<3.05$.}\n\\newline{$^c$ Fraction of IOEs where $T_{J,i}>3.05$.}\n\\newline{$^d$ Fraction of IOEs where $T_{J,i}>3.10$.}\n\\label{table:intermediate_tj_dist}\n\\end{table*}\n\nIn Figure~\\ref{figure:histograms_tji_all}, we plot histograms of all $T_{J,i}$ values attained by test particles in each $T_{J,s}$ bin in order to further investigate their distribution.  In Table~\\ref{table:intermediate_tj_dist}, we also list the fractions of $T_{J,i}$ values on either side of the ostensible $T_J=3.05$ asteroid-comet boundary attained by test particles in each $T_{J,s}$ bin.  While the majority of test particles with $T_{J,s}<3.00$ remain below the $T_J=3.05$ boundary throughout the integration period, some do reach $T_{J,i}>3.05$ (and even $T_{J,i}>3.10$) for at least a portion of the time covered by our integrations.  Similarly, while the majority of test particles with $T_{J,s}>3.10$ remain above the $T_J=3.05$ boundary, some reach $T_{J,i}<3.05$ (and even $T_{J,i}<3.00$) during a portion of the integration period.  By comparison, test particles with $3.00<T_{J,s}<3.10$ spend substantial amounts of time on the opposite side of the $T_J=3.05$ boundary from where they originated.\n\n\n\\subsection{Transfer of Objects from Comet-Like Orbits to Main-Belt Orbits\\label{section:mbc_origins}}\n\n\\subsubsection{Results of Initial Integrations\\label{section:mbc_origins_initial}}\n\nIn Figure~\\ref{figure:start_cometlike_sometimes_mb_all}, we plot IOEs of test particles with comet-like SOEs that attain main-belt-like IOEs at any time during the integration period, as well as the current orbital elements of the known MBCs.  As seen before, we find that test particles with comet-like SOEs reach a significant portion of main-belt orbital element space.   Specifically, of the 1727 test particles in our integrations that had comet-like SOEs, 57 test particles ($\\sim$3.5\\% of the total sample) reached main-belt-like orbits at some point during the integrations, while 8 of those test particles actually had main-belt-like FOEs.  We caution that since our sample of comet-like test particles is not a realistic representation of the known comet population, these rates are not expected to accurately reflect the real-world rates of JFCs evolving onto main-belt-like orbits.\n\nWe note that 29 clones of real comets with $T_{J,i}$$\\,<\\,$3.05 (out of 2630 such clones, or $\\sim\\,$1\\%) also attained main-belt-like IOEs at some point when integrated for 2~Myr (Section~\\ref{section:expdesign}), and two of those objects (clones of 249P and P\/2005 JQ5; $\\sim$0.1\\% of the total sample of comet clones) had main-belt-like FOEs.  Like our test particle set, due to historical discovery biases, this set of comet clones likewise does not necessarily accurately represent the current steady-state population of JFCs.  Nonetheless, we conclude from these results that the fraction of comet-like objects that may attain main-belt-like orbits at some point during their dynamical lifetimes, albeit perhaps only temporarily, is non-zero, and estimate that it may be on the order of $\\sim$0.1-1\\%.\n\nThat said, we see that in our test particle integrations, particles with comet-like SOEs that reach main-belt-like orbits do not attain very low $e$ and very low $i$ simultaneously.  We can approximately define a region of $e$-$i$ space, shaded in orange in Figure~\\ref{figure:start_cometlike_sometimes_mb_all}, into which test particles with comet-like SOEs do not enter over the course of our integrations, and therefore appears to be ``protected'' from comet-like interlopers.  This region is approximately empirically defined by\n\\begin{equation}\n0.775e+\\sin(i)<0.31\n\\label{equation:protected_region}\n\\end{equation}\nand includes the current orbital elements of (1) Ceres, 133P, 176P, 238P, 288P, and P\/2013 R3.\n\nFor reference, we also plot $a$ vs.\\ $T_J$ for the IOEs of the test particles from Figure~\\ref{figure:start_cometlike_sometimes_mb_all}, and identify an analogous ``protected'' region in $a$-$T_J$ space into which test particles with comet-like SOEs do not enter (Figure~\\ref{figure:start_cometlike_sometimes_mb_atj}).  This region can be approximately empirically described by\n\\begin{equation}\nT_J > 55\\cdot\\exp({-2a})+3.05\n\\label{equation:protected_region_atj}\n\\end{equation}\nand contains the orbital elements of the same MBCs in the protected region in $e$-$i$ space identified in Figure~\\ref{figure:start_cometlike_sometimes_mb_all}.\nNotably, the protected region in $a$-$T_J$ space appears to extend to lower $T_J$ values in the outer main belt than the inner main belt.  Practically speaking, for example, this means that having $T_J$$\\,>\\,$3.15 is sufficient for an object to be located in the protected region in the outer main belt ($a>3$~AU), while for $a<2.8$~AU, values of $T_J>3.25$ or even higher are needed for an object to be in the protected zone.  This may perhaps be related to weaker dynamical coupling of objects with Jupiter with increasing average distance from the planet, but more detailed theoretical analysis of this issue (that is beyond the scope of the general study presented here) will be needed to ascertain the exact causes of this behavior.\n\n\\setlength{\\tabcolsep}{5pt}\n\\begin{table*}[ht]\n\\normalsize\n\\caption{\\normalsize Test Particles with Comet-Like SOEs and Main-Belt-Like FOEs}\n\\smallskip\n\\begin{tabular}{ccccccccccccccc}\n\\hline\\hline\n \\multicolumn{1}{c}{Particle} &&\n \\multicolumn{1}{c}{$a_s^a$} &\n \\multicolumn{1}{c}{$e_s$} &\n \\multicolumn{1}{c}{$i_s^b$} &\n \\multicolumn{1}{c}{$T_{J,s}$} & &\n \\multicolumn{1}{c}{$a_f^c$} &\n \\multicolumn{1}{c}{$e_f$} &\n \\multicolumn{1}{c}{$i_f^d$} &\n \\multicolumn{1}{c}{$T_{J,f}$} &&\n \\multicolumn{1}{c}{MMR$^e$} &\n \\multicolumn{1}{c}{$a_{\\rm MMR}^f$} &\n \\multicolumn{1}{c}{$\\Delta a_f^g$} \\\\\n\\hline\nA && 2.505 & 0.826 & 21.018 & 2.806 & & 2.517 & 0.289 & 19.278 & 3.324 & &  3:1 & 2.501 & 0.016 \\\\\nB && 2.855 & 0.661 & 25.457 & 2.827 & & 2.703 & 0.208 & 23.910 & 3.214 & &  8:3 & 2.705 & 0.002 \\\\\nC && 2.955 & 0.561 & 17.465 & 2.951 & & 2.900 & 0.424 & 21.228 & 3.055 & & 12:5 & 2.902 & 0.002 \\\\\nD && 2.911 & 0.593 & 14.106 & 2.956 & & 2.967 & 0.334 & 23.785 & 3.056 & &  7:3 & 2.957 & 0.010 \\\\\nE && 2.634 & 0.716 & 31.668 & 2.821 & & 2.844 & 0.221 & 23.369 & 3.153 & &  5:2 & 2.824 & 0.020 \\\\\nF && 2.988 & 0.525 & 22.045 & 2.937 & & 2.889 & 0.159 & 21.654 & 3.168 & & 12:5 & 2.902 & 0.013 \\\\\nG && 3.075 & 0.470 & 18.323 & 2.981 & & 3.068 & 0.372 & 12.779 & 3.086 & &  9:4 & 3.029 & 0.039 \\\\\nH && 3.207 & 0.432 & 13.874 & 2.997 & & 3.239 & 0.149 & 16.636 & 3.101 & &  2:1 & 3.277 & 0.038 \\\\\n\\hline\\hline\n\\end{tabular}\n\\newline{$^a$ Starting semimajor axis, in AU.}\n\\newline{$^b$ Starting inclination, in degrees.}\n\\newline{$^c$ Final semimajor axis, in AU.}\n\\newline{$^d$ Final inclination, in degrees.}\n\\newline{$^e$ Nearest major or moderate-order MMR}\n\\newline{$^f$ Semimajor axis, in AU, of nearest major or moderate-order MMR}\n\\newline{$^g$ Absolute value of the difference, in AU, between $a_f$ for each particle and the semimajor axis of the nearest major or moderate-order MMR.}\n\\label{table:comet_to_mainbelt}\n\\end{table*}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.5in]{fig_start_cometlike_sometimes_mb_all2.pdf}}\n\\caption{\\small Plots of (a) $a$ vs.\\ $e$, (b) $a$ vs.\\ $i$, and (c) $e$ vs.\\ $i$ in 10\\,000-year intervals for test particles that have comet-like SOEs and that reach main-belt-like IOEs at any point during the integration period.  SOEs are plotted with red dots, while main-belt-like IOEs and FOEs that meet main-belt criteria are plotted with light blue dots and dark blue X's, respectively.  All other IOEs are plotted with small grey dots.  Orbital elements of the known MBCs are plotted with yellow stars.  In (a) and (b), $a_M$ and $a_J$ are marked with solid vertical lines, while the semimajor axes of the 4:1, 3:1, 5:2, 7:3, and 2:1 MMRs with Jupiter are marked with dashed vertical lines. In (a), the loci of Mars-crossing orbits (where $q=Q_{M}$) and Jupiter-crossing orbits (where $Q=q_{J}$) are marked with light green and dark green curved solid lines, respectively, and the loci of orbits for which objects can potentially come within 1.5$R_H$ of Jupiter ($Q=q_{J}-1.5R_H$) are marked with a dark green dashed line.  In (c), the approximate region of $e$-$i$ space into which comet-like test particles never enter is shaded in orange.\n}\n\\label{figure:start_cometlike_sometimes_mb_all}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=4.3in]{fig_start_cometlike_sometimes_mb_atj.pdf}}\n\\caption{\\small Plot of $a$ vs.\\ $T_J$ in 10\\,000-year intervals for test particles that have comet-like SOEs and that reach main-belt-like IOEs at any point during the integration period.  SOEs are plotted with red dots, while main-belt-like IOEs and FOEs that meet main-belt criteria are plotted with light blue dots and dark blue X's, respectively.  All other IOEs are plotted with small grey dots.  Orbital elements of the known MBCs are plotted with yellow stars.  The semimajor axes of the 3:1, 5:2, 7:3, and 2:1 MMRs with Jupiter (from left to right) are marked with dashed vertical lines.  The approximate region of $a$-$T_J$ space into which comet-like test particles never enter, corresponding to the analogous highlighted region of $e$-$i$ space in Figure~\\ref{figure:start_cometlike_sometimes_mb_all}, is shaded in orange.\n}\n\\label{figure:start_cometlike_sometimes_mb_atj}\n\\end{figure*}\n\n\n\nSpecifically examining the eight test particles that are seen to have comet-like SOEs and main-belt-like FOEs, we assign labels (``A''-``H'') to each test particle and list their SOEs and FOEs in Table~\\ref{table:comet_to_mainbelt}.  First, we note that in all cases, $a_s$ and $a_f$ for each particle differ by relatively small amounts (five of the eight particles undergo net changes in $a$ of $<0.1$~AU, while three undergo net changes in $a$ of $\\sim\\,$0.1$\\,-\\,$0.2~AU). This appears to place a practical limit of $a_s$$\\,>\\,$2.25~AU for most particles with comet-like SOEs entering the main-belt, considering the initial requirement of $Q_{i}>(q_{J}-1.5R_{H,{J}})$ for a particle to be considered to have comet-like SOEs.\nIt should also be noted that by only considering test particles with comet-like SOEs and main-belt-like FOEs here, we are focusing on a very select group of test particles that follow a very specific dynamical evolutionary path.  Other particles that experience much larger semimajor axis changes are more likely to be ejected prior to the end of the integrations, leaving just those particles that happen to dynamically evolve in ways that do not change their semimajor axes too drastically.\nMeanwhile, substantial decreases in $e$ are seen for all particles highlighted here, while $i$ is seen to vary inconsistently.  For two of the eight particles (E and G), $i$ declines significantly ($>5^{\\circ}$) between the beginning and end of our integrations, but for three other particles (A, B, and F), $i$ decreases by less than 2$^{\\circ}$, and for the final three particles (C, D, and H), $i$ actually increases.\n\nWhile each of these particles have $T_{J,s}$$\\,<\\,$3.00, five of the eight particles have final $T_J$ values ($T_{J,f}$) of $T_{J,f}$$\\,>\\,$3.10 at the end of our 2~Myr integrations, meaning that $T_{J,f}$$\\,\\gg\\,$$T_{J,s}$ for all of these particles.  This means that all of these particles begin on unambiguously comet-like orbits, and five of the eight particles have transitioned (at least temporarily) to unambiguously main-belt-like orbits at the end of 2~Myr, with only three particles (C, D, and G) ending in the somewhat ambiguous 3.05$\\,<\\,$$T_J$$\\,<\\,$3.10 bin between the two extremes (cf.\\ Figure~\\ref{figure:timeejec_comets}).  Finally, we note that essentially all of these particles have $a_f$ values placing them extremely close to a major, or at least moderate-order (i.e., low-integer), MMR (Table~\\ref{table:comet_to_mainbelt}), suggesting that these resonances may be responsible for helping to temporarily trap these objects in the main belt during the integration period, presumably by providing protection against close encounters with Jupiter \\citep[e.g.,][]{gla97,mal99,gab03,pit04,bro05,car08,jfer14}.\n\n\nIntriguingly, some of these eight particles have SOEs similar to those of currently known JFCs, while their FOEs are similar to those of currently known MBCs, suggesting that it may in fact be possible for JFCs to at least occasionally take on MBC-like orbits.  In Table~\\ref{table:jfcs_mbcs_highlighted}, we list known JFCs with current orbital elements similar to the SOEs of these test particles, and MBCs with current orbital elements similar to the FOEs of these test particles. These results suggest that there is a non-zero probability that the listed MBCs could have JFC-like origins.  Of the JFCs listed in this table, we note that 197P and one of its dynamical clones do in fact temporarily reach a main-belt-like orbit during our initial 2~Myr test integrations (Section~\\ref{section:expdesign}), where the clone remains on such an orbit for almost 1~Myr. However, neither of them remain on a main-belt-like orbit until the end of those integrations.  Of the 2168 comets or dynamical clones of comets with JFC-like SOEs integrated in Section~\\ref{section:expdesign}, 15 objects ($<\\,$1\\% of the total sample) have main-belt-like IOEs at some point during the 2-Myr test integrations, albeit most only very briefly (i.e., for only a few timesteps at a time).  In addition to 197P and one of its clones, other exceptions include one clone each of 249P, P\/2004 T1, and P\/2005 JQ5, which attain main-belt-like IOEs for relatively long periods of time (i.e., 0.5$\\,-\\,$1~million yrs), where the clones of 249P and P\/2005 JQ5 actually have main-belt-like FOEs.  We emphasize though that having main-belt-like FOEs after just 2~Myr is no guarantee of long-term stability, and further note that none of the actual comets in these cases reach main-belt-like IOEs at any time during the same integrations.\n\n\\begin{table*}[ht]\n\\caption{\\small JFCs and MBCs with Orbital Similarities to Particles with Comet-Like SOEs and Main-Belt-Like FOEs}\n\\smallskip\n\\small\n\\begin{tabular}{lcccccccc}\n\\hline\\hline\n \\multicolumn{1}{c}{Object} &&\n \\multicolumn{1}{c}{$a^a$} &\n \\multicolumn{1}{c}{$e$} &\n \\multicolumn{1}{c}{$i^b$} &\n \\multicolumn{1}{c}{$T_{J}$} &\n \\multicolumn{1}{c}{MMR$^c$} &\n \\multicolumn{1}{c}{$a_{\\rm MMR}^d$} &\n \\multicolumn{1}{c}{$\\Delta a^e$} \\\\\n\\hline\n{\\it JFCs} \\\\\n197P\/LINEAR           && 2.866 & 0.630 & 25.54 & 2.856 & 5:2 & 2.824 & 0.042 \\\\\n189P\/NEAT             && 2.921 & 0.597 & 20.38 & 2.909 & 7:3 & 2.957 & 0.036 \\\\\n182P\/LONEOS           && 2.931 & 0.666 & 16.91 & 2.846 & 7:3 & 2.957 & 0.026 \\\\\n26P\/Grigg-Skjellerup  && 3.017 & 0.640 & 22.43 & 2.806 & 9:4 & 3.029 & 0.012 \\\\\n294P\/LINEAR           && 3.200 & 0.595 & 19.09 & 2.818 & 2:1 & 3.277 & 0.077 \\\\\n\\hline\n{\\it MBCs} \\\\\n259P\/Garradd          && 2.726 & 0.342 & 15.90 & 3.217 &  8:3 & 2.705 & 0.021 \\\\\n324P\/La Sagra         && 3.099 & 0.154 & 21.40 & 3.099 & 13:6 & 3.106 & 0.007 \\\\\nP\/2012 T1             && 3.154 & 0.236 & 11.06 & 3.135 &  2:1 & 3.277 & 0.123 \\\\\n313P\/Gibbs            && 3.156 & 0.242 & 10.97 & 3.132 &  2:1 & 3.277 & 0.121 \\\\\n\\hline\\hline\n\\end{tabular}\n\\newline{$^a$ Semimajor axis, in AU.}\n\\newline{$^b$ Inclination, in degrees.}\n\\newline{$^c$ Nearest major or moderate-order MMR}\n\\newline{$^d$ Semimajor axis, in AU, of nearest major or moderate-order MMR}\n\\newline{$^e$ Difference, in AU, between $a_f$ for each object and the semimajor axis of the nearest major or moderate-order MMR.}\n\\label{table:jfcs_mbcs_highlighted}\n\\end{table*}\n\n\\subsubsection{Detailed Orbital Evolution Analysis\\label{section:mbc_origins_detailed}}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.5in]{fig_evolution_ABCD.pdf}}\n\\caption{\\small Plots of orbital parameter evolution (black lines) over the course of our integrations for particles A-D in Table~\\ref{table:comet_to_mainbelt}, as labeled.  Orbital parameters plotted in the first six sub-panels for each particle are, from top to bottom, $a$ (in AU), $q$ (in AU), $Q$ (in AU), $e$, $i$ (in degrees), and $T_J$, where red dots indicate where a particle's orbital elements are comet-like and blue dots indicate where a particle's orbital elements are main-belt-like.  The semimajor axis corresponding to the nearest major or moderate-order MMR to each particle's FOEs (cf.\\ Table~\\ref{table:comet_to_mainbelt}) is marked by a horizontal dotted line in the first sub-panel for each particle.  The final plot in each panel shows the distances of close planetary encounters in units of $R_H$ for each respective planet, where pink, dark red, light blue, and yellow dots show encounters with Jupiter, Mars, Earth, and Venus, respectively.  Grey shaded regions indicate $a$, $q$, $Q$, and $T_J$ ranges that do not meet main-belt criteria (cf.\\ Section~\\ref{section:tisserand_reliability}).\n}\n\\label{figure:mbparticles_evolution_ABCD}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.5in]{fig_evolution_EFGH.pdf}}\n\\caption{\\small Same as Figure~\\ref{figure:mbparticles_evolution_ABCD}, but for particles E-H in Table~\\ref{table:comet_to_mainbelt}, as labeled.\n}\n\\label{figure:mbparticles_evolution_EFGH}\n\\end{figure*}\n\nWe plot the evolution of key orbital parameters, as well as the times and distances to planets of encounters at distances of $<\\,$$3R_H$ (for each planet's respective $R_H$) for particles A-H over the course of our 2-Myr integrations in Figures~\\ref{figure:mbparticles_evolution_ABCD} and \\ref{figure:mbparticles_evolution_EFGH}, marking intermediate times at which their orbits are comet-like and main-belt-like.  As we noted in Section~\\ref{section:mbc_origins_initial}, $a$ varies by relatively small amounts for each particle over the course of the integrations.  Values of $e$ (and therefore $q$ and $Q$) and $i$, however, are seen to change dramatically for most particles, though the timescales of these changes varies from particle to particle.  Notably, we see that while the semimajor axes of most of these highlighted particles appear to be strongly associated with the nearby MMRs listed in Table~\\ref{table:comet_to_mainbelt} over at least some portion of the integrations, some exhibit irregular fluctuations (e.g., particles C, D, and G) or small consistent offsets from the suspected associated MMR (e.g., particles E, F, and H), suggesting that some of these particles are additionally influenced by other nearby and possibly overlapping two- and three-body MMRs \\citep[where overlapping MMRs can actually impart additional short-term stability in certain cases;][]{gab03}, or other secular effects.  For reference, we show more detailed plots (Figures~\\ref{figure:mbparticles_evolution_ABCD_100yr} and \\ref{figure:mbparticles_evolution_EFGH_100yr}) of each particle's evolution in 100~yr intervals over the final $\\sim$50\\,000 years of our integrations (over which most of these particles have attained consistently main-belt-like orbits) of $a$, $e$, $i$, the longitude of perihelion, $\\varpi$, and the relevant resonant angle, $\\theta$ (corresponding to the suspected associated MMR for each particle listed in Table~\\ref{table:comet_to_mainbelt}), where $\\theta$ is given by\n\\begin{equation}\n\\theta = (p+q)\\lambda_J - p\\lambda - q\\varpi\n\\end{equation}\nfor an internal two-body $(p+q):p$ MMR with Jupiter, and $\\lambda_J$ and $\\lambda$ are the mean longitudes of Jupiter and the resonant object, respectively.  A detailed analysis of the resonant dynamical behavior of each particle is beyond the scope of the study presented here, but should certainly be a focus of follow-up studies exploring the range of dynamical behaviors while in the main belt exhibited by initially comet-like objects that attain main-belt-like orbits (ideally involving a larger number of independent particles meeting those criteria than we study here).  The efficiency of various MMRs (or combinations of MMRs) in the temporary stabilization of initially comet-like objects that transition onto main-belt-like orbits and the typical lifetimes of such objects in different MMRs would also be extremely interesting topics to explore in the future.\n\nIn addition to being on Jupiter-approaching or Jupiter-crossing orbits, almost all of these particles have initial orbits that approach or cross the orbits of the terrestrial planets.  Close encounters with the terrestrial planets have been suggested as a possible mechanism for producing the orbit of 2P\/Encke from a JFC-like orbit \\citep[e.g.,][]{val95,lev06}, although in those particular studies, the timescales required to reproduce 2P's orbit greatly exceeded the object's expected active lifetime.  In our integrations, almost every particle's transition from a comet-like orbit to a main-belt-like orbit (in some cases, back and forth multiple times) is accompanied by a large number of close encounters (as defined above) with Mars, Earth, and even Venus (Figures~\\ref{figure:mbparticles_evolution_ABCD} and \\ref{figure:mbparticles_evolution_EFGH}; bottom panels), strongly suggesting that such close interactions with terrestrial planets play a crucial role in dynamically decoupling these objects from Jupiter's gravity and enabling them to transition onto high-$T_J$, main-belt-like orbits.\n\nAn exception to this rule is particle H, for which no close encounters within 3$R_H$ with any terrestrial planets, or even Jupiter, are found.  Despite this lack of close planetary encounters to explain this particle's evolution onto a main-belt-like orbit, we note that besides having the largest $T_{J,s}$ of the eight particles, placing it very close to the ostensible boundary between asteroids and comets at the outset of the integrations, it begins (and ends) very close to the strongly chaotic 2:1 MMR with Jupiter, known for being capable of causing large fluctuations in eccentricities \\citep[cf.][]{mur86,moo97,nes97}, and may also be subject to secular resonances \\citep[cf.][]{wil81} and three-body MMRs \\citep[cf.][]{nes98,gal14}.  Thus, it is not unreasonable to expect that, at least in a small number of cases, the eccentricity of a particle within or close to this MMR could random walk to lower $e$, effectively transitioning from a comet-like orbit to a main-belt-like one, at least temporarily.\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.5in]{fig_evolution_ABCD_100yr.pdf}}\n\\caption{\\small Plots of the time evolution of $a$, $e$, $i$, $\\varpi$, and the resonant angle, $\\theta$, corresponding to the suspected associated MMR for each particle listed in Table~\\ref{table:comet_to_mainbelt} in 100~yr intervals over the final $\\sim$50\\,000 years of our integrations for particles A-D, as labeled.  The semimajor axis corresponding to the nearest major or moderate-order MMR to each particle's FOEs (cf.\\ Table~\\ref{table:comet_to_mainbelt}) is marked by a horizontal dotted line in the first sub-panel for each particle.\n}\n\\label{figure:mbparticles_evolution_ABCD_100yr}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.5in]{fig_evolution_EFGH_100yr.pdf}}\n\\caption{\\small Same as Figure~\\ref{figure:mbparticles_evolution_ABCD_100yr}, but for particles E-H in Table~\\ref{table:comet_to_mainbelt}, as labeled.  Grey shaded regions indicate $a$ ranges that do not meet main-belt criteria (cf.\\ Section~\\ref{section:tisserand_reliability}).\n}\n\\label{figure:mbparticles_evolution_EFGH_100yr}\n\\end{figure*}\n\n\n\n\\subsubsection{Extended Integrations\\label{section:mbc_origins_extended}}\n\nIn order to probe possible outcomes of real objects similar to particles A-H, we perform a simple follow-up study in which we generate a set of 100 clones for each particle centered on its FOEs and with Gaussian distributions in orbital element space characterized by $\\sigma$ values for $a$, $e$, and $i$ of $\\sigma_a=0.001$~AU, $\\sigma_e=0.001$, and $\\sigma_i=0.01^{\\circ}$, respectively, and perform extended integrations to study their long-term dynamical evolution.  This procedure is intended simply to investigate the orbital parameter space in the immediate vicinity of the final orbital elements of our test particles of interest in order to ascertain whether small orbital perturbations (due to any cause) produce interesting dynamical behaviors.  Nonetheless, our chosen $\\sigma$ values give us sets of clones with orbital element ranges approximately similar to those of the extremely young Schulhof and P\/2012 F5 (Gibbs) asteroid families, which have ages of $\\sim$0.8~Myr and $\\sim$1.5~Myr, respectively \\citep[][]{vok11,nov14}.  As such, these clones could be interpreted as a crude representation of a situation where a comet-like object evolves onto a main-belt-like orbit and then undergoes a catastrophic collisional disruption, resulting in a cluster of fragments with similar but slightly varying orbital elements (i.e., a young asteroid family).  Alternatively, these sets of clones could be interpreted as real objects similar to particles A-H that experience a range of random non-gravitational perturbations from the Yarkovsky effect or even outgassing.\n\nWe integrate these new sets of test particles forward for 100~Myr using the same experimental setup as before, and plot the resulting dynamical lifetimes, $t_{\\rm dyn}$, for each particle's set of clones in Figure~\\ref{figure:mbparticles}.  We also list the fractions of clones in each set with dynamical lifetimes in various $t_{\\rm dyn}$ bins in Table~\\ref{table:mbparticles_stability_timescales}.\n\n\\setlength{\\tabcolsep}{5pt}\n\\begin{table}[ht]\n\\small\n\\caption{\\small Dynamical Lifetimes in Extended Integrations of Particles with Comet-Like SOEs and Main-Belt-Like FOEs}\n\\smallskip\n\\begin{tabular}{ccccccccccccccc}\n\\hline\\hline\n \\multicolumn{1}{c}{Particle} &\n \\multicolumn{5}{c}{$t_{\\rm dyn}$ (Myr)} \\\\\n \\multicolumn{1}{c}{Sets} &\n \\multicolumn{1}{c}{$<$10$^a$} &\n \\multicolumn{1}{c}{10-20$^b$} &\n \\multicolumn{1}{c}{20-50$^c$} &\n \\multicolumn{1}{c}{50-100$^d$} &\n \\multicolumn{1}{c}{$>$100$^e$} \\\\\n\\hline\nA     & 0.90 & 0.07 & 0.01 & 0.01 & 0.01 \\\\\nB     & 0.73 & 0.15 & 0.07 & 0.01 & 0.04 \\\\\nC     & 0.69 & 0.15 & 0.11 & 0.03 & 0.02 \\\\\nD     & 0.78 & 0.09 & 0.05 & 0.02 & 0.06 \\\\\nE     & 0.41 & 0.11 & 0.07 & 0.05 & 0.36 \\\\\nF     & 0.16 & 0.02 & 0.06 & 0.07 & 0.69 \\\\\nG     & 0.65 & 0.16 & 0.11 & 0.04 & 0.04 \\\\\nH     & 0.26 & 0.08 & 0.30 & 0.06 & 0.30 \\\\\nTotal & 0.57 & 0.10 & 0.10 & 0.04 & 0.19 \\\\\n\\hline\\hline\n\\end{tabular}\n\\newline{$^a$ Fraction of particles with $t_{\\rm dyn}<10$~Myr.}\n\\newline{$^b$ Fraction of particles with $10$~Myr~$<t_{\\rm dyn}<20$~Myr.}\n\\newline{$^c$ Fraction of particles with $20$~Myr~$<t_{\\rm dyn}<50$~Myr.}\n\\newline{$^d$ Fraction of particles with $50$~Myr~$<t_{\\rm dyn}<100$~Myr.}\n\\newline{$^e$ Fraction of particles with $t_{\\rm dyn}>100$~Myr.}\n\\label{table:mbparticles_stability_timescales}\n\\end{table}\n\nAll original test particles are found to be unstable over our extended integration period, with particle F remaining stable the longest at $\\sim$72~Myr, particle C persisting for $\\sim$26~Myr, and all other particles only remaining stable for $<$15~Myr.  However, while $>$80\\% of the test particles in five of these sets of clones have $t_{\\rm dyn}<20$~Myr, we find that $\\geq$30\\% of the test particles in three of these sets of clones (E, F, and H) have $t_{\\rm dyn}>100$~Myr, placing their stability on par with previously studied MBCs \\citep[e.g., 288P, 324P, P\/2012 T1;][]{hsi12b,hsi12c,hsi13}.  Additionally, we note that even those clones with $t_{\\rm dyn}\\sim20-30$~Myr exhibit dynamical stability on par with certain other MBCs \\citep[e.g., 238P, 259P;][]{hag09,jew09}.\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.0in]{fig_mbparticles.pdf}}\n\\caption{\\small (a) Same as Figure~\\ref{figure:start_cometlike_sometimes_mb_all}a, but cropped and enlarged to focus on the main-belt region.  The FOEs of particles that have comet-like SOEs and main-belt-like FOEs are labeled A-H in Table~\\ref{table:comet_to_mainbelt}, as specified in Table~\\ref{table:comet_to_mainbelt}.  (b) Same as Figure~\\ref{figure:start_cometlike_sometimes_mb_all}c, but cropped and enlarged to focus on the main-belt region.  (c) Histograms indicating the distribution of extended dynamical lifetimes for the sets of clones for particles A-H, as labeled.\n}\n\\label{figure:mbparticles}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=5.8in]{fig_extended_ABCD.pdf}}\n\\caption{\\small Plots of $a$ vs.\\ $e$ (left panels) and $e$ versus $i$ (right panels) plots for IOEs in extended 100-Myr integrations for clones of particles A-D in Table~\\ref{table:comet_to_mainbelt} (as labeled) found to be stable for 100 Myr.  SOEs of clones in each set are marked with red dots, while FOEs are marked with dark blue X's.  Light blue dots indicate main-belt-like IOEs, while gray dots indicate non-main-belt-like IOEs. The same region of $e$-$i$ space as in Figure~\\ref{figure:start_cometlike_sometimes_mb_all} into which comet-like test particles never enter in our original set of integrations is shaded in orange.\n}\n\\label{figure:mbparticles_extended_ABCD}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=5.8in]{fig_extended_EFGH.pdf}}\n\\caption{\\small Same as Figure~\\ref{figure:mbparticles_extended_ABCD}, but for particles E-H in Table~\\ref{table:comet_to_mainbelt}, as labeled.\n}\n\\label{figure:mbparticles_extended_EFGH}\n\\end{figure*}\n\nPlots of IOEs of the clones in each test particle set that remain stable for the full 100~Myr of our extended integrations (Figures~\\ref{figure:mbparticles_extended_ABCD} and \\ref{figure:mbparticles_extended_EFGH}) indicate that almost all of the stable clones of our highlighted test particles continue to remain outside the orange-shaded protected region of $e$-$i$ space (cf.\\ Figure~\\ref{figure:start_cometlike_sometimes_mb_all}; Equation~\\ref{equation:protected_region}) throughout the entirety of our extended integrations.  However, five stable clones of particle H actually intermittently stray into this protected zone for significant total portions of the integrations, although none have FOEs found in that zone.  Specifically, one clone spends a total of $\\sim$1~Myr in the protected zone, two clones each spend $\\sim$15~Myr in the zone, one clone spends $\\sim$40~Myr in the zone, and one clone spends $\\sim$60~Myr in the zone.  However, all of these particles oscillate into and out of the specified protected zone on short timescales of $<$1~Myr each time.  Similar behavior is also observed for clones of particle H that are found to be unstable over 100~Myr (including particle H itself): a small number of clones intermittently enter the protected zone but only do so for $<$1~Myr at one time.\n\nAt this time, we are unable to identify any particular distinguishing dynamical characteristics of these types of interlopers based on their orbital elements alone, but note that their short individual residence times themselves could be a potential way to distinguish them from native objects in this region of orbital element space.  \nMuch more detailed studies of this issue are clearly needed before any firm conclusions can be drawn about how outer solar system interlopers of the low-$i$, low-$e$ main-belt population might be reliably identified in practice, and also just how significant this interloper population is expected to be in the first place.  In particular, future studies could consider more realistic ejection velocity fields and fragment size distributions for fragmentation events given various impact circumstances, impactor properties, and target properties \\citep[e.g.,][]{mic04,mic15,nes06}, or more realistic perturbations from the Yarkovsky effect \\citep[e.g.,][]{bot06,vok15} or outgassing \\citep[e.g.,][]{sek93,maq12}.  Because this work was focused on studying the diagnostic value of $T_J$ derived from osculating orbital elements observed for objects at arbitrary times during their dynamical evolution, we did not include the calculation of proper elements in our analysis.  However, future efforts to identify more reliable distinguishing characteristics between previously JFC-like interlopers in the main belt and native objects would likely benefit from calculations of proper elements and Lyapunov times for simulated interlopers and comparison of the results to those of real-world asteroids in the same regions of osculating orbital element space.\n\n\n\\subsection{Transfer of Other Objects to Main-Belt-Like Orbits}\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.5in]{fig_start_uncometlike_sometimes_mb_all2.pdf}}\n\\caption{\\small Same as Figure~\\ref{figure:start_cometlike_sometimes_mb_all}, but for test particles that do not have comet-like or main-belt-like SOEs (plotted with purple dots) and that reach main-belt-like orbital parameters at any point during the integration period.\n}\n\\label{figure:start_uncometlike_sometimes_mb_all}\n\\end{figure*}\n\nWhile a primary motivation of this study is to investigate whether objects from the outer solar system (i.e., on comet-like orbits) can dynamically evolve onto main-belt-like orbits and thus masquerade as native-born MBCs, we can also use our integrations to see whether objects now found in the main belt may have also potentially originated from elsewhere in the inner solar system.  In Figure~\\ref{figure:start_uncometlike_sometimes_mb_all}, we plot IOEs for 992 test particles that do not have comet-like or main-belt-like SOEs that reach main-belt-like IOEs at any point during our initial 2~Myr integration period, 122 of which have main-belt-like FOEs.\n\nWe find that most of these initially non-main-belt-like particles have $a_s$ mostly within the boundaries of the canonical main belt, but either have $e$ that cause them to be Mars-crossers or $i$ that cause them to have $T_{J,s}<3.05$.\nA total of fourteen particles have IOEs that enter the region of $e$-$i$ space that was found to be largely protected against comet-like interlopers in Section~\\ref{section:mbc_origins}, where two of these particles spend as much as $\\sim$1~Myr of total time in the region, and two other particles actually have FOEs in the region (each spending a total of 600-700~kyr in the protected zone).  The remaining particles with IOEs that reach the protected zone each spend $\\lesssim$250~kyr in the region.  In all cases, the longest continuous period that any of these particles remains in the protected zone, however, is 150-200~kyr, where most only stay for periods of $<$50~kyr at any one time.  As also found in Section~\\ref{section:mbc_origins_extended}, short individual residence times could therefore be a means for distinguishing these types of interlopers from native objects also found in this region of orbital element space.  In any case, however, given that most of the source regions considered in this section are only sparsely populated in the real solar system, the real-world impact of this contamination is likely to be of minimal significance.\n\n\n\\subsection{Transfer of Objects with Main-Belt-Like Orbits to Comet-Like Orbits}\n\nOne consequence of the discovery that some main-belt objects may still contain present-day ice is the possibility that any of these objects that are ejected from the main belt via resonances or other mechanisms may actually mimic ``classical'' JFCs by appearing to be currently icy bodies from the outer solar system when they are not, similar to how the JFC population may contain a component consisting of escaped Hilda asteroids \\citep{dis05}, effectively ``contaminating'' this population of icy objects presumed to be from the outer solar system with icy objects that are actually from the inner solar system.  In essence, this represents the inverse of the question of whether JFC-like interlopers could masquerade as native MBCs.  Such a scenario is problematic because it raises the possibility that the composition of an interloper with a JFC orbit could be erroneously considered representative of objects formed in the Kuiper belt region of the early solar system, when in fact it actually formed in a much higher-temperature region in the main asteroid belt.\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=6.5in]{fig_start_in_mb_sometimes_cometlike_all2.pdf}}\n\\caption{\\small (a) Same as Figure~\\ref{figure:start_cometlike_sometimes_mb_all}a, but for particles with main-belt-like SOEs that have comet-like IOEs at any time during our initial 2~Myr integrations.  SOEs are marked with dark blue circles, comet-like IOEs are marked with pale red dots, comet-like FOEs are marked with bright red X's, and all other IOEs are marked with gray dots.  (b) Same as Figure~\\ref{figure:start_cometlike_sometimes_mb_all}c, but for particles with main-belt-like SOEs that have comet-like IOEs at any time during our initial 2~Myr integrations.  For reference, the orbital elements of comet 81P\/Wild are indicated with an orange star in each panel, and the orbital elements of 103P\/Hartley 2 are indicated with a yellow star in each panel. (c) Histograms showing the normalized distribution of $T_{J,s}$ for particles plotted in panels (a) and (b) (light blue bars), and the normalized distribution of $T_{J,i}$ at time steps at which these particles have comet-like orbital elements over the course of our 2~Myr integrations (light red bars).\n}\n\\label{figure:mb_to_comet}\n\\end{figure*}\n\nTo briefly investigate the possibility of this scenario, we identify particles that have main-belt-like SOEs that have comet-like IOEs at any time during our initial 2~Myr integrations, and plot their SOEs, IOEs, and FOEs (Figure~\\ref{figure:mb_to_comet}).  We also plot normalized histograms of $T_{J,s}$ values for these particles and $T_{J,i}$ values at all time steps at which these particles have comet-like orbital elements.  We see that main-belt-like particles can reach a wide range of comet-like $T_J$ values (Figure~\\ref{figure:mb_to_comet}c), indicating that a low $T_J$ value may not actually be a guarantee of an outer solar system origin in all cases.  This is consistent with previous studies indicating that MMRs with the giant planets are capable of driving main-belt asteroids onto orbits with $T_J$$\\,<\\,$3 or even $T_J<2$ \\citep{far94,gla97,bot02}.  We also see though that $a$ for these escaped main-belt-like particles remain largely unchanged (Figure~\\ref{figure:mb_to_comet}a). As such, the population of comet-like objects with $a$ beyond the 2:1 MMR with Jupiter is likely to be largely free of interlopers from the main belt, though a more detailed study of this problem would be useful for confirming (or rejecting) and refining this preliminary observation.\n\n\n\\section{DISCUSSION\\label{discussion}}\n\n\\subsection{MBC Origins and Reliability as Compositional Tracers}\n\nThe work presented here is intended as an exploratory study to capture the general flavor of the dynamical behavior of objects with $T_J$ values close to the canonical $T_J=3$ dividing line between asteroids and comets.  We note that these results only consider the configuration of the major planets in the modern solar system, i.e., after the end of any and all major planet migration \\citep[e.g.,][]{fer84,tsi05,min09,mor10,wal11,agn12}.  In particular, we are interested in determining whether, given a set of osculating orbital elements observed at an arbitrary point in time during an object's dynamical evolution, $T_J$$\\,>\\,$3 (or $T_J$$\\,>\\,$3.05) is a suitable criterion on its own for reliably identifying objects with inner solar system origins, or if additional dynamical criteria are required.\n\nEven without considering non-gravitational forces like the Yarkovsky effect \\citep[cf.][]{rub95} or cometary outgassing \\citep[cf.][]{mar73,yeo04}, or mutual gravitational interactions among asteroids \\citep[e.g.,][]{nov15}, our results show that considering only the major planets and the Sun, purely gravitational dynamical pathways exist in our solar system via which objects with comet-like orbits (with $T_{J,s}<3$) can evolve onto main-belt-like, and even MBC-like, orbits (with $T_J$ values of $>\\,$3.05), apparently via the influence of MMRs with Jupiter and gravitational interactions with terrestrial planets, consistent with the findings of \\citet{gab03}.  Secular perturbations may also contribute to the dynamical evolution of these objects \\citep[e.g.,][]{kne91,mor91,bai96,mic10,mac12}, though we did not explicitly consider them in this work.  We find that initially comet-like objects that take on main-belt-like orbits in this way do not appear to be stable on long ($\\gtrsim$100~Myr) timescales, likely due to their continued interactions with MMRs while in the main belt, and so probably cannot account for MBCs found to be stable over such long timescales \\citep[e.g.,][]{hag09,hsi12b,hsi12c,hsi13}.  Even so, some actually have similar dynamical lifetimes as have been found for other less stable MBCs ($\\sim$20-30~Myr; Section~\\ref{section:mbc_origins_extended}).  In one of those cases, \\citet{jew09} concluded that the relative instability of 259P indicated that it was only recently transported to its current location in the inner main belt, and considering its high $T_J$ value, suggested that it could have originated from the outer main belt.  However, \\citet{dee07} found that collisional processes, the Yarkovsky effect, and other dynamical interactions produce only minimal mixing of main-belt material between the different major regions of the asteroid belt (as delineated by the $\\nu_6$, 3:1, and 5:2 resonances).  The work presented here provides new support to the possibility that these less stable MBCs could in fact have originated outside the asteroid belt.\n\nOur results suggest a possible mechanism by which some interlopers, or at least their fragments, could actually attain orbits that are stable for longer periods of time by entering the main belt via MMRs and then undergoing catastrophic collisional fragmentations (i.e., the scenario mimicked in Section~\\ref{section:mbc_origins_extended}).  If some of the pieces from such a break-up were then able to gain sufficient separation from the associated MMR due to the velocity kicks imparted by the fragmentation event, they might be able to move onto substantially more stable orbits, free from the destabilizing influence of the MMR (cf.\\ Figure~\\ref{figure:mbparticles}).  This would essentially represent the opposite mechanism suggested for the {\\it ejection} of main-belt asteroids near MMRs onto near-Earth orbits \\citep{far93}.  Additional work accounting for catastrophic collision rates and realistic ejection velocity fields are certainly needed though to determine the efficiency of this process and also the expected rate of such events in the modern (i.e., post-planetary-migration) solar system.\n\nIn our integrations, the transition from a comet-like orbit to a main-belt-like orbit occurs on timescales well in excess of the typical physical lifetime of a JFC (i.e., the length of the period over which sublimation-driven cometary activity is observed before mantling or depletion of volatile material causes observable activity to stop), estimated by \\citet{lev97} to be on the order of tens of kyr.  However, ice could still be preserved in subsurface reservoirs on these ostensibly inert objects even after sustained cometary activity has stopped \\citep[e.g.,][]{sch08}.\nAs such, the long dynamical timescales involved for a comet-like object to transition to a main-belt-like one is not at odds with our current understanding of MBCs as objects with subsurface ice that only sublimates occasionally, for example, upon excavation by an impact \\citep[e.g.,][]{hsi04,cap12}.\n\nThese results could potentially account for the origin of D-type asteroids found throughout the main belt \\citep{car10,dem13,dem14a,dem14b}.  D-type objects are more typically found at distances larger than in the main asteroid belt, particularly among the Hilda asteroids \\citep[e.g.,][]{dah95} and the Jovian Trojans \\citep[e.g.,][]{gra80}.  Some cometary nuclei, presumably originating in the even more distant outer solar system, have also been found to have D-type-like spectra \\citep[cf.][]{fit94,lam04}.  \\citet{dem14b} speculated that the parent bodies of present-day D-type main-belt asteroids could have been implanted into the outer asteroid belt during the era of planetary migration \\citep[cf.][]{lev09} and then distributed throughout the asteroid belt via a combination of catastrophic fragmentation of the parent bodies and Yarkovsky-driven transport across the major main-belt MMRs. Alternatively, they could have been implanted during the early inward and outward migration of Jupiter proposed under the Grand Tack model \\citep{wal11}.  Our results show, however, that it may be possible for such objects to be implanted in the main asteroid belt even in the present day from gravitational interactions alone.\n\nThese results of our integrations are significant in that they indicate that a non-Mars-crossing and non-Jupiter-crossing orbit with $T_J$$\\,>\\,$3.05 observed at some particular point in time may not be a definitive indication of in situ formation in the inner solar system.  In the context of using MBCs as compositional tracers \\citep[cf.][]{hsi14}, this means that care must be taken when considering what portion of the MBC population can be used to infer the primordial distribution of water ice in the early solar system.  Our results indicate that MBCs observed to currently have both low $e$ and low $i$ may be considered reasonably likely to have formed in situ, but there is a non-negligible possibility that some MBCs observed to currently have both large $e$ and large $i$ could actually be JFC-like interlopers \\citep[consistent with the findings of][]{jfer02}.  As such, we suggest that this segment of the MBC population may not be completely reliable compositional tracers of the early solar system at their current locations, and should only be used as such with caution.\n\nOn the other hand, the reliability of MBCs with both low $e$ and low $i$ as compositional tracers appears to be more secure.  Our integrations show that even this segment of the MBC population could be occasionally infiltrated by comet-like interlopers, but that these interlopers may be identifiable by short individual residence times in that region of orbital element space (though additional work is needed to give this preliminary conclusion more detailed context).\n\nOne point is clear: MBCs should not be considered to be a monolithic population with similar origins.  We are now seeing hints of distinct dynamical classes of MBCs with distinct dynamical origins emerge, and need to account for these different origins when attempting to use them to infer conditions in the early solar system.\n\n\n\\subsection{Future Work}\n\nIn this study, we sought to explore the full orbital parameter space of possible inner solar system objects close to the dynamical boundary between asteroids and comets, meaning that test particle set we considered here does not reflect the real distribution of small bodies in the inner solar system.  As such, while our results have revealed possible dynamical pathways via which JFCs might transition, at least temporarily, from comet-like orbits to main-belt-like orbits, we cannot use these particular integrations to ascertain the real-world rate of JFCs undergoing such transitions.\nFollow-up studies using test particle sets that more accurately represent the real-world comet population (e.g., in terms of both orbital element distribution and size distribution, and perhaps also the real-world distribution of longitudes of perihelion with respect to Jupiter's) would be extremely useful for clarifying this issue.  When attempting to determine the fraction of previously JFC-like interlopers in the main belt at any given time, such studies should also take into account the shorter residence times that many of these interlopers appear to have relative to other more stable main-belt objects.\n\nMore realistic representations of fragmentation events in the asteroid belt (i.e., including realistic ejection velocity fields and fragment size distributions for given impact, impactor, and target properties) would also be very useful for determining the rate at which interlopers in the main belt might produce a cluster of dynamically similar objects, some of which might find their way onto stable orbits similar to those of known MBCs or known D-type main-belt asteroids, despite the instability of the original interlopers themselves (cf.\\ Section~\\ref{section:mbc_origins_extended}).\nCalculations of proper elements and Lyapunov times for simulated interlopers and comparison of the results to those of real-world asteroids in the same regions of osculating orbital element space may also aid efforts to identify more reliable distinguishing characteristics between previously JFC-like interlopers in the main belt and native objects.\nAdditionally, as discussed in Section~\\ref{section:mbc_origins_detailed}, detailed analyses of the dynamical behaviors exhibited while in the main belt by objects with initially comet-like orbits that attain main-belt-like orbits (ideally involving a larger number of independent particles meeting those criteria than we find in this work), as well as studies of the efficiency of various MMRs (or combinations of MMRs) in the temporary stabilization of initially comet-like objects that transition onto main-belt-like orbits and the typical lifetimes of objects captured by different MMRs would be extremely valuable.\n\nAnother area for improvement for this work would be the inclusion of non-gravitational forces.  We do not expect that including either the Yarkovsky effect or outgassing forces will negate the main result of this work, that objects with comet-like orbits could occasionally evolve onto main-belt-like orbits, since there is no reason to expect that those effects would {\\it prevent} the dynamical behavior we have already observed with purely gravitational integrations.\nIf anything, non-gravitational forces would likely cause such evolution to occur on even shorter timescales \\citep[cf.][]{ste96,jfer02,pit04}, increase the number of objects undergoing such evolution, or both.  Non-gravitational effects could also increase the rate of interlopers that first enter the main belt via MMRs and then escape the influence of those MMRs to attain more stable main-belt orbits (cf.\\ Section~\\ref{section:mbc_origins_extended}), since we do not expect that non-gravitational forces would preferentially confine objects within MMRs.  Actual numerical integrations including non-gravitational forces would quantify the degree to which all of these effects occur.  \n\n\nLastly, while the evolution of main-belt objects onto comet-like orbits is not the main focus of this work, our integrations indicate that there could be a non-zero interloper component in the JFC population consisting of objects from the main asteroid belt.  Given these preliminary dynamical results and the growing evidence that main-belt objects could contain significant quantities of volatile material \\citep[e.g., MBCs, detection of water ice on Themis, detection of outgassing from Ceres;][]{hsi06,riv10,cam10,kup14}, the possibility that some outgassing objects on cometary orbits could actually have originated in the asteroid belt should be investigated in more detail.  This issue is of particular importance given that compositional studies of comets \\citep[e.g.,][]{bro06,har11} are frequently interpreted assuming outer solar system origins for the objects in question, but these interpretations might change if there is instead a non-trivial possibility of an inner solar system origin for any given JFC.  Numerical integrations of a realistic asteroid population focusing on objects that escape the main belt would help quantify the rate at which the JFC population is contaminated by such interlopers, and therefore how much concern we should have for this possibility when interpreting compositional studies of comets.\n\n\n\\section{SUMMARY}\n\nIn this work, we present the results of numerical integrations of 10\\,000 test particles with starting Tisserand parameter values of 2.80$\\,<\\,$$T_{J,s}$$\\,<\\,$3.20 aimed at investigating the dynamical origins of main-belt comets.  Key results are as follows:\n\\begin{enumerate}\n\\item{As expected, we find that the Tisserand parameter with respect to Jupiter, $T_J$, for individual test particles is not always a reliable indicator of their initial orbit types, and for many test particles, is seen to cross the canonical $T_J=3$ (or $T_J=3.05$) line that ostensibly separates asteroids (assumed to originate in the inner solar system) and comets (assumed to originate in the outer solar system).\nTest particles with $3.00<T_{J,s}<3.10$ are found to spend on the order of 30\\% of their time over the course of 2~Myr integrations on the opposite side of the $T_J=3.05$ boundary from where they began.  Meanwhile, even test particles with $T_{J,s}<3.00$ are found to spend $\\sim$5\\% of their time over the course of 2~Myr integrations with $T_J>3.10$, and test particles with $T_{J,s}>3.10$ are found to spend a similar amount of time with $T_J<3.00$.\n}\n\\item{Of the test particles in our sample set with starting orbital elements similar to those of real-world JFCs, a few percent reach main-belt-like orbits at some point in their first 2~Myr of evolution.  As our initial test particle set is not an accurate representation of the real-world JFC population, this rate should not be regarded to accurately reflect reality. Test integrations of dynamical clones of real JFCs showing similar behavior, though, suggests that the fraction of real-world JFCs occasionally reaching main-belt-like orbits may be on the order of $\\sim$0.1-1\\%, although the fraction that remain on such orbits for appreciable lengths of time is certainly far lower.  For this reason, the number of such objects in the main-belt population at any given time is likely to be small, but still non-zero.\n}\n\\item{The main-belt-like orbits that are reached by test particles with comet-like starting orbital elements in our integrations appear to be largely prevented from simultaneously having both low eccentricities and low inclinations.  This suggests that despite our findings that comet-like objects can occasionally infiltrate the main asteroid belt, objects found in this particular region of orbital element space may be largely free of this potential JFC contamination and may be more reliably considered likely to have formed in situ.   Main-belt comets in this region may therefore provide a more reliable means for tracing the primordial ice content of the main asteroid belt than the main-belt comet population (which includes some objects on high-inclination, high-eccentricity orbits) as a whole.\n}\n\\item{Detailed investigation of the orbital evolution of test particles with comet-like starting orbital elements that have main-belt-like orbital elements at the end of our integrations indicates that they may reach those main-belt-like orbits largely via a combination of gravitational interactions with the terrestrial planets and temporary trapping by MMRs. Additional studies are required, however, to confirm this explanation, and also to ascertain the efficacy of this process for real JFCs.\n}\n\\item{Extended 100-Myr integrations of sets of dynamical clones (generated to roughly mimic the orbital element distribution of very young asteroid clusters, or alternatively the results of a random set of orbital perturbations due to non-gravitational effects like the Yarkovsky effect or outgassing) of the test particles that have comet-like starting orbital elements but are found to have main-belt-like orbital elements at the end of our initial 2-Myr integrations show that most of the original test particles become unstable on timescales of $<$15~Myr, though two remain stable for $\\sim$30-70~Myr.  In three of these clone sets, however, $\\geq$30\\% of the cloned test particles are found to remain stable for $>$100~Myr, on par with stability lifetimes found for other MBCs.  Some of these cloned particles are found to attain orbits with simultaneously low eccentricities and low inclinations, but only for $<$1~Myr at a time, suggesting that such short individual residence times could be a way to distinguish such interlopers from native objects in this region of orbital element space.\n}\n\\item{Our results suggest a possible mechanism for delivering outer solar system material onto stable main-belt-like orbits whereby comet-like objects evolve onto unstable main-belt orbits via terrestrial planet interactions and MMR trapping, and then experience catastrophic collisional disruptions, resulting in some portion of the resulting fragments gaining sufficient separation from their associated MMRs and attaining stable main-belt orbits.  However, more work involving test particle sets that better represent the real-world population of JFCs, and also including non-gravitational forces, realistic collision rates, and realistic ejection velocity fields is needed to quantify the nature and degree of this contamination.\n}\n\\item{We briefly consider the potential for contamination of the Jupiter-family comet population by main-belt objects, and find that while such contamination appears to be possible in principle, interlopers in the comet population with main-belt origins appear to be largely confined to the original semimajor axis boundaries of the main belt, meaning that the population of comet-like objects with semimajor axes beyond the 2:1 MMR with Jupiter is likely to be largely free from interlopers from the main belt.  More detailed study is needed to confirm this preliminary observation, however.\n}\n\\end{enumerate}\n\n\\section*{Acknowledgements}\nWe are grateful to R.\\ Brasser, D.\\ Jewitt, B.\\ Bottke, and P.\\ Lacerda for valuable discussions related to this work, and to R.\\ Brasser and L.\\ Dones for helpful reviews of this manuscript.\nSupport for this work was provided to HHH and NH via the NASA Planetary Astronomy program (NNX14AJ38G).\n\n\\bigski\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\setcounter{equation}{0}\n\n\\noindent Tsallis'  q-statistical mechanics yielded\nvariegated applications in the last  25 years\n\\cite{tsallis,web,pre,epjb1,epjb2,epjb3,epjb4,epjb5,epjb6,\nepjb7,epjb8,epjb9}. This statistics is of great\nimportance for  astrophysics, in what respects to self-gravitating systems \\cite{PP93,chava,lb}.\n Further, it was shown to be useful in  diverse scientific fields. It has to its credit several\nthousands  of  papers and authors \\cite{web}. Investigating\nits structural characteristics should be important for astronomy, physics,\nneurology, biology, economic sciences, etc. \\cite{tsallis}. Paradigmatic example is found in\nits  application to high energy physics, where the q-statistics seems to describe well the\ntransverse momentum distributions of different hadrons\n\\cite{tp11o,tp11,phenix}.\n\n\\noindent In this work we use standard mathematical tools\ndescribed in \\cite{epjb, arxiv} to investigate interesting\nproperties of the Tsallis statistics of n harmonic\noscillators.\n\n\\noindent The central point is the fact that the integrals\nused to evaluate the partition function $Z$ and the\nmean energy $<U>$ diverge for specific q-values.\nThese divergences can be overcome as described in\n\\cite{epjb, arxiv}\n\n\\noindent A basic result to be obtained here is that the number of classical oscillators, $n$,\nis strongly limited by the dimensionality $\\nu$ and\nthe Tsallis parameter q. For $<U>>0$ and $Z>0$, i.e.,\nthe conventional theory,  $n$ must be finite and bounded.\n\n\\vskip 3mm\n\n\n\n\\noindent A different panorama emerges by recourse to analytical extension in $\\nu$. Then it is possible\nto have a situation in which\n$Z>0$, $<U><0$, $C<0$, with n finite and bounded.\nThus, our systems are here bound, representing a ''classical crystal'', and also self-gravitating\n\\cite{lb}. Finally, we will study the theory's poles\nby recourse to dimensional regularization \\cite{epjb, arxiv}.\nWe find at the poles, that i) the specific heat  $C$ is temperature ($T$) dependent (classically!), and,\nii) again, gravitational effects. Note the  $C$ can be $T-$dependent only due to internal degrees\nof freedom, and that this is a quantum effect. We detect this dependence here at a purely classical level.\n\n \\vskip 3mm\n\\noindent We are motivated by the need of trying to determine what kind of hidden correlations are entailed by the non additivity of Tsallis' entropy $S_q$  for two independent systems A, B, i.e.,\n\n$$S_q(A,B) = S_q(A) + S_q(B) + (1-q) S_q(A)S:q(B); \\,\\,\\, q \\in {\\cal R}.$$\nThis is conveniently done by appeal to quite simple systems, whose physics is well known. Any divergence from this physics will originate in the hidden correlations. This is why we employ a system of $n$ HOs here.\n\n\\vskip 3mm\n\\noindent Divergences constitute an important theme of   theoretical physics.\nThe study and elimination of these divergences may be one of the most relevant tasks of theoretical\nendeavor. The typical example is the (thus far failed, alas)  attempt to\nquantify the gravitational field. Examples of  divergences-elimination can be found\nin  references \\cite{tq1,tq2,tq3,tq4,tq5}.\n\n\\noindent  We use here  an quite  simplified version (see \\cite{tr1}), of the\nmethodology of \\cite{tq1,tq2,tq3,tq4,tq5} with regards to   Tsallis\nstatistics \\cite{tsallis,web}, focusing on  its\napplicability to self-gravitation \\cite{PP93,chava,lb}.\nDivergence's removal will be seen to yield quite interesting insights.\n\n\\noindent These emerge using mathematics\nwell known for the last  40 years ago. Their  development\nallowed M. Veltman and G. t'Hooft to be  awarded with the\nNobel prize of physics in 1999.\nComfortable acquaintance with these\nmathematics is not a prerequisite to follow this paper.\nHowever, one must  accept that their physical significance\n is not now to kin doubt. In fact, one just needs i) analytical\nextensions and ii) dimensional regularization \\cite{tq1,tq2,tq3,tq4,tq5}.\n\n\\noindent We will here analyze the behavior of\n$Z$ and $<U>$ in connection with  three zones of possible arguments of the\n$\\Gamma$-function thay appears in $Z$ and $<U>$.\nThese arguments of the $\\Gamma$-function\nrule the   $Z$ - $<U>$ behavior, that in turn produces\n three distinct zones,\nfor a given spatial dimension $\\nu$, Tsallis' index\n$q$ and number of particles $N$.\nThe zone's specifics are:\\\\\n$(1)\\;\\;\\;\\frac {1} {1-q}-n\\nu-1>0$\\\\\n$(2)\\;\\;\\;\\frac {1} {1-q}-n\\nu<0\\;\\;\\;\n\\Gamma\\left(\\frac {1} {1-q}-n\\nu\\right)>0$\\\\\n$(3)\\;\\;\\;\\frac {1} {1-q}-n\\nu=-p\n\\;\\;\\;p=0,1,2,3,4.....$\\\\\nNormal behavior is\nfound in zone (1). Something resembling what might constitute gravitational\neffects (GE) are encountered in zone (2).\nIn zone (3) we find  both normal\nbehavior and also GE (Also known as gravotermal effects).\n\n\\noindent {\\bf Remark than in instance (3) we are performing a\nregularization of the corresponding theory, not\na renormalization.}\n\n\\section{The Harmonic Oscillator}\n\n\\setcounter{equation}{0}\nIt has to be noted, from the beginning,  that   we use in this contribution\nnormal (linear in the probability) expectation values. For simplicity reasons, we do not appeal to the weighted\nones, customarily attached to Tsallis-related papers \\cite{tsallis}. In this case one  restricts oneself to\nthe interval $[0< q \\le 1]$, and, consequently, the so-called Tsallis cut-off  problem \\cite{tsallis} is avoided.\n\nFor the q-partition function one has\n\\[Z=V^n\\int\\limits_{-\\infty}^{\\infty}\n\\left[1+\\beta(1-q)(p_1^2+\\cdot\\cdot\\cdot\np_n^2+q_1^2+\\cdot\\cdot\\cdot q_n^2)\n\\right]^{\\frac {1}{q-1}}\\otimes\\]\n\\begin{equation}\n\\label{eq1.1}\nd^{\\nu}p_1\\cdot\\cdot\\cdot d^{\\nu}p_n\nd^{\\nu}q_1\\cdot\\cdot\\cdot d^{\\nu}q_n,\n\\end{equation}\nOr\n\\begin{equation}\n\\label{eq1.2}\nZ=\\frac {2\\pi^{\\nu n}}\n{\\Gamma\\left(\\nu n\\right)}\n\\int\\limits_{-\\infty}^{\\infty}\n\\left[1+\\beta(1-q) p^2\n\\right]^{\\frac {1}{q-1}}p^{2\\nu n-1}dp.\n\\end{equation}\nWe have integrated over the angles and taken\n$p^2=p_1^2+\\cdot\\cdot\\cdot p_n^2+q_1^2+\\cdot\\cdot\\cdot q_n^2$.\nChanging variables in the fashion  $x=p^2$, the\nlast  integral becomes\n\\begin{equation}\n\\label{eq1.3}\nZ=\\frac {\\pi^{\\nu n}}\n{\\Gamma\\left(\\nu n\\right)}\n\\int\\limits_{-\\infty}^{\\infty}\n\\left[1+\\beta(1-q) x\n\\right]^{\\frac {1}{q-1}}x^{\\nu n-1}dx,\n\\end{equation}\nthat evaluated, yields\n\\begin{equation}\n\\label{eq1.4}\nZ=\\left[\\frac {\\pi} {\\beta(1-q)}\\right]^{\\nu n}\n\\frac {\\Gamma\\left(\\frac {1} {1-q}-\\nu n\\right)}\n{\\Gamma\\left(\\frac {1} {1-q}\\right)}.\n\\end{equation}\nSimilarly we have\n\\[Z<U>=\\int\\limits_{-\\infty}^{\\infty}\n\\left[1+\\beta(1-q)(p_1^2+\\cdot\\cdot\\cdot p_n^2+\nq_1^2+\\cdot\\cdot\\cdot q_n^2)\\right]^{\\frac {1}{q-1}}\\]\n\\begin{equation}\n\\label{eq1.5}\n(p_1^2+\\cdot\\cdot\\cdot p_n^2+q_1^2+\\cdot\\cdot\\cdot q_n^2)\nd^{\\nu}p_1\\cdot\\cdot\\cdot d^{\\nu}q_n.\n\\end{equation}\nIn spherical coordinates this becomes\n\\begin{equation}\n\\label{eq1.6}\nZ<U>=\\frac {2\\pi^{\\nu n}}\n{\\Gamma\\left(\\frac {\\nu n} {2}\\right)}\n\\int\\limits_{-\\infty}^{\\infty}\n\\left[1+\\beta(1-q) p^2\n\\right]^{\\frac {1}{q-1}}p^{2\\nu n+1}dp,\n\\end{equation}\nand setting $x=p^2$\nthis is now\n\\begin{equation}\n\\label{eq1.7}\nZ<U>=\\frac {\\pi^{\\nu n}}\n{\\Gamma\\left(\\nu n\\right)}\n\\int\\limits_{-\\infty}^{\\infty}\n\\left[1+\\beta(1-q) x\n\\right]^{\\frac {1}{q-1}}x^{\\nu n}dx,\n\\end{equation}\nthat evaluated yields\n\\begin{equation}\n\\label{eq1.8}\n<U>=\\frac {1} {Z}\\frac {\\nu n} {\\beta(1-q)}\n\\left[\\frac {\\pi} {\\beta(1-q)}\\right]^{\\nu n}\n\\frac {\\Gamma\\left(\\frac {1} {1-q}-\\nu n\n-1\\right)} {\\Gamma\\left(\\frac {1} {1-q}\\right)},\n\\end{equation}\nor\n\\begin{equation}\n\\label{eq1.9}\n<U>=\\frac {\\nu n} {\\beta[q-\\nu n(1-q)]}.\n\\end{equation}\nThe derivative with respect to  $T$ yields for the specific heat $C$\nat constant  volume\n\\begin{equation}\n\\label{eq1.10}\nC=\\frac {\\nu nk} {q-\\nu n(1-q)}.\n\\end{equation}\n\n\\section{Limitations that restrict the particle-number}\n\n\\setcounter{equation}{0}\n\nWe saw in  Ref. \\cite{tr1}, for an ideal q-gas, that its number of particles $N$ becomes restricted due to hidden q-correlations. Some related work by  Livadiotis,  McComas, and Obregon,  should be mentioned\n\\cite{li1,li2,obreg}. \\vskip 3mm \\noindent\n\nOur original  presentation begins here. We detect a similar effect below for our system of $n$ classical HOs.  We analyze first the Gamma functions\ninvolved in evaluating  $Z$ and $<U>$, for the zone $[0 < q \\le 1]$.\nStarting from  (\\ref{eq2.1}) we get, for a positive Gamma-argument\n\\begin{equation}\n\\label{eq2.1}\n\\frac {1} {1-q}-\\nu n>0.\n\\end{equation}\nIn analogous fashion we have from  (\\ref{eq2.2})\n\\begin{equation}\n\\label{eq2.2}\n\\frac {1} {1-q}-\\nu n-1>0.\n\\end{equation}\nWe are confronted  then with  two conditions that  strictly  limit the\nparticle-number $n$, that is,\n\\begin{equation}\n\\label{eq2.3}\n1\\leq n<\\frac {q} {\\nu(1-q)}\n\\end{equation}\nThere is a maximum allowable $n$.\nFor instance, if\n$q=1-10^{-3}, \\nu=3$, we have\n\\begin{equation}\n\\label{eq2.4}\n1\\leq n<333,\n\\end{equation}\nand one can not exceed 332 particles.\n\\section{The dimensional  analytical extension of divergent integrals \\cite{tq1,tq2,tq3,tq4,tq5}}\n\\setcounter{equation}{0}\n\nWe study first  negative Gamma arguments in (\\ref{eq1.4}). They will demand analytical extension\/dimensional regularization of the integrals (1.4) and (1.8). Accordingly,\n\n\\begin{equation}\n\\label{eq3.1}\n\\frac {1} {1-q}-\\nu n<0,\n\\end{equation}\ntogether with\n\\begin{equation}\n\\label{eq3.2}\n\\Gamma\\left(\\frac {1} {1-q}-\\nu n\\right)>0.\n\\end{equation}\nUtilize now\n\\begin{equation}\n\\label{eq3.3}\n\\Gamma(z)\\Gamma(1-z)=\\frac {\\pi} {\\sin(\\pi z)},\n\\end{equation}\nto encounter\n\n\\begin{equation}\n\\label{eq3.4}\n\\Gamma\\left(\\frac {1} {1-q}-\\nu n\\right)=-\n\\frac {\\pi} {\\sin\\pi\\left(\\nu n-\\frac {1} {1-q}\\right)\n\\Gamma\\left(\\nu n+1-\\frac {1} {1-q}\\right)}>0.\n\\end{equation}\nThe above is true if\n\\begin{equation}\n\\label{eq3.5}\n\\sin\\pi\\left(\\nu n-\\frac {1} {1-q}\\right)<0,\n\\end{equation}\nso that\n\\begin{equation}\n\\label{eq3.6}\n2p+1<\\nu n-\\frac {1} {1-q}<2(p+1)\n\\end{equation}\nwhere $p=0,1,2,3,4,5.....$, or equivalently\n\n\\begin{equation}\n\\label{eq3.9}\n\\frac {2p+1} {\\nu}+\\frac {1} {\\nu(1-q)}<n<\n\\frac {2(p+1)} {\\nu}+\\frac {1} {\\nu(1-q)}.\n\\end{equation}\nWe note that, from  (\\ref{eq3.1}), (\\ref{eq3.2}), and  (\\ref{eq3.4})\nwe find (1) $Z>0$, (2) $<U><0$ (Einstein crystal), (3) $C<0$,\nwhich entails bound states, on account of (2) and self-gravitation according to (3) \\cite{lb}.\n\n\n\\section{The poles of the Harmonic Oscillator treatment}\n\n\\setcounter{equation}{0}\n\nIf the Gamma's argument is such that\n\n\\begin{equation}\n\\label{eq4.1}\n\\frac {1} {1-q}-\\nu n=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,......,\n\\end{equation}\n $Z$ exhibits a single pole. \\vskip 3mm\n\\noindent For $\\nu=1$ one has\n\\begin{equation}\n\\label{eq4.2}\n\\frac {1} {1-q}-n=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,.......\n\\end{equation}\n\\noindent Given that   $0\\leq q<1$, the pertinent\n $q$ values become\n\\begin{equation}\n\\label{eq4.3}\nq=\\frac {1} {2},\\frac {2} {3},\\frac {3} {4},\\frac {4} {5},......,\n\\end{equation}\n$n\\geq 2$. \\vskip 3mm\n\\noindent For $\\nu=2$\n\\begin{equation}\n\\label{eq4.4}\n\\frac {1} {1-q}-2n=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,......,\n\\end{equation}\n\\noindent Once more, since  $0\\leq q<1$,\n\\begin{equation}\n\\label{eq4.5}\nq=\\frac {1} {2},\\frac {2} {3},\\frac {3} {4},\\frac {4} {5},......,\n\\end{equation}\n$n\\geq 1$.\n\\vskip 3mm\n\\noindent For $\\nu=3$\n\\begin{equation}\n\\label{eq4.6}\n\\frac {1} {1-q}-3n=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,......,\n\\end{equation}\nand since  $0\\leq q<1$,\n\\begin{equation}\n\\label{eq4.7}\nq=\\frac {1} {2},\\frac {2} {3},\\frac {3} {4},\\frac {4} {5},......,\n\\end{equation}\n$n\\geq 1$. \\vskip 3mm\n\n\\noindent We tackle now poles in $<U>$. They result from\n\n\\begin{equation}\n\\label{eq4.8}\n\\frac {1} {1-q}-\\nu n-1=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,......,\n\\end{equation}\nfor $\\nu=1$.\n\\begin{equation}\n\\label{eq4.9}\n\\frac {1} {1-q}-n-1=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,......,\n\\end{equation}\nSince $0\\leq q<1$, one has\n\\begin{equation}\n\\label{eq4.10}\nq=\\frac {1} {2},\\frac {2} {3},\\frac {3} {4},\\frac {4} {5},......,\n\\end{equation}\n\\noindent for $\\nu=2$.\n\\begin{equation}\n\\label{eq4.11}\n\\frac {1} {1-q}-2n-1=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,......,\n\\end{equation}\n\\begin{equation}\n\\label{eq4.12}\nq=\\frac {1} {2},\\frac {2} {3},\\frac {3} {4},\\frac {4} {5},......,\n\\end{equation}\n\\noindent For $\\nu=3$\n\\begin{equation}\n\\label{eq4.13}\n\\frac {1} {1-q}-3n-1=-p\\;\\;{\\rm for} \\;\\;p=0,1,2,3,......,\n\\end{equation}\n\n\\begin{equation}\n\\label{eq4.14}\nq=\\frac {1} {2},\\frac {2} {3},\\frac {3} {4},\\frac {4} {5},......,\n\\end{equation}\n\n\\section{The three-dimensional scenario}\n\n\\setcounter{equation}{0}\n\nAs an illustration of dimensional regularization  \\cite{tq1,tq2,tq3,tq4,tq5}\nwe discuss into some detail the dealing with  the poles at  $q=\\frac {1} {2}$ and $q=\\frac {2} {3}$.\n\n\\subsection{Pole at $q=1\/2$}\n\nOne has\n\\begin{equation}\n\\label{eq5.1}\nZ=\\left(\\frac {2\\pi} {\\beta}\\right)^{\\nu n}\n\\Gamma\\left(2-\\nu n\\right).\n\\end{equation}\nUsing\n\\begin{equation}\n\\label{eq5.2}\n\\Gamma\\left(2-\\nu n\\right)\n\\Gamma\\left(\\nu n-1\\right)=\n-\\frac {\\pi} {\\sin\\left(\\pi\\nu n\\right)}\n\\end{equation}\nor, equivalently\n\n\\begin{equation}\n\\label{eq5.3}\n\\Gamma\\left(2-\\nu n\\right)\n\\Gamma\\left(\\nu n-1\\right)=\n\\frac {(-1)^{3n+1}\\pi} {\\sin\\left[\n\\pi n (\\nu-3)\\right]},\n\\end{equation}\nso that\n\\begin{equation}\n\\label{eq5.4}\nZ=\\left(\\frac {2\\pi} {\\beta}\\right)^{\\nu n}\n\\frac {(-1)^{3n+1}\\pi}\n{\\sin[\\pi n(\\nu-3)]\n\\Gamma\\left(\\nu n-1\\right)}.\n\\end{equation}\nGiven that\n\\begin{equation}\n\\label{eq5.5}\n\\sin[\\pi n(\\nu-3)]=\\pi n(\\nu-3)\n\\left\\{1+\\sum\\limits_{m=1}^{\\infty}\n\\frac {(-1)^m} {(2m+1)!} \\left[\\pi n\n(\\nu-3)\\right]^{2m}\\right\\}=\n\\end{equation}\n\\begin{equation}\n\\label{eq5.6}\n=\\pi n(\\nu-3)X,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{eq5.7}\nX=\n\\left\\{1+\\sum\\limits_{m=1}^{\\infty}\n\\frac {(-1)^m} {(2m+1)!} \\left[\\pi n\n(\\nu-3)\\right]^{2m}\\right\\},\n\\end{equation}\nwe obtain\n\\begin{equation}\n\\label{eq5.8}\nZ=\\left(\\frac {2\\pi} {\\beta}\\right)^{3 n}\n\\frac {(-1)^{n+1}}\n{\\Gamma\\left(\\nu n-1\\right)\nXn(\\nu-3)}\n\\left[1+n(\\nu-3)\\ln\\left(\\frac {2\\pi} {\\beta}\\right)+\n\\cdot\\cdot\\cdot\\right]\n\\end{equation}\nThe term independent of  $\\nu-3$ is, according to dimensional regularization recipes\n\\cite{tq1,tq2,tq3,tq4,tq5}\n\\begin{equation}\n\\label{eq5.9}\nZ=\\left(\\frac {2\\pi} {\\beta}\\right)^{3 n}\n\\frac {(-1)^{n+1}}\n{\\Gamma\\left(3 n-1\\right)}\n\\ln\\left(\\frac {2\\pi} {\\beta}\\right)\n\\end{equation}\nThis $Z$  is then the physical one at the pole \\cite{tq1,tq2,tq3,tq4,tq5}. Now, for the mean energy one has\n\n\\begin{equation}\n\\label{eq5.10}\nZ<U>=\\frac {2n\\nu} {\\beta}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{\\nu n}\n\\Gamma\\left(1-\\nu n\\right).\n\\end{equation}\nEmploying\n\\begin{equation}\n\\label{eq5.11}\n\\Gamma\\left(1-\\nu n\\right)\n\\Gamma\\left(\\nu n\\right)=\n\\frac {\\pi} {\\sin\\left(\\pi\\nu n\\right)}\n\\end{equation}\nor, equivalently\n\\begin{equation}\n\\label{eq5.12}\n\\Gamma\\left(1-\\nu n\\right)\n\\Gamma\\left(\\nu n\\right)=\n\\frac {(-1)^{3n}\\pi} {\\sin\\left[\n\\pi n (\\nu-3)\\right]}\n\\end{equation}\nwe encounter for  $<U>$\n\n\\begin{equation}\n\\label{eq5.13}\nZ<U>=\\frac {2n\\nu} {\\beta}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{\\nu n}\n\\frac {(-1)^{3n}\\pi}\n{\\sin[\\pi n(\\nu-3)]\n\\Gamma\\left(\\nu n\\right)}.\n\\end{equation}\n$<U>$ can be rewritten in the fashion\n\n\\[Z<U>=\\frac {n(\\nu-3)} {\\beta}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{\\nu n}\n\\frac {(-1)^{3n}\\pi}\n{\\sin[\\pi n(\\nu-3)]\n\\Gamma\\left(\\nu n\\right)} +\\]\n\\begin{equation}\n\\label{eq5.14}\n\\frac {6n} {\\beta}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{3 n}\n\\frac {(-1)^{3n}\\pi}\n{\\sin[\\pi n(\\nu-3)]\n\\Gamma\\left(\\nu n\\right)}.\n\\end{equation}\n Recalling the Z-procedure gives  for  $<U>$\n\n\\[Z<U>=\\frac {2} {\\beta}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{3 n}\n\\frac {(-1)^{3n}}\n{\\Gamma\\left(3 n\\right)}+\\]\n\\begin{equation}\n\\label{eq5.15}\n\\frac {6n} {\\beta}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{3 n}\n\\frac {(-1)^{3n}}\n{\\Gamma\\left(3 n\\right)}\n\\ln\\left(\\frac {2\\pi} {\\beta}\\right)\n\\end{equation}\nor,  equivalently\n\n\\begin{equation}\n\\label{eq5.16}\nZ<U>=\\frac {2} {\\beta}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{3n}\n\\frac {(-1)^{3n}}\n{\\Gamma\\left(3 n\\right)}\n\\left[1+3n\n\\ln\\left(\\frac {2\\pi} {\\beta}\\right)\\right].\n\\end{equation}\nRemembering now  (\\ref{eq5.9}) for the physical $Z$\n on arrives at\n\n\\begin{equation}\n\\label{eq5.17}\n<{\\cal U}>=\n\\frac {2} {\\beta(3n-1)}\n\\left[\\frac {1} {\\ln\\beta-\\ln 2\\pi}-\n3n\\right].\n\\end{equation}\none treats first $(-1)^{3n+1}=-1$, so that   $n=2,4,6,8,......$, and\n\n\\begin{equation}\n\\label{eq5.18}\nZ=\\frac {1}\n{\\Gamma\\left(3 n-1\\right)}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{3 n}\n\\ln\\left(\\frac {\\beta} {2\\pi}\\right)\n\\end{equation}\nIf\n$(-1)^{3n+1}=1$, then\n$n=1,3,5,7......$ and\n\\begin{equation}\n\\label{eq5.19}\nZ=\\frac {1}\n{\\Gamma\\left(3 n-1\\right)}\n\\left(\\frac {2\\pi} {\\beta}\\right)^{3 n}\n\\ln\\left(\\frac {2\\pi} {\\beta}\\right).\n\\end{equation}\nAccording to  (\\ref{eq5.17}) - (\\ref{eq5.18}) and asking\n$Z>0$ and  $<U>>0$ one finds\n\\begin{equation}\n\\label{eq5.20}\n\\frac {1} {2\\pi ke^{\\frac {1} {3n}}}<T<\n\\frac {1} {2\\pi k}.\n\\end{equation}\nFrom  (\\ref{eq5.17}) -  (\\ref{eq5.19}) and requiring\n$Z>0$ y $<U><0$ (Einstein crystal) one encounters\n\\begin{equation}\n\\label{eq5.21}\n0\\leq T<\\frac {1} {2\\pi ke^{\\frac {1} {3n}}}\n\\end{equation}\nThe specific heat is derived  from\n(\\ref{eq5.17}) for $<U>$. We have\n\\begin{equation}\n\\label{eq5.22}\nC=\n\\frac {2k} {3n-1}\n\\left[\\frac {1} {\\ln\\beta-\\ln 2\\pi}+\n\\frac {1} {(\\ln\\beta-\\ln 2\\pi)^2}-\n3n\\right].\n\\end{equation}\n$C$ depends on $T$ and this is a quantum effect, since classically $C$ is a constant. Also, $C$ depends on $T$ because of the excitation of internal degrees of freedom, which the poles somehow detect.\n\n\n\\subsection{The Pole at $q=2\/3$}\n\nNow  $Z$ is\n\\begin{equation}\n\\label{eq5.23}\nZ=\\left(\\frac {3\\pi} {\\beta}\\right)^{\\nu n}\n\\frac {\\Gamma\\left(3-\\nu n\\right)}\n{\\Gamma\\left(3\\right)}.\n\\end{equation}\nEmploying once  again\n\\begin{equation}\n\\label{eq5.24}\n\\Gamma\\left(3-\\nu n\\right)\n\\Gamma\\left(\\nu n-2\\right)=\n\\frac {\\pi} {\\sin\\left(\\pi\\nu n\\right)},\n\\end{equation}\nor, equivalently\n\n\\begin{equation}\n\\label{eq5.25}\n\\Gamma\\left(3-\\nu n\\right)\n\\Gamma\\left(\\nu n-2\\right)=\n\\frac {(-1)^{3n}\\pi} {\\sin\\left[\n\\pi n (\\nu-3)\\right]},\n\\end{equation}\nso that we have\n\\begin{equation}\n\\label{eq5.26}\nZ=\\frac {1} {2}\n\\left(\\frac {3\\pi} {\\beta}\\right)^{\\frac {\\nu n} {2}}\n\\frac {(-1)^{3n}\\pi}\n{\\sin[\\pi n(\\nu-3)]\n\\Gamma\\left(\\nu n-1\\right)}.\n\\end{equation}\nOne then  dimensionally regularizes  $Z$ - $<U>$ as done for the previous pole, to reach\n\n\n\\begin{equation}\n\\label{eq5.27}\nZ=\\frac {1} {2}\n\\left(\\frac {3\\pi} {\\beta}\\right)^{3 n}\n\\frac {(-1)^{3n}}\n{\\Gamma\\left(3 n-2\\right)}\n\\ln\\left(\\frac {3\\pi} {\\beta}\\right),\n\\end{equation}\n\\begin{equation}\n\\label{eq5.28}\n<{\\cal U}>=\n\\frac {3} {2\\beta(3n-2)}\n\\left[\\frac {1} {\\ln\\beta-\\ln 3\\pi}-\n3n\\right].\n\\end{equation}\nWe seal  first with\n$(-1)^{\\frac {3n-1} {2}}=-1$ and then\n$n=1,3,5,7,9......$, so that\n\\begin{equation}\n\\label{eq5.29}\nZ=\\frac {1}\n{2\\Gamma\\left(3n-2\\right)}\n\\left(\\frac {3\\pi} {\\beta}\\right)^{3 n}\n\\ln\\left(\\frac {\\beta} {3\\pi}\\right).\n\\end{equation}\nFor\n$(-1)^{3n}=1$, one has\n$n=2,4,6,8......$ and\n\\begin{equation}\n\\label{eq5.30}\nZ=\\frac {1}\n{2\\Gamma\\left(3n-2\\right)}\n\\left(\\frac {3\\pi} {\\beta}\\right)^{3 n}\n\\ln\\left(\\frac {3\\pi} {\\beta}\\right).\n\\end{equation}\n\nAccording to   (\\ref{eq5.28}) - (\\ref{eq5.29}) and asking\n\n$Z>0$ - $<U>>0$ we encounter\n\\begin{equation}\n\\label{eq5.31}\n\\frac {1} {3\\pi ke^{\\frac {1} {3n}}}<T<\n\\frac {1} {3\\pi k}.\n\\end{equation}\nFrom  (\\ref{eq5.28})-y (\\ref{eq5.30}) and asking\n$Z>0$ - $<U><0$ we obtain\n\n\n\\begin{equation}\n\\label{eq5.32}\n0\\leq T<\\frac {1} {3\\pi ke^{\\frac {1} {3n}}}.\n\\end{equation}\nAs for  $C$ we have\n\\begin{equation}\n\\label{eq5.33}\nC=\n\\frac {3k} {2(3n-2)}\n\\left[\\frac {1} {\\ln\\beta-\\ln 3\\pi}+\n\\frac {1} {(\\ln\\beta-\\ln 3\\pi)^2}-\n3n\\right].\n\\end{equation}\n\n\\section{Conclusions}\n\n\n\\setcounter{equation}{0}\n\n\\noindent Here one has appealed to  an elementary\nregularization method to study  the poles in both thae partition function $Z$ and\nthe mean energy  $<U>$ for particular, discrete  values of Tsallis' parameter q\nin a non additive q-scenario. After investigating  the thermal\nbehavior at the poles, we found interesting features, like what might possibly constitute  self-gravitation or quantum effects. The\nanalysis was made for one, two, three, and $N$ dimensions. We discover\n pole-characteristics that are  unexpected but  true. In particular:\n\n\\begin{itemize}\n\n\n\\item An upper bound to the\ntemperature at the poles, in agreement with the findings  of Ref.\n\\cite{PP94}.\n\n\\item In some circumstances, Tsallis' entropies are positive only for a\nrestricted temperature-range.\n\n\\item Negative specific heats, which might constitute signatures  of\nself-gravitating systems \\cite{lb}, are encountered.\n\n\\item If the system is bound, we can regard it as a  \"classical\" Einstein-crystal. But we have\nfor it a  temperature dependence of the specific\nheat.\n\n\\item Thus, we find at the poles, that i) the specific heat  $C$ is temperature ($T$) dependent (classically!), and, ii) self-gravitational effects. Note that $C$ can become $T-$dependent only due to internal degrees\nof freedom, and that this is a quantum effect. We detect this dependence  here at a purely classical level.\n\n\\end{itemize}\n\n\\noindent These  physical results are collected employing just statistical consideration,  not mechanical ones. This  might perhaps  remind one of  a similar\nfeature associated to  the entropic force conjectured by\nVerlinde \\cite{verlinde}.\n\n\\noindent The Tsalllis' rule\n\n$$S_q(A,B) = S_q(A) + S_q(B) + (1-q) S_q(A)S:q(B); \\,\\,\\, q \\in {\\cal R},$$\nis seen here to erect a far from trivial scenario, in which strange effects take place.\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}