diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzaabh" "b/data_all_eng_slimpj/shuffled/split2/finalzzaabh" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzaabh" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:1}\n\nDetecting a very weak signal which is almost inaccessible within the \nclassical (i.e., non-quantum) regime is one of the most important subjects \nin quantum information science. \nA strong motivation to devise such an ultra-precise sensor stems from \nthe field of gravitational wave detection \n\\cite{BraginskyBook, Caves1980, Braginsky2008, Miao2012, Abbott2016}. \nIn fact, a variety of linear sensors composed of opto-mechanical oscillators \nhave been proposed \\cite{Milburn,Chen2013, Latune2013, Aspelmeyer2014}, \nand several experimental implementations of those systems in various scales \nhave been reported \n\\cite{Corbitt2007, Matsumoto2015, Underwood2015, Thompson2008, Verhagen2012}. \n\n\nIt is well known that in general a linear sensor is subjected to two types of \nfundamental noises, i.e., the {\\it back-action noise} and the {\\it shot noise}. \nAs a consequence, the measurement noise is lower bounded by the {\\it standard \nquantum limit (SQL)} \\cite{BraginskyBook, Caves1980}, which is mainly due \nto the presence of back-action noise. \nHence, high-precision detection of a weak signal requires us to devise a sensor \nthat evades the back-action noise and eventually beats the SQL; \ni.e., we need to have a sensor achieving {\\it back-action evasion (BAE)}. \nIn fact, many BAE methods have been developed especially in the field of \ngravitational wave detection, e.g., the variational measurement technique \n\\cite{Vyatchanin1995, Kimble2001, Khalili2007} or the quantum locking scheme \n\\cite{CourtyEurophys2003, CourtyPhysRev2003, Vitali2004}. \nMoreover, towards more accurate detection, recently we find some high-level \napproaches to design a BAE sensor, based on those specific BAE methods. \nFor instance, Ref.~\\cite{Miao2014} provides a systematic comparison of \nseveral BAE methods and gives an optimal solution. \nAlso systems and control theoretical methods have been developed to synthesize \na BAE sensor for a specific opto-mechanical system \n\\cite{Tsang2010, YamamotoCFvsMF2014}; \nin particular, the synthesis is conducted by connecting an auxiliary system \nto a given plant system by {\\it direct-interaction} \\cite{Tsang2010} or \n{\\it coherent feedback} \\cite{YamamotoCFvsMF2014}. \n\n\nAlong this research direction, therefore, in this paper we set the goal to develop \na general systems and control theory for engineering a sensor achieving BAE, \nfor both the coherent feedback and the direct-interaction configurations. \nThe key tool used here is the {\\it geometric control theory} \n\\cite{Schumacher1980, Schumacher1982, Wonham1985, Marro2010, Otsuka2015}, \nwhich had been developed a long time ago. \nThis is indeed a beautiful theory providing a variety of controller design \nmethods for various purposes such as the non-interacting control and \nthe disturbance decoupling problem, but, to our best knowledge, it has not been \napplied to problems in quantum physics. \nActually in this paper we first demonstrate that the general synthesis problem of \na BAE sensor can be formulated and solved within the framework of geometric control theory, \nparticularly the above-mentioned disturbance decoupling problem. \\vspace{0.7mm}\\\\\n\\hspace{0.2cm}\nThis paper is organized as follows. \nSection~\\ref{sec:2} is devoted to some preliminaries including a review of the \ngeometric control theory, the general model of linear quantum systems, and \nthe idea of BAE. \nThen, in Section~\\ref{sec:3}, we provide the general theory for designing a \ncoherent feedback controller achieving BAE, and demonstrate an example for \nan opto-mechanical system. \nIn Section~\\ref{sec:4}, we discuss the case of direct interaction scheme, also \nbased on the geometric control theory. \nFinally, in Section~\\ref{sec:5}, for a realistic opto-mechanical system subjected \nto a thermal environment (the perfect BAE is impossible in this case), we provide \na convenient method to find an approximated BAE controller and show how \nmuch the designed controller can suppress the noise. \\vspace{0.7mm}\\\\\n\\hspace{0.2cm}\n{\\bf Notation:} \nFor a matrix $A=(a_{ij})$, $A^{\\top}$, $A^{\\dag}$, \nand $A^{\\sharp}$ represent the transpose, Hermitian conjugate, \nand element-wise complex conjugate of $A$, respectively; i.e., \n$A^{\\top}=(a_{ji})$, $A^{\\dag}=(a_{ji}^*)$, and \n$A^{\\sharp}=(a_{ij}^*)=(A^{\\dag})^{\\top}$. \n$\\Re(a)$ and $\\Im(a)$ denote the real and imaginary parts of a complex \nnumber $a$. \n$O$ and $I_{n}$ denote the zero matrix and the $n \\times n$ identity matrix. \n$\\Kernel A$ and $\\Image A$ denote the kernel and the image of \na matrix $A$, i.e., $\\Kernel A=\\{x\\,|\\, Ax=0\\}$ and \n$\\Image A=\\{y\\,|\\, y=Ax, ~\\forall x\\}$. \n\n\n\n\n\n\\section{Preliminaries}\\label{sec:2}\n\n\n\n\\subsection{Geometric control theory for disturbance decoupling}\\label{sec:2-1}\n\nLet us consider the following {\\it classical} linear time-invariant system:\n\\begin{align}\n \\frac{dx(t)}{dt}&=Ax(t)+Bu(t), ~~ \\nonumber\\\\\n y(t)&=Cx(t)+Du(t),\n\\label{eq_linearsystem}\n\\end{align}\nwhere $x(t) \\in \\mathcal{X}:=\\mathbb{R}^{n}$ is a vector of system variables, \n$u(t) \\in \\mathcal{U}:=\\mathbb{R}^{m}$ and \n$y(t) \\in \\mathcal{Y}:=\\mathbb{R}^{l}$ are vectors of input and output, \nrespectively. \n$A, B, C$, and $D$ are real matrices. \nIn the Laplace domain, the input-output relation is represented by \n\\begin{align*}\n Y(s)=\\Xi (s)U(s), ~~ \\Xi (s)=C(sI-A)^{-1}B+D, \n\\end{align*}\nwhere $U(s)$ and $Y(s)$ are the Laplace transforms of $u(t)$ and $y(t)$, \nrespectively. \n$\\Xi(s)$ is called the {\\it transfer function}. \nIn this subsection, we assume $D=0$. \n\n\n\nNow we describe the geometric control theory, for the disturbance \ndecoupling problem \\cite{Schumacher1980, Schumacher1982}. \nThe following {\\it invariant subspaces} play a key role in the theory. \n\n{\\bf Definition 1:}~\nLet $A : \\mathcal{X} \\rightarrow \\mathcal{X}$ be a linear map. \nThen, a subspace $\\mathcal{V} \\subseteq \\mathcal{X}$ is said \nto be $A$-{\\it invariant}, if $A\\mathcal{V} \\subseteq \\mathcal{V}$. \n\n{\\bf Definition 2:}~\nGiven a linear map $A : \\mathcal{X} \\rightarrow \\mathcal{X}$ \nand a subspace \\Image $B$ $\\subseteq \\mathcal{X}$, a subspace \n$\\mathcal{V} \\subseteq \\mathcal{X}$ is said to be $(A, B)$-{\\it invariant}, \nif $A\\mathcal{V} \\subseteq \\mathcal{V} \\oplus \\Image B$. \n\n{\\bf Definition 3:}~ \nGiven a linear map $A : \\mathcal{X} \\rightarrow \\mathcal{X}$ \nand a subspace \\Kernel $C$ $\\subseteq \\mathcal{X}$, a subspace \n$\\mathcal{V}\\subseteq \\mathcal{X}$ is said to be $(C, A)$-{\\it invariant}, \nif $A(\\mathcal{V} \\cap \\Kernel C) \\subseteq \\mathcal{V}$. \n\n{\\bf Definition 4:}~ \nAssume that $\\mathcal{V}_{1}$ is $(C, A)$-invariant, $\\mathcal{V}_{2}$ is \n$(A, B)$-invariant, and $\\mathcal{V}_{1} \\subseteq \\mathcal{V}_{2}$. \nThen, $(\\mathcal{V}_{1},\\, \\mathcal{V}_{2})$ is said to be a $(C,A,B)$-pair.\n\n\nFrom Definitions 2 and 3, we have the following two lemmas.\n\n\n{\\bf Lemma 1:}~ \n$\\mathcal{V} \\subseteq \\mathcal{X}$ is $(A, B)$-invariant if and only if \nthere exists a matrix $F$ such that\n$\n F \\in \n \\mathcal{F}(\\mathcal{V}) :=\n \\{ F : \\mathcal{X} \\rightarrow \\mathcal{U}\\,| \\, \n (A+BF)\\mathcal{V} \\subseteq \\mathcal{V} \\}. \n$\n\n\n{\\bf Lemma 2:}~\n$\\mathcal{V} \\subseteq \\mathcal{X}$ is $(C, A)$-invariant if and only if \nthere exists a matrix $G$ such that\n$\n G \\in \n \\mathcal{G}(\\mathcal{V}) := \n \\{ G : \\mathcal{Y} \\rightarrow \\mathcal{X} \\,| \\, \n (A+GC)\\mathcal{V} \\subseteq \\mathcal{V} \\}. \n$\n\n\nThe disturbance decoupling problem is described as follows. \nThe system of interest is represented, in an extended form of \nEq.~\\eqref{eq_linearsystem}, as \n\\begin{align*}\n \\frac{dx(t)}{dt}&=Ax(t)+Bu(t)+Ed(t), ~\\\\\n y(t)&=Cx(t), ~~\n z(t)=Hx(t), \n\\end{align*}\nwhere $d(t)$ is the disturbance and $z(t)$ is the output to be regulated. \n$E$ and $H$ are real matrices. \nThe other output $y(t)$ may be used for constructing a feedback controller; \nsee Fig.~\\ref{fig_feedbackframework}. \nThe disturbance $d(t)$ can degrade the control performance evaluated on $z(t)$. \nThus it is desirable if we can modify the system structure by some means so \nthat eventually $d(t)$ dose not affect at all on $z(t)$\n\\footnote{This condition is satisfied if the transfer function from $d(s)$ to $z(s)$ is \nzero for all $s$, for the modified system. Or equivalently, the controllable subspace\nwith respect to $d(t)$ is contained in the unobservable subspace with respect to $z(t)$.}.\nThis control goal is called the disturbance decoupling. \nHere we describe a specific feedback control method to achieve this goal; \nnote that, as shown later, the direct-interaction method for linear quantum \nsystems can also be described within this framework. \nThe controller configuration is illustrated in Fig.~\\ref{fig_feedbackframework}; \nthat is, the system modification is carried out by combining an auxiliary \nsystem (controller) with the original system (plant), so that the whole \nclosed-loop system satisfies the disturbance decoupling condition. \nThe controller with variable \n${x}_{{\\scriptscriptstyle K}} \\in \\mathcal{X}_{{\\scriptscriptstyle K}}\n:=\\mathbb{R}^{n_{\\scriptscriptstyle k}}$ is assumed to take the following form: \n\\begin{align*}\n\\hspace{-0.1cm}\n \\frac{dx_{{\\scriptscriptstyle K}}(t)}{dt}\n &=A_{\\scriptscriptstyle K}x_{{\\scriptscriptstyle K}}(t)\n +B_{\\scriptscriptstyle K}y(t), \\, \\\\\n u(t)&=C_{\\scriptscriptstyle K}x_{{\\scriptscriptstyle K}}(t)\n +D_{\\scriptscriptstyle K}y(t), \n\\end{align*}\nwhere $A_{\\scriptscriptstyle K} : \\mathcal{X}_{{\\scriptscriptstyle K}} \\rightarrow \n\\mathcal{X}_{{\\scriptscriptstyle K}}$, \n$B_{\\scriptscriptstyle K} : \n\\mathcal{Y} \\rightarrow \\mathcal{X}_{{\\scriptscriptstyle K}}$,\n$C_{\\scriptscriptstyle K} : \n\\mathcal{X}_{{\\scriptscriptstyle K}} \\rightarrow \\mathcal{U}$, \nand $D_{\\scriptscriptstyle K} : \\mathcal{Y} \\rightarrow \\mathcal{U}$ are real matrices. \n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=4cm,clip]{Figure\/Fig1_feedbackframework.eps}\n\\caption{General configuration for the disturbance decoupling via a dynamical feedback controller.}\n\\label{fig_feedbackframework}\n\\end{center}\n\\end{figure}\nThen, the closed-loop system defined in the augmented space \n$\\mathcal{X}_{{\\scriptscriptstyle E}} := \\mathcal{X} \\oplus \\mathcal{X}_{{\\scriptscriptstyle K}}$ \nis given by \n\\begin{align}\n\\label{eq_extended_system}\n \\frac{d}{dt}\\left[\\begin{array}{c}\n x\\\\\n x_{{\\scriptscriptstyle K}}\\\\\n \\end{array}\\right] \n &=\\left[\\begin{array}{cc}\n A+BD_{\\scriptscriptstyle K}C & BC_{\\scriptscriptstyle K}\\\\\n B_{\\scriptscriptstyle K}C & A_{\\scriptscriptstyle K}\\\\\n \\end{array}\\right]\n \\left[\\begin{array}{c}\n x\\\\\n x_{{\\scriptscriptstyle K}}\\\\\n \\end{array}\\right] \n +\\left[\\begin{array}{c}\n E\\\\\n O\\\\\n \\end{array}\\right] d, ~ \\nonumber \\\\\n z&=\\left[\\begin{array}{cc}\n H & O\\\\\n \\end{array}\\right]\n \\left[\\begin{array}{c}\n x\\\\\n x_{{\\scriptscriptstyle K}}\\\\\n \\end{array}\\right].\n\\end{align}\nThe control goal is to design $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, \nC_{\\scriptscriptstyle K}, D_{\\scriptscriptstyle K})$ so that, in \nEq.~\\eqref{eq_extended_system}, the disturbance signal $d(t)$ dose not \nappear in the output $z(t)$: see the below footnote.\nHere, let us define \n\\begin{align}\n A_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n A+BD_{\\scriptscriptstyle K}C & BC_{\\scriptscriptstyle K}\\\\\n B_{\\scriptscriptstyle K}C & A_{\\scriptscriptstyle K}\\\\\n \\end{array}\\right] , \n\\label{eq_extended_matrixAe}\n\\end{align}\n$\\mathcal{B}= \\Image B$, $\\mathcal{C}= \\Kernel C$, \n$\\mathcal{E}= \\Image E$, and $\\mathcal{H}= \\Kernel H$. \nThen, the following theorem gives the solvability condition for the disturbance \ndecoupling problem.\n\n\n{\\bf Theorem 1:}~\nFor the closed-loop system \\eqref{eq_extended_system}, the disturbance \ndecoupling problem via the dynamical feedback controller has a solution \nif and only if there exists a $(C,A,B)$-pair $(\\mathcal{V}_{1},\\mathcal{V}_{2})$ \nsatisfying \n\\begin{align}\n \\mathcal{E} \\subseteq \\mathcal{V}_{1} \n \\subseteq \\mathcal{V}_{2} \\subseteq \\mathcal{H}.\n\\label{eq_DDP_condition}\n\\end{align}\n\nNote that this condition does not depend on the controller matrices to \nbe designed. \nThe following corollary can be used to check if the solvability condition \nis satisfied. \n\n\n{\\bf Corollary 1:}~ \nFor the closed-loop system \\eqref{eq_extended_system}, the disturbance \ndecoupling problem via the dynamical feedback controller has a solution \nif and only if\n\\begin{align}\n \\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})}\n \\subseteq \\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}, \\nonumber\n\\end{align}\nwhere $\\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}$ is \nthe maximum element of $(A, B)$-invariant subspaces contained in $\\mathcal{H}$, \nand $\\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})}$ is \nthe minimum element of $(C, A)$-invariant subspaces containing $\\mathcal{E}$. \nThese subspaces can be computed by the algorithms given in Appendix~A. \n\n\nOnce the solvability condition described above is satisfied, then we can \nexplicitly construct the controller matrices $(A_{\\scriptscriptstyle K}, \nB_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K}, D_{\\scriptscriptstyle K})$. \nThe following intersection and projection subspaces play a key role for this purpose; \nthat is, for a subspace $\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \n\\mathcal{X}_{{\\scriptscriptstyle E}}= \\mathcal{X} \\oplus \\mathcal{X}_{{\\scriptscriptstyle K}}$, \nlet us define\n\\begin{align*}\n \\mathcal{V}_{{\\scriptscriptstyle I}}:&= \n \\left\\{x \\in \\mathcal{X}~\\Bigg| \\left[\\begin{array}{c}\n x\\\\\n O\\\\\n \\end{array}\\right] \n \\in \\mathcal{V}_{{\\scriptscriptstyle E}} \\right\\}, ~ \\\\\n \\mathcal{V}_{{\\scriptscriptstyle P}}:&= \n \\left\\{x \\in \\mathcal{X}~\\Bigg| \\left[\\begin{array}{c}\n x\\\\ \n x_{{\\scriptscriptstyle K}}\\\\\n \\end{array}\\right] \n \\in \\mathcal{V}_{{\\scriptscriptstyle E}},~ \n \\exists {x}_{{\\scriptscriptstyle K}} \\in \n \\mathcal{X}_{{\\scriptscriptstyle K}} \\right\\}.\n\\end{align*}\nThen, the following theorem is obtained:\n\n\n{\\bf Theorem 2:}~ \nSuppose that $(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ is a $(C, A, B)$-pair. \nThen, there exist $F \\in \\mathcal{F}(\\mathcal{V}_{2})$, \n$G \\in \\mathcal{G}(\\mathcal{V}_{1})$,\nand $D_{\\scriptscriptstyle K} : \\mathcal{Y} \\rightarrow \\mathcal{U}$ \nsuch that $\\Kernel F_{0} \\supseteq \\mathcal{V}_{1}$ and \n$\\Image G_{0} \\subseteq \\mathcal{V}_{2}$ hold, where\n$F_{0}=F-D_{\\scriptscriptstyle K}C,~G_{0}=G-BD_{\\scriptscriptstyle K}$.\n\n\nMoreover, there exists $\\mathcal{X}_{{\\scriptscriptstyle K}}$ with \n${\\rm dim}\\,\\mathcal{X}_{{\\scriptscriptstyle K}}={\\rm dim}\\,\n\\mathcal{V}_{2} - {\\rm dim}\\,\\mathcal{V}_{1}$, and $A_{{\\scriptscriptstyle E}}$ \nhas an invariant subspace $\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \n\\mathcal{X}_{{\\scriptscriptstyle E}}$ such that $\\mathcal{V}_{1}=\\mathcal{V}_{{\\scriptscriptstyle I}}$ \nand $\\mathcal{V}_{2}=\\mathcal{V}_{{\\scriptscriptstyle P}}$. \nAlso, $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$ \nsatisfies \n\\begin{align}\n\\label{eq_DFC_characterize}\n &C_{\\scriptscriptstyle K}N=F_{0}, ~~ \\nonumber \\\\\n &B_{\\scriptscriptstyle K}=-NG_{0}, ~~ \\nonumber \\\\\n &A_{\\scriptscriptstyle K}N=N(A+BF_{0}+GC),\n\\end{align}\nwhere $N : \\mathcal{V}_{2} \\rightarrow \\mathcal{X}_{{\\scriptscriptstyle K}}$ \nis a linear map satisfying $\\Kernel N=\\mathcal{V}_{1}$.\n\n\nIn fact, under the condition given in Theorem 2, let us define the following \naugmented subspace $\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \\mathcal{X}_{{\\scriptscriptstyle E}}$:\n\\begin{align*}\n \\mathcal{V}_{{\\scriptscriptstyle E}} :=& \\left\\{\n \\left[\\begin{array}{c}\n x\\\\\n Nx\\\\\n \\end{array}\\right] \n \\Bigg|~ x \\in \\mathcal{V}_{2} \\right\\}.\n\\end{align*} \nThen, $\\mathcal{V}_{1}=\\mathcal{V}_{{\\scriptscriptstyle I}}$ and \n$\\mathcal{V}_{2}=\\mathcal{V}_{{\\scriptscriptstyle P}}$ \nhold, and we have \n\\begin{align*}\n A_{{\\scriptscriptstyle E}}\\left[\\begin{array}{c}\n x\\\\\n Nx\\\\\n \\end{array}\\right] \n &=\\left[\\begin{array}{cc}\n A+BD_{\\scriptscriptstyle K}C & BC_{\\scriptscriptstyle K}\\\\\n B_{\\scriptscriptstyle K}C & A_{\\scriptscriptstyle K}\\\\\n \\end{array}\\right]\n \\left[\\begin{array}{c}\n x\\\\\n Nx\\\\\n \\end{array}\\right] \\nonumber\\\\\n &=\\left[\\begin{array}{c}\n (A+BF)x\\\\\n N(A+BF)x\\\\\n \\end{array}\\right] \n \\in \\mathcal{V}_{{\\scriptscriptstyle E}},\n\\end{align*} \nimplying that $\\mathcal{V}_{{\\scriptscriptstyle E}}$ is actually $A_{{\\scriptscriptstyle E}}$-invariant. \nNow suppose that Theorem~1 holds, and let us take the $(C,A,B)$-pair \n$(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ satisfying Eq.~\\eqref{eq_DDP_condition}. \nThen, together with the above result \n($A_{{\\scriptscriptstyle E}}\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \\mathcal{V}_{{\\scriptscriptstyle E}}$), \nwe have $\\Image [E^{\\top}~O]^{\\top} \\subseteq \\mathcal{V}_{{\\scriptscriptstyle E}} \n\\subseteq \\Kernel [H~O]$. \nThis implies that $d(t)$ must be contained in the unobservable subspace \nwith respect to $z(t)$, and thus the disturbance decoupling is realized.\n\n\n\n\n\n\\subsection{Linear quantum systems}\\label{sec:2-2}\n\n\nHere we describe a general linear quantum system composed of $n$ bosonic \nsubsystems. \nThe $j$-th mode can be modeled as a harmonic oscillator with the canonical \nconjugate pairs (or quadratures) $\\hat{q}_{j}$ and $\\hat{p}_{j}$ satisfying the \ncanonical commutation relation (CCR) \n$\\hat{q}_{j}\\hat{p}_{k}-\\hat{p}_{k}\\hat{q}_{j}=i \\delta_{jk}$. \nLet us define the vector of quadratures as \n$\\hat{x}=[\\hat{q}_{1}, \\hat{p}_{1}, \\ldots, \\hat{q}_{n}, \\hat{p}_{n}]^{\\top}$. \nThen, the CCRs are summarized as \n\\begin{align*} \n &\\hat{x}\\hat{x}^{\\top}-(\\hat{x}\\hat{x}^{\\top})^{\\top} \n = i \\Sigma_{n}, ~~\\\\\n &\\Sigma_{n} \n = {\\rm diag}\\{\\Sigma, \\ldots, \\Sigma\\},~~~\n \\Sigma=\\left[\\begin{array}{cc}\n 0 & 1 \\\\\n -1 & 0 \\\\\n \\end{array}\\right]. \n\\end{align*}\nNote that $\\Sigma_{n}$ is a $2n \\times 2n$ block diagonal matrix. \nThe linear quantum system is an open system coupled to $m$ environment \nfields via the interaction Hamiltonian \n$\\hat{H}_{\\rm int}=i\\sum_{j=1}^m (\\hat{L}_j \\hat{A}_j^*-\\hat{L}_j^* \\hat{A}_j)$, \nwhere $\\hat{A}_j(t)$ is the field annihilation operator satisfying \n$\\hat{A}_j(t)\\hat{A}_k^{*}(t')-\\hat{A}_k^{*}(t')\\hat{A}_j(t)=\\delta_{jk}\\delta(t-t')$. \nAlso $\\hat{L}_j$ is given by $\\hat{L}_j=c_j^\\top\\hat{x}$ with \n$c_j\\in{\\mathbb C}^{2n}$. \nIn addition, the system is driven by the Hamiltonian \n$\\hat{H}=\\hat{x}^\\top R\\hat{x}\/2$ with \n$R=R^\\top \\in \\mathbb{R}^{2n \\times 2n}$. \nThen, the Heisenberg equation of $\\hat{x}$ is given by\n\\begin{align}\n \\frac{d\\hat{x}(t)}{dt}=A\\hat{x}(t)+\\sum_{j=1}^m B_j\\hat{W}_j(t), \n\\label{eq_LQS_dynamics}\n\\end{align}\nwhere $\\hat{W}_j(t)$ is defined by \n\\[\n \\hat{W}_j\n = \\left[\\begin{array}{cc}\n \\hat{Q}_{j} \\\\\n \\hat{P}_{j} \\\\\n \\end{array}\\right]\n = \\left[\\begin{array}{cc}\n (\\hat{A}_j+\\hat{A}_j^*)\/\\sqrt{2} \\\\\n (\\hat{A}_j-\\hat{A}_j^*)\/\\sqrt{2} i \\\\\n \\end{array}\\right].\n\\]\nThe matrices are given by \n$A=\\Sigma_n(R + \\sum_{j=1}^m C_j^\\top\\Sigma C_j\/2)$ and \n$B_j=\\Sigma_n C_j^\\top\\Sigma$ \nwith $C_j=\\sqrt{2}[\\Re(c_j), \\Im(c_j)]^\\top \\in \\mathbb{R}^{2 \\times 2n}$. \nAlso, the instantaneous change of the field operator $\\hat{W}_j(t)$ via the \nsystem-field coupling is given by \n\\begin{align}\n \\hat{W}_j^{\\rm out}(t)=C_j\\hat{x}(t)+\\hat{W}_j(t). \n\\label{eq_LQS_output}\n\\end{align}\nSummarizing, the linear quantum system is characterized by the dynamics \n\\eqref{eq_LQS_dynamics} and the output \\eqref{eq_LQS_output}, which are \nexactly of the same form as those in Eq.~\\eqref{eq_linearsystem} ($l=m$ in this case). \nHowever note that the system matrices have to satisfy the above-described \nspecial structure, which is equivalently converted to the following \n{\\it physical realizability condition} \\cite{James2008}: \n\\begin{equation}\n\\label{phys real condition}\n A\\Sigma_n+\\Sigma_n A^{\\top}\n + \\sum_{j=1}^m B_j \\Sigma B_j^{\\top}=O,~\n B_j=\\Sigma_n C_j^{\\top}\\Sigma. \n\\end{equation}\n\n\n\n\n\\subsection{Weak signal sensing, SQL, and BAE}\\label{sec:2-3}\n\n\nThe opto-mechanical oscillator illustrated in Fig.~\\ref{fig_opticalsensor} is a \nlinear quantum system, which serves as a sensor for a very weak signal. \nLet $\\hat{q}_{1}$ and $\\hat{p}_{1}$ be the oscillator's position and momentum \noperators, and $\\hat{a}_{2}=(\\hat{q}_{2}+i \\hat{p}_{2})\/\\sqrt{2}$ represents \nthe annihilation operator of the cavity mode. \nThe system Hamiltonian is given by \n$\\hat{H}=\\omega_{\\scriptscriptstyle m}(\\hat{q}_1^2+\\hat{p}_1^2)\/2\n-g\\hat{q}_1\\hat{q}_2$; that is, the oscillator's free evolution with resonant \nfrequency $\\omega_{\\scriptscriptstyle m}$ plus the linearized radiation \npressure interaction between the oscillator and the cavity field with coupling \nstrength $g$. \n\\begin{figure}[!t]\n \\begin{center}\n \\includegraphics[width=5.2cm,clip]{Figure\/Fig2_opticalsensor.eps}\n \\end{center} \n \\caption{Opto-mechanical system for weak signal sensing.}\n \\label{fig_opticalsensor}\n\\end{figure} \nThe system couples to an external probe field (thus $m=1$) via the coupling \noperator $\\hat{L}_1=\\sqrt{\\kappa}\\hat{a}_2$, with $\\kappa$ the coupling \nconstant between the cavity and probe fields. \nThe corresponding matrix $R$ and vector $c_1$ are then given by \n\\begin{align*}\n R=\\left[\\begin{array}{cccc}\n \\omega_{\\scriptscriptstyle m} &0&-g&0\\\\\n 0&\\omega_{\\scriptscriptstyle m} &0&0\\\\\n -g&0&0&0\\\\\n 0&0&0&0\\\\\n \\end{array}\\right],~~\n c_{1}=\\sqrt{\\frac{\\kappa}{2}}\n \\left[\\begin{array}{c}\n 0 \\\\ 0 \\\\ 1 \\\\ i \\\\\n \\end{array}\\right].\n\\end{align*}\nThe oscillator is driven by an unknown force $\\hat{f}(t)$ with coupling constant $\\gamma$ ; \nthen the vector of system variables $\\hat{x}=[\\hat{q}_1, \\hat{p}_1, \\hat{q}_2, \\hat{p}_2]^\\top$ satisfies \n\\begin{equation*}\n \\frac{d\\hat{x}}{dt}\n =A\\hat{x}+B_1\\hat{W}_1+b \\hat{f},~~\n \\hat{W}_{1}^{\\rm out}\n =C_1\\hat{x}+\\hat{W}_1,\n\\end{equation*}\nwhere\n\\begin{align}\n A&=\\left[\\begin{array}{cccc}\n 0&\\omega_{\\scriptscriptstyle m}&0&0\\\\\n -\\omega_{\\scriptscriptstyle m} &0 &g&0\\\\\n 0&0&-\\kappa\/2 &0\\\\\n g&0&0&-\\kappa\/2 \\\\\n \\end{array}\\right], \\,\n b=\\sqrt{\\gamma}\\left[\\begin{array}{c}\n 0 \\\\\n 1 \\\\\n 0 \\\\\n 0 \\\\\n \\end{array}\\right], \\nonumber \\\\\n C_1&=-B_1^{\\top}\n =\\left[\\begin{array}{cccc}\n 0 & 0 & \\sqrt{\\kappa} & 0\\\\\n 0 & 0 & 0 & \\sqrt{\\kappa}\\\\\n \\end{array}\\right], \\nonumber \\\\\n \\hat{W}_{1}\n &=[\\hat{Q}_{1}, \\hat{P}_{1}]^{\\top}, ~~\n \\hat{W}_{1}^{\\rm out}\n =[\\hat{Q}_{1}^{\\rm out}, \\hat{P}_{1}^{\\rm out}]^{\\top}.\n\\label{eq_matrix_originalplant}\n\\end{align}\nNote that we are in the rotating frame at the frequency of the probe field. \nThese equations indicate that the information about $\\hat{f}$ can be extracted \nby measuring $\\hat{P}_1^{\\rm out}$ by a homodyne detector. \nActually the measurement output in the Laplace domain is given by\n\\begin{align}\n \\hat{P}_{1}^{\\rm out}(s)\n =\\Xi_{f}(s) \\hat{f}(s)+\\Xi_{Q}(s) \\hat{Q}_{1}(s)\n +\\Xi_{P}(s) \\hat{P}_{1}(s),\n\\label{eq_measurement_output}\n\\end{align}\nwhere $\\Xi_{f}$, $\\Xi_{Q}$, and $\\Xi_{P}$ are transfer functions given by \n\\begin{align*}\n\\hspace{-0.8cm}\n \\Xi_{f}(s)&=\\frac{g\\omega_{\\scriptscriptstyle m}\\sqrt{\\gamma\\kappa}}\n {(s^2+\\omega_{\\scriptscriptstyle m}^2)(s+\\kappa\/2)}, ~~\\\\\n \\Xi_{Q}(s)&=-\\frac{g^2 \\omega_{\\scriptscriptstyle m}\\kappa}\n {(s^2+\\omega_{\\scriptscriptstyle m}^2)(s+\\kappa\/2)^2}, ~~\n \\Xi_{P}(s)=\\frac{s-\\kappa\/2}{s+\\kappa\/2}. \n\\end{align*}\nThus, $\\hat{P}_1^{\\rm out}$ certainly contains $\\hat{f}$. \nNote however that it is subjected to two noises. \nThe first one, $\\hat{Q}_{1}$, is the back-action noise, which is due to the \ninteraction between the oscillator and the cavity. \nThe second one, $\\hat{P}_{1}$, is the shot noise, which inevitably appears. \nNow, the normalized output is given by\n\\[\n y_{1}(s)\n =\\frac{\\hat{P}_{1}^{\\rm out}(s)}{\\Xi_{f}(s)}\n =\\hat{f}(s) + \\frac{\\Xi_{Q}(s)}{\\Xi_{f}(s)}\\hat{Q}_{1}(s)\n + \\frac{\\Xi_{P}(s)}{\\Xi_{f}(s)}\\hat{P}_{1}(s),\n\\]\nand the normalized noise power spectral density of $y_{1}$ in the Fourier domain \n$(s=i\\omega)$ is calculated as follows:\n\\begin{align*}\n &S(\\omega) =\\langle |y_{1}-\\hat{f}|^2 \\rangle\n =\\left| \\frac{\\Xi_{Q}}{\\Xi_{f}} \\right| ^2 \n \\langle|\\hat{Q}_1|^2 \\rangle \n + \\left| \\frac{\\Xi_{P}}{\\Xi_{f}} \\right| ^2 \n \\langle |\\hat{P}_1|^2 \\rangle \\\\\n &\\geq 2\\sqrt{\\frac{|\\Xi_{Q}|^2|\\Xi_{P}|^2}{|\\Xi_{f}|^4} \n \\langle |\\hat{Q}_1|^2 \\rangle \n \\langle |\\hat{P}_1|^2 \\rangle }\n \\geq\\frac{| \\omega^2-\\omega_{\\scriptscriptstyle m}^2 | }\n {\\gamma\\omega_{\\scriptscriptstyle m}}\n =S_{\\scriptscriptstyle \\rm SQL}(\\omega).\n\\end{align*}\nThe lower bound is called the SQL. \nNote that the last inequality is due to the Heisenberg uncertainty relation of the normalized noise power, i.e.,\n$\\langle|\\hat{Q}_1|^2 \\rangle \\langle|\\hat{P}_1|^2 \\rangle \\geq 1\/4$. \nHence, the essential reason why SQL appears is that $\\hat{P}_{1}^{\\rm out}$ \ncontains both the back-action noise $\\hat{Q}_1$ and the shot noise $\\hat{P}_1$. \nTherefore, toward the high-precision detection of $\\hat{f}$, we need BAE; \nthat is, the system structure should be modified by some means so that the \nback-action noise is completely evaded in the output signal (note that the shot noise can never be evaded). \nThe condition for BAE can be expressed in terms of the transfer function \nas follows \\cite{Tsang2010, YamamotoCFvsMF2014}; \ni.e., for the modified (controlled) sensor, the transfer function from \nthe back-action noise to the measurement output must satisfy \n\\begin{align}\n \\Xi_{Q}(s)=0, ~~~\\forall s\n\\label{eq_perfectBAE1}.\n\\end{align}\nEquivalently, $\\hat{P}_{1}^{\\rm out}$ contains only the shot noise $\\hat{P}_{1}$; \nhence, in this case the signal to noise ratio can be further improved by \ninjecting a $\\hat{P}_{1}$-squeezed (meaning $\\langle|\\hat{P}_{1}|^2 \\rangle < 1\/2$) \nprobe field into the system. \n\n\n\n\n\n\\section{Coherent feedback control for back-action evasion}\\label{sec:3}\n\n\n\n\n\\subsection{Coherent and measurement-based feedback control}\\label{sec:3-1}\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=7.7cm,clip]{Figure\/Fig3_MFvsCF.eps}\n\\caption{General configurations of feedback control for a given plant quantum system: \n(a) measurement-based feedback and (b) coherent feedback.}\n\\label{fig_MFvsCF}\n\\end{center}\n\\end{figure}\n\n\nThere are two schemes for controlling a quantum system via feedback. \nThe first one is the {\\it measurement-based feedback} \n\\cite{Belavkin1999, Bouten2009, WisemanBook, KurtBook} illustrated in \nFig.~\\ref{fig_MFvsCF}~(a). \nIn this scheme, we measure the output fields and feed the measurement \nresults back to control the plant system. \nOn the other hand, in the {\\it coherent feedback} scheme \n\\cite{James2008,Wiseman1994, Yanagisawa-2003,Mabuchi2008, Iida2012} \nshown in Fig.~\\ref{fig_MFvsCF}~(b), the feedback loop dose not contain \nany measurement component and the plant system is controlled by another quantum system. \nRecently we find several works comparing the performance of these two schemes \n\\cite{Wiseman1994, NurdinLQG2009, Hamerly2012, Jacobs2014, Devoret2016}. \nIn particular, it was shown in \\cite{YamamotoCFvsMF2014} that there are \nsome control tasks that cannot be achieved by any measurement-based feedback \nbut can be done by a coherent one. \nMore specifically, those tasks are realizing BAE measurement, generating \na quantum non-demolished variable, and generating a decoherence-free \nsubsystem; \nin our case, of course, the first one is crucial. \nHence, here we aim to develop a theory for designing a coherent feedback \ncontroller such that the whole controlled system accomplishes BAE. \n\n\n\n\n\\subsection{Coherent feedback for BAE}\\label{sec:3-2}\n\n\nAs discussed in Section \\ref{sec:2-1}, the geometric control theory for \ndisturbance decoupling problem is formulated for the controlled system \nwith special structure \\eqref{eq_extended_system}; \nin particular, the coefficient matrix of the disturbance $d(t)$ is of the form $[E^\\top, O]^\\top$ \nand that of the state vector in the output $z(t)$ is $[H, O]$. \nHere we consider a class of coherent feedback configuration such that \nthe whole closed-loop system dynamics has this structure, in order for \nthe geometric control theory to be directly applicable. \n\n\n\nFirst, for the plant system given by Eqs.~\\eqref{eq_LQS_dynamics} and \n\\eqref{eq_LQS_output}, we assume that the system couples to all the probe fields \nin the same way; i.e., \n\\begin{align}\n B_{j}=B~~~\\forall j. \n\\label{eq_plant_symmetry}\n\\end{align}\nThis immediately leads to $C_{j}=C~\\forall j$. \nNext, as the controller, we take the following special linear quantum system \nwith $(m-1)$ input-output fields: \n\\begin{align}\n\\label{eq_dynamical_controller} \n \\frac{d\\hat{x}_{\\scriptscriptstyle K}}{dt}\n &=A_{\\scriptscriptstyle K} \\hat{x}_{\\scriptscriptstyle K}\n +\\sum_{j=1}^{m-1} B_{{\\scriptscriptstyle K}} \\hat{w}_{j}, \\nonumber\\\\\n \\hat{w}_{j}^{\\rm out}\n &=C_{{\\scriptscriptstyle K}} \\hat{x}_{\\scriptscriptstyle K}\n +\\hat{w}_{j}~~~(j=1,\\,2,\\, \\ldots, m-1),\n\\end{align}\nwhere the matrices $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, \nC_{\\scriptscriptstyle K})$ satisfy the physical realizability condition \n\\eqref{phys real condition}. \nNote that, corresponding to the plant structure, we assumed that the controller \ncouples to all the fields in the same way, specified by $C_{\\scriptscriptstyle K}$. \nHere we emphasize that the number of channels, $m$, should be as small as \npossible from a viewpoint of implementation; \nhence in this paper let us consider the case $m=3$. \nNow, we consider the coherent feedback connection illustrated in \nFig.~\\ref{fig_3I3O_general}, i.e., \n\\begin{align*}\n &\\hat{w}_{1}=S_{1}\\hat{W}_{1}^{\\rm out},~~ \n \\hat{w}_{2}=S_{2}\\hat{W}_{2}^{\\rm out}, ~~ \\\\\n &\\hat{W}_{2}=T_1\\hat{w}_{1}^{\\rm out},~~ \n \\hat{W}_{3}=T_{2}\\hat{w}_{2}^{\\rm out},\n\\end{align*}\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=5cm,clip]{Figure\/Fig4_3I3O_general.eps}\n\\caption{Coherent feedback control of the 3 input-output plant system via the 2 input-output controller.}\n\\label{fig_3I3O_general}\n\\end{center}\n\\end{figure}\nwhere $S_{j}$ and $T_{j}$ are $2 \\times 2$ unitary matrices representing \nthe scattering process of the fields; \nrecall that the scattering process $\\hat{A}^{\\rm out}=e^{i\\theta}\\hat{A}$ \nwith $\\theta \\in \\mathbb{R}$ the phase shift can be represented in the \nquadrature form as \n\\begin{equation*}\n \\left[\\begin{array}{cc}\n \\hat{Q}^{\\rm out} \\\\\n \\hat{P}^{\\rm out} \\\\\n \\end{array}\\right]\n = S(\\theta)\n \\left[\\begin{array}{cc}\n \\hat{Q} \\\\\n \\hat{P} \\\\\n \\end{array}\\right]\n =\\left[\\begin{array}{cc}\n \\cos \\theta &-\\sin \\theta \\\\\n \\sin \\theta &\\cos \\theta \\\\\n \\end{array}\\right]\n \\left[\\begin{array}{cc}\n \\hat{Q} \\\\\n \\hat{P} \\\\\n \\end{array}\\right].\n\\label{eq_phaseshifter}\n\\end{equation*}\n\n\n\nCombining the above equations, we find that the whole closed-loop system \nwith the augmented variable \n$\\hat{x}_{{\\scriptscriptstyle E}}=[\\hat{x}^{\\top},\\hat{x}_{\\scriptscriptstyle K}^{\\top}]^{\\top}$ \nis given by\n\\begin{align}\n\\label{eq_3I3Oclosed2}\n \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n &=A_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n +B_{{\\scriptscriptstyle E}}\\hat{W}_{1}+b_{{\\scriptscriptstyle E}}\\hat{f}, \\nonumber\\\\\n \\hat{W}_{3}^{\\rm out}\n &=C_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n +D_{{\\scriptscriptstyle E}}\\hat{W}_{1},\n\\end{align}\nwhere\n\\begin{align*}\nA_{{\\scriptscriptstyle E}}\n&=\\left[\\begin{array}{c}\nA+B\\{ T_{1}S_{1}+T_{2}S_{2}(T_{1}S_{1}+I_{2})\\} C \\\\ [1.0ex]\nB_{\\scriptscriptstyle K}\\{ (I_{2}+S_{2}T_{1})S_{1}+S_{2}\\} C \n\\end{array}\\right. \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~\\left. \\begin{array}{c}\n B\\{ T_{1}+T_{2}(I_{2}+S_{2}T_{1})\\} C_{\\scriptscriptstyle K} \\\\ [1.0ex]\n A_{\\scriptscriptstyle K}+B_{\\scriptscriptstyle K}S_{2}T_{1}C_{\\scriptscriptstyle K}\n\\end{array}\\right]. \\\\\nB_{{\\scriptscriptstyle E}}\n&=\\left[\\begin{array}{c}\n B(I_{2}+T_{1}S_{1}+T_{2}S_{2}T_{1}S_{1}) \\\\ [1.0ex]\n B_{{\\scriptscriptstyle K}}(I_{2}+S_{2}T_{1})S_{1} \\\\\n \\end{array}\\right] \\\\\nC_{{\\scriptscriptstyle E}}\n&=\\left[\\begin{array}{cc}\n (T_{2}S_{2}T_{1}S_{1}+T_{2}S_{2}+I_{2})C ~\n & ~T_{2}(S_{2}T_{1}+I_{2})C_{\\scriptscriptstyle K}\\\\\n \\end{array}\\right] \\\\\nD_{{\\scriptscriptstyle E}}&=T_{2}S_{2}T_{1}S_{1}, ~~\nb_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n b^{\\top} & O\\\\\n \\end{array}\\right]^{\\top}.\n\\end{align*}\nTherefore, the desired system structure of the form \\eqref{eq_extended_system} \nis realized if we take \n\\begin{align}\n S_{2}T_{1}=-I_{2}.\n\\label{eq_special_structure1}\n\\end{align}\nIn addition, it is required that the back-action noise $\\hat{Q}_{1}$ \ndose not appear directly in $\\hat{P}_{3}^{\\rm out}$, which can be realized by \ntaking \n\\begin{align}\n D_{{\\scriptscriptstyle E}}=-T_{2}S_{1}=\\pm I_{2}.\n\\label{eq_special_structure2}\n\\end{align}\nHere we set $S_j$ and $T_j$ to be the $\\pi\/2$-phase shifter \n(see Fig.~\\ref{fig_3I3OBAE}) to satisfy the above conditions \\eqref{eq_special_structure1} and \n\\eqref{eq_special_structure2};\n\\begin{align}\n\\label{eq_phaseshifter_matching}\n S_{j}=T_{j}=S\n =\\left[\\begin{array}{cc}\n 0 & -1 \\\\\n 1 & 0 \\\\\n \\end{array}\\right] ~~(j=1,2). \n\\end{align}\nAs a consequence, we end up with \n\\begin{align}\n &A_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n A-BC & BSC_{\\scriptscriptstyle K} \\\\ \n B_{\\scriptscriptstyle K}SC & \n A_{\\scriptscriptstyle K}-B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K} \n \\end{array}\\right], ~~\n B_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{c}\n B\\\\\n O \\\\\n \\end{array}\\right], \n\\nonumber \\\\\n &\n C_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n C & O \\\\\n \\end{array}\\right], ~~\n D_{{\\scriptscriptstyle E}}=I_{2}, ~~\n b_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n b^{\\top} & O \\\\\n \\end{array}\\right]^{\\top}.\n\\label{extended_matrices}\n\\end{align}\nThis is certainly of the form \\eqref{eq_extended_system} with \n$D_{\\scriptscriptstyle K}=-I_2$. \nHence, we can now directly apply the geometric control theory to design \na coherent feedback controller achieving BAE; \nthat is, our aim is to find $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, \nC_{\\scriptscriptstyle K})$ such that, for the closed-loop system \n\\eqref{eq_3I3Oclosed2}, the back-action noise $\\hat{Q}_{1}$ (the first element \nof $\\hat W_1$) does not appear in the measurement output $\\hat P_3^{\\rm out}$ \n(the second element of $\\hat W_3^{\\rm out}$). \nNote that those matrices must satisfy the physical \nrealizability condition \\eqref{phys real condition}, and thus they cannot be \nfreely chosen. \nWe need to take into account this additional constraint when applying \nthe geometric control theory to determine the controller matrices.\n\n\n\n\n\n\\subsection{Coherent feedback realization of BAE in the opto-mechanical system}\\label{sec:3-3}\n\n\nHere we apply the coherent feedback scheme elaborated in Section~\\ref{sec:3-2} to \nthe opto-mechanical system studied in Section~\\ref{sec:2-3}. \nThe goal is, as mentioned before, to determine the controller matrices \n$(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$ \nsuch that the closed-loop system achieves BAE. \nHere, we provide a step-by-step procedure to solve this problem; \nthe relationships of the class of controllers determined in each step is depicted \nin Fig.~\\ref{fig_controllerconstraint}. \n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=7cm,clip]{Figure\/Fig5_3I3OBAE.eps}\n\\caption{\nCoherent feedback controlled system composed of the opto-mechanical oscillator, for realizing BAE. \nThe triangle represents the $\\pi\/2$-phase shifter corresponding to Eq.~\\eqref{eq_phaseshifter_matching}. \n}\n\\label{fig_3I3OBAE}\n\\end{center}\n\\end{figure}\n\n{\\bf (i)} \nFirst, to apply the geometric control theory developed above, we need to modify \nthe plant system so that it is a 3 input-output linear quantum system; \nhere we consider the plant composed of a mechanical oscillator and a 3-ports optical cavity, \nshown in Fig.~\\ref{fig_3I3OBAE}. \nAs assumed before, those ports have the same coupling constant $\\kappa$. \nIn this case the matrix $A$ given in Eq.~\\eqref{eq_matrix_originalplant} is \nreplaced by \n\\begin{align}\n A=\\left[\\begin{array}{cccc}\n 0&\\omega_{\\scriptscriptstyle m} &0&0\\\\\n -\\omega_{\\scriptscriptstyle m}& 0&g &0\\\\\n 0&0&-3\\kappa\/2 &0\\\\\n g &0&0&-3\\kappa\/2 \\\\\n \\end{array}\\right] .\n\\nonumber \n\\end{align} \nNow we focus only on the back-action noise $\\hat{Q}_{1}$ and the measurement output \n$\\hat{P}_{3}^{\\rm out}$; hence the closed-loop system \\eqref{eq_3I3Oclosed2} and\n\\eqref{extended_matrices}, which ignores the shot noise term in the dynamical \nequation, is given by \n\\begin{align*}\n\\hspace{-0.8cm}\n \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n &=\\left[\\begin{array}{cc}\n A-BC & BSC_{\\scriptscriptstyle K} \\\\ \n B_{\\scriptscriptstyle K}SC & \n A_{\\scriptscriptstyle K}\n -B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K} \n \\end{array}\\right]\n \\hat{x}_{{\\scriptscriptstyle E}} \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \\left[\\begin{array}{c}\n E \\\\\n O \\\\\n \\end{array}\\right] \\hat{Q}_{1}\n + \\left[\\begin{array}{c}\n b\\\\\n O\\\\\n \\end{array}\\right] \\hat{f}, \\\\\n \\hat{P}_{3}^{\\rm out}\n &=\\left[\\begin{array}{cc}\n H & O \\\\\n \\end{array}\\right] \\hat{x}_{{\\scriptscriptstyle E}}\n +\\hat{P}_{1}, \n\\end{align*}\nwhere $B=B_1$, $C=C_1$, and $b$ are given in \nEq.~\\eqref{eq_matrix_originalplant}, and \n\\begin{align*}\n E=-\\sqrt{\\kappa}\\left[\\begin{array}{cccc}\n 0&0&1&0 \\\\\n \\end{array}\\right]^{\\top},~~~ \n H=\\sqrt{\\kappa}\\left[\\begin{array}{cccc}\n 0&0&0&1 \\\\\n \\end{array}\\right].\n\\end{align*}\nThis system is certainly of the form \\eqref{eq_extended_system}, where now $D_{\\scriptscriptstyle K}=-I_2$.\n\n\n\n{\\bf (ii)} \nIn the next step we apply Theorem~1 to check if there exists a feedback \ncontroller such that the above closed-loop system achieves BAE; \nrecall that the necessary and sufficient condition is Eq.~\\eqref{eq_DDP_condition}, \ni.e., $\\mathcal{E} \\subseteq \\mathcal{V}_{1} \\subseteq \\mathcal{V}_{2} \n\\subseteq \\mathcal{H}$, where now \n\\begin{align*}\n \\mathcal{E}&=\\Image E={\\rm span} \\left\\{\n \\left[\\begin{array}{c}\n 0\\\\\n 0\\\\\n 1\\\\\n 0\\\\\n \\end{array}\\right] \\right\\}, \\\\\n \\mathcal{H}&=\\Kernel H={\\rm span} \\left\\{\n \\left[\\begin{array}{c}\n 1\\\\\n 0\\\\\n 0\\\\\n 0\\\\\n \\end{array}\\right] ,\n \\left[\\begin{array}{c}\n 0\\\\\n 1\\\\\n 0\\\\\n 0\\\\\n \\end{array}\\right] ,\n \\left[\\begin{array}{c}\n 0\\\\\n 0\\\\\n 1\\\\\n 0\\\\\n \\end{array}\\right] \\right\\}. \n\\end{align*}\nTo check if this solvability condition is satisfied, we use Corollary 1; \nfrom\n$\\mathcal{E} \\cap \\mathcal{C}=\\Image E \\cap \\Kernel C =\\phi$ and \n$\\mathcal{H} \\oplus \\mathcal{B} = \\Kernel H \\oplus \\Image B =\\mathbb{R}^{4}$,\nthe algorithms given in Appendix~A yield \n\\begin{align}\n \\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})}\n =\\mathcal{E},~~~\n \\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}\n =\\mathcal{H}, \n\\label{eq_CABpairs}\n\\end{align}\nimplying that the condition in Corollary 1, i.e., \n$\\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})} \\subseteq \n\\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}$, is satisfied. \nThus, we now see that the BAE problem is solvable, as long as there is no \nconstraint on the controller parameters. \n\n\n\nNext we aim to determine the controller matrices \n$(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$, \nusing Theorem~2. First we set \n$\\mathcal{V}_{1}=\\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})} = \\mathcal{E}$ and \n$\\mathcal{V}_{2}=\\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}= \\mathcal{H}$;\nnote that $(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ is a $(C, A, B)$-pair.\nThen, from Theorem 2, there exists a feedback controller with dimension\n${\\rm dim}\\,\\mathcal{X}_{\\scriptscriptstyle K}={\\rm dim}\\,\\mathcal{V}_{2}-{\\rm dim}\\,\\mathcal{V}_{1}=2$. \nMoreover, noting again that $D_{\\scriptscriptstyle K}=-I_2$, there exist matrices \n$F \\in \\mathcal{F}(\\mathcal{V}_{2})$, $G \\in \\mathcal{G}(\\mathcal{V}_{1})$, \nand $N$ \nsuch that\n\\begin{align*}\n \\Kernel F_{0}&=\\Kernel (F+C) \\supseteq \\mathcal{V}_{1}, \\\\\n \\Image G_{0}&=\\Image (G+B) \\subseteq \\mathcal{V}_{2}, ~~\n \\Kernel N=\\mathcal{V}_{1}. \n\\end{align*}\nThese conditions lead to\n\\begin{align*}\n\\hspace{-0.7cm}\n F&=\\left[\\begin{array}{cccc}\n f_{\\scriptscriptstyle 11} & f_{\\scriptscriptstyle 12} \n & -\\sqrt{\\kappa} & f_{\\scriptscriptstyle 14} \\\\\n \\frac{g}{\\sqrt{\\kappa}}&0&0&f_{\\scriptscriptstyle 24} \\\\\n \\end{array}\\right], ~~\n G=\\left[\\begin{array}{cc}\n 0&g_{\\scriptscriptstyle 12}\\\\\n -\\frac{g}{\\sqrt{\\kappa}}&g_{\\scriptscriptstyle 22}\\\\\n g_{\\scriptscriptstyle 31}&g_{\\scriptscriptstyle 32}\\\\\n 0&\\sqrt{\\kappa}\\\\\n \\end{array}\\right], \\\\\n N&=\\left[\\begin{array}{cccc}\n n_{\\scriptscriptstyle 11} & n_{\\scriptscriptstyle 12} \n & 0 & n_{\\scriptscriptstyle 14} \\\\\n n_{\\scriptscriptstyle 21} & n_{\\scriptscriptstyle 22}\n & 0 & n_{\\scriptscriptstyle 24}\\\\\n \\end{array}\\right], \n\\end{align*}\nwhere $f_{ij}, g_{ij}$, and $ n_{ij}$ are free parameters. Then the controller matrices \n$(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$ \ncan be identified by Eq.~\\eqref{eq_DFC_characterize} with the above matrices $(F, G, N)$;\nspecifically, by substituting $C_{\\scriptscriptstyle K} \\to SC_{\\scriptscriptstyle K}$, \n$B_{\\scriptscriptstyle K} \\to B_{\\scriptscriptstyle K}S$, and \n$A_{\\scriptscriptstyle K} \\to A_{\\scriptscriptstyle K}-B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K}$\nin Eq.~\\eqref{eq_DFC_characterize}, we have \n\\begin{align*}\n &SC_{\\scriptscriptstyle K}N=F+C,\\\\\n &B_{\\scriptscriptstyle K}S=-N(G+B), \\\\\n &(A_{\\scriptscriptstyle K}\n -B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K})N \n = N(A+BF_{0}+GC),\n\\end{align*}\nwhich yield \n\\begin{align}\n\\label{controller matrices example 1}\n &A_{\\scriptscriptstyle K}=N(A+BF_{0}+GC+G_{0}F_{0})N^{+}, \\nonumber \\\\\n &B_{\\scriptscriptstyle K}=-NG_{0}\\Sigma, \\nonumber \\\\\n &C_{\\scriptscriptstyle K}=\\Sigma F_{0}N^{+}, \n\\end{align}\nwhere $N^{+}$ is the right inverse to $N$, i.e., $NN^{+}=I_{2}$.\n\n\n{\\bf (iii)}\nNote again that the controller \\eqref{eq_dynamical_controller} has to satisfy \nthe physical realizability condition \\eqref{phys real condition}, which is now \n$A_{\\scriptscriptstyle K}\\Sigma+\\Sigma A_{\\scriptscriptstyle K}^{\\top}\n+2B_{\\scriptscriptstyle K}\\Sigma B_{\\scriptscriptstyle K}^{\\top}=O$ and \n$B_{\\scriptscriptstyle K}=\\Sigma C_{\\scriptscriptstyle K}^{\\top}\\Sigma$. \nThese constraints are represented in terms of the parameters as follows: \\vspace{-1mm}\n\\begin{align}\n\\label{controller parameter constraint}\n&f_{\\scriptscriptstyle 12}=-g_{\\scriptscriptstyle 12},~~\nf_{\\scriptscriptstyle 11}=g_{\\scriptscriptstyle 22},~~\nn_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 22}-n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 21}=-1,~ \n\\nonumber \\\\\n&f_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 1}=f_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 2}-f_{\\scriptscriptstyle 14},~~\nf_{\\scriptscriptstyle 24}+\\sqrt{\\kappa}=\\frac{g}{\\sqrt{\\kappa}}n_{\\scriptscriptstyle 2}, \n\\nonumber \\\\\n& \\hspace{-0.15cm}\n\\left( \\frac{3}{2}\\kappa+\\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right) n_{\\scriptscriptstyle 1}+\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 2}\n=-\\sqrt{\\kappa}f_{\\scriptscriptstyle 11}, \n\\nonumber \\\\\n&\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 1} - \\left ( \\frac{3}{2}\\kappa + \\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right )n_{\\scriptscriptstyle 2}\n=\\sqrt{\\kappa}f_{\\scriptscriptstyle 12}, \n\\end{align}\nwhere \n$n_{\\scriptscriptstyle 1}=n_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 21}$ and \n$n_{\\scriptscriptstyle 2}=n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 22}$. \nThis is one of our main results; \nthe linear controller \\eqref{eq_dynamical_controller} achieving BAE \nfor the opto-mechanical oscillator can be fully parametrized by \nEq.~\\eqref{controller matrices example 1} satisfying the condition \n\\eqref{controller parameter constraint}. \nWe emphasize that this full parametrization of the controller can be \nobtained thanks to the general problem formulation based on the geometric \ncontrol theory. \n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=6.5cm,clip]{Figure\/Fig6_controllerconstraint.eps}\n\\caption{\nThe set of controllers satisfying the condition in each step. For the controller to be a quantum system, \nit must be included in the set (iii). In the set (iv), all the controllers are equivalent up to the phase shift. \n}\n\\label{fig_controllerconstraint}\n\\end{center}\n\\end{figure}\n{\\bf (iv)}\nIn practice, of course, we need to determine a concrete set of parameters to construct the controller. \nEspecially here let us consider a {\\it passive system}; \nthis is a static quantum system such as an empty optical cavity. \nThe main reason for choosing a passive system rather than a non-passive \n(or {\\it active}) one such as an optical parametric oscillator is that, due to the \nexternal pumping energy, the latter could become fragile and also its physical \nimplementation must be more involved compared to a passive system \\cite{Walls2008}. \nNow the condition for the system \n$(A_{\\scriptscriptstyle K}, \\,B_{\\scriptscriptstyle K}, \\,C_{\\scriptscriptstyle K})$ \nto be passive is given by \n$\\Sigma A_{\\scriptscriptstyle K}\\Sigma=-A_{\\scriptscriptstyle K}$ and \n$\\Sigma B_{\\scriptscriptstyle K}\\Sigma=-B_{\\scriptscriptstyle K}$; \nthe general result of this fact is given in Theorem~3 in Appendix~B. \nFrom these conditions, the system parameters are imposed to satisfy, in addition \nto Eq.~\\eqref{controller parameter constraint}, the following equalities: \n\\begin{align}\n\\label{passive para cond}\n f_{\\scriptscriptstyle 12}=\\frac{g}{\\sqrt{\\kappa}}, ~~\n f_{\\scriptscriptstyle 11}=0, ~~ \n n_{\\scriptscriptstyle 11}=-n_{\\scriptscriptstyle 22}, ~~ \n n_{\\scriptscriptstyle 12}=n_{\\scriptscriptstyle 21}.\n\\end{align}\nThere is still some freedom in determining $n_{ij}$, which however corresponds \nto simply the phase shift at the input-output ports of the controller, as indicated \nfrom Eq.~\\eqref{controller matrices example 1}. \nThus, the passive controller achieving BAE in this example is unique up to the \nphase shift. \nHere particularly we chose $n_{\\scriptscriptstyle 11}=1$ and \n$n_{\\scriptscriptstyle 12}=0$. \nThen the controller matrices \\eqref{controller matrices example 1} satisfying \nEqs.~\\eqref{controller parameter constraint} and \\eqref{passive para cond} \nare determined as \n\\begin{align*}\n A_{\\scriptscriptstyle K}\n =\\left[\\begin{array}{cc}\n -\\frac{~g^2}{\\kappa} &-\\omega_{\\scriptscriptstyle m} \\\\\n \\omega_{\\scriptscriptstyle m} & -\\frac{~g^2}{\\kappa} \\\\\n \\end{array}\\right],~~\n C_{\\scriptscriptstyle K}=-B_{\\scriptscriptstyle K}^{\\top}\n =\\left[\\begin{array}{cc}\n \\frac{g}{\\sqrt{\\kappa}} & 0 \\\\\n 0 & \\frac{g}{\\sqrt{\\kappa}} \\\\\n \\end{array}\\right]. \n\\end{align*}\nAs illustrated in Fig.~\\ref{fig_3I3OBAE}, the controller specified by these \nmatrices can be realized as a single-mode, 2-inputs and 2-outputs optical cavity \nwith decay rate $g^2\/\\kappa$ and detuning $-\\omega_{\\scriptscriptstyle m}$. \nIn other words, if we take the cavity with the following Hamiltonian and \nthe coupling operator ($\\hat{a}_{3}=(\\hat{q}_{3}+i \\hat{p}_{3})\/\\sqrt{2}$ is the \ncavity mode) \n\\begin{align}\n \\hat{H}_{\\scriptscriptstyle K}\n &=\\Delta \\hat{a}^{\\ast}_{3}\\hat{a}_{3}\n =\\frac{\\Delta}{2}(\\hat{q}_{3}^2+\\hat{p}_{3}^2), \\nonumber \\\\\n \\hat{L}_{\\scriptscriptstyle K}\n &=\\sqrt{\\kappa_{\\scriptscriptstyle K}}\\hat{a}_{3}\n =\\sqrt{\\frac{\\kappa_{\\scriptscriptstyle K}}{2}}\n (\\hat{q}_{3}+i \\hat{p}_{3}), \n\\label{eq_controller_H}\n\\end{align}\nthen to satisfy the BAE condition the controller parameters \n$(\\Delta, \\kappa_{\\scriptscriptstyle K})$ must satisfy \n\\begin{equation}\n\\label{perfect BAE condition example}\n \\Delta=-\\omega_{\\scriptscriptstyle m}, ~~~\n \\kappa_{\\scriptscriptstyle K}=g^2\/\\kappa. \n\\end{equation}\nSummarizing, the above-designed sensing system composed of the opto-mechanical \noscillator (plant) and the optical cavity (controller), which are combined via coherent \nfeedback, satisfies the BAE condition. \nHence, it can work as a high-precision detector of the force $\\hat{f}$ below the SQL, \nparticularly when the $\\hat P_1$-squeezed probe input field is used; \nthis fact will be demonstrated in Section~\\ref{sec:5}. \n\n\n\n\n\n\\section{Direct interaction scheme}\\label{sec:4}\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=8cm,clip]{Figure\/Fig7_directinteraction.eps}\n\\caption{(a) General configuration of direct interaction scheme. \n(b) Physical implementation of the passive direct interaction controller for the \nopto-mechanical oscillator.}\n\\label{fig_Mankei_scheme}\n\\end{center}\n\\end{figure}\n\n\nIn this section, we study another control scheme for achieving BAE. \nAs illustrated in Fig. \\ref{fig_Mankei_scheme}~(a), the controller in this case \nis directly connected to the plant, not through a coherent feedback; \nhence this scheme is called the {\\it direct interaction}. \nThe controller is characterized by the following two Hamiltonians: \n\\begin{align} \n\\label{eq_direct_intHamiltonian}\n \\hat{H}_{\\scriptscriptstyle K}\n =\\frac{1}{2}\\hat{x}_{\\scriptscriptstyle K}^{\\top}\n R_{\\scriptscriptstyle K}\\hat{x}_{\\scriptscriptstyle K}, ~\n \\hat{H}_{\\rm int}\n =\\frac{1}{2}(\\hat{x}^{\\top}R_1\\hat{x}_{\\scriptscriptstyle K}\n +\\hat{x}_{\\scriptscriptstyle K}^{\\top}R_2\\hat{x}),\n\\end{align}\nwhere $\\hat{x}_{\\scriptscriptstyle K}=[\\hat{q}^{\\scriptscriptstyle \\prime}_{1}, \n\\hat{p}^{\\scriptscriptstyle \\prime}_{1}, \\ldots , \n\\hat{q}^{\\scriptscriptstyle \\prime}_{n_{\\scriptscriptstyle k}}, \n\\hat{p}^{\\scriptscriptstyle \\prime}_{n_{\\scriptscriptstyle k}}]^{\\top}$ is the \nvector of controller variables with $n_{\\scriptscriptstyle k}$ the number of \nmodes of the controller. \n$\\hat{H}_{\\scriptscriptstyle K}$ is the controller's self Hamiltonian with \n$R_{\\scriptscriptstyle K}\\in \n\\mathbb{R}^{2n_{\\scriptscriptstyle k} \\times 2n_{\\scriptscriptstyle k}}$. \nAlso $\\hat{H}_{\\rm int}$ with \n$R_1 \\in \\mathbb{R}^{2n \\times 2n_{\\scriptscriptstyle k}}$, \n$R_2 \\in \\mathbb{R}^{2n_{\\scriptscriptstyle k} \\times 2n}$ represents the \ncoupling between the plant and the controller. \nNote that, for the Hamiltonians $\\hat{H}_{\\scriptscriptstyle K}$ and \n$\\hat{H}_{\\rm int}$ to be Hermitian, the matrices must satisfy \n$R_{\\scriptscriptstyle K}=R_{\\scriptscriptstyle K}^\\top$ and $R_1^\\top=R_2$; \nthese are the physical realizability conditions in the scenario of direct \ninteraction. \nIn particular, here we consider a plant system interacting with a single probe \nfield $\\hat W_1$, with coupling matrices $B_1=B$ and $C_1=C$. \nThen, the whole dynamics of the augmented system with variable \n$\\hat{x}_{{\\scriptscriptstyle E}}=[\\hat{x}^{\\top}, \\hat{x}_{\\scriptscriptstyle K}^{\\top}]^{\\top}$ \nis given by\n\\begin{align}\n\\label{direct whole system}\n \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n &=A_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n +B_{{\\scriptscriptstyle E}}\\hat{W}_{1}+b_{{\\scriptscriptstyle E}}\\hat{f}, \\nonumber \\\\\n \\hat{W}_{1}^{\\rm out}&=C_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}+\\hat{W}_{1} ,\n\\end{align}\nwhere\n\\begin{align}\n\\label{eq_direct_matrixAe}\n A_{{\\scriptscriptstyle E}}\n &=\\left[\\begin{array}{cc}\n ~A ~~& \\Sigma_{n}R_1~~ \\\\\n \\Sigma_{n_{\\scriptscriptstyle k}}R_2 \n & \\Sigma_{n_{\\scriptscriptstyle k}}R_{\\scriptscriptstyle K}\\\\\n \\end{array}\\right], ~~\n B_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{c}\n B\\\\\n O \\\\\n \\end{array}\\right], \\nonumber \\\\\n C_{{\\scriptscriptstyle E}}&=\\left[\\begin{array}{cc}\n C & O \\\\\n \\end{array}\\right], ~~\n b_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n b^{\\top} & O\\\\ \n \\end{array}\\right] ^{\\top}. \n\\end{align}\nNote that $B_{{\\scriptscriptstyle E}}$, $C_{{\\scriptscriptstyle E}}$, and $b_{{\\scriptscriptstyle E}}$ \nare the same matrices as those in Eq.~\\eqref{extended_matrices}. \nAlso, comparing the matrices \\eqref{eq_extended_matrixAe} and \n\\eqref{eq_direct_matrixAe}, we have that $D_{\\scriptscriptstyle K}=O$, which \nthus leads to $F=F_{0}$ and $G=G_{0}$ in Theorem~2. \nNow, again for the opto-mechanical system illustrated in Fig.~\\ref{fig_opticalsensor}, \nlet us aim to design the direct interaction controller, so that the whole system \n\\eqref{direct whole system} achieves BAE; \nthat is, the problem is to determine the matrices \n$(R_{\\scriptscriptstyle K}, R_1, R_2)$ so that the back-action noise $\\hat Q_1$ \ndoes not appear in the measurement output $\\hat P_1^{\\rm out}$. \nFor this purpose, we go through the same procedure as that taken in Section~\\ref{sec:3-3}.\n\n{\\bf (i)}\nBecause of the structure of the matrices $B_{{\\scriptscriptstyle E}}$ and $C_{{\\scriptscriptstyle E}}$, \nthe system is already of the form \\eqref{eq_extended_system}, \nwhere the geometric control theory is directly applicable. \n\n\n{\\bf (ii)}\nBecause we now focus on the same plant system as that in Section \\ref{sec:3-3}, \nthe same conclusion is obtained; \nthat is, the BAE problem is solvable as long as there is no constraint on the \ncontroller matrices $(R_{\\scriptscriptstyle K}, R_1, R_2)$. \n\n\nThe controller matrices can be determined in a similar way to Section \\ref{sec:3-3} as follows. \nFirst, because the $(C,A,B)$-pair $(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ is the \nsame as before, it follows that ${\\rm dim} \\, \\mathcal{X}_{{\\scriptscriptstyle K}}=2$, i.e., \n$n_{\\scriptscriptstyle k}=1$. \nThen, from Theorem~2 with the fact that $F=F_{0}$ and $G=G_{0}$, \nwe find that the direct interaction controller can be parameterized as follows: \n\\begin{align}\n\\label{direct controller matrices example}\n &R_{\\scriptscriptstyle K}=-\\Sigma N(A+BF+GC)N^{+}, \\nonumber \\\\\n &R_1=-\\Sigma_2 BFN^{+}, \\nonumber \\\\\n &R_2=\\Sigma NGC,\n\\end{align}\nThe matrices $F$, $G$, and $N$ satisfy\n$\\Kernel F \\supseteq \\mathcal{V}_{1}, \\Image G \\subseteq \\mathcal{V}_{2}$, and \n$\\Kernel N=\\mathcal{V}_{1}$, \nwhich lead to\n\\begin{align}\n\\label{eq_parameterization2}\n\\hspace{-0.8cm}\n F&=\\left[\\begin{array}{cccc}\n f_{\\scriptscriptstyle 11}&f_{\\scriptscriptstyle 12}\n &0&f_{\\scriptscriptstyle 14}\\\\\n \\frac{g}{\\sqrt{\\kappa}}&0&0&f_{\\scriptscriptstyle 24}\\\\\n \\end{array}\\right], ~~\n G=\\left[\\begin{array}{cc}\n 0&g_{\\scriptscriptstyle 12}\\\\\n -\\frac{g}{\\sqrt{\\kappa}}&g_{\\scriptscriptstyle 22}\\\\\n g_{\\scriptscriptstyle 31}&g_{\\scriptscriptstyle 32}\\\\\n 0&0\\\\\n \\end{array}\\right], \\nonumber \\\\\n N&=\\left[\\begin{array}{cccc}\n n_{\\scriptscriptstyle 11}&n_{\\scriptscriptstyle 12}\n &0&n_{\\scriptscriptstyle 14}\\\\\n n_{\\scriptscriptstyle 21}&n_{\\scriptscriptstyle 22}\n &0&n_{\\scriptscriptstyle 24}\\\\\n \\end{array}\\right],\n\\end{align}\nwhere $f_{ij}, g_{ij}$, and $ n_{ij}$ are free parameters. \n\n{\\bf (iii)}\nThe controller matrices have to satisfy the physical \nrealizability conditions $R_{\\scriptscriptstyle K}=R_{\\scriptscriptstyle K}^\\top$ \nand $R_1^\\top=R_2$; \nthese constraints impose the parameters to satisfy \n\\begin{align}\n\\label{direct controller parameter constraint}\n&f_{\\scriptscriptstyle 12}=-g_{\\scriptscriptstyle 12},~~\nf_{\\scriptscriptstyle 11}=g_{\\scriptscriptstyle 22}, ~~\nn_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 22}-n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 21}=-1, \\nonumber \\\\\n&f_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 1}=f_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 2}-f_{\\scriptscriptstyle 14}, ~~ \nf_{\\scriptscriptstyle 24}=\\frac{g}{\\sqrt{\\kappa}}n_{\\scriptscriptstyle 2}, \n\\nonumber \\\\\n& \\hspace{-0.15cm}\n\\left( \\frac{\\kappa}{2}+\\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right) \n n_{\\scriptscriptstyle 1}+\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 2}\n=-\\sqrt{\\kappa}f_{\\scriptscriptstyle 11}, ~~\\nonumber \\\\\n&\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 1} - \\left ( \\frac{\\kappa}{2} + \\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right )n_{\\scriptscriptstyle 2}\n=\\sqrt{\\kappa}f_{\\scriptscriptstyle 12}, \n\\end{align}\nwhere \n$n_{\\scriptscriptstyle 1}=n_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 21}$ and \n$n_{\\scriptscriptstyle 2}=n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 22}$. \n\\\\\nEquations \\eqref{direct controller matrices example}, \n\\eqref{eq_parameterization2}, and \\eqref{direct controller parameter constraint} \nprovide the full parametrization of the direct interaction controller. \n\n\n{\\bf (iv)}\nTo specify a set of parameters, as in the case of Section \\ref{sec:3-3}, let us aim \nto design a passive controller. \nFrom Theorem~4 in Appendix~B, $R_{\\scriptscriptstyle K}$ and \n$R_2=R_1^\\top$ satisfy the condition \n$\\Sigma R_{\\scriptscriptstyle K}\\Sigma=-R_{\\scriptscriptstyle K}$ and \n$\\Sigma R_2 \\Sigma_{2}=-R_2$, which lead to the same equalities given in \nEq.~\\eqref{passive para cond}. \nThen, setting the parameters to be $n_{\\scriptscriptstyle 11}=1$ and $n_{\\scriptscriptstyle 12}=0$, \nwe can determine the matrices $R_{\\scriptscriptstyle K}$ and $R_2$ as follows: \n\\begin{align*}\n R_{\\scriptscriptstyle K}\n =\\left[\\begin{array}{cc}\n -\\omega_{\\scriptscriptstyle m} &0\\\\\n 0&-\\omega_{\\scriptscriptstyle m} \\\\\n \\end{array}\\right] , ~~\n R_2=R_1^\\top\n =\\left[\\begin{array}{cccc}\n 0&0&g&0 \\\\\n 0&0&0&g \\\\\n \\end{array}\\right] . \n\\end{align*}\nThe controller specified by these matrices can be physically implemented \nas illustrated in Fig.~\\ref{fig_Mankei_scheme}~(b); that is, it is a single-mode \ndetuned cavity with Hamiltonian $\\hat{H}_{\\scriptscriptstyle K}\n= -\\omega_{\\scriptscriptstyle m}\\hat{a}^{\\ast}_{3}\\hat{a}_{3}$, which couples \nto the plant through a beam-splitter (BS) represented by\n$\\hat{H}_{\\rm int}=g(\\hat{a}_{3}\\hat{a}^{*}_{2}+\\hat{a}^{*}_{3}\\hat{a}_{2})$. \n\n\n\n{\\bf Remark:} \nWe can employ an active controller, as proposed in \\cite{Tsang2010}. \nIn this case the interaction Hamiltonian is given by \n$\\hat{H}_{\\rm int}=g_{\\scriptscriptstyle \\rm B}(\\hat{a}_{3}\\hat{a}^{*}_{2}\n+\\hat{a}^{*}_{3}\\hat{a}_{2})+g_{\\scriptscriptstyle \\rm D}(\\hat{a}_{3}\\hat{a}_{2}\n+\\hat{a}^{*}_{3}\\hat{a}^{*}_{2})$, \nwhile the system's self-Hamiltonian is the same as above; \n$\\hat{H}_{\\scriptscriptstyle K}\n= -\\omega_{\\scriptscriptstyle m}\\hat{a}^{\\ast}_{3}\\hat{a}_{3}$. \nThat is, the controller couples to the plant through a non-degenerate optical \nparametric amplification process in addition to the BS interaction. \nTo satisfy the BAE condition, the parameters must satisfy \n$g_{\\scriptscriptstyle \\rm B}+g_{\\scriptscriptstyle \\rm D}=g$. \nNote that this direct interaction controller can be specified, in the \nfull-parameterization \\eqref{direct controller matrices example}, \n\\eqref{eq_parameterization2}, and \\eqref{direct controller parameter constraint}, \nby \n\\[\n f_{\\scriptscriptstyle 11}=f_{\\scriptscriptstyle 12}=f_{\\scriptscriptstyle 14}=0,~~\n n_{\\scriptscriptstyle 11}=-n_{\\scriptscriptstyle 22}=1,~~ \n n_{\\scriptscriptstyle 12}=n_{\\scriptscriptstyle 21}=0.\n\\]\n\n\n\n\n\n\\section{Approximate Back-Action Evasion}\\label{sec:5}\n\n\nWe have demonstrated in Sections \\ref{sec:3-3} and \\ref{sec:4} that the \nBAE condition can be achieved by engineering an appropriate auxiliary \nsystem and connecting it to the plant. \nHowever, in a practical situation, it cannot be expected to realize such perfect \nBAE due to several experimental imperfections. \nHence, in a realistic setup, we should modify our strategy for engineering \na sensor so that it would accomplish {\\it approximate BAE}. \nThen, looking back into Section \\ref{sec:2-3} where the BAE condition, \n$\\Xi_{Q}(s)=0~\\forall s$, was obtained, we are naturally led to consider \nthe following optimization problem to design an auxiliary system achieving \nthe approximate BAE: \n\\begin{align}\n\\label{eq_approximateBAE}\n \\min \\Big\\|\\frac{\\Xi_{Q}(s)}{\\Xi_{f}(s)}\\Big\\|, \n\\end{align}\nwhere $\\|\\bullet\\|$ denotes a valid norm of a complex function. \nIn particular, in the field of robust control theory, the following $H_2$ norm \nand the $H_\\infty$ norm are often used \\cite{Zhou1996}: \n\\[\n \\|\\Xi\\|_2 \n = \\sqrt{\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}|\\Xi(i\\omega )|^{2}d\\omega},~~~\n \\|\\Xi\\|_\\infty\n = \\max_{\\omega}|\\Xi(i\\omega )|. \n\\]\nThat is, the $H_2$ or $H_\\infty$ control theory provides a general procedure \nfor synthesizing a feedback controller that minimizes the above norm. \nIn this paper, we take the $H_2$ norm, mainly owing to the broadband \nnoise-reduction nature of the $H_2$ controller. \nThen, rather than pursuing an optimal quantum $H_2$ controller based \non the quantum $H_2$ control theory \\cite{NurdinLQG2009, Hamerly2012}, \nhere we take the following geometric-control-theoretical approach to solve the \nproblem \\eqref{eq_approximateBAE}. \nThat is, first we apply the method developed in Section \\ref{sec:3} or \n\\ref{sec:4} to the idealized system and obtain the controller achieving BAE; \nthen, in the practical setup containing some unwanted noise, we make a local \nmodification of the controller parameters obtained in the first step, to minimize \nthe cost $\\|\\Xi_{Q}(s)\/\\Xi_{f}(s)\\|_2$. \n\n\n\nAs a demonstration, here we consider the coherent feedback control for the \nopto-mechanical system studied in Section \\ref{sec:2-3}, which is now \nsubjected to the thermal noise $\\hat{f}_{\\rm th}$. \nFollowing the above-described policy, we employ the coherent feedback \ncontroller constructed for the idealized system that ignores $\\hat{f}_{\\rm th}$, \nleading to the controller given by Eqs.~\\eqref{eq_controller_H} and \n\\eqref{perfect BAE condition example}, illustrated in Fig.~\\ref{fig_3I3OBAE}. \nThe closed-loop system with variable $\\hat{x}_{{\\scriptscriptstyle E}}=\n[\\hat{x}^{\\top},\\hat{x}_{\\scriptscriptstyle K}^{\\top}]^{\\top}$, \nwhich now takes into account the realistic imperfections, \nthen obeys the following dynamics:\n\\begin{align}\n\\label{eq_stateout2}\n \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n &=\\widetilde{A}_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n +B_{{\\scriptscriptstyle E}}\\hat{W}_{1} \n +b_{{\\scriptscriptstyle E}}(\\hat{f}_{\\rm th}+\\hat{f}), \\nonumber \\\\\n \\hat{W}_{3}^{\\rm out}\n &=C_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}+\\hat{W}_{1}, \n\\end{align}\nwhere\n\\[\n \\widetilde{A}_{{\\scriptscriptstyle E}}\n =\\left[\\begin{array}{cccc|cc}\n 0&\\omega_{\\scriptscriptstyle m} &0&0&0&0\\\\\n -\\omega_{\\scriptscriptstyle m} &-\\gamma &g&0&0&0\\\\\n 0&0&-\\kappa\/2 &0&0& \\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}} \\\\\n g&0&0&-\\kappa\/2 &-\\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}}&0\\\\\n \\hline 0&0&0&\\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}}&0&\\Delta \\\\\n 0&0&-\\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}}&0&-\\Delta &0\\\\\n \\end{array}\\right] .\n\n\\]\n$B_{{\\scriptscriptstyle E}}$, $C_{{\\scriptscriptstyle E}}$, and $b_{{\\scriptscriptstyle E}}$ are the same matrices \ngiven in Eq.~\\eqref{extended_matrices}. \n$\\hat{f}_{\\rm th}$ is the thermal noise satisfying \n$\\langle \\hat{f}_{\\rm th}(t)\\hat{f}_{\\rm th}(t') \\rangle \\simeq \\bar{n}\\delta (t-t')$,\nwhere $\\bar{n}$ is the mean phonon number at thermal equilibrium \n\\cite{Giovannetti2001, Wimmer2014}. \nNote that the damping effect appears in the $(2, 2)$ component of \n$\\widetilde{A}_{{\\scriptscriptstyle E}}$ due to the stochastic nature of $\\hat{f}_{\\rm th}$.\nAlso, again, $\\kappa_{\\scriptscriptstyle K}$ and $\\Delta$ are the \ndecay rate and the detuning of the controller cavity, respectively. \nIn the idealized setting where $\\hat{f}_{\\rm th}$ is negligible, the perfect BAE is achieved \nby choosing the parameters satisfying Eq.~\\eqref{perfect BAE condition example}. \nThe measurement output of this closed-loop system is, in the Laplace domain, \nrepresented by \n\\[\n \\hat{P}_{3}^{\\rm out}(s)\n =\\widetilde{\\Xi}_{f} (\\hat{f}_{\\rm th}(s)+\\hat{f}(s))\n +\\widetilde{\\Xi}_{Q}\\hat{Q}_{1}(s)+\\widetilde{\\Xi}_{P} \\hat{P}_{1}(s).\n\\]\nThe normalized noise power spectral density of \n$y_{3}(s)=\\hat{P}_{3}^{\\rm out}(s)\/\\widetilde{\\Xi}_{f}(s)$ is calculated as \n\\begin{align}\n \\widetilde{S} (\\omega ) \n &=\\langle | y_{3}(i\\omega )-\\hat{f}(i\\omega )|^2 \\rangle \\nonumber \\\\\n &=\\langle|\\hat{f}_{\\rm th}|^2 \\rangle \n + \\left| \\frac{\\widetilde{\\Xi}_{Q}}{\\widetilde{\\Xi}_{f}} \\right| ^2 \n \\langle |\\hat{Q}_{1}|^2 \\rangle \n + \\left| \\frac{\\widetilde{\\Xi}_{P}}{\\widetilde{\\Xi}_{f}} \\right| ^2 \n \\langle |\\hat{P}_{1}|^2 \\rangle.\n\\label{eq_noisepower_thermal}\n\\end{align}\nThe coefficient of the back-action noise is given by \n\\begin{align}\n \\frac{\\widetilde{\\Xi}_{Q}(s)}\n {\\widetilde{\\Xi}_{f}(s)}=\n -\\frac{ \\sqrt{\\kappa}\\{ \\kappa \\kappa_{\\scriptscriptstyle K}\\Delta \n (s^2+\\gamma s + \\omega_{\\scriptscriptstyle m}^2 )\n + g^2\\omega_{\\scriptscriptstyle m} (s^2+\\Delta^2 )\\} }\n {g\\omega_{\\scriptscriptstyle m}\\sqrt{\\gamma}\n \\{ (s+\\kappa\/2)(s^2+\\Delta^2 ) \n +\\kappa \\kappa_{\\scriptscriptstyle K} s\\} }\n\\label{eq_normalizedtrans_thermal}.\n\\end{align}\nOur goal is to find the optimal parameters $(\\kappa_{\\scriptscriptstyle K}, \\Delta)$ \nthat minimize the $H_2$ norm of the transfer function, \n$\\widetilde{\\Xi}_{Q}\/\\widetilde{\\Xi}_{f}$. \n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=7cm,clip]{Figure\/Fig8_H2norm.eps}\n\\caption{\n$H_{2}$ norm $\\|\\widetilde{\\Xi}_{Q}\/\\widetilde{\\Xi}_{f}\\|_{2}$ \nversus the coupling constant $\\kappa_{\\scriptscriptstyle K}$ and \nthe detuning $\\Delta$.}\n\\label{fig_h2norm}\n\\end{center}\n\\end{figure}\n\n\nThe system parameters are taken as follows \\cite{Wimmer2014}: \n$\\omega_{\\scriptscriptstyle m}\/2\\pi=0.5$ {\\rm MHz}, \n$\\kappa\/2\\pi=1.0$ {\\rm MHz}, \n$\\gamma\/2\\pi=5.0$ {\\rm kHz}, \n$g\/2\\pi=0.3$ {\\rm MHz}, \n$\\bar{n} \\simeq 8.33 \\times 10^{2}$, \nand the effective mass is $1.0 \\times 10^{-12}$ {\\rm kg}. \nWe then have Fig.~\\ref{fig_h2norm}, showing \n$\\|\\widetilde{\\Xi}_{Q}\/\\widetilde{\\Xi}_{f}\\|_2$ as a function of \n$\\kappa_{\\scriptscriptstyle K}$ and $\\Delta$. \nThis figure shows that there exists a unique pair of \n$(\\kappa_{\\scriptscriptstyle K}^{\\rm opt}, \\Delta^{\\rm opt})$ that \nminimizes the norm, and they are given by \n$\\kappa_{\\scriptscriptstyle K}^{\\rm opt} \/2\\pi=0.093$ {\\rm MHz} and\n$\\Delta^{\\rm opt}\/2\\pi=-0.5$ {\\rm MHz},\nwhich are actually close to the ideal values \\eqref{perfect BAE condition example}. \nFig.~\\ref{fig_noisepower} shows the value of Eq.~\\eqref{eq_noisepower_thermal} \nwith these optimal parameters \n$(\\kappa_{\\scriptscriptstyle K}^{\\rm opt}, \\Delta^{\\rm opt})$, where \nthe noise floor $\\langle|\\hat{f}_{\\rm th}|^2 \\rangle$ is subtracted. \nThe solid black line represents the SQL, which is now given by \n\\begin{equation}\n \\widetilde{S}_{\\scriptscriptstyle \\rm SQL} (\\omega ) \n =\\frac{|(\\omega^2 - \\omega_{\\scriptscriptstyle m}^2)-i\\gamma\\omega|}\n {\\gamma \\omega_{\\scriptscriptstyle m}}.\n\\label{eq_SQL_practical}\n\\end{equation}\nThen the dot-dashed blue and dotted green lines indicate that, in the low frequency range, the \ncoherent feedback controller can suppress the noise below the SQL, while, by \ndefinition, the noise power of the autonomous (i.e., uncontrolled) plant system \nis above the SQL. \nMoreover, this effect can be enhanced by injecting a $\\hat{P}_{1}$-squeezed \nprobe field (meaning $\\langle |\\hat{Q}_{1}|^2 \\rangle=e^{r}\/2$ and \n$\\langle |\\hat{P}_{1}|^2 \\rangle=e^{-r}\/2$) into the system. \nIn fact the dashed red line in the figure illustrates the case $r=2$ (about 9 dB squeezing), \nshowing the significant reduction of the noise power. \n\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=8.5cm,clip]{Figure\/Fig9_noisepower.eps}\n\\caption{Normalized power spectral densities of the noise. \nThe black solid line represents the SQL \\eqref{eq_SQL_practical}, and \nthe dot-dashed blue line does the case without feedback. \nThe dotted green and dashed red lines show the cases for the feedback controlled system, \nwith coherent and squeezed probe field, respectively. }\n\\label{fig_noisepower}\n\\end{center}\n\\end{figure}\n\n\n\n \n\n\n\n\\section{Conclusion}\n\\label{sec:6}\n\n\nThe main contribution of this paper is in that it first provides the general \ntheory for constructing a back-action evading sensor for linear quantum \nsystems, based on the well-developed classical geometric control theory. \nThe power of the theory has been demonstrated by showing that, for the \ntypical opto-mechanical oscillator, a full parametrization of the auxiliary \ncoherent-feedback and direct interaction controller achieving BAE was \nderived, which contains the result of \\cite{Tsang2010}. \nNote that, although we have studied a simple example for the purpose of \ndemonstration, the real advantage of the theory developed in this paper \nwill appear when dealing with more complicated multi-mode systems such \nas an opto-mechanical system containing a membrane \n\\cite{Plenio2008, Meystre2008, Nakamura2016, Nielsen2016}. \nAnother contribution of this paper is to provide a general procedure for designing \nan approximate BAE sensor under realistic imperfections; \nthat is, an optimal approximate BAE system can be obtained by solving \nthe minimization problem of the transfer function from the back-action noise \nto the measurement output. \nWhile in Section~\\ref{sec:5} we have provided a simple approach based on \nthe geometric control theory for solving this problem, the $H_2$ or $H_\\infty$ control theory could be \nemployed for systematic design of an approximate BAE controller even \nfor the above-mentioned complicated system. \nThis is also an important future research direction of this work. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn \\cite{KhR1} Khovanov and Rozansky $(KR)$ constructed for each $n >0$ a bigraded rational (co)homology theory categorifying the $\\mathfrak{sl}(n)$-link polynomial (the case $n=0$ is treated in \\cite{KhR2}). Their construction uses matrix factorizations with potential $x^{n+1}$ associated to certain planar graphs, and for $n=2$, the corresponding homology is equivalent to the Khovanov homology defined in \\cite{Kh1}. Gornik \\cite{G} carried out a deformation of the $KR$-theory with potential $x^{n+1} - (n+1)\\beta^n x$, for $\\beta \\in \\mathbb{C},$ and Rasmussen \\cite{Ras2} and Wu ~\\cite{W} investigated $KR$-homologies given by a general non-homogeneous monic potential with degree $n+1$ and complex coefficients. $KR$-construction can be generalized to give the \\textit{universal} matrix factorization link homologies for all $n >0,$ by working with a general homogeneous potential. Recently, Mackaay and Vaz \\cite{MV2} worked out this generalization for $n=3,$ and proved that the universal rational $\\mathfrak{sl}(3)$-matrix factorization link homology is equivalent to the foam link homology in \\cite{MV1} tensored with $\\mathbb{Q}.$ \n\nIn \\cite{CC2} the author constructed the universal $\\mathfrak{sl}(2)$-link cohomology via foams modulo local relations, in the spirit of \\cite{BN} and \\cite{Kh2} (see also \\cite{CC1}). In this paper we introduce the universal rational Khovanov-Rozansky link cohomology for $n=2,$ and show that it is isomorphic to the foam $\\mathfrak{sl}(2)$-link cohomology in \\cite{CC2}, after both theories are tensored with appropriate rings. To obtain the universal $\\mathfrak{sl}(2)$-matrix factorization theory, we consider a potential $p$ that depends on two parameters $a$ and $h,$ and that satisfies $ \\partial p \/\\partial x = 3 (x^2 - hx - a).$\n\n\n$KR$-construction starts from a certain version of the calculus developed by Murakami, Ohtsuki and Yamada \\cite{MOY}, calculus which involves planar trivalent graphs. $KR$-graphs contain two types of edges, namely oriented edges and unoriented thick edges. For our purpose, we consider graphs that are obtained from the latter ones by ``erasing\" all unoriented thick edges. \n\n\n\\section {Webs and matrix factorizations}\\label{factorizations}\n\n\\subsection{The web space}\n\nA \\textit{web} with boundary $B$ is a planar graph with univalent and bivalent vertices. The univalent vertices correspond to boundary points, such as the boundary $\\partial{T}$ of a tangle, and bivalent vertices have either indegree 2 or outdegree 2. Specifically, the two arcs incident with a bivalent vertex are either oriented ``in'' or ``out'' as shown below: \n\\[ \\raisebox{-5pt}{\\includegraphics[height=.2in]{in.pdf}} \\quad \\text {or} \\quad \\raisebox{-5pt}{\\includegraphics[height=.2in]{out.pdf}}\\]\nWe call bivalent vertices \\textit{singular points}. A \\textit{closed web} is a web with empty boundary. We also allow webs with no bivalent vertices, thus oriented arcs or loops. We denote by $\\textit{Foams}(B)$ the category whose objects are web diagrams with boundary $B,$ and whose morphisms are \\textit{singular cobordisms}---called \\textit{foams}---between such webs, regarded up to boundary-preserving isotopies. As morphisms, we read cobordisms from bottom to top, and we compose them by stacking one on top the other. \n\nLet $L$ be a link in $S^3.$ We fix a generic planar diagram $D$ of $L$ and resolve each crossing in two ways, as in Figure~\\ref{fig:resolutions}. We refer to the diagram on the right as the \\textit{oriented resolution}, and to the one on the left as the \\textit{singular resolution}. We remark that the dotted lines should not be considered as edges---we prefer to draw them in order to record that there was a crossing before. \n\\begin{figure}[ht]\n$$\\xymatrix@C=20mm@R=1.5mm{\n & \\includegraphics[height=0.5in]{poscros.pdf} \\ar[ld]_0\\ar[rd]^1& \\\\\n\\includegraphics[height=0.5in]{singresol.pdf} & & \n\\includegraphics[height=0.5in]{orienresol.pdf} \\\\\n & \\includegraphics[height=0.5in]{negcros.pdf} \\ar[ul]^1 \\ar[ur]_0&\n}$$\n\\caption{The two ways of resolving crossings}\n\\label{fig:resolutions}\n\\end{figure}\n\nAs webs have singular points, foams have \\textit{singular arcs} (or \\textit{singular circles}) where orientations disagree. Basic cobordisms, as those between the two different resolutions of a crossing are depicted in Figure~\\ref{fig:saddles}, where the arc colored red is a singular arc.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=.65in]{saddle.pdf}\\end{center}\n\\caption{Singular saddles}\n\\label{fig:saddles}\n\\end{figure}\n\nA diagram $\\Gamma$ obtained by resolving all crossings of $D$---in one of the two possible ways explaned above---is a disjoint union of closed webs. There is a unique way to assign a Laurent polynomial $\\brak{\\Gamma} \\in \\mathbb {Z} [q, q^{-1}]$ to each closed web $\\Gamma$ so that it satisfies the skein relations explained in Figure~\\ref{fig:skein relations}. (We remark that these are exactly the graph skein relations for $n= 2$ given in~\\cite[Figure 3]{KhR1}, where all thick edges are erased.)\n\n\\begin{figure}[ht]\n$$\\xymatrix@R = 2mm{\n\\raisebox{-10pt}{\\includegraphics[height=.35in]{orienloop.pdf}}\\,\\,\\, = \\,\\,\\, q + q^{-1}\\hspace{1cm}\n\\raisebox{-30pt}{\\includegraphics[height=.9in]{removedweb1.pdf}}\\, = \\,\\,(q+ q^{-1})\\,\\raisebox{-30pt}{\\includegraphics[height=.9in]{removedweb2.pdf}} \\\\ \\raisebox{-15pt}{\\includegraphics[height=.5in]{singresclosed.pdf}} = \\raisebox{-15pt}{\\includegraphics[height=.5in]{arc.pdf}} \\hspace{2cm}\\raisebox{-15pt}{\\includegraphics[height=.5in]{2singresol.pdf}}\\,\\, =\\,\\, \\raisebox{-15pt}{\\includegraphics[height=.45in]{2arcs.pdf}} \\\\\n\\raisebox{-30pt}{\\includegraphics[height=.9in]{skein1.pdf}}\\,\\, +\\,\\, \\raisebox{-30pt}{\\includegraphics[height=.9in]{skein2.pdf}} \\,\\,=\\,\\, \\raisebox{-30pt}{\\includegraphics[height=.9in]{skein3.pdf}}\\,\\, + \\,\\,\\raisebox{-30pt}{\\includegraphics[height=.9in]{skein4.pdf}} \n}$$\n\\caption{Skein relations}\n\\label{fig:skein relations}\n\\end{figure}\n\nThe \\textit{bracket} of $D$ is defined by $\\brak{D} = \\sum_{\\Gamma} \\pm q^{\\alpha(\\Gamma)}\\brak{\\Gamma},$ where the sum is over all resolutions of $D$ and $\\alpha(\\Gamma)$ is determined by the rules in Figure~\\ref{fig:decomposition of crossings}.\n\n\\begin{figure}[ht]\n$$\\xymatrix@R=2mm{\n\\raisebox{-12pt}{\\includegraphics[height=0.4in]{poscros.pdf}}\\,\\, = \\,\\,q\\,\\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{orienresol.pdf}}\\,\\, -\\,\\, q^2\\,\\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{singresol.pdf}}\\\\\n\\raisebox{-12pt}{\\includegraphics[height=0.4in]{negcros.pdf}} \\,\\,= \\,\\,q^{-1}\\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{orienresol.pdf}}\\, -\\,q^{-2} \\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{singresol.pdf}}\n}$$\n\\caption{Decomposition of crossings}\n\\label{fig:decomposition of crossings}\n\\end{figure}\n\nIt is well know that $\\brak{D_1} = \\brak{D_2}$ whenever $D_1$ and $D_2$ are related by a Reidemeister move, hence $\\brak{L}:= \\brak{D}$ is an invariant of the oriented link $L.$ Excluding the rightmost terms in Figure~\\ref{fig:decomposition of crossings}, we obtain the skein relation $$q^2 \\raisebox{-8pt}{\\includegraphics[height=.32in]{negcros.pdf}} - q^{-2} \\raisebox{-8pt}{\\includegraphics[height=.32in]{poscros.pdf}} = (q - q^{-1}) \\raisebox{-8pt}{\\includegraphics[height=.32in]{orienresol.pdf}}$$ which yields the quantum $sl(2)$-polynomial of the link $L$ (thus the unnormalized Jones polynomial---its categorification is introduced in~\\cite{Kh1}).\n\n\\subsection{From webs to matrix factorizations}\n\nWe mimic Khovanov's and Rozansky's work in ~\\cite{KhR1} for $n =2$ case, by replacing the polynomial $p(x) = x^3$ with $p(a, h, x) = x^3 -\\frac{3}{2} h x^2- 3ax,$ where $a$ and $h$ are formal variables. Note that $p(a, h, x)$ was chosen in such a way that $ \\frac{1}{3} \\partial p\/ \\partial x = x^2 -hx -a.$ We assume that the reader is somewhat familiar with~\\cite{KhR1}, and we only briefly recall some of its content.\n\nGiven a graded ring $R$ (we consider only polynomial rings) and a homogeneous element $\\omega \\in R,$ the category $\\textit{mf}_{\\omega}$ of graded matrix factorizations with potential $\\omega$ has objects 2-periodic chains\n\\[ M_0 \\stackrel{d_0}{\\rightarrow} M_1 \\stackrel{d_1}{\\rightarrow} M_0, \\] \nwhere $M_0, M_1$ are graded free $R$-modules and $d_0, d_1$ are homomorphisms such that $d_0d_1 = \\omega = d_1d_0.$ We call such an object an $(R, \\omega)$-factorization. Morphisms in $\\textit{mf}_{\\omega}$ are degree-preserving maps of $R$-modules $M_0 \\to N_0, M_1 \\to N_1$ that commute with differentials. In this paper, the morphisms $d_0$ and $d_1$ have degree $3$ (thus $\\deg(\\omega) = 6$) and a homotopy has degree $-3.$ \n\nThe category $\\textit{hmf}_\\omega$ of graded matrix factorizations up to chain homotopies has the same objects as $\\textit{mf}_\\omega$ but fewer morphisms, as homotopic morphisms in $\\textit{mf}_\\omega$ are declared the same in $\\textit{hmf}_\\omega.$ We denote by $\\{r\\}$ the grading shift up by $r,$ and by $\\brak{s}$ the cohomological shift functor for matrix factorizations. Specifically, $M \\brak{1}$ has the form $M_1 \\stackrel {-d_1}{\\rightarrow} M_0 \\stackrel{-d_0}{\\rightarrow} M_1.$ The cohomological shift functor for chain complexes is denoted by $[s].$ \n\nGiven a pair of elements $a_1, b_1 \\in R,$ we denote by $(a_1, b_1)$ the factorization with potential $a_1b_1$\n\\[R \\stackrel{a_1}{\\longrightarrow} R \\{3 -\\deg a \\} \\stackrel{b_1}{\\longrightarrow} R, \\]\nwhere $a_1$ and $b_1$ act on $R$ by multiplication. The middle $R$ was shifted so that the differentials would have degree $3$. Given a finite set of pairs $(a_i, b_i), 1 \\leq i \\leq n$, we denote their tensor product over $R$ by $(\\textbf{a, b}) : = \\otimes_i(a_i, b_i)$, where $\\textbf{a} = (a_1, \\cdots, a_n)$ and $ \\textbf{b} = (b_1,\\cdots, b_n).$ Note that $(\\textbf{a, b})$ is the factorization with potential $\\omega = \\sum_i a_i b_i,$ and we sometimes prefer to write it in the \\textit{Koszul matrix} form:\n\\[ (\\textbf{a, b}) = \\left ( \\begin{array}{cc} a_1 & b_1 \\\\ a_2 & b_2 \\\\ \\cdots & \\cdots \\\\ a_n & b_n \\end{array} \\right ). \\]\nLet $c \\in R$ and denote by $[ij]_c$ the \\textit{elementary row operation} which transforms\n\\[ \\left ( \\begin{array}{cc} a_i & b_i \\\\ a_j & b_j \\end{array} \\right ) \\stackrel{[ij]_c}{\\longrightarrow} \\left ( \\begin{array}{cc} a_i+ca_j & b_i \\\\ a_j & b_j -cb_i\\end{array} \\right )\\]\n and leaves the remaining rows of the Koszul matrix unchanged. An elementary row operation corresponds to a change of the basis vector of the free $R$-module underlying the corresponding factorization, thus it takes a Koszul factorization $(\\textbf{a, b})$ to an isomorphic factorization in $\\textit{hmf}_\\omega.$\n\nAs noted by Rasmussen~\\cite{Ras2}, if $b_i$ and $b_j$ are relatively prime, we can apply a \\textit{twist} via the map $R \\to R$ which sends $x \\to kx,$ for some $k \\in R,$ and obtain an isomorphism of factorizations\n \\[ (a_i, b_i) \\otimes (a_j, b_j) \\cong (a_i + k b_j, b_i) \\otimes (a_j - kb_i, b_j).\\]\n\nUsing Koszul matrix form, the above isomorphism is written as\n \\[ \\left ( \\begin{array}{cc} a_i & b_i \\\\ a_j & b_j \\end{array} \\right ) \\stackrel{\\cong}{\\longrightarrow} \\left ( \\begin{array}{cc} a_i + kb_j & b_i \\\\ a_j - kb_i & b_j \\end{array} \\right).\\]\n \nStarting with a link or tangle diagram $D,$ we put marks on each of its arcs. Resolving all of its crossings as explained in Figure~\\ref{fig:resolutions}, we obtain web diagrams $\\Gamma$ with marked arcs, and we denote by $m(\\Gamma)$ the set of all marks of $\\Gamma.$ We associate to each mark $i$ the polynomial $p(a, h, x_i) = x_i^3 -\\frac{3}{2}h x_i^2 -3ax_i.$ When working with a tangle diagram, each of its resolutions has internal and external marks, where the later ones are the boundary points $i \\in B$ of the tangle diagram. \n\nLet $R = \\mathbb{Q}[a,h, x_i], i \\in m(\\Gamma),$ be the polynomial ring with rational coefficients and variables $a, h$ and $x_i$ (over all marks $i$), and let $R'$ be its subring $\\mathbb{Q}[a,h, x_i], i \\in B.$ We introduce a grading on $R$ and $R'$ by letting $\\deg(x_i) = 2, i \\in m(\\Gamma), \\deg(a) = 4$ and $\\deg(h) =2.$ To a web diagram $\\Gamma$ we assign a graded factorization $\\overline{C}(\\Gamma)$ with (degree 6) homogeneous potential $\\omega(\\Gamma) = \\sum_{i \\in B} o(i)p(a, h, x_i),$ where $o(i) \\in \\{\\pm1\\} $ are ``orientations'' of boundary points, given by the orientation of $\\Gamma$ at these points. \n\nTo an oriented arc $l$ between two neighboring marks $i, j$ and oriented from $i$ to $j$ we assign the factorization $\\overline{L}_i^j$ over the ring $R = \\mathbb{Q}[a, h, x_i, x_j],$ and with potential $$\\omega (\\overline{L}_i^j)= p (a, h, x_j) -p (a, h, x_i) = x_j^3 -x_i^3 -\\frac{3}{2} h (x_j^2 -x_i^2) -3a (x_j -x_i).$$\n \\[\\overline{L}_i^j \\co \\quad \\mathbb{Q}[a,h,x_i,x_j]\\stackrel{\\overline{\\pi}_{ij}}{\\longrightarrow} \\mathbb{Q}[a,h,x_i,x_j]\\{-1\\} \\stackrel{x_j-x_i}{\\longrightarrow} \\mathbb{Q}[a,h,x_i,x_j] \\]\n\nwhere $\\overline{\\pi}_{ij} =\\displaystyle\\frac {\\omega(\\overline{L}_i^j)}{x_j-x_i} = x_i^2 + x_ix_j + x_j^2 -\\frac{3}{2}h(x_i + x_j) -3a$.\n\nTo an oriented circle with one mark $i$ we assign the factorization $\\overline{L}_i^i,$ which is the quotient of $\\overline{L}_i^j$ by the relation $x_j = x_i.$ We obtain a 2-periodic chain complex of $\\mathbb{Q}[a,h,x_i]$-modules\n \\[\n\\mathbb{Q}[a,h,x_i]\\stackrel{\\overline{\\pi}_{ii}}{\\longrightarrow} \\mathbb{Q}[a,h,x_i]\\{-1\\} \\stackrel{0}{\\longrightarrow} \\mathbb{Q}[a,h,x_i], \\] \nwhere $\\overline{\\pi}_{ii} = 3(x_i^2 - h x_i -a)$. This complex has cohomology only in degree 1, namely $$H^1(\\overline{L}_i^i) = \\mathbb{Q}[a,h,x_i]\/(x_i^2 -hx_i - a) \\{-1\\}, \\quad H^0(\\overline{L}_i^i) = 0.$$ \n\nLet $\\mathcal{A} : = \\mathbb{Q}[a, h, X]\/(X^2 - hX -a)$ and $\\iota \\co \\mathbb{Q}[a, h] \\to \\mathcal{A}$ be the inclusion map $\\iota(1) = 1.$ We identify $\\mathbb{Q}[a, h,x_i]\/(x_i^2 -hx_i- a)$ with $ \\mathcal{A}$ by taking $x_i^k \\in \\mathbb{Q}[a, h, x_i]\/(x_i^2 - hx_i -a)$ to $X^k \\in \\mathcal{A}.$ As a module over $\\mathbb{Q}[a, h], \\mathcal{A}$ is free with generators 1 and $X.$ We make $\\mathcal{A}$ graded by giving to 1 degree $-1$ and to $X$ degree 1.\n\nTo an oriented circle without marks we associate the 2-periodic chain complex of $\\mathbb{Q}[a,h]$-modules $0 \\rightarrow \\mathcal{A} \\rightarrow 0$ and denote it $\\mathcal{A} \\brak{1},$ following \\cite{KhR1}. Note that $\\mathcal{A}\\brak{1} \\cong \\overline{L}_i^i$ as 2-periodic complexes of $\\mathbb{Q}[a,h]$-modules, up to homotopies. The isomorphism takes $X^i \\in \\mathcal{A}$ to $x^i \\in (\\overline{L}_i^i)^1,$ for $0 \\leq i \\leq 1,$ and graphically it consists of adding a mark to a circle with no marks.\n\nTo diagrams $\\Gamma^0$ and $\\Gamma^1$ as in Figure~\\ref{maps} we associate the factorizations $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1)$ over the ring $R = \\mathbb{Q}[a, h, x_1, x_2, x_3, x_4]$ (note that $R = R'$):\n\\begin{align*}\n\\overline{C}(\\Gamma^0) &= (\\overline{\\pi}_{41}, x_1 -x_4) \\otimes (\\overline{\\pi}_{32}, x_2 -x_3) \\\\\n\\overline{C}(\\Gamma^1) &= (\\overline{u}_{1}, x_1 + x_2 -x_3 -x_4) \\otimes (\\overline{u}_{2}, x_1x_2 -x_3x_4)\n\\end{align*}\n with $\\omega(\\overline{C}(\\Gamma^0)) = p(a, h, x_1) + p(a, h, x_2) - p(a, h, x_3) - p(a, h, x_4) = \\omega(\\overline{C}(\\Gamma^1))$ and \n \\begin{align*} \\overline{u}_1 &= (x_1 + x_2)^2 + (x_1 + x_2)(x_3 + x_4) + (x_3 + x_4)^2 -3x_1x_2 -\\frac{3}{2} h( x_1+x_2+x_3 + x_4) -3a \\\\ \\overline{u}_2 &= -3(x_3+x_4) +3h.\\end{align*} \n Note that \\begin{align*} \\omega &= \\overline{\\pi}_{41}(x_1-x_4) + \\overline{\\pi}_{32}(x_2-x_3) \\\\ &= \\overline{u}_1\\cdot (x_1+x_2 -x_3 -x_4) + \\overline{u}_2 \\cdot (x_1x_2-x_3x_4).\\end{align*} Precisely we have\n \\[\n\\overline{C}(\\Gamma^0)\\co \\quad\n \\left( \\begin{array}{c}\n R \\\\ R\\{-2\\}\n \\end{array}\\right) \\stackrel{P_0}{\\longrightarrow}\n \\left( \\begin{array}{c}\n R\\{-1\\} \\\\ R\\{-1\\}\n \\end{array} \\right) \\stackrel{P_1}{\\longrightarrow}\n \\left(\\begin{array}{c}\n R \\\\ R\\{-2\\} \\end{array} \\right)\n \\]\n \n \\[P_0 = \n \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41} & x_2-x_3 \\\\ \n \\overline{\\pi}_{32} & x_4-x_1\\end{array}\n \\right), \\quad P_1 = \n \\left(\\begin{array}{cc}\n x_1 - x_4 & x_2-x_3 \\\\ \n\\overline {\\pi}_{32} & -\\overline{\\pi}_{41} \\end{array} \\right)\n \\]\n \n \\[ \\overline{C}(\\Gamma^1)\\co \\quad\n \\left( \\begin{array}{c}\n R\\{-1\\} \\\\ R\\{-1\\}\\end{array}\n \\right) \\stackrel{Q_0}{\\longrightarrow}\n \\left(\\begin{array}{c}\n R\\{-2\\} \\\\ R\\end{array}\n \\right) \\stackrel{Q_1}{\\longrightarrow} \\left(\n \\begin{array}{c} R\\{-1\\} \\\\ R\\{-1\\} \\end{array} \\right)\n \\]\n \n \\[Q_0 = \n \\left( \\begin{array}{cc}\n\\overline {u}_1 & x_1x_2 - x_3x_4 \\\\ \n\\overline {u}_2 & x_3 + x_4- x_1 - x_2 \\end{array}\n \\right), \\quad Q_1 =\n \\left(\\begin{array}{cc}\n x_1 + x_2 - x_3 - x_4 & x_1 x_2 - x_3x_4 \\\\ \n \\overline{u}_2 & - \\overline{u}_1 \\end{array}\\right)\n \\]\n\n\\begin{figure}[ht]\n\\includegraphics[height=1in]{maps.pdf} \n\\caption{Webs $\\Gamma^0$ and $\\Gamma^1$}\n\\label{maps}\n\\end{figure}\n\nWritten in Koszul matrix form, $\\overline{C}(\\Gamma^0) = (\\mathbf{a}, \\mathbf{b})$ and $\\overline{C}(\\Gamma^1) = ( \\mathbf{c}, \\mathbf{d})\\{-1\\}$, where\n\\[(\\mathbf{a}, \\mathbf{b}) =\n \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41} & x_1-x_4 \\\\ \n\\overline{\\pi}_{32} & x_2-x_3\\end{array} \n\\right) \\quad (\\mathbf{c}, \\mathbf{d}) = \n\\left( \\begin{array}{cc}\n\\overline{u}_1 & x_1+x_2-x_3-x_4 \\\\ \\overline{u}_2 & x_1x_2-x_3x_4\\end{array}\\right),\\]\nand where the shift $\\{-1\\}$ in $\\overline{C}(\\Gamma^1)$ is applied to the second row \n$$R \\stackrel {\\overline{u}_2}{\\longrightarrow} R\\{1\\} \\stackrel{x_1x_2-x_3x_4}{\\longrightarrow} R.$$\nWe shifted the degrees of $R$ so that each differential above has degree 3.\n\nFinally, we define the $\\overline{C}(\\Gamma)$ as the tensor product of $\\overline{C}(\\Gamma^1)$ over all singular resolution, of $\\overline{L}_i^j$ over all arcs $l,$ and of $\\mathcal{A} \\brak{1}$ over all loops with no mark. The tensor product is considered over appropriate rings, so that $\\overline{C}(\\Gamma)$ is a free module of finite rank over $R$, and we treat it as a graded factorization---with infinite rank---over the subring $R'.$ \n\n\\begin{lemma}\nGiven any web diagram $\\Gamma,$ its associated factorization $\\overline{C}(\\Gamma)$ lies in $\\textit{hmf}_\\omega.$ Moreover, if $\\Gamma'$ is obtained from $\\Gamma$ by placing a different collection of internal marks then there is a canonical isomorphism $\\overline{C}(\\Gamma') \\cong \\overline{C}(\\Gamma)$ in $\\textit{hmf}_\\omega.$\n\\end{lemma}\n\n\\begin{lemma} For any disjoint union of webs $\\Gamma_1 \\cup \\Gamma_2$ there is a canonical isomorphism in $\\textit{hmf}_\\omega,$ namely\n$\\overline{C}(\\Gamma_1 \\cup \\Gamma_2) \\cong \\overline{C}( \\Gamma_1) \\otimes_{\\mathbb{Q}[a,h]}\\overline{C}( \\Gamma_2).$ In particular we have\n$$\\overline{C}(\\Gamma \\cup \\raisebox{-5pt}{\\includegraphics[height=.2in]{orienloop.pdf}}) \\cong \\overline{C}(\\Gamma)\\brak{1} \\otimes_{\\mathbb{Q}[a,h]} \\mathcal{A}.$$\n\\end{lemma}\n\nNote that multiplication by $\\partial_i \\omega : = \\partial \\omega \/ \\partial {x_i}$ endomorphism of $\\overline{C}(\\Gamma)$ is homotopic to zero. Moreover, multiplication by any polynomial in $(\\textbf{a}, \\textbf{b})$ induces a null-homotopic endomorphism of $\\overline{C}(\\Gamma)$ (see~\\cite[Proposition 2]{KhR1}). \n\n\\textit{Excluding a variable.} An important tool introduced in~\\cite{KhR1} is the process of ``excluding a variable''. Supposed that $x_i$ is one of the generators of the polynomial ring$R$ and that $\\omega = \\sum a_i b_i \\in R',$ where $R = R'[x_i].$ We say that $x_i$ is an \\textit{internal} variable. Any $(R, \\omega)$-factorization $(\\textbf{a, b})$ restricts to an infinite rank factorization over $R'$, and we denote it by $(\\textbf{a, b})'.$ Suppose furthermore that for some $j,$ $b_j= x_i -\\alpha$ where $\\alpha \\in R'.$ Denote by $(\\textbf{a'}, \\textbf{b'})$ the factorization over $R'$ obtained from $(\\textbf{a, b})$ by removing the $j$-th row and substituting $\\alpha$ for $x_i$ everywhere in all other rows.\n \\begin{lemma}\n Factorizations $(\\textbf{a, b})'$ and $(\\textbf{a'}, \\textbf{b'})$ are isomorphic in the homotopy category of $(R', \\omega)$-factorizations. \\end{lemma}\n \n If $\\Gamma$ is a closed web, the potential $\\omega = 0$ and $\\overline{C}(\\Gamma)$ is a 2-periodic complex, thus we can take the cohomology of the corresponding complex. In particular, to the basic closed web with two vertices and with arcs labeled by $x_1$ and $x_2$ we assign the factorization over $\\mathbb{Q}[a, h]$ with trivial potential, which is the quotient of $\\overline{C}(\\Gamma^1)$ by the relations $x_1 = x_4$ and $x_2 = x_3.$ We obtain a complex with homology only in degree zero:\n\\[ H^0(\\overline{C}(\\Gamma^1)\/_{x_1 = x_4,x_2 = x_3}) = \\mathbb{Q}[a,h,x_1, x_2]\/(\\overline{u}^*_1, \\overline{u}^*_2) \\{-1\\}, \\, \\text{where}\\]\n\\begin{align*}\n \\overline{u}^*_1 &= \\overline{u}_1\/ _{x_1=x_4, x_2 =x_3} = 3(x_1 + x_2)^2 -3x_1x_2 -3h(x_1 + x_2) -3a \\\\ \\overline{u}^*_2 &= \\overline{u}_2\/ _{x_1=x_4, x_2 =x_3} = -3(x_1 + x_2 -h).\n\\end{align*}\nTherefore $H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1 + x_2 -h,\\, x_1x_2 + a) \\{-1 \\},$ or equivalently\n\\[ H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\cong \\mathbb{Q}[a,h,x_1]\/(x_1^2 -h x_1- a) \\{-1 \\} \\cong \\mathcal{A} \\quad \\text{and} \\quad H^1(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) = 0.\\]\n\nSimilarly, the quotient of $\\overline{C}(\\Gamma^0)$ by the relations $x_1 = x_4$ and $x_2 = x_3$ is a 2-complex with homology only in degree zero:\n\\[ H^0(\\overline{C}(\\Gamma^0)\/_{x_1 = x_4,x_2 = x_3}) = \\mathbb{Q}[a,h,x_1, x_2]\/(\\overline{\\pi}^*_{41}, \\overline{\\pi}^*_{32}) \\{-2\\}, \\, \\text{where}\\]\n\\begin{align*}\n \\overline{\\pi}^*_{41} &= \\overline{\\pi}_{41}\/ _{x_1=x_4, x_2 =x_3} = 3(x_1^2 -h x_1 - a) \\\\ \\overline{\\pi}^*_{32} &= \\overline{u}_2\/ _{x_1=x_4, x_2 =x_3} = -3(x_2^2 -h x_2 -a).\n\\end{align*}\nTherefore $H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1^2 +h x_1 -a,\\, x_2^2 - hx_2 - a) \\{-2 \\}.$\n\nEquivalently, $H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\cong \\mathcal{A} \\otimes \\mathcal{A} \\quad \\text{and} \\quad H^1(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) = 0.$\n\n \\subsection{Maps $\\Lambda_0$ and $\\Lambda_1$}\\label{sec:maps}\nThe maps $\\Lambda_0$ and $\\Lambda_1$ between $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1),$ where $\\Gamma^0$ and $\\Gamma^1$ are the web diagrams from Figure~\\ref{maps}, are given by a pair of $2 \\times 2$ matrices $(U_0, U_1)$ and $(V_0, V_1),$ respectively. \n\\[U_0 = \n\\left(\\begin{array}{cc}x_4 - x_2 & 0 \\\\ x_1 - x_2 + x_3 + 2x_4 -\\frac{3}{2} h & 1\\end{array}\\right), \\quad U_1 = \\left(\n\\begin{array}{cc}x_4 & -x_2 \\\\ -1 & 1\\end{array}\\right)\n\\]\n \n\\[V_0 = \n\\left(\\begin{array}{cc}1 & 0 \\\\ -x_1 + x_2 - x_3 - 2x_4 +\\frac{3}{2}h & x_4-x_2\\end{array}\\right), \\quad V_1 = \\left(\n\\begin{array}{cc}1 & x_2 \\\\ 1 & x_4\\end{array}\\right)\n\\]\n \nThe compositions $\\Lambda _0 \\Lambda_1$ and$\\Lambda_1 \\Lambda_0$ are homotopic to the multiplication by $x_4 - x_2$ endomorphism of $\\overline{C} (\\Gamma^0)$ and $\\overline{C}(\\Gamma^1),$ respectively. This is easily seen from the following relations:\n \\[U_0V_0 = U_1V_1 =(x_4 -x_2)\\id , \\quad V_0U_0 = V_1U_1 = (x_4 -x_2) \\id.\\]\nThus $\\Lambda_1 \\circ \\Lambda_0 = m(x_4-x_2)$ and $\\Lambda_0 \\circ \\Lambda_1 = m(x_4-x_2)$. On the other hand, since the endomorphism of $\\overline{C}(\\Gamma^0)$---or $\\overline{C}(\\Gamma^1)$--- given by the multiplication by $x_1 + x_2 - x_3 -x_4$ is null-homotopic, we also have $\\Lambda_1 \\circ \\Lambda_0 = m(x_1 - x_3)$ and $\\Lambda_0 \\circ \\Lambda_1 = m(x_1 - x_3).$ Both maps $\\Lambda_0$ and $\\Lambda_1$ are maps of degree 1.\n\nConsidering the Koszul matrices for $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1),$ we apply certain row transformation to each of them:\n\n$ \\overline{C}(\\Gamma^0): \\quad \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41} & x_1-x_4 \\\\ \n\\overline{\\pi}_{32} & x_2-x_3\\end{array} \\right) \\stackrel{[21]_{-1}}{\\longrightarrow} \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41} & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{\\pi}_{32} -\\overline{\\pi}_{41} & x_2-x_3\\end{array} \\right) $ \n\n$ \\overline{C}(\\Gamma^1): \\quad \\left(\\begin{array}{cc}\n\\overline{u}_1 & x_1 + x_2 - x_3 -x_4 \\\\ \n\\overline{u}_2 & x_1x_2-x_3x_4\\end{array} \\right) \\stackrel{[12]_{x_2}}{\\longrightarrow} \\left(\\begin{array}{cc}\n\\overline{u}_1 + x_2 u_2 & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{u}_2 &(x_4 - x_2)(x_2-x_3)\\end{array} \\right).$ \n\nWe apply further a twist to obtain \n\\[ \\overline{C}(\\Gamma^0) \\stackrel {\\cong}{\\longrightarrow} \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41} -k (x_2-x_3) & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{\\pi}_{32} -\\overline{\\pi}_{41} + k (x_1 + x_2 -x_3 -x_4 ) & x_2-x_3\\end{array} \\right)\\]\n\n\\[\\overline{C}(\\Gamma^1) \\stackrel{\\cong}{\\longrightarrow} \\left(\\begin{array}{cc}\n\\overline{u}_1 + x_2 u_2 + 2 (x_4 - x_2)(x_2-x_3) & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{u}_2 -2 (x_1 + x_2 -x_3 -x_4) & (x_4 - x_2)(x_2-x_3)\\end{array} \\right) \\]\n\nwhere $k = x_1 + x_2 + x_3 -\\frac{3}{2}h.$ An easy computation shows that the Koszul matrices above, hence $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1)$, have the following equivalent forms \n\n$ \\overline{C}(\\Gamma^0) \\cong \\left(\\begin{array}{cc}\na_1 & x_1 + x_2 -x_3 -x_4 \\\\ \na_2 (x_4 -x_2) & x_2-x_3\\end{array} \\right) \\, \\text{and} \\,\\, \\overline{C}(\\Gamma^1) \\cong \\left(\\begin{array}{cc}\na_1 & x_1 + x_2 -x_3 -x_4 \\\\ \na_2 & (x_4 -x_2)(x_2-x_3)\\end{array} \\right) $\n\nwhere $a_1 = \\overline{\\pi}_{41} - \\overline{\\pi}_{21} + \\overline{\\pi}_{31}$ and $a_2 = -2x_1 -2x_2 -x_3 -x_4 +3h.$ The first rows are identical and the second rows are related. Consider the \\textit{flip} homomorphisms $$\\psi_{x_4-x_2}\\co (a_2, (x_4-x_2)(x_2-x_3)) \\longrightarrow (a_2(x_4-x_2), x_2-x_3)$$\n\\begin{displaymath} \n\\xymatrix @C=23mm@R=13mm{\nR \\ar [r]^{a_2} \\ar [d]_{ 1} & R \\ar [d]_{x_4-x_2} \\ar[r]^{(x_4-x_2)(x_2-x_3)} & R \\ar [d]_1 \\\\\nR \\ar [r]^{a_2(x_4-x_2)} & R \\ar [r]^{x_2-x_3} & R}\n\\end{displaymath}\nand $ \\psi'_{x_4-x_2} \\co (a_2(x_4-x_2), x_2-x_3) \\longrightarrow (a_2, (x_4-x_2)(x_2-x_3))$\n\n\\begin{displaymath} \n\\xymatrix @C=23mm@R=13mm{\nR \\ar [r]^{a_2(x_4-x_2)} \\ar [d]_{ x_4-x_2} & R \\ar [d]_1 \\ar[r]^{x_2-x_3} & R \\ar [d]_{x_4-x_2} \\\\\nR \\ar [r]^{a_2} & R \\ar [r]^{(x_4-x_2)(x_2-x_3)} & R}\n\\end{displaymath}\nUsing the equivalent Koszul matrices for $\\overline{C}(\\Gamma^0), \\overline{C}(\\Gamma^1),$ the following lemma follows. \n\\begin{lemma}\n$\\Lambda_0 = \\id \\otimes \\,\\psi'_{x_4-x_2}$ and $\\Lambda_1 = \\id \\otimes \\,\\psi_{x_4-x_2}.$\n\\end{lemma}\n\n\\subsection{Complexes of factorizations}\\label{sec:complexes}\nWe start with a generic diagram $D$ of a tangle $T$ with boundary points $B$, and put (at least) one mark on each segment bounded by two crossings. We let $m(D)$ be the set of all marks of $D,$ and consider the polynomial ring $R = \\mathbb{Q}[a, h, x_i]$ for all $i \\in m(D),$ and its subring $R' = \\mathbb{Q}[a, h, x_i]$ for all $i \\in B.$ \n\nWe associate to each crossing $p$ in $D$ a complex $\\overline{C}_p$ of matrix factorizations, as explained in Figure~\\ref{fig:crossings to chain complexes}, where the underlined objects are at the cohomological degree $0.$ \n \\begin{figure}[ht]\n\\[ \\raisebox{-13pt}{\\includegraphics[height=.45in]{poscros.pdf}} \\,\\, = \\,\\, \\left[\\,\\, 0 \\longrightarrow \\raisebox{-13pt}{\\includegraphics[height=.45in]{singresol.pdf}}\\,\\,\\{2\\} \\longrightarrow \\underline{\\raisebox{-13pt}{\\includegraphics[height=.45in]{orienresol.pdf}}\\,\\, \\{1\\}}\\longrightarrow 0\\,\\, \\right ]\\]\n\n\\[ \\raisebox{-13pt}{\\includegraphics[height=.45in]{negcros.pdf}} \\,\\, = \\,\\, \\left[\\,\\, 0 \\longrightarrow \\underline{\\raisebox{-13pt}{\\includegraphics[height=.45in]{orienresol.pdf}}\\,\\,\\{-1\\}} \\longrightarrow\\raisebox{-13pt}{\\includegraphics[height=.45in]{singresol.pdf}}\\,\\, \\{-2\\}\\longrightarrow 0\\,\\, \\right ]\\]\n\\caption{Complex applied to a crossing}\n\\label{fig:crossings to chain complexes}\n\\end{figure}\n\nPrecisely, we have\n\\[ \\raisebox{-8pt}{\\includegraphics[height=.3in]{poscros.pdf}} = [\\,0 \\longrightarrow \\overline{C}(\\Gamma^1)\\{2\\} \\stackrel{\\Lambda_1}{\\longrightarrow} \\overline{C}(\\Gamma^0)\\{1\\} \\longrightarrow 0\\, ]\\]\n\\[ \\raisebox{-8pt}{\\includegraphics[height=.3in]{negcros.pdf}} = [\\, 0 \\longrightarrow \\overline{C}(\\Gamma^0)\\{-1\\} \\stackrel{\\Lambda_0}{\\longrightarrow} \\overline{C}(\\Gamma^1)\\{-2\\} \\longrightarrow 0\\,] \\]\nwhere $\\Gamma^0, \\Gamma^1$ are the oriented and the singular resolutions from Figure~\\ref{fig:resolutions}, and where the matrix factorization $\\overline{C}(\\Gamma^0)$ is at the cohomological degree $0.$\n\nWe associate to $D$ a complex of factorizations, which we denote it by $C(D),$ and which is the tensor product of $\\overline{C}_p,$ over all crossings $p$ in $D,$ of $\\overline{L}_i^j,$ over all arcs $i \\to j,$ and of $\\mathcal{A} \\brak{1},$ over all oriented loops in $D$ with no crossings and no marks. The tensoring is done over appropriate polynomial rings so that $C(D)$ is a free $R$-module of finite rank.\n\n$C(D)$ is a complex of graded ($R, \\omega$)-factorizations, where $\\omega = \\sum_{i \\in B} o(i)p(a, h, x_i).$ We regard $C(D)$ as an object in $K_\\omega : = Kom(\\textit{hmf}_\\omega),$ the homotopy category of complexes over $\\textit{hmf}_\\omega.$ Note that the differentials of $C(D)$ are grading-preserving. In Section \\ref{sec:invariance} we show that the isomorphism class of $C(D)$ is a tangle invariant.\n\nIf $T$ is a link, the set $B$ of boundary points is empty, $R' = \\mathbb{Q}[a, h],$ and $C(D)$ is a complex of graded $\\mathbb{Q}[a, h]$-modules. \n\n\\subsection{Isomorphisms}\\label{sec:isomorphisms}\nWe show that $\\overline{C}(\\Gamma)$ mimics the skein relations of Figure~\\ref{fig:skein relations}. \n\\begin{proposition}\\label{isom 1}(\\textit{First Isomorphism}) There is an isomorphism in $hmf_\\omega$:\n\\[\\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=.3in]{singresclosed.pdf}}\\,) \\cong \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=.3in]{arc.pdf}}\\,)\\brak{1}.\\]\n\\end{proposition}\n\n\\begin{proof} Consider the webs $\\Gamma$ and $\\Gamma_1$ given in Figure~\\ref{fig:direct sum decomposition 1-webs}. Factorizations $\\overline{C}(\\Gamma),\\overline{C}(\\Gamma_1)$ are $(R', \\omega)$-factorizations (the later has infinite rank over $R'$), where $R' = \\mathbb{Q}[a, h, x_2,x_3]$ and $\\omega = p(a, h, x_2) - p(a, h, x_3).$\n\\begin{figure}[h]\n\\[ \\Gamma = \\raisebox{-27pt}{\\includegraphics[height=.8in]{larc.pdf}} \\hspace{3cm} \\Gamma_1 = \\raisebox{-27pt}{\\includegraphics[height=.8in]{lsingresclosed.pdf}}\\]\n\\caption{Webs $\\Gamma$ and $\\Gamma_1$}\n\\label{fig:direct sum decomposition 1-webs}\n\\end{figure}\n\nIn Koszul form, $\\overline{C}(\\Gamma_1) = \\left (\\begin{array}{cc}\\overline{u}'_1 & x_2-x_3 \\\\\n\\overline{u}'_2 & x_1(x_2 - x_3) \\end{array} \\right)$ where $\\overline{u}'_1 = \\overline{u}_1(x_2, x_1,x_1, x_3)$ and $\\overline{u}'_2 = \\overline{u}_1(x_2, x_1,x_1, x_3).$ We apply the elementary row operation $[12]_{x_1}$ and obtain:\n$$\\left (\\begin{array}{cc}\\overline{u}'_1 & x_2-x_3 \\\\\n\\overline{u}'_2 & x_1(x_2 - x_3) \\end{array} \\right) \\stackrel{\\cong}{\\longrightarrow} \\left (\\begin{array}{cc}\\overline{u}'_1+ x_1 \\overline{u}'_2 & x_2-x_3 \\\\\n\\overline{u}'_2 & 0 \\end{array} \\right)$$\nAn easy computation shows that $\\overline{u}'_1+ x_1 \\overline{u}'_2= \\overline{\\pi}_{23}$ and $\\overline{u}'_2 = -3(x_1+x_3) + 3h,$ thus we have $\\overline{C}(\\Gamma_1) \\brak{1}\\cong \\left (\\begin{array}{cc}\\overline{\\pi}_{23} & x_2-x_3 \\\\\n0 & -\\overline{u}'_2 \\end{array} \\right).$ Since $x_1$ is an internal variable, we can eliminate it by removing the row $(0 \\quad -\\overline{u}'_2) .$ Therefore, $\\overline{C}(\\Gamma_1) \\brak {1} \\cong (\\overline{\\pi}_{23}, \\, x_2-x_3 ) = \\overline{C}(\\Gamma).$ \\end{proof}\n\n\\begin{proposition}\\label{isom 2}(\\textit{Second Isomorphism})\nThere is an isomorphism in the category $hmf_{\\omega}$:\n\\[\\overline{C}(\\raisebox{-13pt}{\\includegraphics[height=.45in]{removedweb1.pdf}}) \\cong \\overline{C}(\\raisebox{-13pt}{\\includegraphics[height=.45in]{removedweb2.pdf}})\\{-1\\} \\oplus \\overline{C}(\\raisebox{-13pt}{\\includegraphics[height=.45in]{removedweb2.pdf}})\\{1\\}.\\]\n\\end{proposition}\n\n\\begin{proof} The proof is the same as that of the ``Direct sum decomposition II'' in~\\cite{KhR1}.\n\\end{proof}\n\n\\begin{proposition}\\label{isom 3}(\\textit{Third Isomorphism})\nThere is an isomorphism in the category $hmf_{\\omega}$:\n\\[\\overline{C}(\\, \\raisebox{-8pt}{\\includegraphics[height=.3in]{2singresol.pdf}}\\,) \\cong \\overline{C}(\\,\\raisebox{-5pt}{\\includegraphics[height=.25in]{2arcs.pdf}}\\,).\\]\n\\end{proposition}\n\\begin{proof} Consider the webs $\\Gamma$ and $\\Gamma'$ depicted in Figure~\\ref{fig:direct sum decomposition 3-webs}. Factorizations $\\overline{C}(\\Gamma),\\overline{C}(\\Gamma')$ are $(R', \\omega)$-factorizations, where $R' = \\mathbb{Q}[a, h, x_1, x_2, x_3, x_4]$ and $\\omega = p(a, h, x_1) - p(a, h, x_2) + p(a, h, x_3) - p(a, h, x_4).$\n\\begin{figure}[h]\n\\[ \\Gamma = \\raisebox{-20pt}{\\includegraphics[height=.6in]{l2singres.pdf}} \\hspace{3cm} \\Gamma' = \\raisebox{-20pt}{\\includegraphics[height=.6in]{larcsleftright.pdf}}\\]\n\\caption{Webs $\\Gamma$ and $\\Gamma'$}\n\\label{fig:direct sum decomposition 3-webs}\n\\end{figure}\n\nIn Koszul form, $\\overline{C}(\\Gamma) = (\\textbf{a, b})\\{-2\\} = \\left(\\begin{array}{cc}\\overline{u}_1' & x_1 + x_5 -x_4 -x_6 \\\\ \\overline {u}_2' & x_1x_5 -x_4x_6\\\\ \\overline{u}_1'' & x_3 + x_6 -x_2 -x_5 \\\\ \\overline{u}_2'' & x_3x_6 -x_2x_5 \\end{array} \\right )\\{-2\\},$ \n\nwhere $\\overline{u}_i' = \\overline{u}_i(x_1, x_5, x_6, x_4)$ and $\\overline{u}_i''= \\overline{u}_i(x_3, x_6, x_5, x_2), i = 1, 2.$ A shift of $\\{-1\\}$ corresponds to the second and fourth row. The potential $\\omega$ lives in $R',$ thus $x_5$ and $x_6$ are internal variables. Knowing that $\\overline{u}_2' = -3(x_4 + x_6) + 3h$ and $\\overline{u}_2''= -3(x_2 + x_5) + 3h,$ we can exclude $x_5, x_6$ by crossing out the second and fourth row and replacing $x_5 = h-x_2$ and $x_6 = h-x_4$ in the first and third row of $(\\textbf{a, b}).$ In particular, $\\overline{C}(\\Gamma)$ is isomorphic to the matrix factorization with Koszul form\n$$(\\textbf{c, d}) = \\left(\\begin{array}{cc}\\overline{u}_1(x_1, h -x_2, h-x_4, x_4) & x_1 -x_2 \\\\ \\overline{u}_1(x_3, h-x_4,h- x_2, x_2) & x_3 -x_4\\end{array} \\right). $$\n\nAn easy computations shows that $\\overline{u}_1(x_1, h -x_2,h- x_4, x_4) = \\overline{\\pi}_{21}$ and $\\overline{u}_1(x_3, h -x_3, h-x_2, x_2) = \\overline{\\pi}_{43}$, thus $\\{\\textbf{c, d}\\} = \\overline{C}(\\Gamma'),$ which implies that $\\overline{C}(\\Gamma) \\cong \\overline{C}(\\Gamma').$\n\\end{proof}\n\n\\begin{proposition}\\label{isom 4}(\\textit{Fourth Isomorphism})\nThere are isomorphisms in the category $hmf_{\\omega}$:\n\\[\\overline{C}(\\, \\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-4.pdf}}\\,) \\cong \\overline{C}(\\,\\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-1.pdf}}\\,) \\quad \\text{and} \\quad \\overline{C}(\\, \\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-2.pdf}}\\,) \\cong \\overline{C}(\\,\\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-3.pdf}}\\,).\\]\n\\end{proposition}\n\n\\begin{proof}\nConsider the webs $\\Gamma_1$ and $\\Gamma_2$ depicted in Figure~\\ref{fig:direct sum decomposition 4-webs}. This time, the potential has the form $\\omega = \\sum_{i \\in \\{1,2,3\\}}p(a, h, x_i) - \\sum_{j \\in \\{4,5,6\\}}p(a, h, x_j) \\in R',$ where $R' = \\mathbb{Q}[a, h, x_1, x_2, x_3, x_4, x_5, x_6].$\n\\begin{figure}[h]\n\\[ \\Gamma_1 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-4.pdf}} \\hspace{3cm} \\Gamma_2 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-1.pdf}}\\]\n\\caption{Webs $\\Gamma_1$ and $\\Gamma_2$}\n\\label{fig:direct sum decomposition 4-webs}\n\\end{figure}\n\nIn Koszul form, $\\overline{C}(\\Gamma_2) = \\left(\\begin{array}{cc}\\overline{u}_1' & x_1 + x_2 -x_7 -x_8 \\\\ \\overline {u}_2' & x_1x_2 -x_7x_8\\\\ \\overline{u}_1'' & x_3 + x_8 -x_6 -x_9 \\\\ \\overline{u}_2'' & x_3x_8 -x_6x_9 \\\\ \\overline{u}_1''' & x_7 + x_9 -x_4 -x_5 \\\\ \\overline{u}_2''' & x_7x_9 -x_4x_5 \\end{array} \\right )\\{-3\\},$ \n\nwhere $\\overline{u}_i' = \\overline{u}_i(x_1, x_2, x_8, x_7), \\overline{u}_i''= \\overline{u}_i(x_8, x_3, x_6, x_9)$ and $\\overline{u}_i'''= \\overline{u}_i(x_7, x_9, x_5, x_4)$ for $i = 1, 2.$ A shift by $\\{-1\\}$ was applied to the rows of $\\overline{u}_2', \\overline{u}_2''$ and $\\overline{u}_2'''.$ Variables $x_7, x_8$ and $x_9$ are internal variables and we can use $\\overline{u}_2' = -3(x_7 + x_8) + 3h$ and $\\overline{u}_2''= -3(x_6 + x_9) + 3h$ to exclude $x_8$ and $x_9,$ by crossing out the second and fourth row and substituting in every other row $x_8 = h-x_7$ and $x_9 = h-x_6.$ To exclude the internal variable $x_7,$ we use the right-hand entry of the third row, which now has the form $-x_7 + x_3.$ After these operations, we obtain\n$$\\overline{C}(\\Gamma_2) \\cong \\left(\\begin{aligned}a_1 &= \\overline{u}_1(x_1, x_2 ,h-x_3, x_3) \\\\ a_2 &= \\overline{u}_1(x_3, h-x_6, x_5, x_4) \\\\ a_3 &= \\overline{u}_2(x_3, h-x_6, x_5, x_4) \\end{aligned} \\quad \\begin{aligned} b_1 &= x_1 + x_2 -h \\\\ b_2 &= x_3 +h - x_6 -x_4 -x_5\\\\ b_3 &= x_3(h-x_6) - x_4 x_5 \\end{aligned} \\right )\\{-1\\}.$$\n We perform the row operation $[21]_{-1}$ and arrive at \n\\[ \\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a_1 & b_1 + b_2 \\\\ a_2 -a_1 & b_2 \\\\ a_3 & b_3 \\end{array} \\right).\\] \nNow we apply a twist to the left-hand entries of the first two rows above, for $k= \\frac{h}{2} -x_1 -x_2 -x_4 -x_5 + x_3 -x_6,$ to obtain\n \\[ \\left(\\begin{array}{cc}a_1+ kb_2 & b_1 + b_2 \\\\ a_2 -a_1 -k(b_1 + b_2) & b_2\\\\ a_3 & b_3 \\end{array} \\right). \\] \n Performing the row operation $[23]_{x_3 -x_6}$ we get\n \\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a_1+ kb_2 & b_1 + b_2 \\\\ a_2 -a_1 -k(b_1 + b_2)+(x_3-x_6)a_3 & b_2\\\\ a_3 & b_3 - (x_3 -x_6)b_2 \\end{array} \\right ) = \\left(\\begin{array}{cc}a'_1& b'_1 \\\\ a'_2& b'_2\\\\ a'_3 & b'_3 \\end{array} \\right).\\]\n The later Koszul factorization $(\\textbf{a}', \\textbf{b}')$ is isomorphic in $hmf_{\\omega}$ to $\\overline{C}(\\Gamma_2)\\{1\\},$ and we apply to it a twist with $k' =-\\frac{1}{3}$\n \\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a'_1& b'_1 \\\\ a'_2& b'_2 - \\frac{1}{3} a'_3\\\\ a'_3 & b'_3 + \\frac{1}{3} a'_2 \\end{array} \\right),\\]\nfollowed by the row operation $[23]_{-(x_3 - x_6 - x_4 - x_5)}$ \n \\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a'_1& b'_1 \\\\ a'_2 -(x_3 -x_6 -x_4- x_5)a'_3& b'_2 - \\frac{1}{3} a'_3\\\\ a'_3 & b'_3 + \\frac{1}{3} a'_2 + (x_3 -x_6 -x_4- x_5)(b'_2 -\\frac{1}{3}a'_3) \\end{array}\\right).\\]\n \n Let's denote the previous Koszul matrix by $(\\textbf{a}'', \\textbf{b}'').$ Replacing the entries of $\\textbf{b}''$ we have\n\\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a''_1& x_1+ x_2 +x_3 -x_4-x_5-x_6 \\\\ a''_2 & x_3-x_6\\\\ a''_3 & x_1x_2-x_4x_5 \\end{array} \\right).\\]\n Finally, we apply the row operation $[21]_1$ followed by a twist with $k'' = -\\frac{3}{2}h+x_1 + x_2 + 2x_4 + 2x_5 - x_3 + x_6,$ and we get\n \\begin{align*}\\overline{C}(\\Gamma_2)\\{1\\} &\\cong \\left(\\begin{array}{cc}a''_1& x_1+ x_2 -x_4-x_5 \\\\ a''_2 + a''_1 & x_3-x_6\\\\ a''_3 & x_1x_2-x_4x_5 \\end{array} \\right)\\\\\n &\\cong \\left(\\begin{array}{cc}a''_1 +k''(x_3-x_6) & x_1+ x_2 -x_4-x_5 \\\\ a''_2 + a''_1 - k''(x_1 + x_2 -x_4 -x_5) & x_3-x_6\\\\ a''_3 & x_1x_2-x_4x_5\\end{array}\\right).\\end{align*}\n Computing the entries in the left column above, one founds\n \\begin{align*}\\overline{C}(\\Gamma_2)\\{1\\} &\\cong \\left(\\begin{array}{cc}\\overline{u}_1(x_1, x_2, x_4, x_5) & x_1+ x_2 -x_4-x_5 \\\\ \\overline{\\pi}_{63} & x_3-x_6 \\\\ \\overline{u}_2(x_1 ,x_2, x_4, x_5) & x_1x_2-x_4x_5 \\end{array} \\right)\\\\\n &\\cong \\left(\\begin{array}{cc} \\overline{\\pi}_{63} & x_3-x_6 \\\\ \\overline{u}_1(x_1, x_2, x_4, x_5) & x_1+ x_2 -x_4-x_5 \\\\ \\overline{u}_2(x_1 ,x_2, x_4, x_5) & x_1x_2-x_4x_5 \\end{array} \\right) \\cong \\overline{C}(\\Gamma_1)\\{1\\}.\\end{align*}\nThe later matrix is the Koszul form of the factorization $\\overline{C}(\\Gamma_1)\\{1\\},$ thus $\\overline{C}(\\Gamma_2) \\cong \\overline{C}(\\Gamma_1).$ \n \\begin{figure}[h]\n\\[ \\Gamma_3 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-2.pdf}} \\hspace{3cm} \\Gamma_4 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-3.pdf}}\\]\n\\caption{Webs $\\Gamma_3$ and $\\Gamma_4$}\n\\label{fig:direct sum decomposition 4bis-webs}\n\\end{figure}\n\nWith the labeling of the diagrams $\\Gamma_3, \\Gamma_4$ given in Figure~\\ref{fig:direct sum decomposition 4bis-webs}, the previous proof also implies that $\\overline{C}(\\Gamma_4) \\cong \\overline{C}(\\Gamma_3)$ in $hmf_\\omega.$\n\\end{proof}\n\n\\begin{definition}\nLet $\\Gamma$ be a closed web and $p(\\Gamma)$ be the mod 2 number of circles in the modification of $\\Gamma$ obtained by replacing all singular resolutions with the oriented resolution. Factorization $\\overline{C}(\\Gamma)$ is a 2-complex and has cohomology only in degree $p(\\Gamma).$ Define the cohomology groups of $\\overline{C}(\\Gamma)$ as\n\\[ \\overline{H}(\\Gamma) : = H^{p(\\Gamma)}(\\overline{C}(\\Gamma)). \\]\n\\end{definition}\n\n$\\overline{H}(\\Gamma)$ is a $\\mathbb{Z}$-graded module over $\\mathbb{Q}[a, h].$ The isomorphisms obtained in this section together with the fact that the skein relations in Figure~\\ref{fig:skein relations} determine the evaluation of $\\brak{\\Gamma},$ for any web $\\Gamma,$ imply the following result.\n\n\\begin{proposition}\nFor any closed web $\\Gamma,$ the graded dimension of $\\overline{H}(\\Gamma)$ is $\\brak {\\Gamma},$ namely\n\\[ \\sum _{j \\in \\mathbb{Z}} q^j \\rk_{\\mathbb{Q}[a,h]} \\overline{H}^j(\\Gamma) = \\brak{\\Gamma}.\\]\n\\end{proposition}\n\n\\begin{remark}\nNote that any resolution $\\Gamma$ of a link diagram consists of a disjoint union of closed webs $\\cup_k \\Gamma_k,$ and its ``homology'' satisfies $\\overline{H}(\\Gamma) \\cong \\mathcal{A}^k.$ In Section~\\ref{sec:TQFT} we show that $\\overline{H}$ can be regarded as a $(1+1)$--dimensional TQFT functor.\n\\end{remark}\n\n\n\n\\section{Invariance under Reidemeister moves}\\label{sec:invariance}\n\n\\begin{theorem}\\label{invariance}\nIf $D$ and $D'$ are two diagrams representing the same tangle $T,$ then complexes $C(D)$ and $C(D')$ are isomorphic in $K_{\\omega}.$\n\\end{theorem}\n\\begin{proof}\n\\textbf{Reidemeister I.}\nConsider diagrams $D$ and $\\Gamma$ that differ only in a circular region as in the figure below:\n$$D=\\raisebox{-11pt}{\\includegraphics[height=0.45in]{leftkink.pdf}}\\qquad\n\\Gamma=\\raisebox{-13pt}{\\includegraphics[height=0.45in]{arc.pdf}}$$\nThe complex $C(D)$ has the form\n$$0\\longrightarrow \\underline{\\overline{C}(\\Gamma_2) \\{-1\\}} \\stackrel{\\Lambda_0}{\\longrightarrow} \\overline{C}(\\Gamma_1) \\{-2\\} \\longrightarrow 0$$ \nwhere\n$$\\Gamma_2 = \\raisebox{-20pt}{\\includegraphics[height=.6in]{lreid1orien.pdf}}\\quad \\mbox{and} \\quad\\raisebox{-22pt}{\\includegraphics[height=0.65in]{lsingresclosed.pdf}} = \\Gamma_1$$\nLet $Y_1 \\subset \\overline{C}(\\Gamma_2)\\{-1\\}$ be the inclusion of $\\overline{C}(\\Gamma)\\brak{1}\\{-2\\}$ onto the first summand of $\\overline{C}(\\Gamma_2)\\{-1\\},$ and $Y_2 = f(\\overline {C}(\\Gamma)\\brak{1}\\{-2\\}) \\subset \\overline{C}(\\Gamma_2)\\{-1\\},$ where $f \\co \\overline {C}(\\Gamma)\\brak{1}\\{-2\\} \\to \\overline{C}(\\Gamma_2)\\{-1\\}, f = \\id \\otimes[ m(x_1) \\iota]$. Here we used that $\\overline{C}(\\Gamma_2) \\cong \\overline{C}(\\Gamma)\\brak{1} \\otimes_{\\mathbb{Q}[a,h]} \\mathcal{A}.$\n\nThus $\\overline{C}(\\Gamma_2)\\{-1\\} \\cong Y_1 \\oplus Y_2$ in $hmf_\\omega.$ On the other hand, from the First Isomorphism we know that $\\overline{C}(\\Gamma)\\{-2\\} \\cong \\overline{C}(\\Gamma) \\brak{1}\\{-2\\},$ therefore\n the complex $\\overline{C}(D)$ is isomorphic in $K_{\\omega}$ to the direct sum\n\\begin{align*} \n0 \\longrightarrow &\\underline{Y_2} \\longrightarrow 0 \\\\ \n0 \\longrightarrow &\\underline{Y_1} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma)\\brak{1}\\{-2\\}. \\end{align*} \nSince $Y_2 \\cong \\overline{C}(\\Gamma)$ and the second summand is contractible, it implies that $C(D) \\cong (\\Gamma),$ in the category $K_{\\omega}.$\n\nA similar approach is used to prove the invariance under Reidemeister I involving a positive kink.\n\n\\textbf{Reidemeister IIa.} Consider diagrams $D_1$ and $\\Gamma$ that differ only in a circular region as in the figure below:\n$$D_1=\\raisebox{-11pt}{\\includegraphics[height=0.45in]{tuckedsame.pdf}}\\qquad\n\\Gamma=\\raisebox{-13pt}{\\includegraphics[height=0.45in]{arcsup.pdf}}$$\n\nThe complex $C(D_1)$ has the form\n$$ 0\\longrightarrow \\overline{C}(\\Gamma_{00})\\{1\\} \\stackrel{(f_1, f_3)^t}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{01}) \\oplus \\overline{C}(\\Gamma_{10})} \\stackrel{(f_2, - f_4)}{\\longrightarrow} \\overline{C}(\\Gamma_{11})\\{-1\\} \\longrightarrow 0,$$\nwhose objects are the matrix factorizations corresponding to the four resolutions of $D$ given in Figure~\\ref{fig:reidIIA}, with potential $\\omega = p(a, h, x_1) + p(a, h, x_2) - p(a, h, x_3) - p(a, h, x_4).$\n\n\\begin{figure}[ht]\n\\raisebox{-11pt}{\\includegraphics[height=2.2in]{reidIIA.pdf}}\n\\caption{}\\label{fig:reidIIA}\n\\end{figure}\nUsing the Second Isomorphism and that the marking doesn't matter, we have \n$$\\overline{C}(\\Gamma_{01}) \\cong \\overline{C}(\\Gamma^0)$$ $$ \\overline{C}(\\Gamma_{00})\\cong \\overline{C}(\\Gamma^1) \\cong \\overline{C}(\\Gamma_{11})$$ $$ \\overline{C}(\\Gamma_{10})\\cong \\overline{C}(\\Gamma^1)\\{1\\} \\oplus \\overline{C}(\\Gamma^1)\\{-1\\},$$\n\nwhere $\\Gamma^0$ and $\\Gamma^1$ are the diagrams from figure~\\ref{maps}. Therefore the complex $C(D_1)$ is isomorphic (in $K_{\\omega}$) to the complex\n$$ 0 \\longrightarrow \\overline{C}(\\Gamma^1)\\{1\\} \\stackrel{(f_1, f_{03}, f_{13})^t}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma^0) \\oplus \\overline{C}(\\Gamma^1)\\{1\\} \\oplus \\overline{C}(\\Gamma^1) \\{-1\\}} \\stackrel{(f_2, -f_{04}, -f_{14})}{\\longrightarrow} \\overline{C}(\\Gamma^1)\\{-1\\} \\longrightarrow 0,$$\n\n(where $f_{03}, f_{13}$ and $ f_{04}, f_{14}$ are the components of $f_3$ and $f_4$, respectively, under the Second Isomorphism). The later complex decomposes into the direct sum of complexes\n\\begin{align*}\n0 \\longrightarrow & \\underline{\\overline{C}(\\Gamma^0)} \\longrightarrow 0 \\\\\n0\\longrightarrow \\overline{C}(\\Gamma^1)\\{1\\} \\stackrel{f_{03}}{\\longrightarrow} & \\underline{\\overline{C}(\\Gamma^1) \\{1\\}} \\longrightarrow 0 \\\\\n0 \\longrightarrow &\\underline{\\overline{C}(\\Gamma^1)\\{-1\\}} \\stackrel{f_{14}}{\\longrightarrow} \\overline{C}(\\Gamma^1)\\{-1\\} \\longrightarrow 0.\\end{align*}\n\nThe last two complexes are contractible (this is because the only degree $0$ endomorphisms of $\\overline{C}(\\Gamma^1)$ are rational multiples of the identity endomorphism, thus $f_{03}$ and $f_{14}$ are isomorphisms). Moreover, $\\overline{C}(\\Gamma^0) \\cong \\overline{C}(\\Gamma)$ in $hmf_\\omega^{fd},$ and we conclude that $C(D_1)$ and $C(\\Gamma)$ are isomorphic in $K_\\omega.$\n\n\\textbf{Reidemeister IIb.}\n$$D_2=\\raisebox{-7pt}{\\includegraphics[height=0.25in]{tuckedop.pdf}}\\qquad\n\\Gamma'=\\raisebox{-7pt}{\\includegraphics[height=0.25in]{arcsop.pdf}}$$\n\nThe complex of matrix factorizations $C(D_2)$ is an element of the category $K_{\\omega},$ where $\\omega = p(a, h, x_1) - p(a, h, x_2) + p(a, h, x_3) - p(a, h, x_4),$ and has the form\n$$ 0\\longrightarrow \\overline{C}(\\Gamma_{00})\\{1\\} \\stackrel{(f_1, f_3)^t}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{01}) \\oplus \\overline{C}(\\Gamma_{10})} \\stackrel{(f_2, - f_4)}{\\longrightarrow} \\overline{C}(\\Gamma_{11})\\{-1\\} \\longrightarrow 0.$$\n\nThe resolutions of $D_2$ are given in Figure~\\ref{fig:reidIIB}. \n\\begin{figure}[ht]\n\\raisebox{-11pt}{\\includegraphics[height=2in]{reidIIB.pdf}}\n\\caption{}\\label{fig:reidIIB}\n\\end{figure}\n\nWe know that $ \\overline{C}(\\Gamma_{01}) \\cong \\overline{C}(\\Gamma'') \\brak{1} \\otimes _{\\mathbb{Q}[a,h]} \\mathcal{A} \\cong \\overline{C}(\\Gamma'') \\brak{1} \\{1\\} \\oplus \\overline{C}(\\Gamma'')\\brak{1} \\{-1\\},$\n$$\\overline{C}(\\Gamma_{00}) \\cong \\overline{C}(\\Gamma'') \\brak{1} \\cong \\overline{C}(\\Gamma_{11})\\quad \\mbox{and} \\quad \\overline{C}(\\Gamma_{10}) \\cong \\overline{C}(\\Gamma'),$$\nwhere $\\Gamma''$ is the diagram in Figure~\\ref{fig:Gamma-dprime}. Here we used the First Isomorphism and the Third Isomorphism, and that marking doesn't matter.\n\\begin{figure}[ht]\n\\raisebox{-11pt}{\\includegraphics[height=.9in]{larcsupdown.pdf}}\n\\caption{}\\label{fig:Gamma-dprime}\n\\end{figure}\n\nConsequently, $C(D_2)$ is isomorphic to the following complex\n$$ 0 \\longrightarrow \\overline{C}(\\Gamma'') \\brak{1}\\{1\\} \\longrightarrow \\underline{\\overline{C}(\\Gamma'') \\brak{1} \\{1\\} \\oplus \\overline{C}(\\Gamma'') \\brak{1} \\{-1\\} \\oplus \\overline{C}(\\Gamma')} \\longrightarrow \\overline{C}(\\Gamma'') \\brak{1} \\{-1\\} \\longrightarrow 0,$$\nwhich decomposes into the direct sum of complexes\n\\begin{align*}\n0 \\longrightarrow &\\underline{\\overline{C}(\\Gamma')} \\longrightarrow 0 \\\\\n0 \\longrightarrow \\overline{C}(\\Gamma'')\\brak{1}\\{1\\} \\stackrel{\\cong}{\\longrightarrow} &\\underline{\\overline{C}(\\Gamma'')\\brak{1}\\{1\\}} \\longrightarrow 0 \\\\\n0 \\longrightarrow & \\underline{\\overline{C}(\\Gamma'')\\brak{1}\\{-1\\}} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma'')\\brak{1}\\{-1\\} \\longrightarrow 0.\\end{align*}\nThe last two are contractible, therefore $C(D_2) \\cong C(\\Gamma')$ in the category $K_{\\omega}.$\n\n\\textbf{Reidemeister III.} Given diagrams $D$ and $D'$ below, we show that complexes $C(D)$ and $C(D')$ are isomorphic by showing they are both isomorphic to the same third complex.\n\n$$D \\quad \\raisebox{-13pt}{\\includegraphics[height=.5in]{reid3left.pdf}} \\hspace{2cm} \\raisebox{-13pt}{\\includegraphics[height=.5in]{reid3right.pdf}} \\quad D'$$\n\nThe cube of resolutions corresponding to the diagram $D$ is given in Figure~\\ref{fig:reidIII-left}, and that of $D'$ in Figure~\\ref{fig:reidIII-right}.\n\\begin{figure}[ht]\n\\raisebox{-8pt}{\\includegraphics[height=3.5in]{reidIII-left.pdf}}\n\\caption{The cube of resolutions of $D$}\\label{fig:reidIII-left}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\raisebox{-8pt}{\\includegraphics[height=3.5in]{reidIII-right.pdf}}\n\\caption{The cube of resolutions of $D'$}\\label{fig:reidIII-right}\n\\end{figure}\n\nThe complex $C(D)$ has the form\n\\[0 \\longrightarrow \\overline{C}(\\Gamma_{000}) \\{6\\} \\stackrel{d^{-3}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma_{010})\\{5\\} \\\\ \\overline{C}(\\Gamma_{100})\\{5\\} \\end{array} \\right ) \\stackrel{d^{-2}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{011})\\{4\\} \\\\ \\overline{C}(\\Gamma_{101})\\{4\\} \\\\ \\overline{C}(\\Gamma_{110})\\{4\\} \\end{array} \\right) \\stackrel{d^{-1}}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{111})\\{3\\}} \\longrightarrow 0.\\]\n\nWe know that\n\\begin{align*}\n\\overline{C}(\\Gamma_{110}) & \\cong \\overline{C}(\\Gamma_{011}) \\quad (\\text{the two webs are isotopic})\\\\\n \\overline{C}(\\Gamma_{010}) &\\cong \\overline{C}(\\Gamma_{011}) \\{1\\} \\oplus \\overline{C}(\\Gamma_{011}) \\{-1\\} \\quad (\\text{by the Second Isomorphism})\\\\\n \\overline{C}(\\Gamma_{000}) &\\cong \\overline{C}(\\Gamma_{011}) \\quad (\\text{by the Fourth Isomorphism}).\\end{align*}\n \nThe complex $C(D')$ has the same form as $C(D).$ Moreover, similar isomorphisms of matrix factorizations as those above hold for resolutions of $D',$ with the remark that $\\Gamma_{ijk}$ in $C(D)$ should be replaced by $\\Gamma'_{ijk}$ in $C(D').$\n \n $C(D)$ is isomorphic to the following complex\n \\[0 \\longrightarrow \\overline{C}(\\Gamma_{011}) \\{6\\} \\stackrel{d^{-3}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma_{011})\\{6\\} \\\\ \\overline{C}(\\Gamma_{011})\\{4\\}\\\\ \\overline{C}(\\Gamma_{100})\\{5\\} \\end{array} \\right ) \\stackrel{d^{-2}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{011})\\{4\\} \\\\ \\overline{C}(\\Gamma_{101})\\{4\\} \\\\ \\overline{C}(\\Gamma_{011})\\{4\\} \\end{array} \\right) \\stackrel{d^{-1}}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{111})\\{3\\}} \\longrightarrow 0,\\]\n but the later decomposes into contractible complexes of the form \n\\[0 \\longrightarrow \\overline{C}(\\Gamma_{011}) \\{6\\} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma_{011}) \\{6\\} \\longrightarrow 0 \\]\n\\[0 \\longrightarrow \\overline{C}(\\Gamma_{011}) \\{4\\} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma_{011}) \\{4\\} \\longrightarrow 0 \\]\nand the complex $\\mathcal{C}$\n\\[\\mathcal{C}: \\quad 0 \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma_{100})\\{5\\} \\end{array}\\right ) \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{011})\\{4\\} \\\\ \\overline{C}(\\Gamma_{101})\\{4\\} \\end{array}\\right) \\longrightarrow \\underline{\\overline{C}(\\Gamma_{111})\\{3\\}} \\longrightarrow 0. \\]\nIn other words, complexes $C(D)$ and $\\mathcal{C}$ are isomorphic in $K_\\omega.$\n \nWe apply the same argument as in the case of $C(D)$ to conclude that $C(D')$ is isomorphic in $K_\\omega$ to the complex $\\mathcal{C'}$\n\\[\\mathcal{C'}: \\quad 0 \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma'_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma'_{100})\\{5\\} \\end{array}\\right ) \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma'_{101})\\{4\\} \\\\ \\overline{C}(\\Gamma'_{011})\\{4\\} \\end{array}\\right) \\longrightarrow \\underline{\\overline{C}(\\Gamma'_{111})\\{3\\}} \\longrightarrow 0. \\]\nSince the web diagram $\\Gamma_{ijk}$ from $\\mathcal{C}$ and $\\Gamma'_{ijk}$ from $\\mathcal{C'}$ respectively are isotopic, their matrix factorizations are isomorphic in $hmf_{\\omega}.$ In particular $\\mathcal{C} \\cong \\mathcal {C'}$ in $K_\\omega,$ and the invariance under the third type of Reidemeister move follows. \n\\end{proof}\n\n\\begin{corollary}The isomorphism class of the object $C(D)$ in the category $K_\\omega$ is an invariant of the tangle $T.$\n\\end{corollary}\n\n\\textbf{The case of links.} When $T$ is a link $L,$ for any resolution $\\Gamma$ of $D,$ the corresponding factorization $\\overline{C}(\\Gamma)$ is a 2-periodic complex of graded $R' = \\mathbb{Q}[a,h]$-modules. The category $K_\\omega$ is isomorphic to the category of finite-rank $\\mathbb{Z} \\oplus \\mathbb{Z} \\oplus \\mathbb{Z}_2$-graded $\\mathbb{Q}[a,h]$-modules. For each $\\Gamma,$ the homology groups of $\\overline{C}(\\Gamma)$ are nontrivial only in one degree, thus the cohomology of $C(D)$ is $\\mathbb{Z} \\oplus \\mathbb{Z}$-graded.\n\nWe denote by $KR_{a,h}(D)$ the universal Khovanov-Rozansky complex for $n=2.$ It is obtained from $C(D)$ by replacing each matrix factorization $\\overline{C}(\\Gamma)$ with its homology group $\\overline{H}(\\Gamma).$ Furthermore, we denote by $HKR_{a,h}$ the cohomology of $KR_{a,h}.$ Note that $HKR_{a,h}$ is exactly the $\\mathbb{Z} \\oplus \\mathbb{Z}$-graded cohomology of $C(D).$ We have\n$$ HKR_{a,h}(D) : = \\bigoplus_{i, j \\in \\mathbb{Z}} H^{i,j}(C(D)). $$ \nIt follows from construction that the graded Euler characteristic of $HKR_{a,h}(D)$ is the polynomial $\\brak{L}.$ Specifically, the following holds\n$$ \\brak{L} = \\sum_{i,j \\in \\mathbb{Z}} (-1)^i q^j \\rk_{\\mathbb{Q}[a,h]} H^{i,j} (D).$$\n\n\n\\section{Understanding the differentials}\\label{sec:TQFT}\n\nThe ring $\\mathcal{A} = \\mathbb{Q}[a,h, X]\/(X^2 -hX - a)$ is commutative Frobenius with trace map\n $\\epsilon \\co \\mathcal{A} \\rightarrow \\mathbb{Q}[a, h], \\,\\,\\epsilon(1) = 0, \\,\\epsilon(X) = 1$.\n Multiplication $m \\co \\mathcal{A} \\otimes \\mathcal{A} \\rightarrow \\mathcal{A}$ and comultiplication $\\Delta \\co \\mathcal{A} \\rightarrow \\mathcal{A} \\otimes \\mathcal{A}$ are defined by\n$$ \\begin{cases}\n m(1 \\otimes X) =X, & m(X \\otimes 1) = X\\\\ \nm(1 \\otimes 1) =1, & m(X \\otimes X) =hX+ a\n \\end{cases},\\quad\n \\begin{cases}\n \\Delta(1) = 1 \\otimes X + X \\otimes 1-h1\\otimes 1\\\\ \n \\Delta(X) = X \\otimes X + a 1 \\otimes 1.\\end{cases} $$\n\nRecall that $\\mathcal{A} = \\brak {1, X}_{\\mathbb{Q}[a, h]}$ is graded with $\\deg(1) = -1$ and $\\deg(X) = 1.$ The trace $\\epsilon$ and unit $\\iota$ are maps of degree $-1,$ while multiplication and comultiplication are maps of degree 1.\n\nLet $\\Lambda_0^* \\co \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\to \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}})$ and $\\Lambda_1^*\\co \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\to \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}})$ be the maps induced by $\\Lambda_0$ and $\\Lambda_1,$ respectively, at the homology level of the corresponding factorizations.\n\nWe know that \n\\begin{align*}\n\\overline{C} (\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) &\\stackrel {\\textit{hmf}_\\omega}{\\cong} \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircle}}\\,)\\brak{1}, \\quad \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircle}}\\,) \\cong \\mathcal{A}, \\\\\n\\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) &\\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1 + x_2 -h,\\, x_1x_2 + a) \\{-1 \\} \\\\ & \\cong \\brak{1, x_1}_{\\mathbb{Q}[a,h]}\\{-1\\}, \\\\ \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) &\\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1^2 -hx_1 - a, \\, x_2^2 -hx_2 - a) \\{-2\\} \\\\\n&\\cong \\brak{1, x_1, h-x_2, x_1(h-x_2)}_{\\mathbb{Q}[a,h]} \\{-2\\}.\\end{align*}\n \nThere are $\\mathbb{Q}[a, h]$-module isomorphisms:\n\\begin{align} \\label{isomorphism f}\n \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\stackrel{f}{\\longrightarrow} \\mathcal{A}, \\quad & f \\co \\begin{cases} 1 \\to 1\\\\ x_1 \\to X \\end{cases} \\\\ \\label{isomorphism g}\n \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\stackrel{g}{\\longrightarrow} \\mathcal{A} \\otimes \\mathcal{A}, \\quad & g\\co \\begin{cases} 1\\to 1\\otimes 1,\\,\\, x_1 \\to X \\otimes 1 \\\\ h-x_2 \\to 1 \\otimes X, \\,\\, x_1(h-x_2) \\to X\\otimes X.\\end{cases} \\end{align}\n \n \\begin{lemma}\n $\\Lambda_0^* = m$ and $\\Lambda_1^* = \\Delta.$\n \\end{lemma}\n \\begin{proof}\n We know that $\\Lambda_0\/(x_1 = x_4, x_2 = x_3) = \\id \\otimes \\,\\psi'_{x_1-x_2}$ and $\\Lambda_1\/(x_1 = x_4, x_2 = x_3) = \\id \\otimes \\,\\psi_{x_1-x_2},$ that factorizations $\\overline{C}(\\Gamma^0)\/(x_1 = x_4, x_2 = x_3) = \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}})$ and $\\overline{C}(\\Gamma^1)\/(x_1 = x_4, x_2 = x_3) = \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}})$ have nontrivial homology only in degree 0. \n \n Applying the definition of the morphism $\\psi'_{x_1-x_2}$ we have \n \\begin{align*} \\Lambda^*_0(x_1) &= x_1, &\\Lambda^*_0(h-x_2) &= h-x_2 \\\\\n\\Lambda^*_0(1) &= 1, &\\Lambda^*_0 (x_1(h-x_2)) &= x_1(h-x_2).\\end{align*} But $h-x_2 = x_1$ and $x_1(h-x_2) = hx_1 + a$ in $\\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}),$ thus using the isomorphisms \\ref{isomorphism f} and \\ref{isomorphism g} it follows that $\\Lambda_0^* = m.$ Similarly, using the definition of $\\psi_{x_1-x_2}$ we obtain\n \\begin{align*} \\Lambda^*_1(1) &= x_1-x_2 \\\\ \\Lambda^*_1(x_1) &= (x_1-x_2)x_1.\\end{align*} \n But $(x_1-x_2)x_1 = hx_1 +a -x_1x_2 = a + x_1(h-x_2)$ in $\\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}),$ and $x_1-x_2 = x_1 +(h -x_2) - h.$ By using again the isomorphisms \\ref{isomorphism f} and \\ref{isomorphism g} we obtain $\\Lambda^*_1 = \\Delta.$\n\\end{proof}\n\nIt is well known that the commutative Frobenius algebra $\\mathcal{A}$ gives rise to a TQFT functor---denoted here by $\\mathsf{F}$---from the category $OrCob$ of oriented $(1+1)$--dimensional cobordisms to the category $\\mathbb{Q}[a,h]$-Mod of graded $\\mathbb{Q}[a, h]$-modules and module homomorphisms. The functor assigns the ground ring $\\mathbb{Q}[a, h]$ to the empty 1-manifold, and $\\mathcal{A}^{\\otimes k}$ to the disjoint union of oriented $k$ circles. On the generating morphisms of $OrCob$, the functor associates the structure maps of the algebra $\\mathcal{A}.$ In particular, $\\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.15in]{cuplo.pdf}}) = \\iota,\\, \\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.15in]{caplo.pdf}}) = \\epsilon,\\, \\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.16in]{pair-of-pants.pdf}}) = m$ and\\, $\\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.16in]{upsidedown-pants.pdf}}) = \\Delta$ (note that we read cobordisms from bottom to top). \n\nWe remark that the objects in $OrCob$ are oriented in such a way that a cobordism between two objects is ``properly'' oriented. For that, one needs to orient each nesting (concentric) set of circles so that orientations alternate, giving to the outermost circle the same orientation, for all nesting sets of circles. \n\nGiven a cobordism $S \\in OrCob,$ the homomorphism $\\mathsf{F}(S)$ has degree given by the formula $\\deg(S) = -\\chi(S),$ where $\\chi$ is the Euler charecteristic for $S.$ \n\nConsider a link diagram $D$ with its hypercube of resolutions and associated complex of matrix factorizations $C(D).$ Each resolution $\\Gamma$ of $D$ is a collection of closed webs with an even number of vertices and oriented loops. The homology group $\\overline{H}(\\Gamma)$ of the associated factorization $\\overline{C}(\\Gamma)$ satisfies $\\overline{H}(\\Gamma) \\cong \\mathcal{A}^k,$ where $k$ is the number of connected components of $\\Gamma.$ \n\nWe can replace each resolution of $D$ by a simplified one that has fewer number of vertices, via the First, Third and Fourth Isomorphisms. After this operation we are left with a complex of factorizations isomorphic in $K_\\omega$ to $C(D),$ in which each resolution is a disjoint union of basic closed webs (with exactly two vertices) and oriented loops. Finally, applying the First Isomorphism to all basic closed webs, the latter complex is isomorphic in $K_\\omega$ to a complex of factorizations whose underlying geometric objects are column vectors of nested oriented loops. Each nested set of loops is oriented in such a way that the outermost loop is oriented clockwise, by convention, and as we go inside of the nesting set of loops orientations alternate. In particular, the latter formal complex, call it $\\mathcal{C},$ is an object in the category of complexes over $OrCob.$ Applying the functor $\\mathsf{F}$ to $\\mathcal{C},$ we obtain an ordinary complex $\\mathsf{F}(\\mathcal{C})$ whose objects are $\\mathbb{Q}[a,h]$-modules and whose differentials are $\\mathbb{Q}[a,h]$-module homomorphisms. Moreover, $\\mathsf{F}(\\mathcal{C})$ is homotopy equivalent to $KR_{a,h}(D).$ \nConsequently, the following corollary is implied by the results of this section.\n\n\\begin{corollary}\nThe functor $\\overline{H}$ behaves in the same manner as $\\mathsf{F}$ does. \n\\end{corollary}\n\nRecalling the author's construction and results in~\\cite{CC2}, the previous Corollary implies that the underlying link cohomology and that introduced in~\\cite{CC2} are isomorphic, after tensoring them with appropriate rings. In particular, the functor $\\overline{H}$ is the same as the tautological functor $\\mathcal{F}$ in~\\cite{CC2} (at least when we restrict to the case of links). Moreover, this implies that the theory constructed here is functorial under link cobordisms, relative to boundaries.\n\n\\begin{corollary}\\label{cor:isomorphism}\nFor each oriented link $L$ there is an isomorphism \n$$HKR_{a,h}(L) \\otimes_{\\mathbb{Q}[a,h]} \\mathbb{Z}[i] \\cong \\mathcal{H}(L) \\otimes_{\\mathbb{Z}[i][a,h]} \\mathbb{Q}$$\nwhere $\\mathcal{H}$ is the cohomology link theory explored in~\\cite{CC2}. \\end{corollary}\n \n\\begin{corollary}\nGiven a link cobordism $C \\co L_1 \\to L_2,$ there is a well-defined induced map $HKR_{a,h}(C) \\co HKR_{a,h}(L_1) \\to HKR_{a,h}(L_2)$ between the associated cohomology groups.\n\\end{corollary}\n\n\n\\section{The algebra $\\overline{R}(\\Gamma)$}\n\nLet $\\Gamma$ be a resolution of a link $L$ and denote by $e(\\Gamma)$ the set of all edges in $\\Gamma.$ We introduce an algebra $\\overline{R}(\\Gamma)$ and exhibit the $\\overline{R}(\\Gamma)$ module structure of $\\overline{H}(\\Gamma).$\n\n\\begin{definition}\nWe define $\\overline{R}(\\Gamma) : = R\/(\\textbf{a}, \\textbf{b}),$ where $R = \\mathbb{Q}[a,h][X_i\\vert i \\in e(\\Gamma)]$ and $\\textbf{a}, \\textbf{b}$ are the polynomials that are used to define the factorization $\\overline{C}(\\Gamma).$\n\\end{definition}\n\n$\\overline{H}(\\Gamma)$ is an $\\overline{R}(\\Gamma)$ module, since multiplication by any polynomial in $(\\textbf{a},\\textbf{b})$ induces a null-homotopic endomorphisms of $\\overline{C}(\\Gamma).$\n\n\\begin{proposition}\\label{relations}\nThe algebra $\\overline{R}(\\Gamma)$ is spanned by generators $X_i,$ where $i\\in e(\\Gamma),$ subject to the following relations:\n\\begin{enumerate}\n\\item For every $X_i$ we have $X_i^2 = h X_i + a.$\n\\item For every singular resolution (of the form $\\Gamma^1$) in $\\Gamma,$ the generators $X_1, X_2, X_3, X_4$ satisfy $X_1 + X_2 = h = X_3 + X_4,$ and $X_1 X_2 = -a = X_3 X_4.$\n\\end{enumerate} \n\\end{proposition}\n\\begin{proof}\nFor each closed circle---with one mark $i$---in $\\Gamma,$ the expression $\\overline{\\pi}_{ii} = 3(x_i^2 - hx_i -a)$ lives in $(\\textbf{a},\\textbf{b}).$ For any other $X_i,$ we use that the multiplication by $\\partial_i \\omega= 3(x_i^2 -hx_i -a)$ induces a null-homotopic endomorphism of $\\overline{C}(\\Gamma).$ Thus $X_i^2 = h X_i + a$ holds for any $X_i.$ To prove the second statement we recall that for each singular resolution with arcs labeled as those of $\\Gamma^1,$ the polynomials $x_1 + x_2 -x_3 -x_4$ and $x_1x_2 -x_3x_4$ are in $(\\textbf{a},\\textbf{b}),$ thus $X_1 + X_2 = X_3+X_4$ and $X_1X_2 = X_3X_4$ in $\\overline{R}(\\Gamma),$ near each singular resolution of $\\Gamma.$ Moreover, since $\\overline{u}_2 = 3(x_3 +x_4) -3h$ and $\\overline{u}_1 = (x_1 + x_2)^2 + (x_1 + x_2)(x_3 + x_4) + (x_3 + x_4)^2 -3x_1x_2 -\\frac{3}{2} h( x_1+x_2+x_3 + x_4) -3a$ are also in $(\\textbf{a},\\textbf{b}),$ we have $X_1 + X_2 = h = X_3 + X_4,$ and $X_1 X_2 = -a = X_3 X_4.$\n\\end{proof}\n\n\\begin{remark}\nRelations given in Proposition~\\ref{relations} have a geometric interpretation via a TQFT with dots, where a dot stands for the multiplication by $X$ endomorphism of the algebra $\\mathcal{A}.$ Specifically, the relation $X_i^2 = h X_i + a$ translates into the following skein relation: a disk decorated by two dots equals a disk decorated by one dot times $h$ plus a disk times $a$ (this one is the relation (2D) in~\\cite{CC2}). Given two edges in $\\Gamma$ with labels $i$ and $j$ and which share a vertex, the relation $X_i + X_j = h$ gives a rule for exchanging dots between two neighboring facets of a foam, while relation $X_i X_j = -a$ means that if each of the two neighboring facets have a single dot, we can ``erase'' both dots and multiply the corresponding foam by $-a$ (see relations (ED) in~\\cite[Figure 7]{CC2}).\n\\end{remark}\n\nIt was proved in\\cite{CC2} that if one lets $a$ and $h$ be complex numbers and considers the polynomial $f(X) = X^2 -hX -a \\in \\mathbb{C}[X],$ the isomorphism class of the complex $\\mathfrak{sl}(2)$-foam cohomology $\\mathcal{H}(L, \\mathbb{C})$ is determined by the number of distinct roots of $f[X].$ Corollary~\\ref{cor:isomorphism} implies that this also holds for the complex matrix factorization cohomology $HKR_{a,h}(L, \\mathbb{C}).$ The results are as follows.\n\nIf $f(X) = (X-\\alpha)^2,$ for some $\\alpha \\in \\mathbb{C},$ there is an isomorphism between $HKR_{a,h}(L, \\mathbb{C})$ and the original $\\mathfrak{sl}(2)$ Khovanov-Rozansky cohomology over $\\mathbb{C}.$\n\nIf $f(X) = (X -\\alpha)(X-\\beta),$ for some $\\alpha, \\beta \\in \\mathbb{C},$ for each resolution $\\Gamma$ of a link $L,$ the cohomology $\\overline{H}(\\Gamma)$ is a free module of rank one over the complex algebra $\\overline{R}(\\Gamma).$ \n\n\\begin{proposition}\nFor any $n$-component link $L,$ the dimension of $HKR_{a,h}(L, \\mathbb{C})$ equals $2^n,$ and to each map $\\phi \\co \\{ \\text{components of L} \\} \\to S = \\{ \\alpha, \\beta \\}$ there exists a non-zero element $h_{\\phi} \\in HKR_{a,h}(L, \\mathbb{C})$ which lies in the cohomological degree\n$$ -2 \\sum_{\\substack {(u_1,u_2)\\in S \\times S \\\\ u_1 \\neq u_2}} lk(\\phi^{-1}(u_1), \\phi^{-1}(u_2)),$$\nand all $h_{\\phi}$ generate $HKR_{a,h}(L, \\mathbb{C}).$\n\\end{proposition}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Understanding and Improving Human-Machine Joint Action}\n\nHumans regularly make decisions with and alongside other humans. In what has come to be defined as {\\em joint action}, humans coordinate with other humans to achieve shared goals, sculpting both their actions and their expectations about their partners during ongoing interaction (Sebanz et al. 2006; Knoblich et al. 2011; Pesquita et al. 2018). Humans now also regularly make decisions in partnership with computing machines in order to supplement their abilities to act, perceive, and decide (Pilarski et al. 2017). It is natural to expect that joint action with machine agents might be able to improve both work and play. In situations where someone is limited in their ability to perceive, remember, attend to, respond to, or process stimulus, a machine counterpart's specialized and complementary abilities to monitor, interpret, store, retrieve, synthesize, and relate information can potentially offset or even invert these limitations. Persistent computational processes that extend yet remain part of human cognition, perhaps best described in the words of the author Terry Pratchett as ``third thoughts,'' are evident in common tools like navigation software and calendar reminders.\n\nSpecifically with a focus on joint action, as opposed to more general forms of human-machine interaction, Moon et al. (2013), Bicho et al. (2010), Pilarski et al. (2013), Pezzulo et al. (2011), and others have provided compelling examples of fruitful human-machine coordination wherein a human and a machine work together and co-adapt in real-time joint action environments. One striking characteristic of many of these examples, and what separates them from other examples of human-machine interaction, is that they occur within {\\em peripersonal space} (Knoblich et al. 2011)---i.e., interaction is perceived by the human to unfold continuously in the region of physical space surrounding their body and upon which they can act. While the perception of spatial and temporal proximity between partners has been shown to significantly influence joint decision making (as reviewed by Knoblich et al. 2011), peripersonal joint action settings have to date received less attention than other settings for human-machine interaction. The study of how different machine learning approaches impact human-machine joint action is even less developed, but in our opinion equally important. Our present work therefore aims to extend the discussion on how a human decision maker (here termed a {\\em pilot}) and a machine learning assistant (termed a {\\em copilot}) can learn together and make decisions together in service of a shared goal. We do so by defining a virtual reality environment to probe real-time joint action between humans and learning machines, and, using this environment, contribute a human-machine interaction case study wherein we demonstrate the kinds of changes that might be expected as a pilot interacts with different machine learning copilots in a shared environment. \n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[height=2in]{domain1}\n \\includegraphics[height=2in]{domain2} \n \\includegraphics[height=2in]{domain3} \n \\caption{\\footnotesize The virtual-reality foraging environment used to explore human-agent learning and joint action, comprised of six equidistant fruit objects, background detail, and a repeated cycle of day and night illumination. The human participant (the {\\em pilot}) was able to move about within the virtual world and used their hand-held controllers to both harvest fruit in varying states of ripeness and train their machine-learning assistant (the {\\em copilot}).}\n \\label{fig:world}\n\\end{figure}\n\n\\section{Virtual Reality Environment and Protocol}\n\nA single participant engaged in a foraging task over multiple experimental blocks. This foraging task was designed so as to embed a hard-to-learn sensorimotor skill within a superficially simple protocol. In each block, the participant was asked to interact with a simulated world and a machine assistant via a virtual reality (VR) headset and two hand-held controllers (HTC Vive with deluxe audio strap). The pilot was instructed that in each block they were to move through the world to collect objects, and that these objects would grant them ``points''; they were told that, during the experiment, their total points would be reflected in visual changes to the environment, and that they would receive a unique, momentary audio cue whenever they gained or lost points as a result of their actions (and that they could also be given different audio cues in situations where they might expect to gain or lose points). \n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=5in]{schematic.png}\n \\caption{\\footnotesize Schematic showing the interactions available to both the human pilot and the machine copilot. Using the right and left hand controllers, respectively, the pilot was able to harvest fruit or train their copilot to estimate the value of hue\/saturation combinations, each action resulting in different sound cues as shown. On every time step, the copilot provided sound cues according to its learned predictions $V(h,s)$ and, for the bandit condition, its learned policy $\\pi(V)$.}\n \\label{fig:protocol}\n \\vspace{-1em}\n\\end{figure}\n\nLoosely inspired by the bee foraging example of Schultz et al. (1997), the virtual world presented to the pilot was a simple platform floating in space with six coloured balls (``fruit'') placed at equidistant locations around its perimeter (Fig. \\ref{fig:world}). To tease out behavioural changes in different conditions for pilot\/copilot joint action, over the course of a single trial the light level in the world varied from full sunlight (``day'') to regular periods of darkness lasting roughly 20s (``night'', c.f., Fig. \\ref{fig:results}c, bottom trace). While copilot perception was unaffected by light level, during twilight and night phases the pilot was by design unable to determine the colour of any objects in the environment, seeing only their general shape. The main mechanic of the world was that the pilot could collect fruit to either increase their points or, in some blocks, collect fruit in order to teach their copilot. \n\nTo create conditions for skill learning on the part of the pilot, the environment was designed with a confounding factor (ripeness) that related the appearance of the fruit as observed by the pilot to the reward structure of the task. Over the course of time, each fruit underwent a ``ripening cycle'' wherein it cycled through a progression of colours---hue and saturation levels---and reward that varied in relation to the time since the fruit's appearance. If a fruit was not contacted before a fixed time interval by the pilot, i.e., the end of the ripening cycle, it would disappear and no point gains or losses will be credited to the participant. A short, variable time after the participant collected a fruit or the fruit disappeared due to time, a new fruit would drop from above to the same position as the previous fruit. These new fruit were assigned a random fraction between 0\\% and 95\\% through their ripening cycle. \n\nThe pilot started each trial in the centre of the platform, was able to move short distances and rotate in place, and could make contact with the fruit using either their left or right hand controllers (Fig. \\ref{fig:protocol}). Upon making contact with one of the fruit via their right-hand controller (``harvesting''), the ball would disappear and the pilot would hear an audio cue indicating that they either gained or lost points (one distinct sound for each case, with a small mote of light appearing in place of the fruit when points were gained). Upon contacting a ball with their left-hand controller (``teaching''), the fruit would disappear with no points gained or lost by the pilot, and they would hear a unique tone indicating that they had given information to their copilot (for blocks involving a copilot, otherwise the fruit disappeared and no sound played).\n\nHarvesting fruit was the only way the pilot could gain points. The points gained or lost by the participant for collecting a fruit was determined by a sinusoidal reward function that varied according to the time since the beginning of the ripening cycle (Fig. \\ref{fig:protocol}). All fruit in a given trial were assigned the same, randomly selected reward sinusoid (in terms of phase) and related hue\/saturation progression---i.e., all balls in a given trial would ripen in the same way, and would generate points in the same way, but these ripening mechanics would vary from trial to trial. The frequency of the reward sinusoid (i.e., number of reward maxima and minima until a fruit's disappearance), positive\/negative reward offset, length of the ripening cycle, and time range until reappearance (all akin to difficulty), were predetermined and held constant across all trials and blocks. Difficulty in terms of these parameters was empirically preselected and calibrated so as to provide the pilot with a challenging pattern-learning problem that was still solvable within a single trial.\n\nFollowing an extended period of familiarization with the navigation and control mechanics of the VR environment and the different interaction conditions, the participant (one of the co-authors for this pilot study) experienced three experimental blocks each consisting of three 180s trials; each trial utilized previously unseen ripening mechanics in terms of colour presentation and points phase. There was an approximately 20s break between trials. The blocks related to the three different conditions (Fig. \\ref{fig:protocol}), presented in order, as follows:\n\n\n{\\bf Condition 1, No copilot (NoCP): } The pilot harvested fruit without any signalling or support from a machine copilot. Using the left, training hand to contact fruit had no effect, and did not provide any additional audio cues.\n\n\n{\\bf Condition 2, Copilot communication via Pavlovian control (Pav):} The pilot harvested fruit with support from a machine copilot that {\\em provided audio cues in a fixed way} based on its learned predictions. As described above, the pilot was able to train the copilot by contacting fruit with their left hand controller. Practically, this amounted to updating the value function of the copilot, denoted $V(h,s)$, according to the points value associated with a fruit's current hue $h$ and saturation level $s$ at the time of contact; updates were done via simple supervised learning. At each point in time, the copilot queried its learned value function $V(h,s)$ with the current colour values of each of the six fruits, and, if the value of $V(h,s)$ was positive for a fruit, triggered an audio cue that was unique to that fruit---each fruit had a characteristic sound. In other words, feedback from the copilot to the human pilot was based on a pre-determined function that mapped learned predictions to specific actions (an example of Pavlovian control and communication, c.f., Modayil et al. (2014) and Parker et al. (2014)).\n\n{\\bf Condition 3, Copilot communication learned through trial and error (Bandit):} The pilot harvested fruit with support from a machine copilot that {\\em provided audio cues in an adaptable way} based on collected points (reward) and its learned predictions. This condition was similar to Condition 2 in terms of how the pilot was able to train the copilot. However, instead of deterministically playing an audio cue for the pilot each time the copilot's prediction $V(h,s)$ for a given fruit was positive, the copilot was instead presented with a decision whether or not to play an audio cue for the pilot. The decision to play a cue was based on a stochastic policy $\\pi(V)$ that was updated as in a contextual bandit approach (Sutton and Barto, 2018) according to the points collected by the pilot if and when a fruit was harvested. In essence, if the copilot cued the user and this resulted in the pilot gaining points a short time later, it would reinforce the copilot's probability of playing a sound when it predicted a similar level of points in the future; should the pilot instead harvest the fruit and receive negative points, as when a fruit is harvested after the peak in its ripening cycle and\/or the pilot is consistently slow to react to the copilot's cue, the copilot would decrease its probability of playing a sound when it predicted that level of expected points. The copilot's control policy used learned predictions as state, c.f., prediction in human and robot motor control (Wolpert et al. 2001; Pilarski et al. 2013).\n\n\n\\section{Results and Discussion}\n\n\n\\begin{figure}[t]\n \\centering\n {\\bf(a)} \\includegraphics[height=1.4in]{results1.png} \\hfil\n {\\bf(b)} \\includegraphics[height=1.4in]{results2.png}\\\\ \n {\\bf(c)} \\includegraphics[width=6.6in]{results3.png}\\\\\n \\caption{ \\footnotesize Results from a single pilot participant for the three experimental conditions ({\\em NoCP, Pav,} and {\\em Bandit}). (a) Total score acquired by the pilot in each condition during day and night, as summed over all three trials in a block, along with the total score excluding any events with negative points (+); (b) total Bandit score with respect to fruit location to the front (F), middle (M) or back (B) with respect to the pilot's starting orientation; and (c) representative example of time-series data from the second trial for all three conditions, cross-plotting cumulative score, changes in score ($\\Delta Score$), light level, and bandit learning in terms of human teaching actions ($\\Delta V(h,s)$), copilot decisions to cue or not cue ({\\em up\/down ticks}), and post-cue updates to the copilot's policy as a result of pilot activity ($\\Delta \\pi$).}\n \\label{fig:results}\n \\vspace{-1em}\n\\end{figure}\n\nFigure \\ref{fig:results} presents the aggregate behaviour of the pilot in terms of total score over all three trials per condition, a fruit-by-fruit breakdown of total score, and a detailed presentation of time-series data from the second trial of the experiment. {As a key finding, we observed that interaction with different copilots (Pav and Bandit) led to different foraging behaviours on the part of the pilot during day and night, especially as compared to the no copilot (NoCP) condition.} Interactions with both the Pav and the Bandit copilot led to more foraging behaviour during night-time periods, as compared to the NoCP condition (Fig. \\ref{fig:results}c). Learning to interpret the communication from these copilots appeared to induce multiple foraging mistakes on the part of the pilot, especially during night-time phases (Fig. \\ref{fig:results}a,c) and less familiar fruit locations (Fig. \\ref{fig:results}b). Despite this, the total points collected in the absence of any mistakes (the score without any negative point value events, Fig. \\ref{fig:results}a+) suggest that collaboration with a policy-learning copilot could potentially lead to effective joint action once a good policy has been learned by both the pilot and the copilot. Teaching interactions ($\\Delta V(h,s)$, Fig. \\ref{fig:results}c) also provided a useful window into pilot skill learning. Broadly, the behaviour patterns observed in this preliminary study suggest that gradual addition of cues from a copilot, as in the Bandit policy-learning condition, is likely more appropriate than a strictly Pavlovian control approach. These initial results also indicate that there is room for more complex copilot architectures that can capture the appropriate timing of cues with respect to pilot activity (e.g., a pilot harvesting wrong fruit, or hesitation as per Moon et al. (2013)), and motivate more detailed study into the impact that pilot head position, gaze direction, light level, and other relevant signals have on a copilot's ability to generate good cues. Time delays and credit assignment matter in this joint-action setting and require further thought. \n\n{\\bf Conclusions:} This work demonstrated a complete (though straightforward) cycle of human-agent co-training and learned communication in a VR environment, where closing the loop between human learning (human learns then trains an agent regarding patterns in the world) and agent learning (agent learns to make predictions and provide cues that must be learned by the pilot) appears possible to realize even during brief interactions. The VR fruit foraging protocol presented in this work proved to be an interesting environment to study pilot-copilot interactions in detail, and allowed us to probe the way human-agent behaviour changed as we introduced copilots with different algorithmic capabilities.\n\n\\section*{References}\n\n\\footnotesize\n\n\nBicho, E., et al. (2011). Neuro-cognitive mechanisms of decision making in joint action: A human-robot interaction study. {\\em Human Movement Science} 30, 846--868.\n\nKnoblich, G., et al. (2011). Psychological research on joint action: Theory and data. In WDK2003 (Ed.), {\\em The Psychology of Learning and Motivation} (Vol. 54, pp. 59\u2013-101). Burlington: Academic Press.\n\nModayil, J., Sutton, R. S. (2014). Prediction driven behavior: Learning predictions that drive fixed responses. {\\em AAAI Wkshp. AI Rob}.\n\nMoon, A., et al. (2013). Design and impact of hesitation gestures during human-robot resource conflicts. {\\em Journal of Human-Robot Interaction} 2(3), 18--40.\n\nParker, A. S. R., et al. (2014). Using learned predictions as feedback to improve control and communication with an artificial limb: Preliminary findings. {\\em arXiv}:1408.1913 [cs.AI]\n\nPesquita, A., et al. (2018). Predictive joint-action model: A hierarchical predictive approach to human cooperation. {\\em Psychon. Bull. Rev.} 25, 1751--1769.\n\nPezzulo, G., Dindo, H. (2011). What should I do next? Using shared representations to solve interaction problems. {\\em Exp. Brain. Res.} 211, 613--630.\n\n\nPilarski, P. M., et al. (2013). Real-time prediction learning for the simultaneous actuation of multiple prosthetic joints. {\\em Proc. IEEE Int. Conf. Rehab. Robotics (ICORR)}, Seattle, USA, 1--8.\n\nPilarski, P. M., et al. (2017). Communicative capital for prosthetic agents. {\\em arXiv}:1711.03676 [cs.AI]\n\nSchultz, W., et al. (1997). A neural substrate of prediction and reward. {\\em Science} 275(5306), 1593--9. \n\nSebanz, N., et al. (2006). Joint action: Bodies and minds moving together. {\\em Trends. Cogn. Sci.} 10(2), 70--76.\n\nSutton, R. S., Barto, A. G. (2018). {\\em Reinforcement Learning: An Introduction}. Second Edition. Cambridge: MIT Press.\n\nWolpert, D. M., et al. (2001). Perspectives and problems in motor learning. {\\em Trends Cogn. Sci.} 5(11), 487--494.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nExtreme values of certain spatio-temporal processes, such as wind speeds, are the main cause of severe damage in property, from electricity distribution grid to transport and agricultural infrastructures. Accurate assessment of causal relationships between environmental processes and their effects on risk indicators, are highly important in risk analysis, which in return depends on sound inferential methods as well as on good quality informative data. Often, information on the relevant environmental processes comes from monitoring networks, as well as from numerical-physical models (simulators) that typically solve a large set of partial differential equations, capturing the essence of the physical process under study (Skamarock \\emph{et al.} 2008, Cardoso \\emph{et al.} 2013). In general, monitoring networks are formed by a sparse set of stations, whose instrumentation are vulnerable to disruptions, resulting in data sets with many missing observations, whereas, simulated data from numerical simulators typically supply average yield of the process in grid cells of pre-specified dimensions, often at high resolutions, spanning large spatial domains, with no missing observations. However, simulated data typically mismatch and misaligned observed data, therefore calibrating it and bringing it in line with observed data may supply modellers with more reliable and richer sources of data. Data assimilation methods, namely combining data from multiple sources, are well known in environmental studies, with data often being used to generate initial boundary conditions for the numerical simulators (Kalnay, 2003). There is a very rich statistical literature on data assimilation and data fusion with the objective of enriching the information for inference (Fuentes and Raftery 2005, Berrocal \\emph{et al.} 2012, Zidek \\emph{et al.} 2012, Berrocal \\emph{et al.} 2014, McMillan \\emph{et al.} 2010). These statistical methods are often based on Bayesian hierarchical methods for space time data (Banerjee \\emph{et al.} 2004) and are constructed around the idea of relating the monitoring station data and the simulated data using spatial linear models with spatially varying coefficients (Berrocal 2019). Since these relations involve data measured at different spatial resolutions, the models often are called downscaler models (Berrocal \\emph{et al.} 2012). The principal objective of these downscaler models is to relate observations measured at different space resolutions using spatial linear models. However, as a by-product, these models can be used for calibrating one set of data as a function of the other, as it will be explained later.\n\nThe motivation behind this present work has its roots in a\nconsulting work done for a major electricity producer and distributor.\n The electricity grid constantly faces disruptions due to damages\nin the distribution system, with heavy economic losses. These damages and consequent disruptions occur due to a combination of many\nfactors such as topography and precipitation, however extreme winds and storms are the main cause of such damages. Risk maps that indicate likely places of costly disruptions in electric grids are important decision support tools for administering the power grid and are particularly useful in deciding if costly corrective actions should be taken to improve\nstructures. It is natural that these risk maps should be based primarily on observed wind speeds among other factors and it was decided that daily maximum wind speeds should be used as proxy information. Hence, such risk maps can be interpreted as vulnerability maps of electricity grid to extreme wind speeds, expressed in terms of probability. However generating such maps depends on reliable wind data at fairly high spatial and temporal resolutions.\n\n The data available for this particular study corresponds to simulated wind speeds from a simulator (The WRF model, version 3.1.1) at a regular grid of 81ksq grid cell size, obtained at 10 minutes interval from 2006-2013; however only daily maximum wind speed will be used. Observed daily maximum wind speed is also available during the same period of time, from 117 stations in Portugal mainland, but missing observations reach to 90\\% in some stations. Only around one third of the stations have less than 30\\% missing observations. There is an additional challenge: although simulated and observed data are similar in the bulk\nof the distribution, they quite often mismatch at extreme values. Therefore,\n adequate methods of data\nfusion and calibration, can be used to combine these two different sources of data and may provide information which is more reliable from a spatial point of\nview and produce more accurate probability maps showing the spatial\ndistribution of damage risks.\nSince electricity grid damages are ultimately caused by extreme wind speeds, ultimate aim should be to develop statistical methods for data fusion and calibration that can extrapolate beyond the\nrange of observed data into the tails of a distribution. However, in this manuscript, we make a review of statistical fusion and calibration methods for the whole range of data. Calibration methods that extend beyond the range of data will be reported elsewhere.\nOur objective is to explore several methods to model the relationship between simulated and observed wind speeds at observation sites, so as to extrapolate this relationship in space at grid cell or county level resolution. In other words, more than imputing missing observations, we want to use simulated wind speeds for risk assessment, after being calibrated, i.e., brought in line with observed wind speeds.\n After giving a brief description of standard calibration and data fusion methods to update simulated data based on the observed data, we will propose and describe in detail a specific conditional quantile matching calibration method and show how our wind speed data can be calibrated using this method. We also briefly explain how calibration can be extended specifically to data coming from the tails of simulated and observed data, using asymptotic models and methods suggested by extreme value theory.\n\nThe outline of the paper is as follows: In section \\ref{cali}, we give an\n overview of statistical calibration methods. In section \\ref{NAV}, we report a new approach\n for calibration through a conditional quantile matching calibration method (Pereira \\emph{et al}., 2019), using an extended Generalized Pareto distribution (Naveau \\emph{et al.}, 2016) for the simulated and observed data, adequate for calibrating simultaneously the bulk and the tails of the distribution.\nFinally, in section \\ref{wind},\n this method will be exemplified using a wind speed data. Further discussion and conclusions are in section \\ref{disc}.\n\n \n\n\\section{Statistical Calibration methods; an overview} \\label{cali}\n\nCalibration plays a crucial role in almost all areas of experimental sciences and can be defined in a nutshell as \"the comparison between measurements - one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device \" (Wikipedia). When measurements are obtained under random environments, then statistical methods and models have to be employed to compare such sets of measurements. In the simplest case of random experiments involving linear calibration, linear regression models are used as models. The univariate calibration is then defined as inverse regression problem (Lavagnini and Magno, 2007). These models and consequent calibration techniques are, inevitably, restricted to uncorrelated repeated observations or dependent Gaussian structures, simplifying immensely the problem (Aitchison and Dunsmore, 1976). However, more often than not, even in designed experiments, the response relationships are mostly nonlinear and therefore Gaussian structures are hardly justifiable as models. Nonlinear calibration then typically have to be formulated by conditional specification of distributions, and consequently substantial amount of numerical integration and approximations are needed.\n\nLittle is known on calibration of environmental data sets displaying nonlinear, non-Gaussian structures in a spatio-temporal setting. In these cases, defining calibration through spatial linear models as inverse regression problems will oversimplify the structures and will not be adequate. Often calibrating simulated data based on observed data is done by calibration of the simulator, namely the numerical-physical model, using Bayesian methods (Kennedy and O'Hagan, 2001, and Wilkinson, 2010). These generic methods are based on the following general ideas: The simulator first is approximated by a linear parametric emulator, and using Bayesian arguments, data are used to convert prior knowledge on these model parameters to posterior distributions. The newly generated data from this approximate emulator is then treated as the calibrated data.\n\n\nThere are many statistical calibration methods for different purposes and based on different paradigms. We can classify these methods into\n\\begin{enumerate}\n\\item[(i)] Quantile matching-based approaches,\n\\item[(ii)] Inverse regression,\n\\item[(iii)] Simulator--emulator-based approaches,\n\\item[(iv)] Data assimilation\/data fusion.\n\\end{enumerate}\n\nBefore describing these methods, we give here some basic notation.\n\n We denote by $Y(s,t)$ and $X(s,t)$, respectively the observed and simulated wind speeds at location $s\\in \\mathbb{R}^2$ and time $t$.\nTo simplify notation, often we will use $Y$ and $X$ for observed and simulated wind speeds when data are used without any space-time reference. Typically $X$ are simulated over a regular grid, say $B$, often represented by points $s_B$ which correspond to the centroid of the grid cells, whereas $Y$ are observed in stations located at different spatial points $s$.\n\n\n\n\\subsection{Quantile matching-based approaches}\n\n\n For the time being, if we ignore totally space-time variations and dependence structures, calibration can be seen as a\nsimple scaling making use of marginal distributions fitted respectively to $X$ and $Y$ (CDF transform method, Michelangeli \\emph{et al}, 2009).\n\nSuppose we have a set of $n$ observed $y_i$ and simulated $x_i$ , $i=1,...,n$ data. Let $F_Y$ and $F_X$ be, respectively, the distribution functions of $Y$ and $X$. Then the new calibrated (scaled) data $x_i^*$ is defined as\n \\begin{equation}\n x_i^*=F^{-1}_Y(F_X(x_i)), \\quad i= 1, \\ldots,n. \\label{qmb}\n\\end{equation}\n\n Since\n$$P(X^*\\le z)=F_Y(z),$$\ncalibrated data has the same distribution as the observed data. Note that if $F_X=F_Y$ then $x_i^*=x_i$. Figure \\ref{fig1} depicts the result of applying this calibration method when $Y$ follows a Student distribution with 3 degrees of freedom, and $X$ follows a standard normal distribution.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=10cm]{qm.eps}\n \\caption{Illustration of the quantile matching approach}\\label{fig1}\n\\end{figure}\n\n\n\n\n This calibration method depends on the marginal distributions of the random variables involved $Y$ and $X$ and hence it does not\nmake use of the expected strong dependence between the two sets of data.\n\n\nAn ideal calibration should involve the joint distribution of $Y$ and $X$ defined in some way. A possibility is the use of a conditional quantile matching approach, which will be described in section \\ref{NAV}. Further, in the same section, we also introduce\nan extension to cover space-time non-homogeneity by scaling (calibrating) the data from\n\\begin{equation}\n x^*(s,t)=F^{-1}_{Y(s,t)}(F_{X(s,t)}(x(s,t)),\\label{eqmb}\n\\end{equation}\nassuming marginal distributions of $Y(s,t)$ and $X(s,t)$ for every $s$ and $t$.\n\n These distributions will be estimated by fitting them and considering the\n parameters as smooth functions of spatially and temporarily varying covariates and space-time latent processes as in section \\ref{wind}.\n\n\n\n\n \\subsection{Inverse regression}\n\n Calibration is usually seen as a method of adjusting the scale of a measurement instrument on the basis of an informative experiment. As such, it is seen as an inverse regression problem. However, there are several problems associated with this approach (see, e.g. Kang \\emph{et al.}, 2017).\n Aitchison and Dunsmore (1975) approach the problem from a Bayesian perspective by defining the \\textit{calibrative distribution}. See also (e.g. Racine-Poon, 1988, Osborne, 1991 and Muehleisen and Bergerson, 2016).\n\nAccording to Aitchison and Dunsmore's proposal, a parametric model is fitted to a random vector $(X,Y)$, where, e.g. $X$ is the random variable representing a measurement obtained in a laboratory and $Y$ the random variable representing the measurement obtained in a field experiment. This parametric model is defined through a conditional specification such that\n$$f_{(X,Y)}(x,y\\mid \\psi,\\theta)=f_{X\\mid Y}(x\\mid y,\\psi)f_Y(y\\mid \\theta).$$ There are two parameters involved, namely the arrival parameter $\\psi$ and the structural parameter $\\theta$. A further assumption is that the initial sources of information are stochastically independent, so that\n$$p(\\psi,\\theta)=p(\\psi)p(\\theta).$$\n\n Now suppose that one has a future laboratory experiment resulting in a value $x_f$ and that further experiments follow the same pattern of arrival as the original trials $(\\mathbf{x},\\mathbf{y})=(x_i,y_i, i=1,...,n)$. Now the data available is $(x_i,y_i, x_f), i=1,...,n$ and the unknowns are $(y_f,\\psi,\\theta)$. The objective is to obtain the predictive distribution (called in this case the \\emph{calibrative distribution}) for the corresponding field experiment $Y_f$, which is simply obtained by integrating out $\\psi$ and $\\theta$,\n\n$$p(y_f\\mid \\mathbf{x},\\mathbf{y},x_f)=\\int f_X(x_f\\mid y_f,\\psi)p(y_f,\\psi,\\theta\\mid x_f,\\mathbf{x},\\mathbf{y})d\\psi d\\theta $$\n where\n $$p(y_f,\\psi,\\theta\\mid x_f,\\mathbf{x},\\mathbf{y}) \\propto f_X(x_f\\mid y_f,\\psi) f_Y(y_t\\mid \\theta)\\prod_{i=1}^{n} f_X(x_i\\mid y_i,\\psi)f_Y(y_i\\mid \\theta)p(\\psi)p(\\theta),$$\n assuming that the trial records are independent.\n\n\nGeneralizing to the situation under study, without considering space and time dependence,\nfor a model $[X\\mid Y][Y]$, the objective is to obtain the distribution $$Y(s_0)\\mid x(s_0),x(s^*),y(s^*)$$ for an unknown $Y(s_0)$ based on the observed and simulated data $(x(s^*),y(s^*))$ on $N$ stations and the simulated value $x(s_0)$.\n\n\n \\subsection{Simulator--emulator-based approaches}\n\n Kennedy and O'Hagan (2001) describe calibration as statistical postprocessing of simulator deterministic forecast.\n They assume a computer model describing some physical system, that is a function of variable inputs $x$ that can be measured and calibration inputs $\\nu$ needed to run the model, but whose values are not\nknown in the experiment. The output of the computer model is then assumed to be some function of the inputs, say $\\eta(x,u)$. Observations from the field experiment are assumed to have been observed at $u = \\theta$ and, possibly at different values of $x$. The model for observations for known input variables $x$ is then assumed to be a function of the computer model output $\\eta(x,\\nu)$, of a\ntrue underlying process $\\xi(x)$ and some model inadequacy described by $\\delta(x)$.\nThe objective is to estimate the calibration settings $\\theta$ consistent with the field experiments and the computer model.\n\nSigrist \\emph{et al.} (2015) give detailed description of stochastic versions of space-time advection-diffusion PDE's and their solutions as models for emulators and describe a method of postprocessing simulated data.\n\nHowever simulator--emulator-based approaches assume detailed information of how emulators work in terms of a set of parameters, which is not the case in most situations related to climate models.\n\n\n \\subsection{Data fusion and calibration}\n\n\n Data assimilation or data fusion methods are used to combine different sources of information in order to obtain more accurate results. A recent review on data assimilation is given in Berrocal (2019) together with many references.\n\n\n Among these methods, the interest lies in statistical approaches to data assimilation\/data fusion and\n in particular to hierarchical Bayesian models (HBM) for combining monitoring data and computer model output.\n\n There are basically two different approaches regarding these methods. The Bayesian melding proposed by Fuentes and Raftery (2005) assumes that there is a true latent point-level process $Z(s)$ (\\emph{a GP with spatially varying mean and non-stationary covariance function}) to which both the observed $Y(s)$ and simulated $X(s)$ processes are linked to.\n The observed values are governed by this latent process with a random error and simulated values are expressed as a linearly calibrated integral over a grid cell (scaled by the area of the cell) of the latent point-level process,\n $$Y(s)=Z(s)+\\epsilon(s) \\qquad X(s)=a(s) + b(s)Z(s) + \\delta(s),$$\nwhere $\\delta(s)$ explains the random deviation\nat location $s$ with respect to the underlying true process $Z(s)$.\n The aim is to obtain the posterior\npredictive distribution of the truth $Z$ at a new site $s^*$. However, the misalignment of the two processes involved brings computational difficulties in the implementation. Foley and Fuentes (2008) apply this type of modelling to hurricane surface wind prediction and\nMcMillan \\emph{et al.} (2010) propose a spatio-temporal extension of this Bayesian melding approach.\n\n\n The other approach suggested by Berrocal \\emph{et al.} (2010) is a Bayesian hierarchical downscaler model. They consider a spatial linear model relating the monitoring station data and the computer model output, with spatially varying coefficients which are in turn modeled as Gaussian processes (GPs). These models offer the advantage of local calibration of the numerical model output without incurring in problems due to the dimensionality of the computer model output, since they are only fitted at the grid cells where the monitoring stations reside. An extension to this downscaler model, by borrowing information from neighboring grid cells, was introduced by Berrocal \\emph{et al.} (2012).\n The proposed approach, the downscaler model for the observations and simulated data from the computer model is\n\n$$Y(s) =\\beta_0+\\beta_0(s)+\\beta_1+\\beta_1(s)x(B)+e(s),\\quad e(s)\\quad i.i.d.\\quad N(0, \\tau^2),$$ with $B$ the grid cell containing $s$\n $$X(B)= \\mu + V(B) +\\eta(B), \\quad \\eta(B) \\quad i.i.d. \\quad N(0, \\sigma^2)$$ where $V(B)$ is a GP model with a ICAR structure (Rue and Held, 2005), and $\\beta_0(s)$ is modeled as a mean-zero GP with exponential covariance structure.\n\nA smoothed version is possible considering\n$$Y(s) =\\beta_0+\\beta_0(s)+\\beta_1(\\mu+V(B))+e(s),\\quad e(s)\\quad i.i.d. \\quad N(0, \\tau^2)$$ with $B$ the grid cell containing $s$.\n\n The aim is to obtain the predictive distribution of $Y$ and its expected value at grid cell level.\n\nThey also considered a smoothed downscaler using spatially varying random weights and a\nspace-time extension.\n\n\n\\section{Calibration methods for bulk and tails}\\label{NAV}\n\nPereira \\emph{et al.} (2019) develop a covariate-adjusted version of the quantile matching-based approach as in (\\ref{qmb}) where the distributions of simulated and real data change along a covariate. At the same time they suggest a regression method\nthat simultaneously models the bulk and the (right) tail of the distributions involved\n using the extended Generalized Pareto distribution (EGPD) (Naveau \\emph{et al.}, 2016) as a model for both the simulated and observed data.\n\n Under fairly general conditions, according to the asymptotic theory of extremes, the generalized Pareto distribution (GPD) appears as a natural model for the right tail of a distribution, by focusing on the excesses over a high but fixed threshold. Here, the choice of this threshold plays a very important role in inference, ignoring the part of the data that lie below this threshold. See, for example, Beirland \\emph{et al.} (2004). The EGPD modelling strategy suggested by Naveau \\emph{et al} (2016) avoids this selection problem, as we will see in next section.\n\n In what follows we propose an extension of this conditional quantile matching calibration for the bulk and tails, to spatial temporal data.\n\n\\subsection{ Naveau \\emph{et al} (2016) EGPD models}\nNaveau \\textit{et al.}~(2016) suggest an extension of Generalized Pareto model tailored for both the bulk and tails, and---contrarily to most methods for extremes--- does not require a threshold to be selected. The objective of this extension is to generate a new class of distributions with GPD type tails consistent with extreme value theory, but also flexible enough to model efficiently the main bulk of the observed data without complicated threshold selection procedures.\n\nLet $Y$ be a positive random variable with cumulative distribution function defined as:\n \\begin{equation*}\n F_Y(y \\mid \\theta) = G\\bigg(H(y\\mid\\xi,\\sigma )\\bigg),\n \\label{qr}\n\\end{equation*}\nwhere $G$ is a function obeying some general assumptions (see Naveau \\emph{et al.}, 2016) and $H$ is the cumulative distribution function of a Generalized Pareto distribution (GPD), that is\n\\begin{equation*}\n H(y\\mid\\xi,\\sigma ) =\n\\left\\{\n \\begin{array}{ll}\n 1 - (1 + \\frac{\\xi}{\\sigma} y)_{+}^{-1\/\\xi}, & \\hbox{$\\xi \\neq 0$.} \\\\\n 1 - \\exp(-\\frac{y}{\\sigma}), & \\hbox{$\\xi = 0$ .}\n \\end{array}\n\\right.\n\\end{equation*}\n with $\\sigma>0$, and $y>0$ if $\\xi\\geq 0$ and $y<-\\frac{\\sigma}{\\xi}$ if $\\xi<0$. The parameter $\\sigma$ is a dispersion parameter while $\\xi$ is a shape parameter controlling the rate of decay of the right tail of a distribution (de Zea Bermudez and Kotz, 2010).\n\n Naveau \\emph{et al.} (2016) consider four forms of $G(u)$ resulting in four different classes of distributions.\n\n\\subsection{Spatio temporal conditional quantile matching calibration for the bulk and tails}\nHere we use one of the forms, namely, $G(u)=u^\\kappa$ where $\\kappa$ is a parameter controlling the shape of the lower tail, although the theory can be easily extended to any of the other forms of the $G$ function.\n\nLet us assume that both random variables $X$ and $Y$ are space-time dependent and we want to calibrate $X$ based on $Y$. As in (\\ref{eqmb}) the calibrated data is given as\n\\begin{equation*}\n x^*(s,t)=F^{-1}_{Y(s,t)}(F_{X(s,t)}(x(s,t)),\n\\end{equation*}\nNow assume further that both random variables are distributed as an EGPD with different parameters. In order to better accommodate for the situation $\\xi<0$ we make a transformation $\\delta=-\\frac{\\sigma}{\\xi}$. Hence, for $\\xi_x\\neq 0$\n\\begin{equation}\nF_{X(s,t)}(x(s,t)\\mid \\delta_x(s,t), \\xi,\\kappa_x)=\n \\left(1 - \\left(1 - \\frac{1}{\\delta_x(s,t)} x(s,t)\\right)_{+}^{-1\/\\xi_x}\\right)^{\\kappa_x}, \\label{fx}\n\\end{equation}\nand assuming as well $\\xi_y\\neq 0$\n\\begin{equation}\nF_{Y(s,t)}(y(s,t)\\mid \\delta_y(s,t), \\xi_y,\\kappa_y)=\n \\left(1 - \\left(1 - \\frac{1}{\\delta_y(s,t)} y(s,t)\\right)_{+}^{-1\/\\xi_y}\\right)^{\\kappa_y}. \\label{fy}\n\\end{equation}\n\nAlthough it is assumed that these random variables are conditional independent, a dependence structure is introduced through the transformed space-time dependent parameters $\\delta_x, \\delta_y$ by modelling them as a function of a common random spatio-temporal process, in a Bayesian hierarchical modelling framework.\n\nAs an exemplification of this modelling strategy, in the next section, we will built a Bayesian hierarchical model for the wind speed data.\n\n\\section{Bayesian hierarchical model for the wind speed data}\\label{wind1}\n A preliminary data analysis of the wind speed data used in this study, shows that observed and simulated data are consistent with the case $\\xi<0$ and hence, the distributions for $X$ and $Y$ will have an end-point characterized by the respective parameter $\\delta$.\n\nWe assume that the observed data $\\{Y(s_i,t_j), i=1,...,N; j=1,...,T\\}$, with $N$ the number of stations with observed data in the study period and $T$ the length of the time period, follow a distribution as in (\\ref{fy}), where\n $\\delta_y(i,j)\\sim Exp(\\lambda_y(i,j)) ,$ $\\delta_y(i,j)>\\max(y),$ i.e., follows a shifted exponential distribution with\n $\\log(\\lambda_y(i,j))=\\beta_y+ W(s_i)+Z(t_j),$ and $W$ follows a Multivariate Gaussian process, defined on the space, $W\\sim MVN(0,\\tau_W\\Sigma_W)$.\nThe matrix $\\Sigma_W$ has diagonal elements equal to 1 and off-diagonal elements, $\\Sigma_{i\\ell}=f(d_{i\\ell};\\alpha)$, where $f(.;.)$ is a function of $d_{i\\ell}$, the centroids' distance of every two stations $s_i$ and $s_\\ell$, and $\\alpha$ a parameter representing the radius of the 'disc' centred at each $s$. The $\\tau_W$ is a precision parameter. For the temporal random process we assume a random walk process of order 1, $Z\\sim MVN(0,\\tau_Z\\Sigma_Z)$, where $\\tau_Z$ is a precision parameter and $\\Sigma_Z$ is a matrix with a structure reflecting the fact that any two increments $z_i-z_{i-1}$ are independent (Rue and Held, 2005).\n\n\n\nWe assume, as well, that the simulated data $\\{X(s_i,t_j), i=1,...,N_s; j=1,...,T\\}$ follow a distribution as in (\\ref{fx})\n with $N_s$ total number of stations, where the model for $\\delta$ shares the same latent processes $W$ and $Z$ with the model for the observed data, such that $\\delta_x(i,j)\\sim Exp(\\lambda_x(i,j)),$ $\\delta_x(i,j)>\\max(x),$ with\n $\\log(\\lambda_x(ij))=\\beta_x+ W(s_i)+Z(t_j).$\n\n\nTo complete the Bayesian hierarchical model we consider the following prior specification for the parameters and hyperparameters of the models\n\n $\\beta_y,\\beta_x$ i.i.d. $N(0,0.01)$,\n\n $\\kappa_y,\\kappa_x$ i.i.d. $Ga(0.05,0.05)$,\n\n $\\xi_y, \\xi_x$ i.i.d. $U(-0.5,0)$,\n\n $\\tau_W, \\tau_Z$ i.i.d. $Ga(1,0.1)$,\n\n $\\alpha \\sim U(0.1,0.5)$.\n\nFinally, the calibrated values are obtained as the\nmean of the predictive distribution of $F^{-1}_Y(F_X(x(s_i,t_j))$\n at $s_i, i=1,...,N_s$ and time $t_j,j=1,...,T$.\n\n\n\n\n\n\n\n\\section {Application to wind speed data }\\label{wind}\n\nWe used observed and simulated wind speed data from the period 01\/01\/2013 to 28\/02\/2013, so $T=59$. There are $N=51$ stations where we have both observed and simulated daily maximum wind speeds. Additionally we have extra 66 stations with simulated values for the maximum daily wind speeds, so that $N_s=117$. In Figure \\ref{turkman:fig1} we depict the median of observed and simulated wind speeds for the 51 stations together with the\n2.5\\% and 97.5\\% empirical quantiles (95\\% IQR).\n\n\n \\begin{figure}[!ht]\\centering\n\\includegraphics[width=10cm]{obs_simul.eps}\n\\caption{\\label{turkman:fig1} Median of observed and simulated wind speeds for the 51 stations, and the 95\\% IQR wind speeds by station (dashed lines).}\n\\end{figure}\n\n\nThe model was implemented in \\texttt{R2OpenBUGS} (Sturtz \\emph{et al.} (2005). In Table \\ref{tab1} we show the summary statistics for the marginal posterior distributions of the parameters of the model.\n\n We observe that the posterior mean of $\\kappa_y$ has a much smaller mean than the posterior mean of $\\kappa_x$ which is consistent with the fact that, in general, simulated data are shifted to the right in relation to the observed data, indicating the possible existence of some bias in the simulated data. The posterior mean of the precision (inverse of the variance) parameters for the space model ($\\tau_W$) and for the temporal model ($\\tau_Z$) suggest that time dependence is stronger than space dependence. The posterior mean for $\\beta_y$ is slightly smaller than the posterior mean for $\\beta_x$. This naturally contributes for higher values for $\\sigma_y(i,j)$ relatively to $\\sigma_x(i,j)$ and with greater dispersion, as it can be seen in Figure \\ref{boxplot} where we show daily boxplots of the posterior means of the parameters $\\sigma(i,j),\\forall j$ for both models. In that figure it is marked two dates, 19 of January, a day where it was observed a storm with heavy winds (storm GONG, maximum observed wind 29.6m\/s), particularly in regions close to the littoral, and 14th of February, a very mild day all over the country (Valentine's day; maximum observed wind 8.20m\/s). The variation observed along the days is consistent with the fact that on windy days the maximum wind speed along the stations varies much more than on mild days. Also the temporal dependence is clear in these pictures.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Summary statistics for the marginal posterior distributions} \\label{tab1}\n\\begin{tabular}{cccccccc}\n \\hline\n & mean & standard deviation & 2.5\\% quantile & median & 97.5\\% quantile & min & max \\\\\n \\hline\n$\\alpha$ & 0.451 & 0.042 & 0.346 & 0.462 & 0.499 & 0.241 & 0.500 \\\\\n $\\beta_y$ & -1.094 & 0.149 & -1.376 & -1.093 & -0.806 & -1.541 & -0.595 \\\\\n $\\beta_x$ & -0.854 & 0.134 & -1.105 & -0.849 & -0.598 & -1.243 & -0.365 \\\\\n $\\kappa_y$ & 5.312 & 0.197 & 4.936 & 5.310 & 5.701 & 4.675 & 5.936 \\\\\n $\\kappa_x$ & 18.588 & 0.725 & 17.230 & 18.585 & 20.080 & 16.480 & 20.930 \\\\\n $\\tau_W$ & 4.234 & 0.715 & 2.923 & 4.187 & 5.762 & 2.218 & 7.022 \\\\\n $\\tau_Z$ & 0.396 & 0.095 & 0.241 & 0.384 & 0.618 & 0.191 & 0.850 \\\\\n $\\xi_y$ & -0.070 & 0.002 & -0.074 & -0.070 & -0.067 & -0.077 & -0.065 \\\\\n $\\xi_x$ & -0.081 & 0.001 & -0.084 & -0.081 & -0.078 & -0.085 & -0.076 \\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=7cm]{boxplot_sigmas.eps}\n\\includegraphics[width=7cm]{boxplot_sigmax.eps}\n\\caption{\\label{boxplot} Boxplot of the posterior means of $\\sigma_y(i,j)$ (left) and $\\sigma_x(i,j)$ (right) }\n\\end{figure}\n\n\n\n\nThese two days were studied, in particular, for exemplification of the conditional quantile calibration method proposed. For the purpose of exemplification of the results we represent in Figures \\ref{gong} and \\ref{val}, on the left, a kernel density estimation (considering all the stations) for the observed and simulated maximum wind speed on that day, together with the mean of the predictive distribution of the calibrated data as defined in (\\ref{eqmb}). On the right side we represent the observed and simulated maximum wind speed on that day for each station, together with the mean of the predictive distribution for the calibrated data.\n\nWe observe that, on a storm day (Figure \\ref{gong}) the observed winds have longer tails than simulated winds. The calibration method was able to capture both tails of the distribution for the observed data, although it shifted the bulk of the distribution to the left. Regarding a mild windy day (Figure \\ref{val}), the distribution of the simulated data is shifted to the right relatively to the distribution of the observed data with longer tails, as it was observed in a preliminary study. This bias is corrected with the calibration method.\n\nIn Figures \\ref{mapa_gong} and \\ref{mapa_val} there is a spatial representation of the observed, simulated and calibrated values for each of these two days.\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=5.3cm]{density_full_GONG.eps}\n\\includegraphics[width=5.3cm]{GONG_stations.eps}\n\\caption{\\label{gong} Kernel density estimation (left), observed and simulated maximum wind speed for each station, together with the mean of the predictive distribution for the calibrated data, for a storm day.}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=5.3cm]{density_full_14Fev.eps}\n\\includegraphics[width=5.3cm]{14Fev_stations.eps}\n\\caption{\\label{val} Kernel density estimation (left), observed and simulated maximum wind speed for each station, together with the mean of the predictive distribution for the calibrated data, for a mild day.}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=10cm]{mapas_talk_at_14fev.eps}\n\\caption{\\label{mapa_gong} Storm day: observed, simulated and calibrated maximum wind speeds}\n\\end{figure}\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=10cm]{mapas_talk_at_gong2.eps}\n\\caption{\\label{mapa_val} Mild day: observed, simulated and calibrated maximum wind speeds }\n\\end{figure}\n\n\n\n\n\\section{Discussion and further extensions}\\label{disc}\n\nIn this article we discussed several possible ways of calibrating data obtained from a simulator based on observations at stations. We also proposed and implemented a conditional quantile matching calibration (CQCM) using a space-time extended generalized Pareto distribution.\n\nThe performance of the CQCM method was exemplified with two specific days, a storm day and a mild day. In both cases the calibrated data matched well the observed data on the tails, although on the storm day it did not capture well the bulk of the distribution. Ideally this method should be extended to the grid level, since the simulator produces data at a fine grid level and this is much more interesting if the objective is the construction of a risk map. However this extension is not trivial and some assumptions regarding the model structure have to be assumed.\n\n Damages in electricity grid are basically governed by extreme winds and primarily simulated and observed data coming from the right tail differ.\n Hence adequate calibration methods must be specifically adapted to extreme observations coming from the right tails and\n methods and models to be used in calibration should ideally be compatible with extreme value theory.\n A range of approaches for characterising the extremal behaviour of spatial process have been suggested and a brief comparison of these methods can be found in Tawn \\emph{et al.}~(2018).\n Downscaling method described by Towe {\\it et al.}~(2017)--- based on the conditional extremes process---is more suitable, with adequate modifications, to calibrate extreme simulated data based on observed wind speeds. Work on this approach is under progress.\n \n\\section*{Acknowledgments}\nResearch partially financed by national funds through FCT - Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e a Tecnologia, Portugal, under the projects PTDC\/MAT-STA\/28649\/2017 and UIDB\/00006\/2020.\n\n\n\n\\References\n\n\n\\item[] Aitchison, J. and Dunsmore, I.~R. (1975). \\emph{Statistical Prediction Analysis}. Cambridge: Cambridge University Press.\n\\item[] Beirland, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). \\emph{Statistics of\nExtremes: Theory and applications}. J Wiley, Chichester.\n\\item[] Banerjee, S., Carlin, B.~P. and Gelfand, A.~E. (2004). \\emph{Hierarchical Modeling and Analysis for Spatial Data}, Boca Raton, FL: Chapman and Hall.\n\\item[] Berrocal, V.~J. (2019). Data assimilation. 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Conf. on Nuclear\nPhysics (Munich)} vol\u02dc1 (Amsterdam: North-Holland\/American\nElsevier) p\u02dc517\n\\endrefs\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nShortly after the discovery of pulsars and the realization that they are\nrotating neutron stars, deformations of rotating neutron stars were proposed as\nsources of continuous gravitational radiation~\\cite{Shklovskii1969,\nOstriker1969, Ferrari1969, Melosh1969}; see~\\cite{Press1972} for an early\nreview.\nSearches for such radiation are an ongoing concern of the LIGO and Virgo\ngravitational wave detectors~\\cite{LIGO_psrs2010, LIGO_CasA, LIGO_Vela}; see~\\cite{Owen2009, Pitkin, Astone} for recent\nreviews.\nIt is thus of\ngreat interest to know the maximum\nquadrupolar deformation that a neutron star could sustain, in order to motivate\nfurther searches and help interpret upper limits or detections.\nIn the case of elastic (as opposed to magnetic) deformations, the main factor\ninfluencing the answer is whether the neutron star contains particles more\nexotic than neutrons~\\cite{OwenPRL, Owen2009}.\nHowever, the structure of the star also plays an important role.\n\nWhile there are\nrelativistic calculations of the quadrupole deformations due to magnetic\nfields (e.g.,~\\cite{IS,CFG,YKS, FR,CR}), all the computations involving elastic\ndeformations have used Newtonian gravity. Moreover, all but two of these computations\nhave used the integral expression obtained in the Cowling approximation (i.e.,\nneglecting the self-gravity of the perturbation) by\nUshomirsky, Cutler, and\nBildsten (UCB)~\\cite{UCB}; see \\cite{OwenPRL, Lin, KS, Horowitz}.\nHaskell, Jones, and Andersson (HJA)~\\cite{HJA} dropped the Cowling approximation using\na somewhat different formalism than UCB's; there is a further application of\ntheir results in~\\cite{Haskelletal}.\n\nWe improve these treatments by generalizing the UCB integral to relativistic\ngravity with no Cowling approximation. We also provide a\nsimilar generalization for the Newtonian no-Cowling case, as a warm-up. In\naddition to providing a simpler formalism for performing computations than the\nmore general Newtonian gravity treatment in HJA, the integrals we obtain allow us to verify that a\nmaximal uniform\nstrain continues to yield the maximum quadrupole deformation in the Newtonian\nand relativistic no-Cowling cases. (UCB showed this to be true for an arbitrary equation of state\nin the Newtonian Cowling approximation case; we are able to verify that it is\ntrue in the more general cases for each background stellar model we consider.) \n\nWe then apply our calculation to the standard case of quadrupoles supported by\nshearing the lattice of nuclei in the crust, as well as the cases where the\nquadrupole is supported by the hadron--quark mixed phase lattice in the core, or a crystalline color superconducting phase throughout a solid strange quark star. For the crustal\nquadrupoles, we calculate the shear modulus following HJA, using the equation\nof state (EOS) and composition results of Douchin and Haensel~\\cite{DH} and\nthe effective shear modulus calculated by Ogata and Ichimaru~\\cite{OI}. (There are recent\nimprovements to the Ogata and Ichimaru result~\\cite{HH, Baiko,BaikoCPP}, but these only reduce their\nshear modulus by $<10\\%$.) For the hadron--quark mixed phase, we use our recent calculations of the EOS and\nshear modulus~\\cite{J-MO1} for a variety of parameters. (We also consider the range of surface tensions for\nwhich the mixed phase is favored.) For crystalline quark matter, we use the shear modulus calculated by Mannarelli, Rajagopal, and Sharma~\\cite{MRS}, and the EOS given by\nKurkela, Romatschke, and Vuorinen~\\cite{KRV}.\n\nIn all cases, we use a breaking strain of $0.1$, comparable to that\ncalculated by Horowitz and Kadau~\\cite{HK} using molecular dynamics\nsimulations. (Hoffman and Heyl~\\cite{HoHe} have recently obtained\nvery similar values over more of parameter space.) This result is directly applicable to the crustal lattice, at least for the outer crust, above neutron drip (though see Chugunov and\nHorowitz~\\cite{ChHo} for caveats). We also\nfeel justified in applying it to the inner crust, as well as to the mixed phase and crystalline quark matter, since the primary source of\nthe high breaking strain appears to be the system's large pressure.\nBut one can apply our results to any breaking strain using the linear scaling of\nthe maximum quadrupole with breaking strain.\n\nIn our general relativistic calculation, we use the relativistic theory of elasticity given by Carter and\nQuintana~\\cite{CQ} and placed in a more modern guise by\nKarlovini and Samuelsson~\\cite{KaSaI}.\nHowever, all we need from it is the relativistic form of the elastic stress-energy tensor, which can be obtained by simple\ncovariance arguments, as noted by Schumaker and Thorne~\\cite{ST}. We also use the standard\n\\citet{TC} Regge-Wheeler gauge~\\cite{RW} formalism for perturbations\nof static relativistic stars, following\nHinderer's recent calculation~\\cite{Hinderer} of the quadrupole moment of a\ntidally deformed relativistic star (first discussed in~\\citet{FH}), and the classic calculation by \\citet{Ipser}.\n\nEven though we are interested in the gravitational radiation emitted by\nrotating stars, it is sufficient for us to calculate the static quadrupole deformation.\nAs discussed by\nIpser~\\cite{Ipser}, and then proved for more general situations by\nThorne~\\cite{Thorne}, this static quadrupole (obtained\nfrom the asymptotic\nform of the metric) can be inserted into the quadrupole formula to obtain the emitted\ngravitational radiation in the fully relativistic,\nslow-motion limit. [This\napproximation has uncontrolled remainders of order $(\\omega\/\\omega_K)^2$, where\n$\\omega$ and $\\omega_K$ are the star's angular velocity and its maximum---i.e.,\nKepler---angular velocity, respectively.\nThis ratio is $\\lesssim 10^{-2}$ for the pulsars for which LIGO has been able\nto beat the spin-down limit~\\cite{LIGO_psrs2010}.]\n\nWe shall generally show the gravitational constant $G$ and speed of light $c$\nexplicitly, though we shall take $G = c =1$ in most of Sec.~\\ref{GR}, only restoring them in our\nfinal expressions.\nThe relativistic calculation was aided by use of the computer algebra\nsystem {\\sc{Maple}}\nand the associated tensor manipulation package {\\sc{GRTensorII}}~\\cite{GRTensor}.\nWe used {\\sc{Mathematica}}~7 to perform numerical computations.\n\nThe paper is structured as follows: In Sec.~\\ref{Newt}, we review UCB's formalism\nand extend it by introducing a Green function to compute the maximum Newtonian quadrupole deformation without\nmaking the Cowling approximation. In Sec.~\\ref{GR}, we further generalize to\nthe fully relativistic case, and compare the various approximations for the maximum quadrupole.\nIn Sec.~\\ref{results2}, we show the maximum quadrupoles for three different cases: first crustal quadrupoles,\nthen hadron--quark hybrid quadrupoles, and finally solid strange quark star quadrupoles. We also describe\nthe modifications to our formalism needed to treat solid strange quark stars. We discuss all these\nresults in Sec.~\\ref{discussion}, and summarize and conclude in\nSec.~\\ref{concl2}.\nIn the Appendix, we show that the mixed phase is favored by global energy arguments even for surface tensions large enough that it is disfavored by local energy arguments.\n\n\\section{Newtonian calculation of the maximum quadrupole}\n\\label{Newt}\n\nWe first demonstrate how to compute the maximum Newtonian quadrupole without making the Cowling approximation. This provides\na warm-up before we tackle the full relativistic case, and also allows us to verify some of the\nstatements made by UCB and HJA. We use\nthe basic formalism of UCB, modeling the star as nonrotating, with the\nstress-energy tensor of a perfect fluid plus\nshear terms, and treating the shear contributions as a first-order\nperturbation of hydrostatic equilibrium.\nThis perturbative treatment should be quite a good approximation: The maximum shear stress to energy density ratio we consider in the crustal and hybrid star cases is $\\lesssim 0.05\\%$ (and the maximum shear stress to pressure ratio is\n$\\lesssim 0.3\\%$). (Here we have taken the shear stress to\nbe $\\mu\\bar{\\sigma}_\\mathrm{max}$, which is good up to factors of order unity.) And even in the case of solid strange quark stars, the\nmaximum shear stress to energy density ratio is still only at most $\\sim 0.2\\%$.\n[We have already discussed the effects of rotation in the relativistic case, above; UCB note at\nthe beginning of their Sec.~4 that rotation also only modifies the perturbative Newtonian results for the static deformations we and they consider at the $O([\\omega\/\\omega_K]^2)$ level.] \n\nIt is convenient to start by writing the quadrupole moment\nin terms of the surface value of the perturbation to the star's\nNewtonian potential. We start from UCB's definition of\n\\<\n\\label{Q22}\nQ_{22} := \\int_0^\\infty\\delta\\rho(r)r^4dr\n\\?\n[where the (Eulerian) density perturbation $\\delta\\rho$ and all similar\nperturbed quantities have only an $l = m = 2$ spherical harmonic component].\n[Note that this\nquadrupole moment differs by an overall constant from the one defined by\nThorne~\\cite{Thorne}---e.g., his Eq.~(5.27a).] We then recall that\nthe perturbed Poisson equation for the $l = 2$ part of the perturbed\ngravitational potential is\n\\<\\label{Poisson}\n(\\triangle_2\\delta\\Phi)(r) := \\frac{1}{r^2}[r^2\\delta\\Phi'(r)]' - \\frac{6}{r^2}\\delta\\Phi(r) = 4\\pi G\\delta\\rho\n\\?\n($\\triangle_2$ is the $l = 2$ radial part of the Laplacian), with boundary conditions of\n\\<\n\\label{PoissonBCs}\n\\delta\\Phi(0) = 0, \\qquad R\\delta\\Phi'(R) = -3\\delta\\Phi(R),\n\\?\nwhere $R$ is the radial coordinate of the star's surface.\n[See, e.g., Eqs.~(2.15) and (2.16) in \\cite{LMO}---their $\\Phi_{22}$ is our\n$\\delta\\Phi$. Note also that the primes denote derivatives with respect to $r$. Additionally, we shall\ncontinue to be inconsistent with our inclusion of the functional dependence of quantities---e.g., $\\delta\\rho$\ndepends upon $r$, even though we do not always indicate this explicitly. We will eventually stop displaying $\\delta\\Phi$'s \nexplicit functional dependence on $r$, for instance.]\nIf we now substitute Eq.~\\eqref{Poisson} into Eq.~\\eqref{Q22} and integrate by\nparts using the boundary conditions~\\eqref{PoissonBCs}, we obtain\n\\<\\label{Q22N}\nQ_{22} = -\\frac{5R^3}{4\\pi G}\\delta\\Phi(R).\n\\?\nThis\nsort of expression is more commonly seen in the relativistic case, where it is\nnecessary to obtain the quadrupole in this manner by looking at the\nperturbation's asymptotic behavior---see the discussion in Sec.~\\ref{GR}.\n\nWe now wish to obtain an equation for $\\delta\\Phi$ in terms of the shear\nstresses. We follow UCB in decomposing the perturbed stress tensor as\n[see their Eqs.~(59) and (61)]\n\\<\\label{deltatau}\n\\begin{split}\n\\delta\\tau_{ab} &= -\\delta pY_{lm}g_{ab} + t_{rr}Y_{lm}(\\hat{r}_a\\hat{r}_b - e_{ab}\/2) + \nt_{r\\perp}f_{ab}\\\\\n&\\quad + t_\\Lambda(\\Lambda_{ab} + Y_{lm}e_{ab}\/2).\n\\end{split}\n\\? \nHere $\\delta p$ is the (Eulerian) pressure perturbation; $Y_{lm}$ is a spherical\nharmonic; $\\hat{r}_a$ is the radial unit vector; $t_{rr}$, $t_{r\\perp}$, and\n$t_\\Lambda$ are the components of the shear stresses; and\n$g_{ab}$ denotes the metric of flat, $3$-dimensional Euclidean space.\n(Following UCB, we will generally write out $l$ and $m$ explicitly, even\nthough we only consider $l = m = 2$ here.)\nAlso [Eqs.~(40) in UCB],\n\\begin{subequations}\n\\begin{align}\ne_{ab} &:= g_{ab} - \\hat{r}_a\\hat{r}_b,\n\\\\\nf_{ab} &:= 2r\\hat{r}_{(a}\\nabla_{b)}Y_{lm}\/\\beta,\n\\\\\n\\beta &:= \\sqrt{l(l+1)} = \\sqrt{6},\n\\\\\n\\label{Lambda}\n\\Lambda_{ab} &:= r^2\\nabla_a\\nabla_bY_{lm}\/\\beta^2 + f_{ab}\/\\beta.\n\\end{align}\n\\end{subequations}\n(We have corrected the dropped factor of $\\beta^{-1}$ multiplying $f_{ab}$ in\nUCB's definition of $\\Lambda_{ab}$---this was also noticed by HJA.)\nWe also have\n\\begin{equation}\n\\label{ts}\nt_{ab} = 2\\mu\\sigma_{ab},\n\\end{equation}\nwhere $\\mu$ is the shear modulus and $\\sigma_{ab}$ is the strain tensor.\n(This is a factor-of-$2$ correction to the expression in UCB, as noted in\n\\cite{OwenPRL}.)\nNow, a convenient expression can be obtained from the perturbed\nequation of hydrostatic equilibrium\n\\<\\label{hydro}\n\\nabla^a\\delta\\tau_{ab} = \\delta\\rho g(r)\\hat{r}_b + \\rho\\nabla_b\\delta\\Phi\n\\?\n($\\nabla_a$ denotes the flat-space covariant derivative), by\nsubstituting for $\\delta\\rho$ using the Poisson equation~\\eqref{Poisson}\nand projecting along $\\hat{r}^b$, yielding\n\\<\\label{deltaPhieq1}\n\\begin{split}\n\\frac{\\triangle_2\\delta\\Phi}{4\\pi G} + \\frac{\\rho}{g(r)}\\delta\\Phi' &= \\frac{\\hat{r}^b\\nabla^a\\delta\\tau_{ab}}{g(r)}\\\\\n&= \\frac{1}{g(r)}\\left[-\\delta p' + t_{rr}' + \\frac{3}{r}t_{rr} - \\frac{\\beta}{r}t_{r\\perp}\\right].\n\\end{split}\n\\?\nWe then project Eq.~\\eqref{hydro} along $\\nabla^bY_{lm}$ to express $\\delta p$ in terms of the shear stresses $t_{rr}$, $t_{r\\perp}$, and $t_\\Lambda$,\nalong with $\\rho$ and $\\delta\\Phi$, giving\n\\<\n\\delta p = -\\rho\\delta\\Phi - \\frac{t_{rr}}{2} + \\frac{r}{\\beta}t_{r\\perp}' + \\frac{3}{\\beta}t_{r\\perp} + \\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda.\n\\?\nSubstituting this into Eq.~\\eqref{deltaPhieq1}, we thus obtain\n\\<\\label{deltaPhieq2}\n\\begin{split}\n\\triangle_2\\delta\\Phi - \\frac{4\\pi G}{g(r)}\\rho'\\delta\\Phi &=\n\\frac{4\\pi G}{g(r)}\\biggl[\\frac{3}{2}t_{rr}' - \\frac{4}{\\beta}t_{r\\perp}' - \\frac{r}{\\beta}t_{r\\perp}''\\\\\n&\\quad - \\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda' + \\frac{3}{r}t_{rr} - \\frac{\\beta}{r}t_{r\\perp}\\biggr].\n\\end{split}\n\\?\n\nWe now wish to obtain an integral expression for $Q_{22}$ that generalizes\nUCB's Eq.~(64) to the case where we do not make the Cowling approximation.\nWe shall do this by obtaining the Green function for the left-hand side of\nEq.~\\eqref{deltaPhieq2} and then integrating by parts.\nWe will be able to\ndiscard all of the boundary terms, since the\nstresses vanish at the star's surface (we assume that the shear modulus vanishes there) and the integrand vanishes at the star's center. We can obtain the\nGreen function using the standard Sturm-Liouville expression in terms of the solutions of the\nhomogeneous equation [e.g., Eq.~(10.103) in Arfken and Weber~\\cite{AW}~].\nWe obtain the appropriate solution to the homogeneous equation numerically\nfor a given background stellar model (EOS and mass). The equation for the Green function is\n[multiplying the left-hand side of Eq.~\\eqref{deltaPhieq2} by $r^2$ to improve its regularity]\n\\<\\label{Leq}\n\\begin{split}\n(\\mathcal{L}_N\\mathcal{G})(r,\\bar{r}) &:= \\frac{\\partial}{\\partial r}\\left[r^2\\frac{\\partial}{\\partial r}\\mathcal{G}(r,\\bar{r})\\right] - \\left[6 + \\frac{4\\pi G r^2}{g(r)}\\rho'\\right]\\mathcal{G}(r,\\bar{r})\\\\\n&\\,= \\delta(r - \\bar{r})\n\\end{split}\n\\?\n[$\\delta(r - \\bar{r})$ is the Dirac delta function], with boundary conditions (at the star's center and surface) of\n\\<\\label{Newt_BC}\n\\mathcal{G}(0,\\bar{r})=0, \\qquad R\\partial_1\\mathcal{G}(R,\\bar{r}) = -3\\mathcal{G}(R,\\bar{r}),\n\\?\nwhere $\\partial_1$ denotes a partial derivative taken with respect to the\nfirst ``slot'' of the function.\n\nIf we then write [using Eq.~\\eqref{Q22N}, the factor of $r^2$ from the Green function equation~\\eqref{Leq}, and the\nprefactor on the right-hand side of Eq.~\\eqref{deltaPhieq2}]\n\\begin{equation}\\label{GN}\nG_N} % Changed to save space; was {\\mathrm{Newt}(r) := -5R^3r^2\\mathcal{G}(R,r)\/g(r),\n\\end{equation}\nwe have\n\\begin{widetext}\n\\<\\label{Q22N1}\n\\begin{split}\nQ^N} % Changed to save space; was {\\mathrm{Newt}_{22} &= \\int_0^RG_N} % Changed to save space; was {\\mathrm{Newt}(r)\\left[\\frac{3}{2}t_{rr}' - \\frac{4}{\\beta}t_{r\\perp}' - \\frac{r}{\\beta}t_{r\\perp}'' -\n\\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda' + \\frac{3}{r}t_{rr} - \\frac{\\beta}{r}t_{r\\perp}\\right]dr\\\\\n&= -\\int_0^R\\biggl\\{\\left[\\frac{3}{2}G_N} % Changed to save space; was {\\mathrm{Newt}'(r) - \\frac{3}{r}G_N} % Changed to save space; was {\\mathrm{Newt}(r)\\right]t_{rr} + \\left[\\frac{r}{\\beta}G_N} % Changed to save space; was {\\mathrm{Newt}''(r) - \\frac{2}{\\beta}G_N} % Changed to save space; was {\\mathrm{Newt}'(r) + \\frac{\\beta}{r}G_N} % Changed to save space; was {\\mathrm{Newt}(r)\\right]t_{r\\perp\n+ \\left(\\frac{1}{2} - \\frac{1}{\\beta^2}\\right)G_N} % Changed to save space; was {\\mathrm{Newt}'(r)t_\\Lambda\\biggr\\}dr.\n\\end{split}\n\\?\n\\end{widetext}\nWe have freely integrated by parts in obtaining the second expression, noting that the boundary terms are zero since $G_N} % Changed to save space; was {\\mathrm{Newt}(r)$ vanishes sufficiently rapidly as $r \\to 0$ and\nthe stresses are zero at the surface of the star (since we assume that the shear modulus vanishes at the star's surface).\\footnote{We shall treat the case where the stresses do \\emph{not} vanish at the surface of the star when we consider solid strange quark stars in Sec.~\\ref{SQM_computation}. Also, note that HJA claim that UCB's\nexpression does not include distributional contributions due to sudden changes\nin the shear modulus. This is not the case---these are included due to UCB's\nintegration by parts (cf.\\ the definition of the distributional derivative). All that the UCB derivation requires is,\ne.g., that the shear modulus vanish outside of the crust, not that\nit do so continuously.}\nThis reduces to UCB's Eq.~(64) if we take the Cowling approximation\n\\begin{equation}\n\\label{CowlingGN}\nG_N} % Changed to save space; was {\\mathrm{Newt}(r) \\to r^4\/g(r),\n\\end{equation}\ncorresponding to dropping the second term on the left-hand side of\nEq.~\\eqref{deltaPhieq2}.\n\nTo obtain an analogue of the expression for the maximum quadrupole\ngiven in Eq.~(5) of Owen~\\cite{OwenPRL}, we note that\nUCB's argument about maximum uniform strain leading to the maximum quadrupole\nstill holds here for the stars we consider, since the coefficients of the stress\ncomponents in the integrand are all uniformly positive. (We have checked this numerically for each background\nstellar model we consider.)\nThe strain tensor components are\n\\begin{subequations}\n\\label{sNewt}\n\\begin{align}\n\\label{srr}\n\\sigma_{rr} &= (32\\pi\/15)^{1\/2}\\bar{\\sigma}_\\mathrm{max},\n\\\\\n\\label{srp}\n\\sigma_{r\\perp} &= (3\/2)^{1\/2}\\sigma_{rr},\n\\\\\n\\label{sL}\n\\sigma_\\Lambda &= 3\\sigma_{rr}\n\\end{align}\n\\end{subequations}\nin the case where the\nstar is maximally (and uniformly) strained---see Eqs.~(67) in UCB. The\nbreaking strain $\\bar{\\sigma}_\\mathrm{max}$ is given by the von Mises\nexpression,\n\\begin{equation}\n\\label{vonMises}\n\\sigma_{ab}\\sigma^{ab} = 2\\bar{\\sigma}_\\mathrm{max}^2.\n\\end{equation}\nIt thus\ncorresponds to assuming that the lattice yields when it has stored a certain maximum\nenergy density. We then have\n\\<\\label{Q22N2}\n\\frac{|Q_{22}^{\\mathrm{max}, N}|}{\\bar{\\sigma}_\\mathrm{max}} = \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\mu(r)\\left[rG_N} % Changed to save space; was {\\mathrm{Newt}''(r) + 3G_N} % Changed to save space; was {\\mathrm{Newt}'(r)\\right]dr.\n\\?\nThis reduces to Eq.~(5) in Owen~\\cite{OwenPRL} if we use the Cowling\napproximation~\\eqref{CowlingGN}.\n\nNote that there is no direct contribution from $\\rho'$ to $G_N} % Changed to save space; was {\\mathrm{Newt}''$ in the\nno-Cowling case, despite what one might expect from Eq.~\\eqref{Leq}: Writing\n$\\bar{\\mathcal{G}}(r) := \\mathcal{G}(R,r)$ for notational simplicity, the $\\rho'$ contribution from\n\\<\n\\bar{\\mathcal{G}}''(r) = (2\/r)\\bar{\\mathcal{G}}'(r) + \n[6\/r^2 + 4\\pi G\\rho'(r)\/g(r)]\\bar{\\mathcal{G}}(r)\n\\?\nis exactly canceled by one from \n\\<\ng''(r) = 6Gm(r)\/r^4 - 8\\pi G\\rho(r)\/r + 4\\pi G\\rho'(r)\n\\?\nin\n\\<\n\\begin{split}\nG''_N} % Changed to save space; was {\\mathrm{Newt}(r) &= -5R^3r^2[\\bar{\\mathcal{G}}''(r)\/g(r) - \\bar{\\mathcal{G}}(r)g''(r)\/\\{g(r)\\}^2\\\\\n&\\quad + \\{\\text{terms with no }\\rho'\\}].\n\\end{split}\n\\?\nHowever, there is a direct contribution from\n$\\rho'$ to $G_N} % Changed to save space; was {\\mathrm{Newt}''$ (via $g''$) if we make the Cowling\napproximation [Eq.~\\eqref{CowlingGN}].\nWe shall see that this leads to a\nsignificant\ndifference in the resulting contributions to the quadrupole moment from regions\nof the star surrounding a sudden change in density (e.g., near the crust-core\ninterface, which will be relevant for the quadrupoles supported by crustal\nelasticity considered by UCB and others).\n\nNumerically, we compute $G_N} % Changed to save space; was {\\mathrm{Newt}$ using the standard expression for the Green function in terms of the\ntwo independent solutions to the homogeneous equation [see, e.g., Eq.~(10.103) in Arfken and Weber~\\cite{AW}~]. Since we are solely interested in the Green function evaluated at the star's surface, we can\neliminate one of the homogeneous solutions using the boundary conditions there, and only consider the\nhomogeneous solution that is regular at the origin, which we call $F$.\nIn terms of $F$, the Green function is given by\n\\<\\label{cG_F}\n\\mathcal{G}(R,r) = -\\frac{F(r)}{3RF(R) + R^2F'(R)}.\n\\?\nWe thus solve $\\mathcal{L}_N F = 0$ [with the operator $\\mathcal{L}_N$ given by Eq.~\\eqref{Leq}] with the boundary conditions\n$F(r_0) = 1$ and $F'(r_0) = 2\/r_0$, where $r_0$ is the small inner radius used in the solution of the\nOV equations, as discussed at the end of Sec.~\\ref{GR}. [These boundary conditions come from\nregularity at the origin, which implies that $F(r) = O(r^2)$ there.]\n\nOur Green function method for obtaining the maximum quadrupole numerically may\nseem more complicated than existing methods because it introduces extra steps.\nHowever this method is ideal for showing that maximum stress gives the maximum\nquadrupole and for seeing how much stresses at different radii contribute to\nthe total quadrupole.\nIt also appears to be the simplest way of dealing with any potential\ndistributional contributions from the derivatives of the shear modulus, since\nthey are automatically taken care of by the integration by parts.\n\n\\section{General relativistic calculation of the maximum quadrupole}\n\\label{GR}\n\nHere we compute the maximum quadrupole moment in general relativity, using the Regge-Wheeler\ngauge~\\cite{RW} relativistic stellar perturbation theory developed by \\citet{TC}, as in\nthe similar calculation of the tidal Love number of a relativistic star by\n\\citet{Hinderer}.\nWe\nstart by writing down the line element corresponding to a static, even-parity,\n$l = 2$ first-order perturbation of a static,\nspherical, relativistic star in the Regge-Wheeler gauge [cf.\\ Eq.~(14) in Hinderer~\\cite{Hinderer}~]:\n\\<\\label{ds2}\n\\begin{split}\nds^2 &= -[1 + H_0(r)Y_{lm}]f(r)dt^2 + [1 + H_2(r)Y_{lm}]h(r)dr^2\\\\\n&\\quad + [1 + K(r)Y_{lm}]r^2(d\\theta^2 + \\sin^2\\theta d\\phi^2).\n\\end{split}\n\\?\nHere we have used the notation of Wald~\\cite{Wald} for the background, so that $f$ and $h$ are the standard\nSchwarzschild functions for the unperturbed star, with $f = e^{2\\phi}$, where\n\\<\n\\phi'(r) = \\frac{m(r) + 4\\pi r^3 p}{r[r - 2m(r)]},\n\\?\nwith $\\phi(R) = \\log(1 - 2M\/R)\/2$, and\n\\<\nh(r) = \\left[1 - \\frac{2m(r)}{r}\\right]^{-1}.\n\\?\nIn these expressions,\n\\begin{equation}\nm(r) := 4\\pi \\int_0^r \\rho(\\bar{r})\\bar{r}^2d\\bar{r}.\n\\end{equation}\nAlso, recall that we write our spherical harmonics in terms of $l$ and $m$, following\nUCB, even though we specialized to $l = m = 2$, and that we are now taking $G = c = 1$.\n\nThe metric perturbation is determined by $H_0$, $H_2$, and $K$, which here are sourced by the\nperturbation to the star's stress-energy tensor. The appropriate stress-energy\ntensor can be obtained directly\nfrom the standard Newtonian expression~\\eqref{deltatau} by simple covariance arguments, as in\nSchumaker and Thorne~\\cite{ST}, or from the detailed relativistic elasticity\ntheory of Carter and Quintana~\\cite{CQ} [see their Eq.~(6.19); this is also given in Eq.~(128) of\nKarlovini and Samuelsson~\\cite{KaSaI}~]. All\nwe really need for our purposes is to note that the shear contribution is\ntracefree with respect to the background metric, so that we can use the obvious\ncovariant generalization of the decomposition given by UCB,\\footnote{Of course, this assumes that it is\npossible to obtain \\emph{any} symmetric tracefree tensor from the detailed relativistic expression, but---as would be expected (and can easily be seen from the expressions)---this is indeed the case, at least if one only works to first order in the perturbation, as we\ndo here.\nAlso, it is instructive to note that we do not need to know the specifics of the matter displacements that generate\nthe quadrupoles we consider, only that there is a tracefree contribution to the star's\nstress-energy tensor whose maximum value is given by the material's shear\nmodulus and von Mises breaking strain.} yielding \n\\<\\label{deltaT}\n\\begin{split}\n\\delta T_{ab} &= [\\delta\\rho\\hat{t}_a\\hat{t}_b + \\delta p(g_{ab} + \\hat{t}_a\\hat{t}_b) - \nt_{rr}(\\hat{r}_a\\hat{r}_b - q_{ab}\/2)]Y_{lm}\\\\\n&\\quad - \nt_{r\\perp}f_{ab} - t_\\Lambda(\\tilde{\\Lambda}_{ab} + h^{1\/2}Y_{lm}q_{ab}\/2),\n\\end{split}\n\\?\nwith the full stress-energy tensor given by\n\\<\nT_a{}^b = \\rho\\hat{t}_a\\hat{t}^b + p(\\delta_a{}^b + \\hat{t}_a\\hat{t}^b) + \\delta T_a{}^b.\n\\?\nHere, indices now run over all four spacetime dimensions and $g_{ab}$ denotes\nthe background (spacetime) metric (which we use to raise and lower indices).\nAdditionally, we have introduced the background temporal and radial unit vectors $\\hat{t}_a$ and $\\hat{r}_a$;\n$q_{ab}$ is the induced metric on the unit $2$-sphere;\n$f_{ab} := 2r\\hat{r}_{(a}\\nabla_{b)}Y_{lm}\/\\beta$;\nand\n$\\tilde{\\Lambda}_{ab} := r^2h^{1\/2}\\nabla_a\\nabla_bY_{lm}\/\\beta^2 + f_{ab}\/\\beta$.\nHere $\\hat{r}_a$ and $\\nabla_a$ now have their curved-space meanings.\n\nOur $\\tilde{\\Lambda}_{ab}$\ndiffers from the Newtonian $\\Lambda_{ab}$ [from UCB, given in our Eq.~\\eqref{Lambda}] due to the insertion of $h^{1\/2}$. This insertion is necessary\nfor $\\tilde{\\Lambda}_{ab}$ to be transverse and orthogonal to $f_{ab}$ (with respect to the background spacetime metric).\nThe same logic leads to the introduction of the factor of $h^{1\/2}$ multiplying $q_{ab}$ in the $t_\\Lambda$ term in\nEq.~\\eqref{deltaT}; it is there so that the $t_\\Lambda$ term is orthogonal to the $t_{rr}$ term. We\nhave used UCB's convention\nfor the relative sign between the perfect fluid and shear portions of the stress-energy tensor, though we\nhave reversed the overall sign. (However, we used the UCB convention proper\nin Sec.~\\ref{Newt}.)\nThe factor of $h^{1\/2}$ in the coefficient of $t_\\Lambda$ leads to a factor\nof $h^{-1}$ in the\nstrain $\\sigma_\\Lambda$ that corresponds to the von Mises\nbreaking strain~\\eqref{vonMises}.\nWe thus replace the Newtonian Eq.~\\eqref{sL} with\n\\begin{equation}\n\\label{sLGR}\n\\sigma_\\Lambda = 3\\sigma_{rr}\/h,\n\\end{equation}\nleaving Eqs.~\\eqref{srr} and~\\eqref{srp} unchanged.\n\nOne can now obtain an equation for $H_0$ from the perturbed Einstein equations,\nas in Ipser~\\cite{Ipser}. (The other two metric functions, $H_2$ and $K$, can be expressed in terms of $H_0$;\nthese expressions are given by Ipser.) The concordance for notation\nis $\\nu = 2\\phi$, $e^\\nu = f$, $\\lambda = 2\\psi$, $e^\\lambda = h$, $\\rho_1 = -\\delta\\rho$, $p_1 = -\\delta p$,\n$\\mathfrak{P}_2 = t_{rr}$, $\\mathfrak{Q}_1 = h^{1\/2}t_{r\\perp}\/\\beta$, and $\\mathfrak{S} = h^{1\/2}t_\\Lambda\/\\beta^2$. Additionally, Ipser's $H_0$ is the\nnegative of ours. The relevant result is given in Ipser's Eqs.~(27)--(28), and\nis (in our notation)\n\\<\nH_0'' + \\left(\\frac{2}{r} + \\phi' -\\psi'\\right)H_0' + \\mathcal{P}(r)H_0= 8\\pi h^{1\/2}\\mathcal{S}(r),\n\\?\nwhere\n\\<\\label{cP}\n\\mathcal{P}(r) := 2\\phi'' + 2\\phi'\\left(\\frac{3}{r} - \\phi' -\\psi'\\right) + \\frac{2\\psi'}{r} - \\frac{\\beta^2}{r^2}h\n\\?\nand\n\\<\\label{cS}\n\\begin{split}\n\\mathcal{S}(r) &:=\nh^{1\/2}(\\delta\\rho + \\delta p - t_{rr}) +\n2\\biggl\\{(3 - r\\phi')\\frac{t_{r\\perp}}{\\beta} + r\\frac{t_{r\\perp}'}{\\beta}\\\\\n&\\,\\quad + [r^2\\phi'' + r\\phi'(5 - r\\phi') + r\\psi' -\\beta^2h\/2 + 1]\\frac{t_\\Lambda}{\\beta^2}\\\\\n&\\,\\quad + r^2\\phi'\\frac{t_\\Lambda'}{\\beta^2}\\biggr\\}\\\\\n&=: h^{1\/2}(\\delta\\rho + \\delta p) + \\mathcal{S}_{[t]}(r).\n\\end{split}\n\\?\nHere we have defined $\\psi := (1\/2)\\log h$ and written $\\mathcal{S}_{[t]}$\nfor the contributions from shear stresses. (The ``$=:$'' notation implies that the quantity\nbeing defined is on the right-hand side of the equality.)\n\nWe now wish to eliminate $\\delta\\rho$ and $\\delta p$ in favor of the shear stresses, as in the Newtonian calculation. We use the same projections of stress-energy conservation as in the Newtonian case (projecting onto the quantities defined by\nthe background spacetime, for simplicity) along with the Oppenheimer-Volkov\n(OV) equations, giving\n\\<\n\\begin{split}\n\\delta\\rho + \\delta p &= \\frac{1}{\\phi'}\\biggl[-\\frac{H_0'}{2}(\\rho + p)\n-\\delta p' + t_{rr}' + \\left(\\frac{3}{r} + \\phi'\\right)t_{rr}\\\\\n&\\quad - \\frac{\\beta}{r}h^{1\/2}t_{r\\perp}\\biggr]\n\\end{split}\n\\?\nand\n\\<\n\\begin{split}\n\\delta p &= -\\frac{H_0}{2}(\\rho + p) - \\frac{t_{rr}}{2} + \\frac{1}{\\beta h^{1\/2}}\\left[(3 + r\\phi')t_{r\\perp} + rt_{r\\perp}'\\right]\\\\\n&\\quad + h^{1\/2}\\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda.\n\\end{split}\n\\?\nUsing the second expression to substitute for $\\delta p'$ in the first, we have\n\\<\\label{drplusdp}\n\\begin{split}\n\\delta\\rho + \\delta p &= \\frac{1}{\\phi'}\\biggl\\{\\frac{H_0}{2}(\\rho' + p') + \\left[\\frac{3}{r} + \\phi'\\right]t_{rr} + \\frac{3}{2}t_{rr}'\\\\\n&\\quad - \\frac{1}{\\beta h^{1\/2}}\\biggl[\\left(\\frac{\\beta^2 h}{r} + \\phi' + r\\phi'' - \\psi'[3 + r\\phi'] \\right)t_{r\\perp}\\\\\n&\\quad + (4 + r[\\phi' - \\psi'])t_{r\\perp}' + rt_{r\\perp}''\\biggr] + \n\\left(\\frac{1}{2} - \\frac{1}{\\beta^2}\\right)\\\\\n&\\quad\\times h^{1\/2}(\\psi't_\\Lambda + t_\\Lambda')\\biggr\\}\\\\\n&=: \\frac{H_0}{2\\phi'}(\\rho' + p') + \\frac{\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)}{\\phi'}.\n\\end{split}\n\\?\nThe equation for $H_0$ thus becomes\n\\<\\label{cLGR}\n\\begin{split}\n(\\mathcal{L}_\\mathrm{GR} H_0)(r) &:= H_0'' + \\left(\\frac{2}{r} + \\phi' -\\psi'\\right)H_0'\\\\\n&\\,\\quad + \\left[\\mathcal{P}(r) - 4\\pi h\\frac{\\rho' + p'}{\\phi'}\\right]H_0\\\\\n&\\,= 8\\pi h^{1\/2}[h^{1\/2}\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)\/\\phi' + \\mathcal{S}_{[t]}(r)].\n\\end{split}\n\\?\n[$\\mathcal{P}(r)$ and $\\mathcal{S}_{[t]}(r)$ are given in Eqs.~\\eqref{cP} and~\\eqref{cS}, respectively.]\nAs expected, this\nreduces to Eq.~\\eqref{deltaPhieq2} in the Newtonian\nlimit [where we have $H_0 \\to 2\\delta\\Phi$ and $\\phi' \\to g(r)$].\n\nWe now want to write the equation for $H_0$ in Sturm-Liouville form in order to obtain its Green\nfunction easily. To do this, we note that the appropriate ``integrating factor'' (for the first\ntwo terms) is $r^2(f\/h)^{1\/2}$, which gives\n\\begin{multline}\n\\label{H0_S-L}\n[r^2(f\/h)^{1\/2}H_0']' + r^2(f\/h)^{1\/2}\\left[\\mathcal{P}(r) - 4\\pi h\\frac{\\rho' + p'}{\\phi'}\\right]H_0\\\\\n= 8\\pi r^2f^{1\/2}[h^{1\/2}\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)\/\\phi' + \\mathcal{S}_{[t]}(r)].\n\\end{multline}\nWe also need the boundary conditions, which are given by matching $H_0$ onto a\nvacuum solution at the surface of the star. The vacuum\nsolution that is regular at infinity is given by Eq.~(20) in Hinderer~\\cite{Hinderer} with $c_2 = 0$,\nviz.,\n\\<\\label{H0_BC}\n\\begin{split}\nH_0(R) &= c_1\\biggl[\\left(\\frac{2}{\\mathcal{C}} - 1\\right)\\frac{\\mathcal{C}^2\/2 + 3\\mathcal{C} - 3}{1 - \\mathcal{C}}\\\\\n&\\quad + \\frac{6}{\\mathcal{C}}\\left(1 - \\frac{1}{\\mathcal{C}}\\right)\\log\\left(1 - \\mathcal{C}\\right)\\biggr],\n\\end{split}\n\\?\nwhere we have evaluated this at the star's surface ($r = R$) and\ndefined the star's compactness\n\\<\n\\label{compactness}\n\\mathcal{C} := 2GM\/Rc^2\n\\?\n(now returning to showing factors of $G$ and $c$ explicitly).\nWe require that $H_0$\nand $H_0'$ be continuous at the star's surface. The value of $c_1$ obtained from\nthis matching of the internal and external solutions gives us \nthe quadrupole moment. If we use the quadrupole moment amplitude that reduces\nto the UCB integral [given in our Eq.~\\eqref{Q22}] in the Newtonian limit, we have\n\\<\\label{Q22rel}\nQ_{22} = \\frac{G^2}{c^4}\\frac{M^3c_1}{\\pi}.\n\\?\n[This expression comes from inserting a pure $l = m = 2$ density perturbation\ninto Eq.~(2) in Hinderer~\\cite{Hinderer}, contracting the free indices with\nunit position vectors, performing the angular integral, for\nwhich the expressions in Thorne~\\cite{Thorne} are useful,\nand noting that the result is $(8\\pi\/15)Y_{22}$ times\nour Eq.~\\eqref{Q22}. The\ngiven result then follows immediately from Hinderer's Eqs.~(7), (9), and (22);\nwe reverse the overall sign since we have reversed the UCB sign convention for the\nstress-energy tensor.]\n\nWe then have a Green function for $Q_{22}$ of \n\\<\\label{cG_GR}\n\\begin{split}\n\\mathcal{G}_\\mathrm{GR}(R,r) &= \\left(\\frac{2GM}{c^2}\\right)^3\\left(1 - \\frac{2GM}{Rc^2}\\right)^{-1}\\\\\n&\\quad\\times\\frac{\\mathcal{U}(r)}{c^2R^2[\\mathcal{U}'(R)H_0(R) - \\mathcal{U}(R)H_0'(R)]}\n\\end{split}\n\\?\n(including the overall factor of $8\\pi G\/c^4$ that\nmultiplies the source).\nHere $\\mathcal{U}$ is given by $\\mathcal{L}_\\mathrm{GR}\\mathcal{U} = 0$ [$\\mathcal{L}_\\mathrm{GR}$ is given in\nEq.~\\eqref{cLGR}], with boundary conditions $\\mathcal{U}(r_0) = 1$ and $\\mathcal{U}'(r_0) = 2\/r_0$.\n[Compare Eq.~(10.103) in Arfken and Weber~\\cite{AW}, as well as our Newtonian\nversion above.]\nAdditionally, $H_0(R)$ and $H_0'(R)$ are given by the boundary conditions~\\eqref{H0_BC}\nwith $c_1 \\to 1$. [One obtains this expression by first computing the Green function\nfor $H_0(R)$ following Arfken and Weber, then dividing through by the quantity in brackets in Eq.~\\eqref{H0_BC}\nto obtain $c_1$, and finally using Eq.~\\eqref{Q22rel} to obtain $Q_{22}$. We have also noted that\n$1\/f \\to h \\to 1\/(1 - 2GM\/Rc^2)$ at the star's surface.]\nWe thus define, for notational simplicity, two relativistic generalizations\nof $G_N} % Changed to save space; was {\\mathrm{Newt}(r)$: One,\n\\<\\label{GGR}\nG_\\mathrm{GR}(r) := \\frac{r^2(fh)^{1\/2}\\mathcal{G}_\\mathrm{GR}(R,r)}{\\phi'},\n\\?\nfor\nthe contributions from $\\mathcal{S}_{[\\delta\\rho,\\delta p]}$, and one,\n\\<\\label{GbGR}\n\\bar{G}_\\mathrm{GR}(r) := r^2f^{1\/2} \\mathcal{G}_\\mathrm{GR}(R,r),\n\\?\nfor the contributions\nfrom $\\mathcal{S}_{[t]}$. \n\nWith these definitions, the integral expression for the quadrupole in terms\nof the stresses and the\nstructure of the background star is\n\\<\\label{Q22GR}\n\\begin{split}\nQ_{22} &= \\int_0^R\\left[G_\\mathrm{GR}(r)\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)+\\bar{G}_\\mathrm{GR}(r)\\mathcal{S}_{[t]}(r)\\right]dr\\\\\n&= \\int_0^R(\\mathcal{C}_{rr}t_{rr} + \\mathcal{C}_{t\\perp}t_{r\\perp} + \\mathcal{C}_\\Lambda t_\\Lambda)dr,\n\\end{split}\n\\?\nwhere\n\\begin{subequations}\\label{cCs}\n\\begin{gather}\n\\mathcal{C}_{rr} := \\left(\\frac{3}{r} + \\phi'\\right)G_\\mathrm{GR}(r) - \\frac{3}{2}G_\\mathrm{GR}'(r) -\nh^{1\/2}\\bar{G}_\\mathrm{GR}(r),\\\\\n\\begin{split}\n\\mathcal{C}_{r\\perp} &:= -\\frac{\\beta h^{1\/2}}{r}G_\\mathrm{GR}(r) + \\frac{2 + r(\\phi' + \\psi')}{\\beta h^{1\/2}}G_\\mathrm{GR}'(r)\\\\\n&\\,\\quad - \\frac{r}{\\beta h^{1\/2}}G_\\mathrm{GR}''(r) + \\frac{4 - 2r\\phi'}{\\beta}\\bar{G}_\\mathrm{GR}(r) - 2\\frac{r}{\\beta}\\bar{G}'_\\mathrm{GR}(r),\n\\end{split}\\\\\n\\begin{split}\n\\mathcal{C}_\\Lambda &:= \\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)h^{1\/2}G_\\mathrm{GR}'(r)\\\\\n&\\,\\quad + \\frac{2r\\phi'(3 - r\\phi') + 2r\\psi' - \\beta^2h + 2}{\\beta^2}\\bar{G}_\\mathrm{GR}(r)\\\\\n&\\,\\quad - \\frac{2r^2\\phi'}{\\beta^2}\\bar{G}_\\mathrm{GR}'(r),\n\\end{split}\n\\end{gather}\n\\end{subequations}\nand we have integrated by parts twice to obtain the second equality in\nEq.~\\eqref{Q22GR}, using the same argument as in our Newtonian calculation.\n\nWe now look at the maximum quadrupole. This is still given by the uniformly\nmaximally strained case: We have checked numerically that the\ncoefficients of the three\nstress terms are always negative for all the background stars we consider. We thus\nhave a maximum quadrupole\ngiven by inserting Eqs.~\\eqref{ts}, \\eqref{srr}, \\eqref{srp}, and\n\\eqref{sLGR} into Eq.~\\eqref{Q22GR}, yielding\n\\begin{widetext}\n\\<\\label{Q22GR2}\n\\frac{|Q_{22}^\\mathrm{max, GR}|}{\\bar{\\sigma}_\\mathrm{max}} = \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\mu(r)\\biggl\\{\\left[\\frac{6}{r}(h^{1\/2} - 1) - 2\\phi'\\right]G_\\mathrm{GR}(r) + \\left[3 - \\frac{r}{h^{1\/2}}(\\phi' + \\psi')\\right]G_\\mathrm{GR}'(r\n+ \\frac{r}{h^{1\/2}}G_\\mathrm{GR}''(r) + \\mathcal{Q}^\\mathrm{stress}\\biggr\\}dr,\n\\?\nwhere\n\\<\n\\mathcal{Q}^\\mathrm{stress} := 2\\left[\\frac{r\\phi'(r\\phi' - 3) - r\\psi' - 1}{h} + r\\phi' + h^{1\/2} + 1\\right]\\bar{G}_\\mathrm{GR}(r) + 2r\\left(\\frac{r\\phi'}{h} + 1\\right)\\bar{G}_\\mathrm{GR}'(r)\n\\?\n\\end{widetext}\nis the contribution from the stresses' own gravity. We have split it off both for\nease of notation and because it is negligible except for the\nmost massive and compact stars, as illustrated\nbelow. The contributions from the density and pressure perturbations are so much larger due to the factor of $1\/\\phi'$ present in $G_\\mathrm{GR}$ [cf.\\ Eqs.~\\eqref{GGR} and~\\eqref{GbGR}]. It is easy to see that Eq.~\\eqref{Q22GR2} reduces to\nEq.~\\eqref{Q22N2} in the Newtonian limit, where $h\\to 1$,\nand we can neglect the contributions involving $\\phi'$, $\\psi'$, and $\\mathcal{Q}^\\mathrm{stress}$.\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrands_SLy_h0_7e19.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[$Q_{22}$ integrands for the SLy EOS and an\n$0.500\\,M_{\\odot}$ star]{\\label{Q22_integrands_SLy_2} The $Q_{22}$ integrands (without the factor of $\\mu\\bar{\\sigma}_\\mathrm{max}$) for the SLy EOS and an\n$0.500\\,M_{\\odot}$ star with a compactness of $0.12$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrands_SLy_h0_2e20.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[$Q_{22}$ integrands for the SLy EOS and a\n$1.40\\,M_{\\odot}$ star]{\\label{Q22_integrands_SLy_4} The $Q_{22}$ integrands (without the factor of $\\mu\\bar{\\sigma}_\\mathrm{max}$) for the SLy EOS and a\n$1.40\\,M_{\\odot}$ star with a compactness of $0.35$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrands_SLy_h0_7e20.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[$Q_{22}$ integrands for the SLy EOS and a maximum mass star]{\\label{Q22_integrands_SLy_6} The $Q_{22}$ integrands (without the factor of $\\mu\\bar{\\sigma}_\\mathrm{max}$) for the SLy EOS and a maximum mass,\n$2.05\\,M_{\\odot}$ star, with a compactness of $0.60$.}\n\\end{figure}\n\nWe now show how the relations between the different maximal-strain $Q_{22}$ Green functions\n[given by the integrands in Eqs.~\\eqref{Q22N2} and~\\eqref{Q22GR2} without the factors of\n$\\mu$ (but with the overall prefactor)] vary with\nEOS, as well as with the mass of the star for a given EOS.\nThis gives\nan indication of how much difference the various approximations make in different situations. We start with the unified SLy EOS~\\cite{DH}, obtained by Haensel\nand Potekhin~\\cite{HP}\n(using the table provided by\nthe Ioffe group~\\cite{HPY} at~\\cite{Ioffe}), which is a standard choice for making predictions about crustal\nquadrupoles (e.g., in Horowitz~\\cite{Horowitz}, HJA, and our Sec.~\\ref{results_crust}). Here we illustrate the\nchanges in the Green functions with mass for stars with masses ranging from $0.5\\,M_{\\odot}$ to the EOS's maximum mass of $2.05\\,M_{\\odot}$;\nsee Figs.~\\ref{Q22_integrands_SLy_2}, \\ref{Q22_integrands_SLy_4}, and \\ref{Q22_integrands_SLy_6}. (All three Green functions agree extremely closely for stars around the EOS's minimum mass of $0.094\\,M_{\\odot}$, so we do not show this case, particularly because such\nlow-mass neutron stars are of unclear astrophysical relevance.)\nThese stars' compactnesses [defined in Eq.~\\eqref{compactness}]\nrange from $0.12$ to\n$0.6$. Note that Fig.~\\ref{Q22_integrands_SLy_6} has a different vertical scale than the other two plots, due to the suppression of the quadrupole for massive, compact stars (discussed below). \n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrand_Hy1_h0_max_ratios.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[Ratios of the $Q_{22}$ integrands for the Hy1 EOS and a maximum mass\nstar]{\\label{Q22_integrands_Hy1_ratios} Ratios of the $Q_{22}$ integrands\nwith the Newtonian Cowling approximation integrand for the Hy1 EOS and a maximum mass,\n$2.06\\,M_{\\odot}$ star, with a compactness of $0.49$. Note that the top and bottom plots have completely\nseparate vertical axis scalings.}\n\\end{figure}\n\nWe illustrate the ratios of the various $Q_{22}$ Green functions to the\nNewtonian Cowling approximation one for the maximum mass ($2.05\\,M_{\\odot}$) hybrid\nstar using the Hy1 EOS (see Table~I in~\\cite{J-MO1}) in Fig.~\\ref{Q22_integrands_Hy1_ratios}.\\footnote{As\ndiscussed in~\\cite{J-MO1}, for our low-density EOS, we use the same combination of\nthe Baym, Pethick, and Sutherland (BPS)~\\cite{BPS} EOS for $n_B < 0.001\\text{ fm}^{-3}$ and\nthe Negele and Vautherin~\\cite{NV} EOS for $0.001\\text{ fm}^{-3} < n_B < 0.08\\text{ fm}^{-3}$\nused by Lattimer and Prakash~\\cite{LP2001} ($n_B$ is the baryon number density).\nThese were obtained from the table provided by\nKurkela~\\emph{et al.}~\\cite{Kurkelaetal} at~\\cite{Kurkelaetal_URL}.\nBulk quantities of hybrid stars such as the mass and quadrupole moment (from core deformations) do not depend much on the precise choice of low-density EOS.} We see the overestimate of the Newtonian\nno Cowling approximation calculation for perturbations in the core, particularly compared with the \ngeneral relativistic (GR) version, and also see the\noverestimate of the Newtonian Cowling approximation version for crustal perturbations. (We do not make\nsome sort of similar plot for the solid strange quark star case, since the expressions for the maximum\nquadrupole in this case end up being rather different than the integrated-by-parts ones presented in the previous sections, as we shall see in Sec.~\\ref{SQM_computation}.)\n\nIn all these cases, we compute the stellar background fully relativistically, using\nthe OV equations and identifying the OV equations' Schwarzschild\nradial coordinate\nwith the Newtonian radial coordinate when necessary. We have used the enthalpy form of the OV\nequations given by Lindblom~\\cite{Lindblom} and implemented the inner boundary\ncondition by taking the star to have an inner core of radius $r_0 =\n100\\text{ cm}$, whose mass is given by $(4\/3)\\pi r_0^3\\epsilon_0$, where\n$\\epsilon_0$ is the energy density corresponding to the central enthalpy that\nparametrizes the solution. (The spike near the origin seen in the bottom plot in Fig.~\\ref{Q22_integrands_Hy1_ratios} is due to this implementation of the inner boundary condition and has a\nnegligible effect on the computed maximum quadrupoles.) In all cases, we have used {\\sc{Mathematica}}~7's\ndefault methods to solve the differential equations, find roots, etc. We have computed as many derivatives\nas possible analytically, to aid numerical accuracy, e.g., using the OV equations to substitute for derivatives of the pressure, and also using the Green function equations to\nexpress second derivatives of the Green functions in terms of the functions themselves and their first derivatives.\n\n\n\\section{Results}\n\\label{results2}\n\n\\subsection{Maximum $Q_{22}$ for crustal deformations}\n\\label{results_crust}\n\nHere we consider the maximum quadrupoles from elastic deformations of a\nnonaccreted\ncrust in three possible situations, following HJA. In particular, we use the SLy EOS\n(as do Horowitz~\\cite{Horowitz} and HJA, though they do not refer to it by that name) and\nimpose two comparison crustal thicknesses to ascertain how much this\naffects the maximum quadrupole. Here we use the\nsame rough\nmodel for the crust's shear modulus used by HJA. We\nalso consider the more detailed model for the shear modulus\nobtained using the crustal composition provided by Douchin and\nHaensel~\\cite{DH} (also used by Horowitz~\\cite{Horowitz} and HJA).\nHere the crust's thickness is fixed to the value given in that work. In this case,\nwe also consider a different high-density EOS that yields much less compact stars with larger crusts.\n\nSpecifically, the two comparison crustal thicknesses are given by taking the base of the\ncrust to occur at densities of $2.1\\times10^{14} \\text{ g cm}^{-3}$ (thick\ncrust, for comparison with UCB) or $1.6\\times10^{14} \\text{ g cm}^{-3}$ (thin\ncrust, following a suggestion by Haensel~\\cite{Haensel}), while Douchin and\nHaensel place the bottom of the crust at a density of\n$1.28\\times10^{14} \\text{ g cm}^{-3}$.\nFor the two comparison cases, we take the shear modulus to be\n$10^{16}\\text{ cm}^2\\text{ s}^{-2}$ times the star's density (in $\\text{g cm}^{-3}$). As\nillustrated in HJA's Fig.~2, this is an underestimate of $<50\\%$, except at the very extremes of the density range considered.\\footnote{Note that Fig.~3 in HJA is not in agreement with their Fig.~2. When we reproduce those figures, we\nfind that the ratio $\\mu\/\\rho$ is considerably closer to $10^{16}\\text{ cm}^2\\text{ s}^{-2}$ over all the density range than the trace shown in HJA's Fig.~3, so their approximation is better than it would appear from that figure.}\nWe plot the quadrupole moment and ellipticity for these two cases for masses between\n$\\sim 1.2\\,M_{\\odot}$ (around the minimum observed neutron star mass---see~\\cite{Lattimer_table})\nand the SLy EOS's maximum mass of $2.05\\,M_{\\odot}$ in Fig.~\\ref{Q22s_vs_M_SLy}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_SLy_c16.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_SLy} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nmaximum quadrupole deformations (and fiducial ellipticity) due to crustal\nstresses versus mass for two choices of crustal thickness. These are computed using the SLy EOS with the rough HJA recipe for the shear\nmodulus and a breaking strain of $0.1$.}\n\\end{figure}\n\nIn addition to the quadrupole moments, we also show the fiducial ellipticity\n$\\epsilon_\\mathrm{fid} = \\sqrt{8\\pi\/15}Q_{22}\/I_{zz}$\n[e.g., Eq.~(2) of~\\cite{OwenPRL}~].\nHere $I_{zz}$ is the star's principal moment of\ninertia, for which we use the fiducial value of $I_{zz} = 10^{38} \\text{ kg m}^2 =\n10^{45} \\text{ g cm}^2$ used in the LIGO\/Virgo papers rather than the true value for a given mass and EOS,\nwhich can be greater by a factor of a few.\nWe do this for easy comparison with the observational\npapers, since they frequently quote\nresults in terms of this fiducial ellipticity instead of the quadrupole\nmoment, which is the quantity truly measured.\n\n\\emph{Nota bene} (N.B.):\\ We present these fiducial ellipticities \\emph{only} for comparison with LIGO\/Virgo\nresults, not to give\nany indication of the size of the deformation. While the true ellipticity\ngives a measure of the size of the deformation in the Newtonian case (up to ambiguities from the fact that the true density distribution is nonuniform), it does not do so in\nany obvious way in the relativistic case. Nevertheless, the relativistic shape of the star's surface can be obtained from its quadrupole deformation, as shown in~\\cite{NKJ-M_shape}. However, if one wished to know, for instance, how much the star is deformed as a function of radius, one would need to calculate this using a detailed relativistic theory of elasticity to relate the stresses to the matter displacements, as in Penner~\\emph{et al.}~\\cite{Penneretal}.\n\nIn the more detailed\ncase, we use the HJA version of the Ogata and Ichimaru~\\cite{OI} shear modulus,\ncombined with the Douchin and Haensel~\\cite{DH} results for the crust's\ncomposition.\nThis is [correcting a typo in HJA's Eq.~(20)],\n\\<\\label{mueff}\n\\mu_\\mathrm{eff} = 0.1194\\left(\\frac{4\\pi}{3}\\right)^{1\/3}\\left(\\frac{1-X_n}{A}n_b\\right)^{4\/3}(Ze)^2,\n\\?\nwhere $X_n$ is the fraction of neutrons outside of nuclei, $A$ and $Z$ are the\natomic and proton number of the nuclei, respectively, $n_b$ is the baryon number density, and\n$e$ is the fundamental charge.\n\nSince HJA's study, there have been a few improvements\nto the Ogata and Ichimaru result: Horowitz and Hughto~\\cite{HH} have computed\nthe effects of charge screening, finding a $\\sim 7\\%$ reduction in the shear\nmodulus. Baiko~\\cite{BaikoCPP} has also considered a relativistic model\nfor the electron polarizability and arrived at similar conclusions. Indeed, Baiko's results suggest that screening will yield an even smaller\ncorrection in the innermost portion of the crust, where the shear modulus is the largest, and the\nelectrons are the most relativistic, with a relativity parameter over an order of magnitude larger than the largest\nBaiko considers. (However, the ion charge numbers are also almost always somewhat greater than the largest Baiko considers,\nparticularly at the very innermost portion of the crust, which will tend to increase the effect.) \n\nBaiko~\\cite{Baiko} has also recently computed quantum corrections, and finds\nthat they reduce the shear modulus by up to $\\sim 18\\%$ in some regimes.\nHowever, in our case, the reduction will be much smaller, based on the scaling\nof $\\rho^{1\/6}\/(ZA^{2\/3})$ given near the end of Baiko's Sec.~6. Even though\nour densities are over an order of magnitude greater, the nuclei we\nconsider are also over an order of magnitude more massive than the ${}^{12}$C\ncomposition Baiko considers, so\nthe quantum mechanical effects end up being reduced by about an order of\nmagnitude from the number Baiko quotes.\nWe thus use the same Ogata and Ichimaru result used by HJA, noting that the\nresulting quadrupoles might be reduced by less than $10\\%$ due to charge screening\nand quantum effects---an error which is small compared to other\nuncertainties, such as crust thickness and the composition of dense matter. Indeed, there\nis a factor of $\\sim 2$ uncertainty in the shear modulus due to angle averaging (even\ndisregarding whether the implicit assumption of a polycrystalline structure for the crust is\nwarranted): As shown by\nHill~\\cite{Hill}, the Voigt average used by Ogata and Ichimaru is an upper bound on the\ntrue shear modulus of a polycrystal. A lower bound is given by the Reuss average (also\ndiscussed in Hill~\\cite{Hill}), for which the prefactor in Eq.~\\ref{mueff} would be\n$0.05106$. \n\nNote that there would be even\nfurther corrections to the shear modulus due to pasta phases (see~\\cite{PP}), but such phases are not present in the Douchin and Haensel model~\\cite{DH}. We also note that the Douchin and Haensel results only include the very\ninnermost portion of the outer crust. However, this lack of coverage has a negligible effect on the final\nresults for the quadrupoles, since the neglected region has at most half the radial extent of the inner crust and the shear modulus in this region is orders of magnitude below its maximum value at the bottom of the inner crust. We have checked this explicitly using the detailed calculations\nof the outer crust composition due to R{\\\"u}ster, Hempel, and Schaffner-Bielich~\\cite{RHS-B}, available at~\\cite{HempelURL}.\n\nWe plot the maximum quadrupole and ellipticity in the three approximations for the detailed shear modulus\nmodel in Fig.~\\ref{Q22s_vs_M_SLy_DH_HJA}. Here we show these for the SLy EOS proper, and also for\na high-density EOS that yields much less compact stars (and a crust that is $\\sim 2$ times as thick), and thus larger maximum quadrupoles. For the latter EOS, we have chosen (for simplicity) the LKR1 hybrid EOS from~\\cite{J-MO1}---the maximum compactnesses for the two EOSs are $0.60$ (SLy) and $0.43$ (LKR$1$). (We show the much larger quadrupoles that could be\nsupported by the mixed phase in the core for the LKR$1$ EOS in Fig.~\\ref{Q22_vs_M_EOSs}, but here just show the crustal quadrupoles using the Douchin and Haensel model for the crust.)\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_crustal.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_SLy_DH_HJA} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nmaximum quadrupole deformations (and fiducial ellipticity) due to crustal\nstresses versus mass, for the SLy EOS with the detailed Douchin and Haensel + Ogata and\nIchimaru model for the shear modulus and a breaking strain of $0.1$, plus the crustal quadrupoles for the LKR$1$ EOS with the same crustal model.\n}\n\\end{figure}\n\nIn all of these crustal results, in addition to the expected relativistic suppression of the quadrupole (which\nbecomes quite dramatic for compact, high-mass stars), we also find that the\nNewtonian Cowling approximation slightly overestimates the quadrupole (by\n$\\sim 25$--$50\\%$), as observed by HJA (though they found the overestimate to be\nconsiderably greater, around a factor of at least a few). This overestimate is due to the cancellation of\ncontributions from $\\rho'$ when one drops\nthe Cowling approximation (see the discussion at the end of Sec.~\\ref{Newt}).\nThe overall decrease in the maximum crustal quadrupole with mass is due primarily to the fact that the crust thins by a factor of $\\sim 4$ (SLy) or $\\sim 2$ (LKR1) in going from a $1\\,M_{\\odot}$ star to\nthe maximum mass star, though the quadrupole itself receives even further suppressions with mass due to relativistic\neffects and an increased gravitational field.\n\n\\subsection{Maximum $Q_{22}$ for hybrid stars}\n\\label{results_hybrid}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_Hy1_sigma80_mu_corr2.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_Hy1_sigma80} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nmaximum quadrupole deformations (and fiducial ellipticity) of hybrid stars versus\nmass, using the Hy1 EOS with a surface tension of\n$\\sigma = 80\\text{ MeV fm}^{-2}$ and a breaking strain of $0.1$.}\n\\end{figure}\n\nHere we display the maximum quadrupole deformations as a\nfunction of stellar mass for each of the hybrid EOS parameter sets considered\nin~\\cite{J-MO1}. (N.B.: Most of the results from~\\cite{J-MO1} we use or refer to here were\ncorrected in the erratum to that paper.) We start by showing these values calculated in the various\napproximations using the Hy1 EOS (with a surface tension of\n$\\sigma = 80\\text{ MeV fm}^{-2}$; see Table~I in~\\cite{J-MO1}) in\nFig.~\\ref{Q22s_vs_M_Hy1_sigma80}, and then\nrestrict our attention to the relativistic results. (The relation between the results of the different approximations is roughly\nthe same for all the hybrid EOSs we consider.)\nHere the maximum quadrupoles increase with mass, since the volume of mixed phase increases with mass, and this is more than enough to offset the suppressions due to relativity and the increased gravitational field.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_Hy1_sigmas_mu_corr2.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22_vs_M_Hy1_sigmas} The full relativistic maximum quadrupole deformations (and fiducial ellipticity) of hybrid\nstars versus mass, using the Hy1 EOS with various surface tensions $\\sigma$\nand a breaking strain of $0.1$.}\n\\end{figure}\n\nWe also show how the maximum relativistic quadrupole\nvaries with the surface tension for the Hy1 EOS in\nFig.~\\ref{Q22_vs_M_Hy1_sigmas}. The slightly larger quadrupoles for lower surface tensions\nat low masses are expected, due to a slightly larger shear modulus at low\npressures for lower surface tensions---see Fig.~10 in~\\cite{J-MO1}. In fact, despite differences of\nclose to an order of magnitude in the high-pressure shear modulus for the Hy$1$ EOS\nin going from a surface tension of $20\\text{ MeV fm}^{-2}$ to one of\n$80\\text{ MeV fm}^{-2}$ (see Fig.~10 in~\\cite{J-MO1}), the\ndifferences in the resulting maximum quadrupoles are at most\na factor of a few (for large masses). This is not unexpected: These quantities\nare dominated by the portions of the mixed phase further out in the star, where\nthe shear moduli have a much weaker dependence on the surface tension.\n(Additionally, the fact that larger surface tensions lead to smaller shear moduli at low\npressures helps to minimize the effect, though the maximum\nquadrupoles still increase with increasing surface tension for high masses, as\nexpected.) \n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_EOSs_mu_corr2.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22_vs_M_EOSs} The full relativistic maximum quadrupole deformations (and fiducial ellipticity) of hybrid\nstars versus mass, using the EOSs from Table~I in~\\cite{J-MO1}, all with a surface tension of\n$\\sigma = 80\\text{ MeV fm}^{-2}$\nand a breaking strain of $0.1$.}\n\\end{figure}\n\nFinally, we show the maximum quadrupoles for different\nhybrid EOSs in Fig.~\\ref{Q22_vs_M_EOSs}. (Note that these curves start somewhat above the minimum\nmasses for which the mixed phase is present, since we are mostly interested in the significantly larger\nmaximum quadrupoles possible for larger masses.) The\nconsiderable differences are due primarily to the substantial variations in the extent\nof the mixed phase in stable stars with EOS parameters as well as the\nEOS dependence of the stars'\ncompactnesses (see Table~I in~\\cite{J-MO1}), not to variations in the magnitude of the shear modulus for a\ngiven quark matter fraction (compared in Fig.~12 in~\\cite{J-MO1}). In particular, the LKR1 EOS produces stars with a very large region of mixed\nphase---up to $72.5\\%$ of the star's radius---and a (relatively) small maximum\ncompactness---only $0.433$. (Note that our quadrupole curve for\nthe LKR1 EOS ends slightly short of the EOS's maximum mass of $1.955\\,M_{\\odot}$,\nonly going to $1.948\\,M_{\\odot}$, due to problems with the numerics.)\n\nN.B.:\\ These maximum quadrupoles may all be overly optimistic. First, as was discussed in Sec.~\\ref{results_crust}, the averaging used to obtain the effective shear modulus only gives an upper bound on the true shear modulus of a\npolycrystal. (We do not quote results for the Reuss lower\nbound here, since it is only straightforward to obtain for the three-dimensional droplet phases. However, \nwe shall note that preliminary investigations, using the Reuss bound for the droplet phases, and the\nVoigt bound for the rest, give reductions in the maximum quadrupoles of up to $\\sim 5$ for lower masses.)\n\nSecond, the relatively large value we have chosen for the surface tension also increases the maximum\nquadrupoles, while recent calculations place the surface tension on the low side ($\\sim 10$--$30\\text{ MeV fm}^{-2}$)---see~\\cite{PKR} for the latest results. Nevertheless, as we show in the Appendix, the mixed phase is nevertheless favored by global energy\narguments even for these large surface tensions. The maximum quadrupoles are also affected by the\nmethod of EOS interpolation and the lattice contributions to the EOS, as is illustrated in the Appendix, though\nthe largest change is only $\\sim40\\%$ (at least for the LKR$1$ and Hy$1'$ EOSs, the two EOSs that yield the largest quadrupoles).\n\nNote that LIGO's current upper limits on fiducial ellipticity \nin the most interesting cases (the Crab pulsar, PSR~J0537--6910, and\nCas~A)~\\cite{LIGO_psrs2010, LIGO_CasA} are $\\sim 10^{-4}$, corresponding to a\nquadrupole moment of $\\sim 10^{41} \\text{ g cm}^2$.\nThe first hybrid star estimate by \\citet{OwenPRL} was an order of magnitude\nlower.\nThus our new results here show that current LIGO upper limits are interesting\nnot only for quark stars but also for hybrid stars, at least high-mass ones.\nIndeed, the most extreme case we consider, the LKR$1$ EOS with high surface tensions, gives maximum\nquadrupoles of a $\\text{few} \\times 10^{42}\\text{ g cm}^2$, which are above and therefore relevant to the\nlimits set by Virgo for the Vela pulsar~\\cite{LIGO_Vela}. \n\n\\subsection{Maximum $Q_{22}$ for crystalline color superconducting quark stars}\n\\label{SQM_computation}\n\nHere we consider stars made of crystalline color superconducting quark matter, for which the\nshear modulus has been estimated by Mannarelli, Rajagopal, and Sharma~\\cite{MRS}.\\footnote{This\nestimate is not angle averaged, but Mannarelli, Rajagopal, and Sharma's calculation has relatively large uncontrolled remainders, so we do not worry about the effects of angle averaging here.}\n[See Eq.~(1) in Haskell~\\emph{et al.}~\\cite{Haskelletal} for the expression in cgs units.] Such\nstars have also been treated (with varying degrees of sophistication) by Haskell~\\emph{et al.}~\\cite{Haskelletal}, \\citet{Lin}, and~\\citet{KS}. However, only Lin considers the case of a solid quark star, as we will do here, and does so using quite a rough model. (The others consider crystalline color superconducting\ncores in hybrid stars.)\n\nSince strange quark stars have a nonzero surface density---and solid quark stars have a nonzero\nsurface shear modulus, with the standard density-independent treatment of the superconducting\ngap---we have to make some changes to our previously obtained expressions\nin order to treat them.\n\nFirst, the outer boundary condition changes. The potential (in the Newtonian case) and metric\nperturbation (in the GR case) are no longer continuous at the star's surface, due to the presence\nof $\\rho'$ in both equations [see Eqs.~\\eqref{Leq} and~\\eqref{H0_S-L}]. As discussed in Hinderer~\\emph{et al.}~\\cite{Hindereretal} (following Damour and Nagar~\\cite{DN}), one can obtain the distributional contribution to the boundary conditions [Eqs.~\\eqref{Newt_BC} and~\\eqref{H0_BC}] using the usual procedure of integrating the defining\ndifferential equation over $[R-\\epsilon,R+\\epsilon]$ and taking the limit $\\epsilon\\searrow 0$.\nIn the Newtonian case, this gives [defining $\\rho_-$ as the density immediately inside the star's\nsurface and $R^-$ to mean evaluation at $R-\\epsilon$ in the limit $\\epsilon\\searrow 0$]\n\\<\n\\delta\\Phi'(R^-) = \\left[\\frac{4\\pi G}{g(R)}\\rho_- - \\frac{3}{R}\\right]\\delta\\Phi(R),\n\\?\nand in the GR case, we have (with $G = 1$)\n\\<\nH_0'(R^-) = H_{0,\\mathrm{old}}'(R) + \\frac{4\\pi h}{\\phi'(R)}\\rho_-H_0(R),\n\\?\nwhere $H_{0,\\mathrm{old}}'(R) $ is computed using Eq.~\\eqref{H0_BC}.\nWe thus make the replacement $3RF(R) \\to [3 - 4\\pi G \\rho_- R\/g(R)]RF(R)$ in the expression \nfor the Newtonian Green function [Eq.~\\eqref{cG_F}], and the replacement\n$H_0'(R) \\to H_{0,\\mathrm{old}}'(R) + 4\\pi h\\rho_-H_0(R)\/\\phi'(R)$ in the GR case [Eq.~\\eqref{cG_GR}].\nThese changes in the boundary conditions increase the maximum quadrupole by a factor of $\\lesssim 2$ in the example case considered below; the largest effect is for the least massive stars considered.\n\nSecond, we would have to keep the boundary terms at the outer boundary when integrating by parts to\nobtain the expressions for the maximum quadrupole, since the shear modulus no longer vanishes at the\nstar's surface. However, since here the shear modulus is smooth, it is numerically preferable not to perform any integration by parts, thus avoiding potential problems with large cancellations between the surface and integrated terms. In this case, the expressions for the quadrupole assuming the UCB maximum\nuniform strain are [cf.\\ Eqs.~\\eqref{Q22N2} and \\eqref{Q22GR2}]\n\\<\n\\label{Q22UCBs}\n\\frac{|Q_{22}^{\\text{UCB strain}, N}|}{\\bar{\\sigma}_\\mathrm{max}} = \\sqrt{\\frac{32\\pi}{15}}\\int_0^RG_N(r)[r\\mu''(r) - \\mu'(r)]dr\n\\?\nand\n\\<\n\\begin{split}\n\\frac{|Q_{22}^\\text{UCB strain, GR}|}{\\bar{\\sigma}_\\mathrm{max}} &= \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\bigl[G_\\mathrm{GR}(r)\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{UCB}(r)\\\\\n&\\quad + \\bar{G}_\\mathrm{GR}(r)\\mathcal{I}_{[t]}^\\mathrm{UCB}(r)\\bigr]dr,\n\\end{split}\n\\?\nwhere\n\\begin{widetext}\n\\begin{subequations}\n\\begin{align}\n\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{UCB}(r) &:= \\left[\\frac{6}{r}(h^{1\/2} - 1) - 2\\phi' + \\frac{r\\phi'' + \\phi'(1 - r\\psi') - \\psi'}{h^{1\/2}}\\right]\\mu(r) + \\left[\\frac{2 + r(\\phi' - \\psi')}{h^{1\/2}} - 3\\right]\\mu'(r) + \\frac{r\\mu''(r)}{h^{1\/2}},\\\\\n\\mathcal{I}_{[t]}^\\mathrm{UCB}(r) &:= 2\\left\\{\\frac{r\\phi'[r(\\phi' + 2\\psi') - 5] - r\\psi' - r^2\\phi'' - 1}{h} + r\\phi' + h^{1\/2}\\right\\}\\mu(r) - 2r\\left(\\frac{r\\phi'}{h} + 1\\right)\\mu'(r).\n\\end{align}\n\\end{subequations}\n\\end{widetext}\n\nHowever, these expressions will not actually yield the maximum quadrupole in this case, due to an important\ndifference between the cases where the shear modulus vanishes at the star's surface and those where it does not. It is simplest to see this in the\nNewtonian case for a star with a constant shear modulus: Since the UCB maximum strain expression~\\eqref{Q22UCBs} only\ndepends upon derivatives of the shear modulus, it\npredicts a \\emph{zero} maximum quadrupole, which seems absurd. One can, however, make a small\nadjustment to the form of the maximum strain one considers to yield a nonzero quadrupole in this case. This modification will also yield considerably larger maxima in the realistic case we\nconsider, as well, where the shear modulus is close to constant---it decreases by less than a factor of $2$ in going from\nthe star's center to its surface in the example case we consider below.\n\nSpecifically, in the case of a slowly varying shear modulus, with $\\mu(r)\\gg|r\\mu'(r)|,|r^2\\mu''(r)|$, appropriate for strange quark stars, we want the terms involving $\\mu$ itself to be\nlargest. The appropriate choice for the strain in this case is most readily apparent\nfrom inspection of the Newtonian expression for the maximum quadrupole in terms of the stress tensor components, Eq.~\\eqref{Q22N1}. We want the maximum contribution from the undifferentiated terms, which implies that we want $t_{rr}$ and $-t_{r\\perp}$ to be as large as possible. For $t_\\Lambda$, we note that since $\\mu'(r)<0$,\nwe also want $-t_\\Lambda$ to be as large as possible. Realizing that we can freely change the sign of any of\nthe $\\sigma_\\bullet$ that give maximum uniform strain [given for the Newtonian case in Eqs.~\\eqref{sNewt}; cf.\\ Eq.~(65) in UCB], we thus reverse the sign of $\\sigma_{r\\perp}$ and $\\sigma_\\Lambda$. [The same logic holds for the more involved GR case, as well, where\nthe appropriate expression for $\\sigma_\\Lambda$ will be the negative of Eq.~\\eqref{sLGR}.]\n\nThe resulting expressions for the putative maximum quadrupole in this case are thus\n\\<\\begin{split}\n\\frac{|Q_{22}^{\\text{mod.\\ strain}, N}|}{\\bar{\\sigma}_\\mathrm{max}} &= \\sqrt{\\frac{32\\pi}{15}}\\int_0^RG_N(r)\\biggl[\\frac{12}{r}\\mu(r) + 5\\mu'(r)\\\\\n&\\quad +r\\mu''(r)\\biggr]dr\n\\end{split}\n\\?\nand\n\\<\n\\begin{split}\n\\frac{|Q_{22}^{\\text{mod.\\ strain, GR}}|}{\\bar{\\sigma}_\\mathrm{max}} &= \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\bigl[G_\\mathrm{GR}(r)\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{mod}(r)\\\\\n&\\quad + \\bar{G}_\\mathrm{GR}(r)\\mathcal{I}_{[t]}^\\mathrm{mod}(r)\\bigr]dr,\\end{split}\n\\?\nwhere\n\\begin{widetext}\n\\begin{subequations}\n\\begin{align}\n\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{mod}(r) &:= \\left[\\frac{6}{r}(h^{1\/2} + 1) + 2\\phi' + \\frac{r\\phi'' + \\phi'(1 - r\\psi') - \\psi'}{h^{1\/2}}\\right]\\mu(r) + \\left[\\frac{2 + r(\\phi' - \\psi')}{h^{1\/2}} + 3\\right]\\mu'(r) + \\frac{r\\mu''(r)}{h^{1\/2}},\\\\\n\\mathcal{I}_{[t]}^\\mathrm{mod}(r) &:= - 2\\left\\{\\frac{ r\\phi'[5- r(\\phi'+2\\psi')] + r\\psi' + r^2\\phi'' + 1}{h} - r\\phi' + h^{1\/2}\\right\\}\\mu(r)\n- 2r\\left(\\frac{r\\phi'}{h} + 1\\right)\\mu'(r).\n\\end{align}\n\\end{subequations}\n\n\\end{widetext}\nIn principle, these merely give a lower bound on the\nmaximum quadrupole, unlike the case in which the shear modulus vanishes below the surface, where there is a firm argument that maximum uniform strain maximizes the quadrupole. However, even if they do not give the absolute\nmaximum, they should be quite close for cases like the one we consider here, where the shear modulus varies quite slowly.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_SQM1_final.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_SQM1} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nquadrupole deformations (and fiducial ellipticity) of maximally strained strange quark stars versus\nmass, using the EOS discussed in the text with a breaking strain of $0.1$. We show these both for the\nstandard UCB uniform maximum strain, and our modification that yields significantly larger quadrupoles in this case.}\n\\end{figure}\n\nApplying these expressions to a specific case, we use the strange quark matter EOS calculated by Kurkela,\nRomatschke, and Vuorinen (KRV)~\\cite{KRV}, generating an EOS for the parameter values of\ninterest using the {\\sc{Mathematica}} notebooks available at~\\cite{KRV_URL}.\nThe relevant parameters are the values of the $\\overline{\\mathrm{MS}}$ renormalization point,\n$\\Lambda_{\\overline{\\mathrm{MS}}}$, and the strange quark mass, $m_s$, both at a scale of $2$~GeV, along with the coefficient in the relation between the renormalization scale and the\nquark chemical potential, $X$, the color superconductivity gap parameter, $\\Delta$ (taken to be\nindependent of density),\\footnote{Note that $\\Delta$ enters the KRV EOS through a color flavor locked (CFL)\npressure term. This is not quite appropriate for the crystalline color superconducting phase we consider here, since it assumes that all the quarks pair, while only some of them pair in the crystalline phase. However, as\ndiscussed in Sec.~VI~B of~\\cite{Alford2008}, the condensation energy of the crystalline phases is easily\n$1\/3$ to $1\/2$ that of the CFL phase with zero strange quark mass, which is the pressure\ncontribution used by KRV. We have thus not altered this term in our calculations, since the contribution is\nalready approximate, in that it assumes a density-independent gap parameter.\nMoreover, we only consider a fairly low value of $\\Delta$,\nwhile Knippel and Sedrakian~\\cite{KS} suggest that the crystalline phase might be favored up to $\\Delta = 100$~MeV. Our EOS may thus simply correspond to a slightly larger value of $\\Delta$, which would \nincrease the maximum quadrupole, since the shear modulus scales as $\\Delta^2$.} and the\nminimal quark chemical potential at which strange quark matter exists, $\\mu_{q,\\mathrm{min}}$.\nWe consider the EOS obtained by choosing $\\Lambda_{\\overline{\\mathrm{MS}}} = 355$~MeV, $m_s = 70$~MeV, $X=4$, $\\Delta = 10$~MeV, and $\\mu_{q,\\mathrm{min}} = 280$~MeV.\nThis parameter set yields a maximum mass of $2.45\\,M_{\\odot}$, with a maximum compactness of $0.467$. \n\nThese parameter choices were generally inspired by those considered at~\\cite{KRV_URL}, though with a smaller value of $\\Delta$, to place us well within the crystalline\nsuperconducting regime. However, as Knippel and Sedrakian~\\cite{KS} suggest, the crystalline phase\ncould still be favored for considerably larger $\\Delta$s, up to $\\sim 100$~MeV, for the low-temperature\ncase relevant for neutron stars. We thus note that increasing $\\Delta$ decreases the maximum mass, and increases the\nmaximum quadrupole, though the latter is increased by considerably less than the na{\\\"\\i}ve scaling of $\\Delta^2$ one would\nexpect from the scaling of the shear modulus, likely due to the increased compactness of the stars with\nlarger $\\Delta$s: For $\\Delta = 100$~MeV, we have a maximum mass and compactness of $2.12\\,M_{\\odot}$ and $0.508$, respectively, and a maximum\nquadrupole of $\\sim 3.5\\times 10^{45}\\text{ g cm}^2$ for a $1.4\\,M_{\\odot}$ star, $\\sim 20$ times that for $\\Delta = 10$~MeV. However, one must bear in\nmind that our perturbative treatment starts to become questionable with such large gap parameters, for which the maximum shear stresses are more than $10\\%$ of the background's energy density. The uncontrolled remainders in the Mannarelli,\nRajagopal, and Sharma~\\cite{MRS} calculation of the shear modulus also increase as the gap parameter\nincreases.\n\nWe show the quadrupole for a maximally uniformly strained star in the three approximations (Newtonian\nCowling, Newtonian no Cowling, and GR) for both the UCB and modified maximum strain choices for this EOS in Fig.~\\ref{Q22s_vs_M_SQM1}. Here we have used a breaking strain of $0.1$, by the same high\npressure argument as in the mixed phase case. (While the very outermost portions of the star are at low\npressure, the parts that are at a lower pressure than the crustal case for which the $0.1$ breaking strain\nwas calculated make negligible\ncontributions to the quadrupole.)\n\nN.B.:\\ To obtain the EOS used for this figure, we made some slight modifications to the KRV {{\\sc{EoScalc}}} {\\sc{Mathematica}} notebook so that it would output particle number densities on a denser mesh for low strange quark chemical potentials. This then gave an EOS table with better low-pressure coverage than their default settings produced. We still needed to perform\nan extrapolation of the EOS to zero pressure, where we found that a linear extrapolation of the energy\ndensity and quark chemical potential in terms of the pressure using the lowest two entries of the table provided a good fit.\n(More involved approaches involving fitting to more points and\/or a quadratic extrapolation produce very similar results.)\n\nAdditionally, it is worth pointing out that the applying the KRV results to compact stars\npushes their second-order perturbative calculation towards the edge of its domain of validity. However,\nin our case, the smallest value of the quantum chromodynamics (QCD) renormalization scale we consider is\n$1.12$~GeV, at which value the QCD coupling constant is $\\sim 0.45$. Thus, the uncontrolled remainders in\nthe expansion are suppressed by at least a factor of $\\sim 0.1$. (While Rajagopal and Shuster~\\cite{RS} find that perturbative QCD calculations of the color superconducting gap are only reliable at energy scales of\n$\\gtrsim 10^5$~GeV, the specifics of this calculation are rather different from the calculation of the EOS we\nare considering here, where the gap is taken as an input parameter.) While it is unreasonable to expect\nthis calculation to be a truly accurate description of strange quark matter, it is not clear that any of the alternative descriptions of strange quark matter are \\emph{a priori} guaranteed to be a better description of the physics, given the very considerable uncertainties associated\nwith this phase of matter.\n\n\\section{Discussion}\n\\label{discussion}\n\nPrevious studies of the tidal and magnetic deformations of compact\nstars have found similar relativistic suppressions of quadrupole moments with\ncompactness.\nIn the tidal case, see the Love number computations in~\\cite{Hinderer, BP, DN, Hindereretal, PPL}.\nIn the case of magnetic deformations, the expected suppressions are seen in,\ne.g., \\cite{IS, CFG, YKS, FR}.\nIn fact, since the largest compactness considered in these latter papers is only\n$0.48$ (in~\\cite{IS}), one imagines that they overestimate the maximum quadrupoles\nby at least a factor of a few for more compact stars (for a fixed magnitude of magnetic\nfield).\n\nAs was argued by Damour and Nagar~\\cite{DN} in the tidal case, all these suppressions are primarily related to the ``no-hair'' property of black holes: The largest relativistic suppression we find comes from the boundary conditions\n[through the $H_0(R)$ and $H_0'(R)$ in the Green function's denominator---see Eq.~\\eqref{cG_GR}], where\none matches on to the external vacuum spacetime. For instance, for the SLy EOS's maximum\ncompactness of $0.6$, $H_0(R)$ and $H_0'(R)$ are $\\sim 3.5$ and $\\sim 6$ times their\nNewtonian values\n[which can be obtained from the first term of Eq.~(21) in Hinderer~\\cite{Hinderer}~].\nIn fact, these ratios go to infinity in the formal black hole limit, where the compactness\napproaches unity, as required by the\nno-hair property, and discussed by Damour and Nagar~\\cite{DN} (see\ntheir Secs.~IV~C and VII~A, but note that their definition for the compactness is half of ours). This implies that the stiffness of spherically symmetric\ncurved vacuum spacetime suppresses the quadrupole. The quadrupole is also\nsuppressed by a larger effective gravitational acceleration (given by $\\phi'$), which appears\nin the denominator of $G_\\mathrm{GR}$, replacing the Newtonian $g(r)$ [cf.\\ Eqs.~\\eqref{GN} and~\\eqref{GGR}]. (But\nrecall that we always compute the background stellar structure relativistically, so this\nlarger acceleration \\emph{only} affects the perturbation equations, and not, e.g., the thickness of\nthe crust for a given mass and EOS, which is the same in both the Newtonian and\nrelativistic calculations of the quadrupole.) \n\nOur results imply that nearly all of the Newtonian computations of quadrupoles due to elastic\ndeformations of relativistic stars overestimate the quadrupole moment, often by at\nleast a factor of a few. The only exceptions we have found are\nfor low-to-mid mass strange quark stars and for elastic stresses in the cores of neutron stars around $0.5\\,M_{\\odot}$. In\nboth of these cases, the Newtonian Cowling\napproximation is a slight underestimate for contributions to the quadrupole, though the Newtonian no Cowling\nversion is still an overestimate. See Fig.~\\ref{Q22_integrands_SLy_2} for an illustration in the core case;\nbut note that neutron stars with such low masses are not known to exist in nature.\nThe overestimate from performing a Newtonian Cowling approximation calculation \ncan be $\\sim 6$ for massive stars whose quadrupole is\nbeing generated by an elastic deformation near the\ncrust-core interface, as considered by UCB and others. This is due in part to the\nsudden changes in density at that interface entering directly through $g''$, as\ndiscussed at the end of Sec.~\\ref{Newt}.\n\nHowever, the calculations by Horowitz~\\cite{Horowitz} for crustal deformations of very low mass stars only receive negligible\ncorrections (of $\\lesssim5\\%$), since he considers\ncompactnesses of $\\sim 0.01$. In fact, one makes even\nsmaller errors in using the Cowling approximation to treat these stars,\nsince the changes in density in the crust (times $4\\pi G r^2$) are much smaller than the star's gravitational field there.\n\nNo neutron stars with such low masses have ever been observed (nor is there\na compelling mechanism for forming them). Nevertheless, Horowitz remarks that\ngravitational wave detection of\ngravitational waves from elastically deformed neutron stars will,\n\\emph{ceteris paribus}, be biased towards low(er) mass neutron stars, if one\nconsiders deformations generated by crustal stresses. This is an important \npoint, particularly when considering the astronomical interpretation of\ndetections (or even upper limits), and the results we present here make the\nbias against high-mass stars even stronger. (This bias also applies to solid\nquark stars, though there it is rather weak. It does \\emph{not} apply to hybrid\nstars, however, where it is high-mass stars that can sustain the largest quadrupoles.)\n\nOf course, one must remember that all of these values are maxima, assuming a\nmaximally strained star, while there is no reason, \\emph{a priori}, for a given star to be\nmaximally strained.\nMoreover, as UCB and HJA note, these calculations assume\nthat all the strain goes into the $l = m = 2$ perturbation, though strain in\nother modes (e.g., the $l = 2$, $m = 0$ mode due to rotation) can push the\nlattice closer to its breaking strain while not\nincreasing the $l = m = 2$ quadrupole.\n\n\\section{Conclusions and outlook}\n\\label{concl2}\n\nWe have presented a method for calculating the maximum elastic quadrupole\ndeformation of a relativistic star with a known shear modulus and breaking strain.\nWe then applied this method to stars whose elastic deformations are\nsupported by a shear modulus either from the Coulomb lattice of nuclei in the crust, a hadron--quark mixed phase in the core, or crystalline superconducting strange quark matter throughout the star. (In the last case, we have made the requisite changes to the method so that it is valid when the star has a nonzero surface\ndensity and the shear modulus does not vanish at the star's surface.) In all but the strange quark case, we find that the\nrelativistic quadrupole is suppressed, compared with the standard,\nNewtonian Cowling approximation calculation of the quadrupole, at least for\nstars with masses of $\\gtrsim 1\\,M_{\\odot}$ (corresponding to the observed masses\nof neutron stars) and the EOSs we have investigated. These suppressions can be\nup to $\\sim 4$ in the hybrid case, and\n$\\sim 6$ in the crustal case. In the strange quark star case, the Newtonian Cowling approximation\ncalculation slightly underestimates the quadrupole (by tens of percent) for low-to-standard mass stars, but is still an overestimate of $\\sim 2$ at higher\nmasses.\n\nThese suppressions strengthen the Horowitz~\\cite{Horowitz} argument that\nsearches for gravitational waves from elastically deformed neutron stars\nsupported by crustal stresses are biased towards lower-mass stars. The same argument also\napplies to strange quark stars, though there the suppressions with increasing mass are less\nsevere (and the maximum quadrupoles are all considerably larger). However,\nthis argument does not apply to quadrupole deformations of hybrid stars, since\nthe increase in the size of the region of mixed phase with increasing mass\ndominates the various suppressions.\n\nOur results also imply that many of the\nprevious calculations of elastic quadrupoles (e.g.,~\\cite{Lin, Haskelletal,\nKS, UCB, HJA}) will need their results revised downwards. (While we find much larger maximum quadrupoles for solid strange quark stars than did Lin~\\cite{Lin}, this is only because we assume a breaking\nstrain $10$ times that assumed by Lin. If we take the same $10^{-2}$ breaking strain as does Lin, then we find a suppression of a factor of a few, though this is very likely within the uncertainties of Lin's calculation,\nwhich assumed a uniform density, incompressible star with a uniform shear modulus.)\n\nIt is instructive to compare our results with the numbers quoted in Pitkin's review~\\cite{Pitkin}. All of these were obtained by Pitkin using scalings given in the aforementioned papers,\nsometimes updating to the Horowitz and Kadau~\\cite{HK} breaking strain, and provide a good overview of the standard Newtonian predictions.\nNone of our detailed calculations for maximum crustal quadrupoles approach the\nhigh values Pitkin obtained using UCB's fitting formula (as corrected by\n\\citet{OwenPRL}). However, our very largest hybrid star quadrupoles are an order of magnitude\nabove Pitkin's quoted maximum, even if one only assumes a breaking strain of $10^{-2}$, as does Pitkin. Additionally, our estimates for maximum solid quark star quadrupoles ($\\sim 10^{44}\\text{ g cm}^2$ for $1.4\\,M_{\\odot}$ stars) are considerably larger than the ones quoted by Pitkin (based on a different shear modulus model), even if we reduce them by an order of magnitude due to scaling the breaking strain to Pitkin's $10^{-2}$. In fact, they\nare in the same range as those Pitkin quotes for a model for crystalline superconducting hybrid stars (with an optimistic gap parameter $5$ times the one we used for solid quark stars, leading to a shear modulus $\\sim 40$ times our shear modulus's maximum value).\n\nEven with the relativistic suppressions, we obtain maximum quadrupole deformations of\n$\\text{a few}\\times10^{42}\\text{ g cm}^2$ \nin the hybrid case for a very stiff hadronic EOS, and\n$\\text{a few}\\times10^{41}\\text{ g cm}^2$ for more realistic cases. In both\nsituations, the largest maximum quadrupoles are given by the most massive stars. These\nvalues are proportional to the breaking strain and assume that the Horowitz and Kadau~\\cite{HK}\nbreaking strain of about $0.1$ is applicable to the mixed phase. Such large\nquadrupole deformations were previously thought only to be possible for solid\nquark stars (see~\\cite{OwenPRL, Lin, Haskelletal, KS}), or from crustal deformations in the very\nlow-mass neutron stars considered by Horowitz~\\cite{Horowitz}. These large\ndeformations (corresponding to fiducial ellipticities of\n$\\text{a few}\\times10^{-3}$ in the extreme case, and\n$\\sim 5\\times10^{-4}$ in a more realistic case)\nwould be able to be\ndetected by current LIGO searches for gravitational waves from certain known\nneutron stars~\\cite{LIGO_psrs2010, LIGO_CasA, LIGO_Vela}. (However, we must note that there is no reason to assume that\nsuch isolated stars are anywhere near maximally strained, even neglecting the\nuncertainties in the description of their interiors.)\n\nThe prospects for\ncrustal quadrupoles are now somewhat less optimistic, and definitely favor\nlower-mass stars. However, for a canonical $1.4\\,M_{\\odot}$ neutron star, we find that the maximum\nrelativistic crustal quadrupole is in the range $\\sim\\text{(1--6)}\\times 10^{39}\\text{ g cm}^2$ [corresponding to\nfiducial ellipticities of $\\sim\\text{(1--8)}\\times 10^{-6}$], depending on the model used for the crust and the\nhigh-density EOS. (Note that the fully consistent Douchin and Haensel model with its associated high-density EOS yields the lowest numbers. Additionally, there is the possibility of a further reduction of up to $\\sim 2$ due to the angle averaging procedure used to obtain the shear modulus.) On the high side, these numbers are consistent\nwith those given previously for breaking strains of $0.1$ by Horowitz~\\cite{HK, Horowitz},\\footnote{But recall\nthat the results from Horowitz~\\cite{Horowitz} were obtained using the SLy EOS and crustal composition results, so they\nare the same as our Newtonian Cowling approximation SLy predictions, given in Fig.~\\ref{Q22s_vs_M_SLy_DH_HJA}, except $\\sim7\\%$ lower, since Horowitz is using the Horowitz and Hughto~\\cite{HH} result for the shear modulus. In the fully relativistic case, one requires a thicker crust than provided by the pure SLy results to obtain values for the maximum quadrupole comparable to those given by Horowitz.} though they are a factor of $\\sim 5$ lower than the maximum Pitkin~\\cite{Pitkin} obtained using scalings of previous results and the maximum\nvalue given by HJA (scaled to this breaking strain). For stars around $2\\,M_{\\odot}$, the relativistic suppressions\nlead to maximum quadrupoles that are nearly an order of magnitude smaller than those for a $1.4\\,M_{\\odot}$ star in the compact SLy case: ${\\sim\\text{(1--5)}}\\times 10^{38}\\text{ g cm}^2$ [corresponding to\nfiducial ellipticities of $\\sim\\text{(1--6)}\\times 10^{-7}$]; and even in the much less compact LKR$1$ case, there is a suppression of $\\sim 5$. Previous Newtonian studies (see Fig.~3 in~\\cite{Horowitz}) had only\nfound suppressions of around a factor of $4$, due to the thinning of the crust and the increase\nin Newtonian gravity with increasing mass. It will be interesting to consider further models for the crustal\ncomposition and EOS in this case, particularly the large\nsuite of crustal models including the pasta phases recently calculated by Newton, Gearheart, and Li~\\cite{NGL}. (See~\\cite{GNHL} for order-of-magnitude estimates of the maximum quadrupole for these models,\nillustrating the sensitive dependence on the slope of the symmetry energy.)\n\nOne can also compare these maximum elastic quadrupoles with those generated by an internal magnetic field. Here the values depend,\nof course, upon the equation of state, compactness, and---perhaps most crucially---magnetic field topology, as well as the quantity one\nchooses to use to measure the magnitude of the magnetic field. But\nsticking to order-of-magnitude numbers, and considering a canonical $1.4\\,M_{\\odot}$ neutron star, Frieben and\nRezzolla~\\cite{FR} show that a toroidal internal\nfield of $\\sim10^{15}$~G would generate a quadrupole of $\\sim 10^{39}$--$10^{40}\\text{ g cm}^2$, comparable to the\nmaxima we find for crustal quadrupoles. Similarly, quadrupoles of $\\sim 10^{41}$--$10^{42}\\text{ g cm}^2$, around the maxima we find for hybrid\nstars, could come from magnetic fields of $\\sim 10^{16}$~G, while the maximum quadrupoles of $\\sim 10^{44}\\text{ g cm}^2$ we find for crystalline strange quark stars could also be\ngenerated by magnetic fields of $\\sim 10^{17}$~G, close to the maximum allowed field strength. (But note that these magnetic deformations are all computed for ordinary, purely hadronic neutron stars. Additionally,\nthe quoted maximum elastic quadrupoles in the hybrid case are attained only for more massive stars than the $1.4\\,M_{\\odot}$\nstars for which we are quoting the magnetic deformation results.)\nThe quoted values for magnetic quadrupoles come from the fits given in Sec.~7 of Frieben and Rezzolla~\\cite{FR}, except for the final ones, which are obtained from inspection of their Fig.~5 and Table~3. All these values agree in order of magnitude with the\npredictions for the twisted torus topology given by Ciolfi, Ferrari, and Gualtieri~\\cite{CFG}, and with many other studies for various topologies cited in Frieben and Rezzolla~\\cite{FR}. But note that very recent calculations by Ciolfi and Rezzolla~\\cite{CR} show that the magnetic field required to obtain a given quadrupole deformation with the twisted torus topology could be reduced by about an order of magnitude if the toroidal contribution dominates.\n\nOne would also like to make relativistic calculations of the maximum energy that could be\nstored in an elastic deformation.\nThis would be useful in properly computing the available\nenergy for magnetar flares, for instance.\n(Using Newtonian scalings, \\citet{CO} estimated that the hybrid case was especially\ninteresting compared to existing LIGO upper limits for gravitational wave emission from such flares.)\nThe basic expressions (at least in\nthe perfect fluid case) appear to be readily available in the literature\n(see, e.g.,~\\cite{Schutz2,DI}; \\cite{ST, Finn} give related results including\nelasticity). However, one cannot apply these directly to the crustal and hybrid cases, even in the Newtonian\nlimit, due to the\ndistributional nature of the density and pressure perturbations. Specifically, the\nsudden change in shear modulus at the phase transitions gives delta functions in the\nderivatives of the density and pressure perturbations. Since the energy expressions involve\nsquares of these derivatives, one would have to invoke some sort of regularization procedure, or apply a different\nmethod. Developing appropriate expressions for this case\nwill be the subject of future work.\n\nReturning to the quadrupoles, one might also want to consider the shape of the deformed star, particularly\nin the relativistic case---the ellipticity is already only a rough indicator of the shape of the\ndeformation in the Newtonian case---as has now been done in~\\cite{NKJ-M_shape}.\nBut the effects of the star's magnetic field are surely the most interesting to consider, from\nits influence on the lattices that support elastic deformations, to the changes to the boundary conditions at the star's surface from an external magnetic field (particularly for magnetars), to the internal magnetic field's own contribution to the star's deformation.\nOne might also want to consider the lattice's full elastic modulus tensor in this case, instead of simply assuming a polycrystalline structure and angle averaging to obtain an effective isotropic shear modulus, as was\ndone here. (And even if one assumes a polycrystalline structure, one could use more involved, sharper\nbounds on the shear modulus than the ones considered here---see~\\cite{WDO} for a classic review of such bounds.)\n\n\\acknowledgments\n\nWe wish to thank S.~Bernuzzi, D.~I.~Jones, A.~Maas, R.~O'Shaughnessy, and the anonymous referee for helpful suggestions.\nThis work was supported \nby NSF grants PHY-0855589 and PHY-1206027, the Eberly research funds of Penn State, and the DFG SFB\/Transregio 7.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}