diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfgnl" "b/data_all_eng_slimpj/shuffled/split2/finalzzfgnl" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfgnl" @@ -0,0 +1,5 @@ +{"text":"\\section{I. Introduction}\nSince the early description of quantum mechanics, the effect of quantum interference led to fascinating and peculiar predictions, one being the appearance of quantum beats caused by interference of emission paths from excited atoms \\cite{Breit1933}. \nNowadays, quantum interference is an essential feature in a quantum network based on quantum optical systems with single atoms as nodes and single photons transferring information between them \\cite{Cirac1997,Kimble2008}. It allows for entanglement between two nodes \\cite{Moehring2007,Slodicka2013,Hofmann2012} and employing coherent phenomena at atom-photon interfaces to faithfully convert quantum information between photonic communication channels and atomic quantum processors \\cite{Ritter2012}. In this context, the controlled emission \\cite{Blinov2004,Maunz2007,Almendros2009,Kurz2013} and absorption \\cite{Piro2011,Huwer2013} of single photons by a single atom is crucial for schemes proposing the heralded mapping of a photonic polarization state into an atomic quantum memory \\cite{Muller2013}. Here we show the quantum coherent character of the absorption and emission of single photons through the interference between indistinguishable quantum channels in a single atom.\n \nFirst experimental observations of quantum beats were induced by pulsed optical excitation of atoms with two upper states which decay to the same ground state \\cite{Dodd1967, Aleksandrov1964}. In continuously excited atomic ensembles, the observation of quantum beats in the correlation of two photons was demonstrated for a coherent superposition state with short lifetime in a calcium cascade \\cite{Aspect1984} and in cavity mediated systems with coherent ground states \\cite{Norris2010}. For a single ion \\cite{Schubert1995}, transient effects of the internal dynamics showed photonic oscillations by interference in absorption. \n\nHere we generate controlled, high-contrast quantum beats in a spontaneous Raman scattering process which consists of the emission of single 393-nm photons after absorption of 854-nm laser photons in a single trapped $^{40}\\text{Ca}^+$ ion. We consider two distinct excitation schemes, called $\\Lambda$ and V. For both schemes we employ the controlled generation of an initial coherent superposition state of two Zeeman sublevels in the metastable D$_{5\/2}$ manifold with phase-coherent laser pulses. Quantum beats are then observed in the arrival-time distribution of the detected 393-nm photons after onset of the 854-nm laser pulse that induces their emission. We show that the observed quantum beats are fully controlled through changes of the phase in the atomic superposition and the photonic polarization input states. In particular, we highlight two distinct physical origins of the quantum beats, namely, quantum interference of two 854-nm absorption amplitudes and two 393-nm emission amplitudes, respectively. The two absorption-emission pathways resemble a which-way experiment where indistinguishability is afforded by a quantum eraser \\cite{Scully1982}. \n\n\\section{II. Experimental sequence}\n\nThe experimental setup is illustrated in Fig.~\\ref{setup}(a). A single $^{40}$Ca$^+$ ion is trapped in a linear Paul trap and excited by various laser beams. A magnetic field $B$ parallel to the trap axis defines the quantization axis and lifts the degeneracy of the atomic levels [Fig.~\\ref{setup}(b)]. The experimental sequence starts with Doppler cooling on the S$_{1\/2}$-P$_{1\/2}$ transition with a 397-nm laser, aided by an 866-nm laser that repumps the population from the metastable D$_{3\/2}$ state. After cooling, a circularly polarized laser pulse optically pumps the ion to $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$. Then it is excited to two selected Zeeman sublevels in the metastable D$_{5\/2}$ manifold, employing a laser at 729~nm which is locked to an ultra-stable high-finesse cavity. Efficient population transfer up to 99.6(3)$\\%$ on each of the two transitions is achieved by frequency-selective coherent pulses. From the difference of the two frequencies we extract directly the Larmor frequency $\\nu_{L}$ in D$_{5\/2}$ [see Eq.~(\\ref{Zeeman-splitting}) below]. The superposition state is excited to the $\\text{P}_{3\/2}$ level by a laser at 854-nm wavelength with controlled polarization and frequency. Its detuning $\\Delta$ from the line center is previously calibrated by a spectroscopic measurement and it is adjusted to provide identical pumping rates for the two initial Zeeman sublevels. The 854-nm beam enters parallel to the quantization axis and its polarization is adjustable to any defined linear or circular state by a combination of $\\lambda\/4$ and $\\lambda\/2$ waveplates. The absorption of the 854-nm laser photons leads to the emission of a 393-nm Raman-scattered photon that is collected perpendicular to the quantization axis by an in-vacuum high-numerical-aperture laser objective (HALO) with a numerical aperture of 0.4 \\cite{Gerber2009}. The photons are selected by their polarization with a polarizing beam splitter (PBS) cube and detected by a photomultiplier tube (PMT). The PMT pulses are fed into a time-correlated single-photon counting module (Pico Harp 300) for temporally correlating them with the sequence trigger, i.e., with the onset of the 854-nm pulse. \n\nAn inherent requirement for quantum interference between two scattering paths is to keep indistinguishability in all degrees of freedom of the involved quantum channels. In the $\\Lambda$- and V-type level configurations, the two absorption channels exhibit a frequency difference originating from the differential Zeeman shift of the two D$_{5\/2}$ sublevels, up to $\\sim$20~MHz for typical magnetic fields of $\\sim$3~G. The PMT time resolution of 300~ps sets a much lower frequency resolution and thus erases the information of this frequency splitting \\cite{Togan2010}. The indistinguishability concerning the polarization is attained by the detection perpendicular to the magnetic field of only $\\left\\lvert\\rm{H}\\right\\rangle$-polarized or only $\\left\\lvert\\rm{V}\\right\\rangle$-polarized 393-nm photons (i.e.,\\ polarization parallel or orthogonal to the quantization axis, respectively).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.6cm]{setup}\n\\caption{(Color online) (a) Schematic of the experimental setup: excitation of a single trapped ion with an 854-nm laser beam parallel to the magnetic field axis and polarization-selective detection of 393-nm photons perpendicular to it. HALO stands for high-numerical-aperture laser objective \\cite{Gerber2009}, PBS for polarizing beam splitter, and PMT for photomultiplier tube. (b) Level scheme and relevant transitions of the $^{40}$Ca$^+$ ion.}\n\\label{setup} \n\\end{figure}\n\n\n\n\\section{III. Theoretical analysis}\n\nA brief and simplified theoretical description of the population transfer from D$_{5\/2}$ to S$_{1\/2}$ is presented which emphasizes the coherent evolution of the internal states for the two different level configurations.\n\nFigure~\\ref{levelscheme_both} shows the $\\Lambda$- and V-type level configurations with the relevant transitions and the squared Clebsch-Gordan coefficients (CGCs).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.4cm]{levelscheme_both}\n\\caption{(Color online) (a) The $\\Lambda$-shaped three-level system consisting of\n$\\left\\lvert\\rm{D}_{5\/2},-\\frac{3}{2}\\right\\rangle$, $\\left\\lvert\\rm{D}_{5\/2},+\\frac{1}{2}\\right\\rangle$ and $\\left\\lvert\\rm{P}_{3\/2},-\\frac{1}{2}\\right\\rangle$ including the relevant transitions with their squared Clebsch-Gordan coefficients. (b) The V-shaped three-level system consisting of\n$\\left\\lvert\\rm{P}_{3\/2},-\\frac{3}{2}\\right\\rangle$, $\\left\\lvert\\rm{P}_{3\/2},+\\frac{1}{2}\\right\\rangle$ and $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$ including the relevant transitions with their squared Clebsch-Gordan coefficients. Red arrows, absorption of 854-nm photons ($\\sigma^+$ and $\\sigma^-$); blue wavy arrows, emission of 393-nm photons; gray arrows, parasitic absorption (854~nm) and emission channels (393~nm).}\n\\label{levelscheme_both}\n\\end{figure}\n\nBoth schemes are initialized in a coherent superposition state, \n\\begin{equation}\n\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle = \\sqrt{\\rho_1} \\left\\lvert \\text{D}_{5\/2},m_\\text{D}\\right\\rangle + \\text{e}^{i\\Phi_{\\text{D}}(t)} \\sqrt{\\rho_2}\\left\\lvert \\text{D}_{5\/2},m_{\\text{D'}}\\right\\rangle, \n\\label{initial_state}\n\\end{equation}\nwith populations $\\rho_1$ and $\\rho_2$ adjusted by two consecutive Rabi pulses from $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$. The phase $\\Phi_{\\rm{D}}(t)=\\Phi_{\\rm{D}}(0)+2\\pi\\nu_{L}t$ is composed of a starting phase $\\Phi_{\\rm{D}}(0)$ which is set by the sequence control as the relative phase between the two 729-nm pulses and a part oscillating with the Larmor frequency,\n\\begin{equation}\n \\nu_{L}=\\frac{\\mu_B}{h}\\Delta m_j \\,g_j\\,B,\n \\label{Zeeman-splitting}\n\\end{equation} \nincluding the magnetic field $B$, the Land$\\acute{\\rm{e}}$ factor $g_j=\\frac{6}{5}$, and the Bohr magneton $\\mu_B$.\n\nThe polarization state of the laser photons at 854~nm is defined as a superposition of two orthogonal states, namely, right ($\\left\\lvert\\rm{R}\\right\\rangle$) and left ($\\left\\lvert\\rm{L}\\right\\rangle$) circularly polarized light \n\\begin{equation}\n\\left\\lvert\\psi_{854}\\right\\rangle=\\rm{cos}\\tfrac{\\vartheta}{2}\\left\\lvert\\rm{R}\\right\\rangle+\\rm{sin}\\tfrac{\\vartheta}{2}\\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{L}\\right\\rangle.\n\\label{state854}\n\\end{equation}\nAny linear photonic polarization state, such as horizontal $\\left\\lvert\\rm{H}\\right\\rangle$, vertical $\\left\\lvert\\rm{V}\\right\\rangle$, diagonal $\\left\\lvert\\rm{D}\\right\\rangle$, and antidiagonal $\\left\\lvert\\rm{A}\\right\\rangle$ is adjusted by rotating the two wave plates [see Fig.~\\ref{setup}(a)] to change $\\Phi_{854}$, while $\\vartheta$ is fixed to $\\pi\/2$. As the propagation direction of the incoming laser photons is chosen parallel to the applied magnetic field, the photon polarization translates to the reference frame of the atom according to \\begin{equation}\n\\left\\lvert\\psi_{854}\\right\\rangle = \\rm{cos}\\tfrac{\\vartheta}{2}\\left\\lvert+1\\right\\rangle + \\rm{sin}\\tfrac{\\vartheta}{2}\\rm{e}^{i\\Phi_{854}}\\left\\lvert-1\\right\\rangle,\n\\end{equation}\nwhereby $\\left\\lvert m_{854} \\right\\rangle = \\left\\lvert \\pm1 \\right\\rangle$ stand for the photon polarizations that effect $\\Delta m = \\pm1$ (i.e., $\\sigma^{\\pm}$) transitions, respectively, between the Zeeman sublevels of D$_{5\/2}$ and P$_{3\/2}$. \n\nThe coupled quantum system is represented by a joint state between photon and atom,\n\\begin{equation}\n\\left\\vert\\psi(t)\\right\\rangle=\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle\\otimes \\left\\lvert\\psi_{854}\\right\\rangle. \n\\label{joint}\n\\end{equation}\nBased on \\cite{Muller2013} the absorption process is described by\n\\begin{equation}\n\\hat{A}=\\sum_{m_{\\rm{D}},m_{\\rm{P}}}C_{m_{\\rm{D}},m_{854},m_{\\rm{P}}} ~ \\tilde{c}_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta) \\left\\lvert m_{\\rm{P}}\\right\\rangle\\left\\langle m_{\\rm{D}}\\right\\lvert\\left\\langle m_{854}\\right\\lvert\n\\label{absop}\n\\end{equation}\nwhere $m_{854}=0,\\pm1$, and $C_{m_{\\rm{D}},m_{854},m_{\\rm{P}}}$ are the CGCs \\cite{footnoteCGC}. The detuning-dependent atomic response $\\tilde{c}_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta)=\\lvert \\tilde{c}(\\Delta)\\rvert\\rm{e}^{i\\phi(\\Delta)}$ has a complex Lorentzian lineshape with linewidth $\\Gamma_{\\rm{P}_{3\/2}}$. We combine these coefficients according to \n\\begin{equation}\nC_{m_{\\rm{D}},m_{854},m_{\\rm{P}}} ~ \\tilde{c}_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta) = c_{m_{\\rm{D}},m_{\\rm{P}}}(\\Delta)\n\\end{equation}\nusing $m_{\\rm{P}}=m_{\\rm{D}}+m_{854}$. The detuning $\\Delta=\\omega_{\\rm{L}}-\\omega_{0}$ between the laser frequency $\\omega_{\\rm{L}}$ and the D$_{5\/2}$ to P$_{3\/2}$ line center $\\omega_{0}$ will be used as a control knob to compensate for the different CGCs in the absorption channels.\n\nAfter the absorption, the ion decays in a spontaneous (Raman) emission process to the S$_{1\/2}$ ground state, described by an emission operator,\n\\begin{equation}\n\\hat{E}=\\sum_{m_{\\rm{S}},m_{\\rm{P}}}C_{m_{\\rm{P}},m_{393},m_{\\rm{S}}}\\left\\lvert m_{393}\\right\\rangle\\left\\lvert m_{\\rm{S}}\\right\\rangle\\left\\langle m_{\\rm{P}}\\right\\lvert,\n\\label{emop}\n\\end{equation}\nwith $m_{393}=0,\\pm1$. Applying the absorption and emission operator to the joint state of Eq.~(\\ref{joint}) gives $\\hat{E}\\hat{A}\\left\\vert\\psi(t)\\right\\rangle$, a new joint state between an atom in S$_{1\/2}$ and a single photon in the 393-nm mode, which is projected onto $\\left\\lvert \\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle$ conditioned on the detection of a 393-nm photon with certain polarization. \n\n\\subsection{A. The \\texorpdfstring{$\\Lambda$}{Lambda} system\n\nIn the case of the $\\Lambda$ system, the ion is prepared in the coherent superposition state \n\\begin{equation}\n\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle=\\sqrt{\\tfrac{1}{2}}\\left(\\left\\lvert\\rm{D}_{5\/2},-\\tfrac{3}{2}\\right\\rangle+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert\\rm{D}_{5\/2},+\\tfrac{1}{2}\\right\\rangle\\right),\n\\label{L-superposition}\n\\end{equation}\nsuch that two absorption paths share the same emission channel [Fig.~\\ref{levelscheme_both}(a)]. Applying the absorption operator to the joint state gives\n\\begin{equation}\n\\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\sqrt{\\tfrac{1}{2}}\\cos\\tfrac{\\vartheta}{2}~c_{-\\nicefrac{3}{2},-\\nicefrac{1}{2}}(\\Delta) \\left\\lvert\\rm{P}_{3\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{2}}\\sin\\tfrac{\\vartheta}{2}~c_{+\\nicefrac{1}{2},-\\nicefrac{1}{2}}(\\Delta)~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{2}}\\cos\\tfrac{\\vartheta}{2}~c_{+\\nicefrac{1}{2},+\\nicefrac{3}{2}}(\\Delta)~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{3}{2}\\right\\rangle.\n\\end{split}\n\\end{equation}\nHighest visibility of the quantum beats is expected when the interfering transitions have equal weights. The amplitudes of the two absorbing paths are determined by the detuning $\\Delta$, which is adjusted in order to compensate for the two different CGCs, i.e., such that $|c_{-\\nicefrac{3}{2},-\\nicefrac{1}{2}}(\\Delta)| = |c_{+\\nicefrac{1}{2},-\\nicefrac{1}{2}}(\\Delta)| = c$. \nFor a linear photonic polarization state ($\\vartheta=\\tfrac{\\pi}{2}$) it follows that \n\\begin{equation}\n\\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\tfrac{1}{2}c\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n& \\quad\n+\\tfrac{1}{2}c'\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{3}{2}\\right\\rangle.\n\\end{split}\n\\label{absorbH}\n\\end{equation}\nwith $c' = c_{+\\nicefrac{1}{2},+\\nicefrac{3}{2}}(\\Delta)$. Here the term $\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)$ already shows the interference of the two absorption paths in the amplitude of $\\left\\lvert\\rm{P}_{3\/2},-\\frac{1}{2}\\right\\rangle$.\nThe transfer of this oscillation to the emitted 393-nm photons is obtained by applying the emission operator $\\hat{E}$,\n\\begin{equation}\n\\begin{split}\n\\hat{E}\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\sqrt{\\tfrac{1}{6}}c\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)\\left\\lvert 0 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{12}}c\\left(1+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\right)\\left\\lvert -1 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},+\\tfrac{1}{2}\\right\\rangle\\\\\n&+\\sqrt{\\tfrac{1}{8}}c'\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\left\\lvert +1 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},+\\tfrac{1}{2}\\right\\rangle.\n\\end{split}\n\\label{state1}\n\\end{equation}\nOf these three different decay channels, selection of $\\Delta m = 0$, i.e., of $\\pi$ photons, is favorable for keeping indistinguishability between the two interfering scattering channels. A $\\pi$ photon detected perpendicular to the magnetic field transforms to an $\\left\\lvert\\rm{H}\\right\\rangle$-polarized photon in the photonic reference frame. The detection of these photons projects the joint state of Eq.~(\\ref{state1}) onto $\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle$. The intensity of the emitted light is proportional to \n\\begin{equation}\nI \\propto \\tfrac{1}{3} c^2(1+\\text{cos}\\,(\\Phi_{\\text{D}}(t)+\\Phi_{854}))\n\\label{intensity1}\n\\end{equation}\nand shows the possibility to be controlled by changing the photonic input phase $\\Phi_{854}$ or the atomic superposition phase $\\Phi_{\\rm{D}}(t)$ through the offset phase $\\Phi_{\\rm{D}}(0)$. We note again that interference happens in the absorption process, since two pathways lead to the same excited intermediate state before the emission process takes place.\n\n\\subsection{B. The V system}\nThe V-type configuration and the corresponding excitation scheme for the generation of 393-nm photons is shown in Fig.~\\ref{levelscheme_both}(b). Starting in the eigenstate $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$, the first resonant 729-nm pulse transfers 75$\\%$ of the population to $\\left\\lvert\\rm{D}_{5\/2},+\\frac{3}{2}\\right\\rangle$. The remaining population is subsequently transferred to $\\left\\lvert\\rm{D}_{5\/2},-\\frac{5}{2}\\right\\rangle$ with a $\\pi$ pulse, resulting in the coherent superposition state\n\\begin{equation}\n\\left\\lvert\\psi_{\\rm{D}}(t)\\right\\rangle=\\sqrt{\\tfrac{1}{4}}\\left\\lvert\\rm{D}_{5\/2},-\\tfrac{5}{2}\\right\\rangle+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\sqrt{\\tfrac{3}{4}}\\left\\lvert\\rm{D}_{5\/2},+\\tfrac{3}{2}\\right\\rangle. \n\\label{V-superposition}\n\\end{equation}\nThe unequal squared CGCs of the two 393-nm decay channels, i.e., the spurious decay into unwanted channels [see Fig.~\\ref{levelscheme_both}(b)], is compensated for by the unequal initial population distribution in D$_{5\/2}$. Applying the absorption operator (\\ref{absop}) to the joint state gives\n\\begin{equation}\n \\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle\n &=\\sqrt{\\tfrac{1}{4}}\\cos\\tfrac{\\vartheta}{2}~c_{-\\nicefrac{5}{2},-\\nicefrac{3}{2}}(\\Delta)\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{3}{2}\\right\\rangle\\\\\n& +\\sqrt{\\tfrac{3}{4}}\\sin\\tfrac{\\vartheta}{2}~c_{+\\nicefrac{3}{2},+\\nicefrac{1}{2}}(\\Delta)~\\rm{e}^{i\\Phi_{\\rm{D}}(t)} \\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{1}{2}\\right\\rangle.\n \\end{split}\n\\end{equation}\nThe two different CGCs on the 854-nm absorption channels are again compensated by two different 854-nm repumping rates, set by adjusting the detuning $\\Delta$ such that $|c_{-\\nicefrac{5}{2},-\\nicefrac{3}{2}}(\\Delta)| = |c_{+\\nicefrac{3}{2},+\\nicefrac{1}{2}}(\\Delta)| = c$. For a linear 854-nm polarization with phase $\\Phi_{854}$ we get \n\\begin{equation}\n \\begin{split}\n\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle&=\\sqrt{\\tfrac{1}{8}}c\\left\\lvert\\rm{P}_{3\/2},-\\tfrac{3}{2}\\right\\rangle+\\sqrt{\\tfrac{3}{8}}c~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert\\rm{P}_{3\/2},+\\tfrac{1}{2}\\right\\rangle.\\\\\n \\end{split}\n\\end{equation}\n\nApplying the emission operator (\\ref{emop}) results in\n\\begin{equation}\n \\begin{split}\n\\hat{E}\\hat{A}\\left\\lvert\\psi(t)\\right\\rangle &= \\left(\\left\\lvert -1 \\right\\rangle+\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert +1 \\right\\rangle\\right)\\sqrt{\\tfrac{1}{8}}c\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle\\\\\n&+ \\tfrac{1}{2}c~\\rm{e}^{i\\Phi_{\\rm{D}}(t)}\\rm{e}^{i\\Phi_{854}}\\left\\lvert 0 \\right\\rangle\\left\\lvert\\rm{S}_{1\/2},+\\tfrac{1}{2}\\right\\rangle.\n\\label{superpos}\n \\end{split}\n\\end{equation}\nThe first term in this state shows the oscillating phase between two atomic ($\\sigma^{\\pm}$) transitions leading to interference of two emission channels when they are projected on the same axis. More precisely, the superposition of $\\sigma^-$ and $\\sigma^+$ emission causes the dipolar emission pattern to rotate with the Larmor frequency about the quantization axis, which leads to a temporal modulation of the detected photons in the direction perpendicular to that axis. Measurement of the photonic superposition $\\left\\lvert\\rm{V}\\right\\rangle=\\sqrt{\\frac{1}{2}}\\left(\\left\\lvert\\sigma^+\\right\\rangle-\\left\\lvert\\sigma^-\\right\\rangle\\right)$ projects the joint state of Eq.~(\\ref{superpos}) onto $\\left\\lvert\\rm{S}_{1\/2},-\\tfrac{1}{2}\\right\\rangle$, which results in an intensity \n\\begin{equation}\nI \\propto \\tfrac{1}{8}c^2(1-\\rm{cos}\\,(\\Phi_{\\rm{D}}(t)+\\Phi_{854})),\n\\label{intensity}\n\\end{equation}\nshowing again oscillations with the Larmor frequency.\nThe second term in (\\ref{superpos}) describes the emission of parasitic $\\pi$ photons which transforms in the photonic reference frame to $\\left\\lvert\\rm{H}\\right\\rangle$. They are suppressed by rotating the PBS by $90^{\\circ}$ with respect to the $\\Lambda$ case.\n\n\\section{IV. Experimental Results}\n\\subsection{A. Quantum beats in arrival-time distributions}\n\nWe first discuss the experimental results for the $\\Lambda$ scheme. The blue circles in Fig.~$\\ref{L-incoherent}$ show the arrival-time distribution of the 393-nm Raman-scattered photons for the case of an initial coherent superposition between $\\left\\lvert\\rm{D}_{5\/2},-\\frac{3}{2}\\right\\rangle$ and $\\left\\lvert\\rm{D}_{5\/2},+\\frac{1}{2}\\right\\rangle$. The exponential decay of the photon wave packet is modulated with a period of 106 ns, corresponding to a Larmor frequency in D$_{5\/2}$ of 9.4 MHz, which is in agreement with the frequency difference of the initially populated Zeeman sublevels, as determined from 729-nm spectroscopy. The data points are fitted by numerically solving the optical Bloch equations, including all relevant Zeeman sublevels and the projection of the final joint state [Eq.~(\\ref{state1})] according to the detection of an $\\left\\lvert\\rm{H}\\right\\rangle$-polarized photon. The visibility of the oscillation, determined from the envelope of the fit (gray dotted line), is 93.1(6)$\\%$. For comparison, we also show the photon arrival-time distribution for the case of an initial statistical mixture in D$_{5\/2}$ (black dots). The data set is generated by averaging the two individually recorded arrival-time distributions for the $\\left\\lvert\\rm{D}_{5\/2},-\\frac{3}{2}\\right\\rangle$ to $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$ and $\\left\\lvert\\rm{D}_{5\/2},+\\frac{1}{2}\\right\\rangle$ to $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle$ scattering processes. From the fit of the exponential decay (green line) we get an 1\/e decay time of 461(2) ns. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.4cm]{L-incoherent}\n\\caption{(Color online) Arrival-time distribution of single 393 nm photons from an initial coherent superposition (blue circles) and statistical mixture (black dots) in D$_{5\/2}$. Red solid line, fit to the data by numerically solving the 18-level optical Bloch equations; green dash-dotted line, exponential fit to the data for the mixture; gray dotted line, exponential fit to the envelope of the oscillation, from which the visibility is determined.}\n\\label{L-incoherent}\n\\end{figure}\n\nIn the following we show that we have full control over the quantum phases that enter into the quantum beats of Fig.~\\ref{L-incoherent}. First we study their dependence on the linear polarization of the absorbed 854-nm photons. Rotation of the wave plates allows adjustment of the incoming linear polarization, i.e., of the photonic phase $\\Phi_{854}$ in Eq.~(\\ref{intensity1}), to the canonical basis states $\\left\\lvert\\rm{H}\\right\\rangle$, $\\left\\lvert\\rm{V}\\right\\rangle$, $\\left\\lvert\\rm{D}\\right\\rangle$, and $\\left\\lvert\\rm{A}\\right\\rangle$ (and any value in between). Figure~\\ref{L-osc-pol+phase}(a) shows arrival-time distributions for the polarization input states $\\left\\lvert\\text{D}\\right\\rangle$ and $\\left\\lvert\\text{A}\\right\\rangle$ (for the sake of clarity, we only show these two). The phase difference of $\\Delta \\Phi_{854}=180^{\\circ}$ set by the waveplates is revealed in the two oscillations; the value derived from fitting the Bloch equations is $178.2(1.6)^{\\circ}$. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{L-osc-pol+phase}\n\\caption{(Color online) $\\Lambda$ scheme. (a) Arrival-time distributions of the 393-nm photons showing quantum beats for two different 854-nm input polarization states, $\\left\\lvert\\rm{D}\\right\\rangle$ and $\\left\\lvert\\rm{A}\\right\\rangle$. (b) Arrival-time distributions showing quantum beats for two values, 0 and $\\pi$, of the phase $\\Phi_{\\rm D}(0)$ of the initial atomic superposition. In (a) and (b), the bin size is 2~ns for 6~min measurement time. }\n\\label{L-osc-pol+phase}\n\\end{figure}\n\nSimilarly, according to Eq.~(\\ref{intensity1}), also the phase of the initial atomic superposition state $\\Phi_{\\rm D}(0)$ enters directly into the quantum beats. We control this phase via the radio-frequency source that drives the acousto-optic modulator setting the amplitude of the 729-nm laser. Figure~\\ref{L-osc-pol+phase}(b) displays the change in the phase of the quantum beats effected by changing $\\Phi_{\\rm D}(0)$ by $180^{\\circ}$, while keeping $\\Phi_{854}$ constant; the Bloch equation fits yield a phase difference of $181.1(1.1)^{\\circ}$. The $<1$\\% deviation between the set values and the fitted values highlights the precise control that we have over the quantum beats. \nWith the V scheme, we achieve analogous control through changes of both the photonic phase, i.e.\\ the polarization of the incoming 854-nm photons, and the atomic phase, i.e.\\ the difference phase of the preparing 729-nm pulses. Figure~\\ref{V-osc-pol+phase}(a) displays the arrival-time distributions when the 854-nm polarization is adjusted to the orthogonal basis states $\\left\\lvert\\rm{D}\\right\\rangle$ and $\\left\\lvert\\rm{A}\\right\\rangle$. The Bloch equation fits yield a phase difference of 176.5(1.7)$^{\\circ}$. In Fig.~\\ref{V-osc-pol+phase}(b) the offset phase $\\Phi_{\\rm D}(0)$ of the coherent superposition in D$_{5\/2}$ is set to 0 and $\\pi$. Here the phase difference from the fits is $181.4(1.5)^{\\circ}$. The deviation by 1-2\\% from the ideal values confirms the precise control over the quantum-beat phase also for the case of the V scheme. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{V-osc-pol+phase}\n\\caption{(Color online) V scheme. (a) Quantum beats for two different 854-nm input polarization states, $\\left\\lvert\\rm{D}\\right\\rangle$ and $\\left\\lvert\\rm{A}\\right\\rangle$. (b) Quantum beats for two values, 0 and $\\pi$, of the phase $\\Phi_{\\rm D}(0)$ of the initial atomic superposition. The bin size is 2~ns for 10~min measurement time [6~min in (b)]. }\n\\label{V-osc-pol+phase}\n\\end{figure}\n\nDue to the spurious decay from $|\\rm{P}_{3\/2},+\\frac{1}{2}\\rangle$ to $|\\rm{S}_{1\/2},+\\frac{1}{2}\\rangle$ [see Fig.~\\ref{levelscheme_both}(b)] in the V scheme, maximum visibility of the quantum beats requires an unequal population ratio in the initial superposition state, as can be seen from Eq.~(\\ref{V-superposition}). The data of Fig.~\\ref{V-osc-pol+phase} are acquired with the optimum population ratio, which is experimentally verified by the following procedure: From the two pulses which prepare the coherent superposition, the duration of the first $\\left\\lvert\\rm{S}_{1\/2},-\\frac{1}{2}\\right\\rangle \\to \\left\\lvert\\rm{D}_{5\/2},+\\frac{3}{2}\\right\\rangle$ pulse is varied, thereby adjusting the amount of transferred population. The subsequent $\\pi$ pulse to $\\left\\lvert\\rm{D}_{5\/2},-\\frac{5}{2}\\right\\rangle$ transfers the remaining population. In Fig.~\\ref{best-pop} the quantum-beat visibility is shown as a function of the population in $\\left\\lvert\\rm{D}_{5\/2},+\\frac{3}{2}\\right\\rangle$. The highest visibility of 78.2(9)$\\%$ is obtained for 75$\\%$ population, in agreement with the ratio of the two involved CGCs. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{V-best-pop}\n\\caption{(Color online) Quantum-beat visibility in the V scheme as a function of the initial population in $|\\rm{D}_{5\/2},+\\frac{3}{2}\\rangle$. The data points are calculated from exponential fits to the envelopes of the observed quantum beats.}\n\\label{best-pop}\n\\end{figure}\n\nThe reduced visibility of the quantum beats in the V scheme (Fig.~\\ref{V-osc-pol+phase}) compared to the $\\Lambda$ scheme (Fig.~\\ref{L-osc-pol+phase}) results from the nature of the respective interference phenomena. In the $\\Lambda$ scheme the excitation amplitudes to the P level interfere constructively or destructively; hence, the overall probability for 393-nm photon emission is modulated. In contrast, in the V scheme the photons are emitted with a spatially rotating pattern. Therefore, the HALO that collects the 393-nm photons in $\\sim 4\\%$ solid angle will also pick up a small fraction of the dipolar emission when the preferred emission direction is perpendicular to the HALO axis. \n \n\n\\subsection{B. Phase-dependent photon-scattering probability}\nThe quantum beats in Figs.~\\ref{L-osc-pol+phase} and ~\\ref{V-osc-pol+phase} extend over many periods of the underlying Larmor precession; therefore, the total probability to detect a photon (i.e., time-integrated over the whole wave packet) is nearly independent of the control phases $\\Phi_{854}$ and $\\Phi_{\\rm D}(0)$. This behavior changes, however, when the exciting laser pulse is shorter than the quantum-beat period (or at least of comparable duration). In this case the time-integrated probability of detecting a photon may be significantly enhanced or suppressed by the quantum interference. In order to emphasize this effect, we created quantum beats with high 854-nm laser power, such that the total duration of the generated photon wave packet was as short as $\\sim$70~ns (including about 60~ns of broadening by the acousto-optic modulator rise time); at the same time, we reduced the magnetic field to extend the Larmor period. In Fig.~\\ref{det-eff-both}, the time-integrated photon detection probability for the case of a short excitation pulse is plotted against the atomic control phase $\\Phi_{\\rm D}(0)$ (dots). With the $\\Lambda$-type level scheme [Fig.~\\ref{det-eff-both}(a)] we find a variation by a factor of 7 ($\\sim76\\%$ visibility), while for the V-scheme [Fig.~\\ref{det-eff-both}(b)] the ratio is about 3 ($\\sim48\\%$ visibility). For comparison, the phase-dependent photon detection probability for a long photon, covering many quantum-beat periods, is also shown (crosses). Here the modulation is hardly visible; it disappears asymptotically. Hence, in the regime of short excitation pulses compared to the Larmor-precession period, the atomic control phase serves as a knob to determine the probability with which a photon is scattered into the detector. With faster modulation of the exciting laser than what could be attained in our setup, even much larger enhancement-suppression ratios may be reached. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{det-eff-both}\n\\caption{(Color online) Suppression and enhancement of the generation efficiency of 393~nm photons controlled via the atomic phase $\\Phi_{\\rm D}(0)$. The dots and solid curves (sinusoidal fits) are for short 854~nm excitation pulses (duration $<$ quantum-beat period), while the crosses and dashed curves are for long pulses (duration $\\gg$ quantum-beat period). (a) $\\Lambda$ scheme: The short photon has about 70~ns duration and 553~ns quantum-beat period, the long photon about 850~ns duration and 106.5~ns quantum-beat period. (b) V scheme: The short photon extends over $\\sim$120~ns at 277.4~ns beat period, for the long photon the values are 750 and 53.2~ns. }\n\\label{det-eff-both}\n\\end{figure}\n\n\n\\section{V. PHYSICAL MECHANISM}\n\nIn this final section we highlight the fundamental difference between the interference processes involved in the two schemes. This is done by analyzing the time dependence of the population in D$_{5\/2}$ during the excitation with 854~nm light: In the experiment, the excitation is interrupted after a certain pulse duration and the remaining population in the D level is determined through state-selective fluorescence \\cite{Roos1999}. For the $\\Lambda$ scheme, the result is displayed in Fig.~\\ref{L-stairs}. Here the input polarization is set to $\\left\\lvert\\rm{V}\\right\\rangle$, and the magnetic field has a value of 0.987~G, which results in a Larmor period between the Zeeman sublevels of 302 ns. The pulse length of the 854-nm excitation is varied in steps of 12.5 ns. The figure shows that the depopulation of the D$_{5\/2}$ manifold exhibits a \"stairs\"-like modulation with the Larmor frequency. The population dynamics are very well fitted by 18-level Bloch equations (solid line in Fig.~\\ref{L-stairs}). The inset in Fig.~$\\ref{L-stairs}$ shows the time derivative of the population in D$_{5\/2}$, which is proportional to the population in P$_{3\/2}$. Comparison with Fig.~\\ref{L-osc-pol+phase} confirms that the observed quantum beats in the case of the $\\Lambda$ scheme are due to a corresponding oscillation of the population of the excited P level. This manifests that the interference happens indeed between the two absorption pathways from D$_{5\/2}$ to P$_{3\/2}$: As these require $\\sigma^+$- and $\\sigma^-$-polarized light, respectively, the Larmor-precessing phase of the initial superposition state creates an oscillatory excitation probability for any fixed linear incoming polarization.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{L-stairs}\n\\caption{(Color online) Change of the D$_{5\/2}$ population for different 854-nm pulse lengths in the $\\Lambda$ scheme. The stairs-like behavior results from interference of two excitation amplitudes to $\\left\\lvert\\rm{P}_{3\/2},-\\frac{1}{2}\\right\\rangle$. The pulse length is varied in steps of 12.5~ns. The solid line is calculated with the 18-level Bloch equations including experimental parameters. (Inset) Time derivative of the calculated solid line showing the oscillation of the population in P$_{3\/2}$, which leads to suppression and enhancement of the photon emission at 393~nm.}\n\\label{L-stairs}\n\\end{figure} \n\nIn contrast, a similar measurement for the case of the V scheme, shown in Fig.~\\ref{V-nostairs}, does not exhibit any modulation on the exponential depopulation curve of the D$_{5\/2}$ level. The Bloch equation fit describes excitation from D$_{5\/2}$ to P$_{3\/2}$ at a constant rate, i.e., with no interference in the excitation pathways. The observed quantum beats in the emitted 393-nm photon wave packet are therefore due to interference of emission amplitudes: The Larmor precession of the initial state enters into the emitted superposition of $\\sigma^+$- and $\\sigma^-$-polarized components and causes a spatially rotating emission pattern, which leads to an oscillatory detection probability. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{V-nostairs}\n\\caption{(Color online) Depopulation of D$_{5\/2}$ for different 854-nm pulse lengths in the V scheme, showing simple exponential decay without modulation. The points are measured data; the line is a fit by Bloch equation dynamics. (Inset) Time derivative of the calculated curve \\cite{footnoteV}.}\n\\label{V-nostairs}\n\\end{figure} \n\n \nFurther evidence for our explanation of the physical mechanism is provided by measuring the depletion of the D$_{5\/2}$ level as a function of the initial phase, $\\Phi_{\\rm D}(0)$, after an excitation pulse of fixed duration. A comparison of the results for the two distinct level schemes, under otherwise equal conditions, is displayed in Fig.~\\ref{depletion}. The phase enters into the depletion in the $\\Lambda$ case, while the V case is practically insensitive to it. \n \nThe two types of quantum beats of Figs.~\\ref{L-osc-pol+phase} and \\ref{V-osc-pol+phase}, while looking very similar, are hence identified to have manifestly different physical origins. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.3cm]{depletion}\n\\caption{(Color online) Depopulation of D$_{5\/2}$ for different intial phases $\\Phi_{\\rm D}(0)$ of the initial superposition, after an excitation pulse of 12.5~ns. The blue dots (solid curve) are measured (calculated) for the $\\Lambda$ scheme, the red crosses (dashed curve) for the V scheme. The Larmor period was 300~ns ($\\Lambda$) and 150~ns (V).}\n\\label{depletion}\n\\end{figure} \n\n\\section{V. Summary}\nWe investigated experimentally two cases of single-photon quantum beats originating from interference of photon-scattering amplitudes. A single trapped Ca$^+$ ion is prepared in a coherent superposition state in its D$_{5\/2}$ manifold and then excited on the D$_{5\/2}$ to P$_{3\/2}$ transition (854~nm), whereby it releases a single photon on the P$_{3\/2}$ to S$_{1\/2}$ transition (393~nm). Quantum beats with the frequency of the atomic Larmor precession are manifested as oscillations in the photon arrival-time distribution with $>93\\%$ visibility. The phase of the quantum beats is controlled through setting the phase of the initial atomic superposition and through the polarization of the exciting 854-nm light. For a $\\Lambda$- and a V-type atomic level scheme, we identified interference of absorption and emission amplitudes, respectively, to be the physical mechanisms behind the quantum beats. The two mechanisms are fundamentally different in that the remaining population in the D$_{5\/2}$ level reveals the interference in the $\\Lambda$ case but not in the V case. As a consequence, for small ratios between the excitation pulse length and the Larmor period, excitation out of D$_{5\/2}$ can be efficiently suppressed and enhanced in the $\\Lambda$ scheme through the choice of the control phases, while in the V case the excitation probability is phase insensitive. The presented experimental techniques and the physical mechanisms are essential ingredients for mapping arbitrary polarization states of photons onto a single ion \\cite{Kurz2014}.\n\n\n\\vspace{0.2cm}\n\\begin{acknowledgments}\nWe acknowledge support by the BMBF (Verbundprojekt QuOReP, CHIST-ERA project QScale), the German Scholars Organization \/ Alfried Krupp von Bohlen und Halbach-Stiftung, and the ESF (IOTA COST Action).\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n Since the first observations of baryonic decays of $B$ mesons\nby ARGUS~\\cite{1} and CLEO~\\cite{2}, many three-body baryonic $B$ \ndecays have been found~\\cite{3}. \nAlthough the general pattern of these decays \ncan be understood intuitively from heavy $b$ quark decays~\\cite{4}, \nmany specific details cannot be explained by this simple picture. \n\nUsing a generalized factorization \napproach, Ref.~\\cite{5} predicts rather large branching fractions \n($\\sim$$10^{-5}$) for the Cabibbo-suppressed processes\n$B \\to \\overline{p} \\Lambda D^{(*)} $. \nThe branching fractions of other related \nbaryonic decays such as $B^0 \\to p\\overline{p}D^0$~\\cite{6,7},\n~$B^0 \\to p\\overline{p}K^{*0}$~\\cite{8}, $B^- \\to p\\overline{p}K^{*-}$~\\cite{9,10} and \n$B^- \\to p\\overline{p}\\pi^-$~\\cite{9} are used as\ninputs in such estimates because baryon form factors entering\nthe decay amplitudes are difficult to calculate from first principles. \nThe expected values of the branching fractions for $B^- \\to \\overline{p}\\Lambda D^0$ and \n$B^- \\to \\overline{p}\\Lambda D^{*0}$\nare already within reach with the data sample accumulated at Belle.\n\nNearly all baryonic $B$ decays into three- and\nfour-body final states possess a common feature: baryon-antibaryon \ninvariant masses that peak near threshold. This \nthreshold enhancement is found both in charmed and charmless \ncases~\\cite{3}. A similar effect has been observed \nin $J\/\\psi \\to p\\overline{p}\\gamma$ decays by BES~\\cite{bes1,bes2} \nand CLEO~\\cite{cleo1}, but is not seen in \n$J\/\\psi \\to p \\overline{p} \\pi^0 $~\\cite{bes1} and \n$\\Upsilon(1S) \\to p\\overline{p}\\gamma$~\\cite{cleo2}. One of the possible \nexplanations of this phenomenon suggested in the literature is a final state \n$N\\overline{N}$ interaction~\\cite{int}.\n\nIn this paper, we present results on the $B^- \\to \\overline{p}\\Lambda D^{(*)0}$ decays in order to test the factorization hypothesis\nand study the $\\overline{p}\\Lambda$ threshold enhancement effect.\n\n\n\nThe data sample used in the study corresponds to an integrated luminosity of 605 \nfb$^{-1}$, containing 657 $ \\times 10^6~B\\overline{B}$ pairs, collected at the $\\Upsilon(4S)$ resonance\nwith the Belle detector at the KEKB asymmetric-energy\n$e^+e^-$ (3.5~GeV and 8~GeV) collider~\\cite{KEKB}.\nThe Belle detector~\\cite{Belle} is a large-solid-angle magnetic spectrometer that consists of\n a silicon vertex detector (SVD),\na 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov\ncounters (ACC), a barrel-like arrangement of time-of-flight scintillation counters \n(TOF), and an electromagnetic calorimeter (ECL) composed of CsI(Tl) crystals \nlocated inside a superconducting solenoid coil that provides a 1.5~T magnetic field.\n\n\nThe selection criteria for the final state charged particles in $B^{-} \\to \\pld$ and $B^{-} \\to \\pldst$ are \nbased on information obtained from the tracking system (SVD and CDC) and \nthe hadron identification system (CDC, ACC, and TOF). \nThe primary and $D^0$ daughter charged tracks\nare required to have a point of\nclosest approach to the interaction point (IP) that is within\n$\\pm$0.3 cm in the transverse ($x$--$y$) plane, and within $\\pm$3.0 cm\nin the $z$ direction, where the $+z$ axis is opposite to the\npositron beam direction.\nFor each track, the likelihood values $L_p$, $L_K$, or $L_\\pi$ that it is a proton, kaon, or pion,\nrespectively, are determined from the information provided by the hadron identification system. \nA track is identified as a proton if $L_{p}\/(L_p +L_{K}) > 0.6$ and $L_{p}\/(L_p +L_{\\pi}) > 0.6$, as a kaon\nif $L_{K}\/(L_K +L_{\\pi}) > 0.6$, or as a pion if $L_{\\pi}\/(L_K +L_{\\pi}) > 0.6$.\n The efficiency for identifying a kaon (pion) is 85$-$95\\% depending on the momentum of the track, while the probability for a pion\n(kaon) to be misidentified as a kaon (pion) is 10$-$20\\%.\nThe proton identification efficiency is 84\\% while the probability for a kaon or a pion to be misidentified as a proton is less than 10\\%.\n\nWe reconstruct $\\Lambda$'s from their decays to $p \\pi^-$. \nEach $\\Lambda$ candidate must have a displaced vertex and the direction of its momentum vector must be\nconsistent with an origin at the IP.\nThe proton-like daughter is required to satisfy the proton criteria described above, and no further selections are applied\nto the daughter tracks. \nThe reconstructed $\\Lambda$ mass is required to be in the range 1.111 GeV\/$c^2 < M_{p\\pi^-} <1.121 $ GeV\/$c^2$~\\cite{3}.\n\nCandidate $D^0$ mesons are reconstructed in the following two sub-decay channels: \n$D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$, $\\pi^0 \\to \\gamma\\gamma$. \nThe $\\gamma$'s that constitute $\\pi^0$ candidates are required to have \nenergies greater than 50 MeV if the $\\gamma$ is reconstructed from the barrel ECL\nand greater than 100 MeV for the endcap ECL, and not be associated with any charged\ntracks in CDC.\nThe energy asymmetry of $\\gamma$'s from a $\\pi^0 $, $\\frac{|E_{\\gamma1}-E_{\\gamma2}|}{E_{\\gamma1}+E_{\\gamma2}}$, \nis required to be less than 0.9.\nThe mass of a $\\pi^0$ candidate is required to be within the range \n$0.118$ GeV\/$c^2 < M_{\\gamma\\gamma}<0.150$ GeV\/$c^2$ before a mass-constrained fit is applied to improve the $\\pi^0$ momentum resolution. We impose a cut on the invariant masses of the $D^0$ candidates, $|M_{K^-\\pi^+}-1.865$ GeV\/$c^2|<0.01$ GeV\/$c^2$ and $1.837$ \nGeV\/$c^2 5.27$ GeV\/$c^2$ and of $M_{\\rm bc}$ (b, d) for $|{\\Delta{E}}| <0.05 $ GeV; \nthe top row is the fit result for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+ $ (a, b) and the bottom row for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+ \\pi^0$ (c, d). \nThe points with error bars are data; the solid\ncurve shows the fit; the dashed curve represents the signal, \nand the dotted curve indicates continuum background.}\n\\label{fg:data_fit1}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\begin{sideways} {~~~~~~~~~~~~~~~~~~~~$d\\mathcal{B} \/ dM_{\\overline{p}{\\Lambda}}$ $(10^{-6})$ \/ ( 0.2 GeV \/$c^2)$ } \\end{sideways}\n \\hskip -0.5 cm \n{\\includegraphics[width=0.35\\textwidth]{.\/Fig3.eps}}\\\\\n\\vskip -0.9cm \n\\hskip 3.5 cm \n {\\large $M_{\\overline{p}{\\Lambda}} $ (GeV\/$c^2$)}\n\\caption{ Differential branching fraction ($d\\mathcal{B} \/ dM_{\\overline{p}{\\Lambda}}$) as a function of the $\\overline{p}{\\Lambda}$ mass for $B^{-} \\to \\pld$.\nNote that the last bin with the central value of 3 GeV\/$c^2$ has a bin width of 0.8 GeV\/$c^2$. The solid curve is a fit with a threshold function.\n}\n\\label{fg:pld0_mpl}\n\\end{figure}\n\n\\begin{figure}[htb]\n \\vskip 0.0cm\n \\hskip 2.cm {\\bf{(a)}} \\hskip 3.15 cm {\\bf{(b)}}\n \\vskip -1.2cm\n\\includegraphics[width=0.43\\textwidth]{.\/Fig4ab.eps}\\\\\n \\hskip 2. cm {\\bf{(c)}} \\hskip 3.15 cm {\\bf{(d)}}\n \\vskip -1.2cm\n\\includegraphics[width=0.43\\textwidth]{.\/Fig4cd.eps}\n\\caption{\nDistributions of ${\\Delta{E}}$ (a, c) for $M_{\\rm bc} > 5.27$ GeV\/$c^2$ and of $M_{\\rm bc}$ (b,d) for $|{\\Delta{E}}| <0.05 $ GeV; \nthe top row is the fit result for $B^{-} \\to \\pldst$ in the $\\Delta M$ sideband region (a,b) and the bottom row for $B^{-} \\to \\pldst$ in the $\\Delta M$ signal region (c, d). \nThe points with error bars are data; the solid curve shows the result of the fit; \nthe dot-dashed and dotted curve indicates the CF and continuum background; \nthe dashed curve represents the signal.}\n\\label{fg:data_fit2}\n\\end{figure}\n\n\\begin{table}\n\\caption{ Summary of the results: event yield, significance, efficiency, and branching\nfraction.}\n\\tabcolsep= 6pt\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{clcccc}\n\\hline\\hline\n Mode & $N_{signal}$ &\t$\\mathcal{S}$ & $\\epsilon$(\\%) &\t $\\mathcal{B}( 10^{-5})$ \\\\\n\\hline\\hline \n$\\overline{p} \\Lambda D^0_{K^-\\pi^+}$ \t & $26.5^{+6.3}_{-5.6}$ & 7.4 &\t 11.7 & \t $1.39^{+0.33}_{-0.29} \\pm 0.16$\\\\\n$\\overline{p} \\Lambda D^0_{K^-\\pi^+ \\pi^0}$ & $35.6^{+11.7}_{-10.7}$ & 3.4 &\t 4.0 & $1.54^{+0.50}_{-0.46} \\pm 0.26$\t\\\\ \n\\hline\n$B^{-} \\to \\pld$\t \t\t\t\t\t\t \t & & 8.1 &\t \t\t & $1.43^{+0.28}_{-0.25} \\pm 0.18$\t\\\\ \n\\hline\\hline \n$B^{-} \\to \\pldst$ \t\t\t & $4.3^{+3.2}_{-2.4}$ & 2.1 &\t 2.8 & \t $1.53^{+1.12}_{-0.85} \\pm 0.47$\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\label{data_yields}\n\\end{table}\t\t\t\n\n\nSystematic uncertainties are estimated using high-statistics control samples.\nA track reconstruction efficiency uncertainty of 1.2\\% is assigned for each track.\nFor the proton identification efficiency uncertainty, we use a $\\Lambda \\to p \\pi^-$ sample, and for\n$K-\\pi$ identification uncertainty we use a sample of kinematically identified $D^{*+} \\to D^0\\pi^+$,\n $D^0 \\to K^-\\pi^+$ decays. \nThe average efficiency discrepancy due to hadron identification differences \nbetween data and MC simulations has been corrected for the final branching fraction measurements. \nThe corrections due to the hadron identification are 10.7\\% and 10.6\\% for $B^{-} \\to \\pld$ and $B^{-} \\to \\pldst$, respectively.\nThe uncertainties associated with the hadron identification corrections are 4.2\\% for \ntwo protons (one from $\\Lambda$ decay), 0.5\\% for a charged pion, and 1.0\\% for a charged kaon.\n\nThe $\\pi^0$ selection uncertainty is found to be 5.0\\% by comparing the ratios of efficiencies between $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$ for data and MC samples.\nIn the $\\Lambda$ reconstruction, we find an uncertainty of $4.1\\%$ from the differences between data and MC for the efficiencies of tracks displaced from the interaction point, the $\\Lambda$ proper time distributions, and the $\\Lambda$ mass spectrum.\nThe uncertainty due to the $\\mathcal R$ selection for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+$ is estimated \nfrom the control sample $B^{-} \\to {D}^0 \\pi^-$, $D^0 \\to K^-\\pi^+\\pi^-\\pi^+$ and is determined to be 1.3\\%.\nThe $\\mathcal R$ related uncertainty for $B^{-} \\to \\pld$, $D^0 \\to K^-\\pi^+ \\pi^0$ is 3.0\\% estimated \nfrom $B^{-} \\to {D}^0 \\pi^-$, $D^0 \\to K^-\\pi^+\\pi^0$.\nThe uncertainties due to the $D^0$ mass selection for $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+\\pi^0$ are 1.9\\% and 1.6\\%, respectively.\n\nThe dominant systematic uncertainty for $B^{-} \\to \\pld$ is due to the modeling of PDFs, estimated by including a $B \\to \\pld\\pi$ or a nonresonant $B^{-} \\to \\overline{p} \\Lambda K^-\\pi^+ (K^-\\pi^+ \\pi^0)$ component in the fit, \nmodifying the efficiency after changing the signal ${M_{\\overline{p}\\Lambda}}$ distribution, and varying the parameters of the signal and background PDFs by one standard deviation using MC samples.\nThe modeling uncertainties are 7.5\\% and 12.9\\% for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+$ and $D^0 \\to K^-\\pi^+ \\pi^0$, respectively. \nThe overall modeling uncertainty for $B^{-} \\to \\pldst$ of 28.6\\% is obtained from two kinds of PDF modifications.\nThe parameters of the fixed CF component are varied by their $\\pm 1\\sigma$ statistical uncertainties, which were obtained from the fit to the $\\Delta M$ sideband region.\nWe also include an additional PDF for the \ncombinatorial background based on the \\textsc{Pythia}~\\cite{Pythia} $b$ quark fragmentation process, e.g.,\n$B^{-} \\to \\overline{p} \\Lambda {D}^{0}$, $B^{+} \\to \\overline{p} \\Delta^{++} {D}^{*0}$, $B^{-} \\to \\overline{p} \\Delta^{0} {D}^{*0}$, $B^{-} \\to \\overline{p} \\Sigma^0 {D}^{*0}$, etc.\n\nThe systematic uncertainties from the sub-decay branching fractions are calculated from the corresponding \nbranching uncertainties in~\\cite{3}; they are 1.5\\% (3.7\\%) and 6.0\\% \nfor $B^{-} \\to \\pld,~D^0 \\to K^-\\pi^+$ ($D^0 \\to K^-\\pi^+\\pi^0$) and $B^{-} \\to \\pldst$, respectively.\nThe uncertainty in the number of $B\\overline{B}$ pairs is 1.4\\%.\nThe total systematic uncertainties are 11.6\\% (17.1\\%) and 30.9\\% for $B^{-} \\to \\pld$ with $D^0 \\to K^-\\pi^+$\n ($D^0 \\to K^-\\pi^+ \\pi^0$) and $B^{-} \\to \\pldst$, respectively.\nThe final results are listed in Table~\\ref{data_yields}, where the significance values are\nmodified and include the systematic uncertainty related to PDF modeling.\n\nIn summary, using a sample of $657 \\times 10^6~B\\overline{B}$ events, we report the first \nobservation of $B^{-} \\to \\pld$ with a \nbranching fraction of $(1.43^{+0.28}_{-0.25} \\pm 0.18)\\times 10^{-5}$ and a significance of 8.1$\\sigma$.\nNo significant signal is found for $B^{-} \\to \\pldst$ and the corresponding upper limit is $4.8 \\times 10^{-5}$ at the 90\\% confidence level.\nWe also observe a $\\overline{p}{\\Lambda}$ enhancement near\nthreshold for $B^{-} \\to \\pld$, which is similar to a common feature found in charmless three-body \nbaryonic $B$ decays~\\cite{3}. \nThe measured $B^{-} \\to \\pld$ branching fraction agrees with the theoretical prediction \nof $(1.14 \\pm 0.26) \\times 10 ^{-5} $~\\cite{5}. This indicates that the generalized factorization approach with parameters determined from experimental data gives reasonable estimates for $b \\to c$ decays. \nThis information can be helpful for future theoretical studies of the angular distribution puzzle in the penguin-dominated processes, $B^- \\to p\\overline{p}K^{-}$ and $B^{0} \\to {p\\overline{\\Lambda}\\pi^-}$~\\cite{5}.\nThe measured branching fraction for $B^{-} \\to \\pld$ can also be used to tune the parameters in the event generator, e.g., \n\\textsc{Pythia}, for fragmentation processes involving $b$ quarks. \nAlthough the current statistics for $B^{-} \\to \\pld$ are still too low to perform\nan angular analysis of the baryon-antibaryon system, the proposed super flavor factories~\\cite{SFF1,SFF2}\n offer promising venues for such studies. \n\\\\\n\n\nWe thank the KEKB group for the excellent operation of the\naccelerator, the KEK cryogenics group for the efficient\noperation of the solenoid, and the KEK computer group and\nthe National Institute of Informatics for valuable computing\nand SINET4 network support. We acknowledge support from\nthe Ministry of Education, Culture, Sports, Science, and\nTechnology (MEXT) of Japan, the Japan Society for the \nPromotion of Science (JSPS), and the Tau-Lepton Physics \nResearch Center of Nagoya University; \nthe Australian Research Council and the Australian \nDepartment of Industry, Innovation, Science and Research;\nthe National Natural Science Foundation of China under\ncontract No.~10575109, 10775142, 10875115 and 10825524; \nthe Ministry of Education, Youth and Sports of the Czech \nRepublic under contract No.~LA10033 and MSM0021620859;\nthe Department of Science and Technology of India; \nthe BK21 and WCU program of the Ministry Education Science and\nTechnology, National Research Foundation of Korea,\nand NSDC of the Korea Institute of Science and Technology Information;\nthe Polish Ministry of Science and Higher Education;\nthe Ministry of Education and Science of the Russian\nFederation and the Russian Federal Agency for Atomic Energy;\nthe Slovenian Research Agency; the Swiss\nNational Science Foundation; the National Science Council\nand the Ministry of Education of Taiwan; and the U.S.\\\nDepartment of Energy.\nThis work is supported by a Grant-in-Aid from MEXT for \nScience Research in a Priority Area (``New Development of \nFlavor Physics''), and from JSPS for Creative Scientific \nResearch (``Evolution of Tau-lepton Physics'').\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec1}\n\nIt is generally believed that most of the stars and galaxies can be described \nin good approximation as fluid bodies in thermodynamical equilibrium. In \nthe framework of general relativity, this implies (see e.g.\\ \n\\cite{hartle,lindblom}) that the corresponding spacetimes are stationary and \naxisymmetric. Moreover it is usually assumed (though there is no proof \nknown to us) that they are equatorially symmetric. \nThis stresses the importance of the study of stationary axisymmetric \nspacetimes. A relativistic treatment is necessary for rapidly \nrotating and massive compact objects like pulsars, neutron stars and \nblack-holes. \n\nThough the importance of global solutions describing stationary \naxisymmetric fluid bodies is generally accepted, the complicated structure \nof the Einstein equations with matter gives little hope that such solutions \ncan be found in the near future. Only for special and somewhat unphysical \nequations of state \n\\cite{wahl,kramer,seno}, it was possible to give solutions in the \nmatter region which are discussed as candidates for an interior solution. \nIn the exterior vacuum region, however, powerful solution generating techniques \nare at hand. Since the surface $\\Gamma_z$\nof a compact astrophysical object constitutes a \nnatural boundary at which the metric functions are not continuously \ndifferentiable, one is looking \nfor solutions to the vacuum equations\nthat are analytic outside this contour and can be at least \ncontinuously extended to $\\Gamma_z$. This means that the typical problem \none has to consider for the vacuum Einstein equations is a boundary value \nproblem of Dirichlet, von Neumann or mixed type, see \\cite{hp2}. The \nmatter then enters only in form of boundary conditions for the vacuum \nequations. This is possible if an interior solution is known or if only \ntwo--dimensionally extended bodies like disks or shells are considered. \nIn the latter case,\nthe surfacelike distribution of the matter implies \nthat the matter equations reduce to ordinary differential equations. \nNotice that disks are important models in astrophysics for certain types \nof galaxies.\n\nThe reason why it is much more promising to treat only the vacuum case \nis the equivalence of the stationary axisymmetric Einstein equations \nto a single nonlinear differential equation for a complex \npotential, the so called \nErnst equation \\cite{ernst}. The latter belongs to a family of \ncompletely integrable nonlinear equations that are studied as\nthe integrability conditions for associated linear differential systems. The \ncommon feature of these linear systems is that they contain an additional \nvariable, the so called spectral parameter, which reflects an underlying \nsymmetry of the differential equations under investigation, in the case of \nthe Ernst equation the Geroch group \\cite{geroch}. \nAssociated linear systems for the Ernst equations \nwere given in \\cite{belzak,maison,neuglinear}. \nThe existence of this parameter can be used to construct solutions by\nprescribing the singular structure of the matrix of the linear system with\nrespect to the spectral parameter.\n\nOne of the most successful solution techniques for nonlinear differential\nequations rests on methods of\nalgebraic geometry and leads to the so called finite-gap solutions that can be\nexpressed elegantly in terms of theta functions. Such methods were first used\nto construct periodic and quasiperiodic\nsolutions to nonlinear evolution equations like the Korteweg--de~Vries \n(KdV) and the Sine--Gordon (SG) equation. For a\nsurvey of this subject we refer the reader to~\\cite{soliton1,algebro}. \nHowever, it was only recently that \nalgebro-geometrical methods were applied to the Ernst equation,\nsee~\\cite{korot1}. The found solutions differ from similar \nsolutions of other equations in several aspects, e.g.\\ they are in general \nnot periodic or quasi-periodic. The main difference is that \nthis class is much richer than previously obtained ones. \n\nThe development of solution techniques yields a deeper insight into the\nstructure of nonlinear differential equations. However, from a practical point\nof view, it would also be desirable to solve initial value problems or, for the\nErnst equation, boundary value problems.\nOne approach to solve boundary value problems of the above mentioned\ntype with the help of the linear system is to translate the \nphysical boundary conditions into a Riemann-Hilbert problem which is\nequivalent to a linear integral equation, see~\\cite{soliton1}.\nNeugebauer and Meinel~\\cite{ngbml1}\nsucceeded in doing this in the case of the rigidly rotating dust disk. They\nwere able to reduce the matrix problem on a sphere to a scalar\nRiemann-Hilbert problem\non a hyperelliptic Riemann surface which can be solved explicitly via \nquadratures. By making use of the gauge transformations of the linear system\nwe were able to show~\\cite{prd} that this is possible in\ngeneral if the boundary value problem leads to a Riemann-Hilbert\nproblem with rational jump data. Up to now there is however no direct \nway to infer the jump data from the boundary value problem one wants to solve.\nThe explicit form of the hyperelliptic solutions possibly offers a different\napproach to boundary value problems: one can try to identify the free \nparameters in the solutions, a real valued function and a set of complex\nparameters, the branch points of the hyperelliptic Riemann surface, from \nthe problem one wants to solve. \n\nTo this end we study a class of solutions -- which is essentially\nequivalent to \\cite{korot1} and \\cite{meinelneugebauer} --\nthat is constructed via a \ngeneralized Riemann--Hilbert problem on a hyperelliptic Riemann surface. \nWe present a complete discussion of the singularity\nstructure of these Ernst potentials. It is possible to \nidentify a subclass of solutions that are everywhere regular except \nat some contour, which can possibly be related to the surface of an\nisolated body, where the Ernst potential is bounded. These solutions \nare asymptotically flat and\nequatorially symmetric, and thus show all the features one might expect \nfrom the exterior solution for an isolated relativistic ideal fluid. \nThey can have a Minkowskian and an extreme relativistic limit in \nwhich the body is `hidden' behind a horizon, and the exterior solution \nbecomes the extreme Kerr solution. This provides the hope that further \nsolutions to physically interesting boundary value problems to the Ernst\nequation, besides the rigidly rotating dust disk, can be identified within this\nclass. First results on this subclass where published in \\cite{prl}.\n\nThe paper is organized as follows. In section~\\ref{sec2} we introduce the \nlinear system associated to the Ernst equation and discuss how the matrix \nof the system has to be constructed in order to end up with new solutions \nto the Ernst equation. Using the results of~\\cite{krichever1}, we show how\nRiemann surfaces arise naturally in the context of linear systems with a\nspectral parameter. In the case of the Ernst equation, these are hyperelliptic\nRiemann surfaces with a special structure of the branch points. We will\nrestrict ourselves to regular compact Riemann surfaces and are eventually led\nto consider families of hyperelliptic Riemann surfaces of arbitrary genus,\nparametrized by the physical coordinates.\n\nIn section 3 we recall some basic notions of the theory of Riemann surfaces,\ntheta functions and the solution of Riemann--Hilbert problems on\nRiemann surfaces due to Zverovich, and present the class of solutions. It \nis shown that the solution of the axisymmetric Laplace equation which can \nbe freely prescribed in \\cite{meinelneugebauer}\nis a period of the Abelian integrals \nwhich determine the singularity structure of the matrix of the linear \nsystem. The differential relations between these periods are a subset of \nthe so called Picard--Fuchs equations which we write down for the Ernst \nequation.\nIn section 4 we discuss the singularity structure of these\nsolutions. It is shown that the solutions can have a regular axis and \nare in general asymptotically flat. Using an identity for \ntheta functions, we are able to give in section 5 \ncompact formulas for two metric \nfunctions and a simple condition for the occurrence of ergospheres.\nA subclass of solutions with equatorial symmetry is presented in section 6. \nThe common physical features of this subclass like the \nextreme relativistic limit are discussed. \nIn section 7, we use the equatorial symmetry to give simplified formulae \nfor the potential in the equatorial plane and on the axis. Since the rigidly \nrotating dust disk belongs to the simplest non-static solutions which are\nof genus 2, we consider this case in detail in section 8. In section 9, we \nsummarize the results and add some concluding remarks. \n\n\n\\section{Linear System for the Ernst equation and \nMonodromy matrix}\\label{sec2}\n\\setcounter{equation}{0}\n\nIt is well known (see \\cite{exac}) that the metric of stationary axisymmetric \nvacuum spacetimes can be written in the Weyl--Lewis--Papapetrou form\n\\begin{equation}\\label{3.1}\n\t{\\rm d} s^2 =-{\\rm e}^{2U}({\\rm d} t+a{\\rm d} \\phi)^2+{\\rm e}^{-2U}\n\t\\left({\\rm e}^{2k}({\\rm d} \\rho^2+{\\rm d} \\zeta^2)+\n\t\\rho^2{\\rm d} \\phi^2\\right)\n\t\\label{vac1}\n\\end{equation}\nwhere $\\rho$ and $\\zeta$ are Weyl's canonical coordinates and \n$\\partial_{t}$ and $\\partial_{\\phi}$ are the two commuting asymptotically\ntimelike respectively spacelike Killing vectors. \n\nIn this case the vacuum field equations are equivalent \nto the Ernst equation for the \ncomplex potential $f$ where $f={\\rm e}^{2U}+{\\rm i}b$, and where\nthe real function $b$ \nis related to the metric functions via\n\\begin{equation}\\label{3.2}\n\tb_{,z}=-\\frac{{\\rm i}}{\\rho}{\\rm e}^{4U}a_{,z}\n\t\\label{vac9}.\n\\end{equation}\nHere the complex variable $z$ stands for $z=\\rho+{\\rm i}\\zeta$. With these\nsettings, the Ernst equation reads\n\\begin{equation}\\label{3.3}\n\tf_{z\\bar{z}}+\\frac{1}{2(z+\\bar{z})}(f_{\\bar{z}}+f_z)=\\frac{2 }{f+\\bar{f}}\n\tf_z f_{\\bar{z}}\n \\label{vac10}\\enspace,\n\\end{equation}\nwhere a bar denotes complex conjugation in $\\bigBbb{C}$. With a solution $f$,\nthe metric function $U$ follows directly from the definition of the Ernst \npotential whereas $a$ can be obtained from (\\ref{vac9}) via quadratures. \nThe metric function $k$ can be calculated from the relation\n\\begin{equation}\\label{3.4}\n\t\tk_{,z} = 2\\rho \\left(U_{,z}\\right)^2-\\frac{1}{2\\rho}{\\rm e}^{4U}\n \\left(a_{,z}\\right)^2.\n\t\\label{vac8}\n\\end{equation}\nThe integrability condition of (\\ref{vac9}) and (\\ref{vac8}) is the \nErnst equation.\n\nThe remarkable feature of the Ernst equation is that it is completely \nintegrable. This means that it can be considered as the integrability \ncondition of an overdetermined linear differential system for a \nmatrix valued function $\\Phi$ that contains an additional variable, the so \ncalled spectral parameter $K$. The occurrence of the linear system with a \nspectral parameter is a consequence of the symmetry group of the Ernst \nequation, the Geroch group \\cite{geroch}. Several forms of the linear system \nare known \nin the literature (\\cite{belzak,maison,neuglinear}). They are related \nthrough gauge transformations (see \\cite{cosgrove}). The choice of a \nspecific form of the linear system is equivalent to a gauge fixing. We \nwill use the form of \\cite{neuglinear},\n\\alpheqn\n\\begin{eqnarray}\n\t\\Phi_{,z}(K,\\mu_0;z,\\bar{z})& = & \\left\\{\\left(\n\t\\begin{array}{cc}\n\t\tN & 0 \\\\\n\t\t0 & M\n\t\\end{array}\n\t\\right)\n +\\frac{K-{\\rm i}\\bar{z}}{\\mu_0(K)}\\left(\n\t\\begin{array}{cc}\n\t\t0 & N \\\\\n\t\tM & 0\n\t\\end{array}\n \\right)\\right\\}\\Phi(K,\\mu_0;z,\\bar{z})\\doteq W\\Phi\\enspace,\n\t\\label{lin1} \\\\\n\t\\Phi_{,\\bar{z}}(K,\\mu_0;z,\\bar{z}) & = & \\left\\{\\left(\n\t\\begin{array}{cc}\n\t\t\\bar{M} & 0 \\\\\n\t\t0 & \\bar{N}\n\t\\end{array}\n\t\\right)\n +\\frac{K+{\\rm i}z}{\\mu_0(K)}\\left(\n\t\\begin{array}{cc}\n\t\t0 & \\bar{M} \\\\\n\t\t\\bar{N} & 0\n\t\\end{array}\n \\right)\\right\\}\\Phi(K,\\mu_0;z,\\bar{z})\\doteq V\\Phi\n\t\\label{lin2}\n\\end{eqnarray}\n\\reseteqn\nwhere\n\\begin{equation}\\label{3.6}\n\tM = \\frac{f_z}{f+\\bar{f}}, \\quad\n\tN = \\frac{\\bar{f}_z}{f+\\bar{f}}.\n\\end{equation}\nObviously $M$ and $N$ depend only on the coordinates $z$ and $\\bar{z}$ \nand not on the spectral parameter $K$ that lives on the Riemann surface\n${\\cal L}(z,\\bar{z})={\\cal L}$ given by $\\mu_0^2(K)=(K-{\\rm i}\n\\bar{z})(K+{\\rm i}z)$. Notice that\n${\\cal L}$ is a Riemann surface of genus zero with coordinate dependent \nbranch points. This is a special feature of the family of chiral field \nequations to which the Ernst equation belongs that has no counterpart among \nthe completely integrable nonlinear evolution equations for which \nalgebro-geometric solutions have been constructed first.\n\nOn $\\cal L$ we have an involutive map $\\sigma$, defined by\n\\begin{equation}\\label{3.61}\n{\\cal L}\\ni P=(K,\\pm\\sqrt{(K-{\\rm i}\\bar{z})(K+{\\rm i}z)})\\to\\sigma(P)\\equiv\nP^{\\sigma}=(K,\\mp\\sqrt{(K-{\\rm i}\\bar{z})(K+{\\rm i}z)})\\in{\\cal L}\\enspace,\n\\end{equation}\nand an anti--holomorphic involution $\\tau$, defined by\n\\begin{equation}\\label{3.62}\n{\\cal L}\\ni P=(K,\\pm\\sqrt{(K-{\\rm i}\\bar{z})(K+{\\rm i}z)})\\to\\tau(P)\\equiv\n\\bar{P}=(\\bar{K},\\pm\\sqrt{(\\bar{K}-{\\rm i}\\bar{z})(\\bar{K}+\n{\\rm i}z)})\\in{\\cal L}\\enspace.\n\\end{equation}\n\nIt is possible to use the existence of the above linear system for the \nconstruction of solutions to the Ernst equation. To this end one \ninvestigates the singularity structure of the matrices $\\Phi_{z}\\Phi^{-1}$ and \n$\\Phi_{\\bar{z}}\\Phi^{-1}$ with respect to the spectral parameter and infers \na set of conditions for the matrix $\\Phi$ \n(at least twice differentiable with respect \nto $z$, $\\bar{z}$) that satisfies the linear system \n(\\ref{lin1}) and (\\ref{lin2}). \nThis is done (see e.g.~\\cite{korot1}) in\n\\begin{theorem}\\label{theorem2.1}\n Let $\\Phi(P)$ ($P\\in{\\cal L}$) be a $2\\times 2$--matrix with the following\nproperties:\\\\\nI. $\\Phi(P)$ is holomorphic and invertible at the branch\npoints $P_0=-{\\rm i}z$ and $\\bar{P}_0$ such that the logarithmic derivative\n$\\Phi_{z}\\Phi^{-1}$ diverges as $(K+{\\rm i}z)^{\\frac{1}{2}}$ at $P_0$ and \n$\\Phi_{\\bar{z}}\\Phi^{-1}$ as $(K-{\\rm i}\\bar{z})^{\\frac{1}{2}}$ at\n$\\bar{P}_0$.\\\\\nII. All singularities of $\\Phi$ on ${\\cal L}$ (poles, essential\nsingularities, zeros of the determinant of $\\Phi$, branch cuts and branch\n\tpoints) are regular which means that the logarithmic derivatives \n\t$\\Phi_{z}\\Phi^{-1}$ and $\\Phi_{\\bar{z}}\\Phi^{-1}$ are holomorphic in the \n neigbourhood of the singular points (this implies they have to be\n independent\n of $z$, $\\bar{z}$). In particular $\\Phi(P)$ should have\\\\\n\ta) regular singularities at the points $A_i\\in {\\cal L}$ ($i=1,\\dots,n$)\n which do not depend on $z$, $\\bar{z}$,\\\\\n\tb) regular essential singularities at the points $S_i$ \n ($i=1,\\dots,m$) which do not depend on $z$, $\\bar{z}$,\\\\\n c) boundary values at a set of (orientable, piecewise smooth)\n\tcontours $\\Gamma_i \\subset {\\cal L}$ ($i=1,\\dots,l$) \n independent of $z$, $\\bar{z}$, which are related on both sides of the\n contours via\n\t\\begin{equation}\\label{3.9}\n\\left.\\Phi_- (P)=\\Phi_+ (P) {\\cal G}_i(P)\\right|_{P\\in\\Gamma_i}\\enspace.\n \\label{lin6}\n\t\\end{equation}\n\twhere ${\\cal G}_i(P)$ are matrices independent of $z$, $\\bar{z}$ with \n H\\\"older--continuous components and non--vanishing determinant.\\\\\nIII. $\\Phi$ satisfies the reduction condition\n\t\\begin{equation}\\label{3.10}\n\t \\Phi(P^{\\sigma}) = \\sigma_3 \\Phi(P) \\gamma\\enspace,\n\t\t\\label{lin7}\n\t\\end{equation}\n\twhere $\\sigma_3$ is the third Pauli matrix, and where $\\gamma$ is an \n invertible matrix independent of $z$ and $\\bar{z}$.\\\\\n\tIV. The normalization and reality condition\n\\begin{equation}\n\t\\Phi(P=\\infty^+)=\\left(\n\t\\begin{array}{rr}\n\t\t\\bar{f} & 1 \\\\\n\t\tf & -1\n\t\\end{array}\n\t\\right)\n\t\\label{lin9}.\n\\end{equation}\n\nThen the function $f$ in (\\ref{lin9}) is a solution to the Ernst equation.\n\\end{theorem}\nA proof of this Theorem may be obtained by comparing the above matrix\n$\\Phi$ with the linear system (\\ref{lin1}) and (\\ref{lin2}).\n\\begin{proof}\nBecause of I, $\\Phi$ and $\\Phi^{-1}$ can be expanded in a series in\n$t=\\sqrt{K+iz}$ and $t'=\\sqrt{K-i\\bar{z}}$ in a neighbourhood of $P=P_0$\nand $P=\\bar{P}_0\\neq P_0$ respectively at all points $P_0$, $\\bar{P}_0$ which \ndo not belong to the singularities given in II. This implies that $\\Phi_z\n\\Phi^{-1}=\\alpha_0\/t+\\alpha_1+\\alpha_2t +\\cdots$. We recognize that, because of\nI and II, $\\Phi_z \\Phi^{-1}-\\alpha_0\/t$ is a holomorphic function. The\nnormalization condition IV implies that this quantity is bounded at \ninfinity. According to Liouvilles theorem, it is a constant. \nSince $\\Phi$, $\\Phi^{-1}$ and $\\Phi_z$ are single valued functions on ${\\cal \nL}$, they must be functions of $K$ and $\\mu_0$. Therefore we have $\\Phi_z\n\\Phi^{-1}=\\beta_0 \\sqrt{\\frac{K-\\bar{P}_0}{K-P_0}}+\\beta_1$. The matrix \n$\\beta_0$ must be \nindependent of $K$ and $\\mu$ since $\\Phi_z \\Phi^{-1}$ must have the same \nnumber of zeros and poles on ${\\cal L}$. The structure of the matrices \n$\\beta_0$ and $\\beta_1$ follows from III. From the normalization condition IV,\nit follows that $\\Phi_z \\Phi^{-1}$ has the structure of (\\ref{lin1}).\nThe corresponding equation for $\\Phi_{\\bar{z}}\\Phi^{-1}$ can be obtained \nin the same way.\n\\end{proof}\n\nFor a given Ernst potential $f$, the matrix $\\Phi$ in the above theorem is \nnot uniquely determined. This reflects the fact that the gauge is not uniquely \nfixed in the linear system (\\ref{lin1}) and (\\ref{lin2}). \nIf we choose without loss of generality \n$\\gamma=\\sigma_1$ (the first Pauli matrix), \nthe remaining gauge freedom can be seen from \n\\begin{corollary}\n Let $\\Phi(P)$ be a matrix subject to the conditions of\nTheorem~\\ref{theorem2.1},\nand $C(K)$ be a $2\\times2$-matrix that only depends on $K\\in \\bigBbb{C}$\nwith the properties\n\\begin{eqnarray}\n\tC(K) & = & \\alpha_1(K) \\hat{1}+\\alpha_2(K) \\sigma_1,\n\t\\nonumber \\\\\n\t\\alpha_1(\\infty) &= &1,\\quad \\alpha_2(\\infty)=0.\n\t\\label{c}\n\\end{eqnarray}\nThen the matrix $\\Phi'(P)=\\Phi(P) C(K)$ also satisfies the conditions of\nTheorem~\\ref{theorem2.1} and $\\Phi'(\\infty^+)=\\Phi(\\infty^+)$.\n\\end{corollary}\nIt is this gauge freedom to which we refer when we speak of the gauge \nfreedom of the linear system in the following.\n\nIt is interesting to note that the metric function $a$ can be obtained \nfrom a given matrix $\\Phi$ without solving the equation (\\ref{vac9}), see \n\\cite{korot1}. We get \n\\begin{proposition}\n Let $\\delta$ be a local parameter in the vicinity of $\\infty^-$. Then\n\t\\begin{equation}\n (a-a_0)e^{2U}={\\rm i}(\\Phi_{11}-\\Phi_{12})_{,\\delta}\\enspace,\n\t\t\\label{a}\n\t\\end{equation}\n\twhere $a_0$ is a constant that is fixed by the condition that $a=0$ on \n\tthe regular part of the axis and at spatial infinity, and where \n\t$\\Phi_{,\\delta}$ denotes the linear term in the expansion of $\\Phi$ in \n\t$\\delta$ divided by $\\delta$. \n\\end{proposition}\nThe proof follows from the linear system (\\ref{lin1}) and (\\ref{lin2}).\n\\begin{proof}\n\tIt is straightforward to check the relation \n\t\\begin{equation}\n\t\t(\\Phi^{-1}\\Phi_{,\\delta})_{,z}=\\Phi^{-1}(\\Phi_{z}\\Phi^{-1})_{,\\delta} \n\\Phi\n \\label{a1}\\enspace.\n\t\\end{equation}\nWith (\\ref{lin1}), we get \n\\begin{equation}\n\\left(\\Phi^{-1}\\Phi_{\\delta}\\right)_{21,z}=\\frac{{\\rm i}\\rho}{(f+\\bar{f})^2}\n (\\bar{f}-f)_z\\enspace,\n\t\\label{a2}\n\\end{equation}\nfrom which, together with (\\ref{vac9}), (\\ref{a}) follows.\n\\end{proof}\nNotice that $a_0$ is not gauge independent (in the sense of the above \ncorollary) whereas $a$ is.\n\nTheorem~\\ref{theorem2.1} can be used to construct solutions to the\nErnst equation by determining the structure and the singularities of \n$\\Phi$ in accordance with the conditions I--IV. For nonlinear evolution\nequations, large classes of solutions were constructed with the help of\nalgebro-geometric methods, in particular Riemann surface techniques. A \nkeypoint in this context is the occurrence of Riemann surfaces\nwhich are related to the linear system of the integrable equation under\nconsideration. In this paper we want to show how solutions for the Ernst \nequation\ncan be constructed by making use of the so called monodromy matrix\nof the Ernst system, which -- following~\\cite{krichever1} -- can be introduced\nas follows.\n\nFor a given linear system (\\ref{lin1}) and (\\ref{lin2}), we define the\nmonodromy matrix $L$ as a solution to the system\n\\begin{equation}\\label{3.19}\n L_z=[W,L],\\quad L_{\\bar{z}}=[V,L]\n \\label{mono3}\\enspace.\n\\end{equation}\nFor a known solution $\\Phi$ of (\\ref{lin1}) and (\\ref{lin2}), \n$L$ can be directly constructed in the form\n\\begin{equation}\\label{3.20}\n L(K)=-\\hat{\\mu}(K) \\Phi{\\cal C}\\Phi^{-1}\n \\label{mono5}\n\\end{equation}\nwhere ${\\cal C}$ is an arbitrary constant matrix with $\\det{\\cal C}=-1$ and\n$\\hat{\\mu}$ does not depend on the physical coordinates. Since $\\Phi$\nis analytic in $K$, there is a solution to (\\ref{mono3}) with the same \nproperties.\n\nIt follows from (\\ref{mono3}) that the coefficients of the characteristic \npolynomial $Q(\\mu, K)=\\det (L(K)-\\hat{\\mu}\\hat{1})$ are independent of the\ncoordinates. Without loss of generality we may assume $\\mbox{Tr}L(K)=0$. Then\n$L$ has the structure\n\\begin{equation}\\label{3.21}\n L=\\left(\n \\begin{array}{rr}\n A(K) & B(K) \\\\\n C(K) & -A(K)\n \\end{array}\n \\right)\n \\label{mono4}\\enspace.\n\\end{equation}\nThe equation $Q(\\hat{\\mu},K)=0$, i.e. \n\\begin{equation}\\label{3.21.1}\n \\hat{\\mu}^2=A^2+BC\\enspace,\n \\label{mono6}\n\\end{equation}\nis then the equation of an algebraic curve which \nin general will have infinite genus. We will restrict the analysis in the \nfollowing to the case of a regular curve with finite genus.\n\nIn this case, the Riemann surface $\\hat{{\\cal L}}$ is given by an \nequation of the form \n\\begin{equation}\\label{3.22}\n \\hat{\\mu}^2=\\prod_{i=1}^{g}(K-E_i)(K-F_i)\n \\label{mono20}\n\\end{equation}\nwhere $E_i$ and $F_i$\nare obviously independent of the physical coordinates. This equation represents\na two sheeted covering of the Riemann sphere and thus a four sheeted covering\nof the complex plane. A point $\\hat{P}\\in\\hat{\\cal L}$ can be given by\n$\\hat{P}=(K,\\mu_0(K),\\hat{\\mu}(K))$. The Hurwitz diagram of $\\hat{{\\cal L}}$ \nis shown in figure 1.\n\\begin{figure}[ht]\n\\begin{center}\n\\unitlength1cm\n\\begin{picture}(7,4)\n\\thicklines\n\\put(0.4,4){\\line(1,0){6.6}}\n\\put(0.4,3){\\line(1,0){6.6}}\n\\put(0.4,2){\\line(1,0){6.6}}\n\\put(0.4,1){\\line(1,0){6.6}}\n\\put(0.9,4){\\line(0,-1){2}}\n\\put(1.2,3){\\line(0,-1){2}}\n\\put(2.2,4){\\line(0,-1){2}}\n\\put(2.5,3){\\line(0,-1){2}}\n\\multiput(3.7,4)(0.8,0){2}{\\line(0,-1){1}}\n\\multiput(3.7,2)(0.8,0){2}{\\line(0,-1){1}}\n\\multiput(5.7,4)(0.8,0){2}{\\line(0,-1){1}}\n\\multiput(5.7,2)(0.8,0){2}{\\line(0,-1){1}}\n\\put(0.9,4){\\circle*{0.15}}\n\\put(0.9,2){\\circle*{0.15}}\n\\put(1.2,3){\\circle*{0.15}}\n\\put(1.2,1){\\circle*{0.15}}\n\\put(2.2,4){\\circle*{0.15}}\n\\put(2.2,2){\\circle*{0.15}}\n\\put(2.5,3){\\circle*{0.15}}\n\\put(2.5,1){\\circle*{0.15}}\n\\multiput(3.7,4)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(3.7,3)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(3.7,2)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(3.7,1)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,4)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,3)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,2)(0.8,0){2}{\\circle*{0.15}}\n\\multiput(5.7,1)(0.8,0){2}{\\circle*{0.15}}\n\\put(0,3.9){4}\n\\put(0,2.9){3}\n\\put(0,1.9){2}\n\\put(0,0.9){1}\n\\put(0.9,0.5){${\\rm i}\\bar{z}$}\n\\put(2,0.5){$-{\\rm i}z$}\n\\put(3.5,0.5){$E_1$}\n\\put(4.3,0.5){$F_1$}\n\\put(4.9,0.5){$\\cdots$}\n\\put(5.5,0.5){$E_g$}\n\\put(6.3,0.5){$F_g$}\n\\put(3.5,2.5){$E_1^\\sigma$}\n\\put(4.3,2.5){$F_1^\\sigma$}\n\\put(4.9,2.5){$\\cdots$}\n\\put(5.5,2.5){$E_g^\\sigma$}\n\\put(6.3,2.5){$F_g^\\sigma$}\n\\end{picture}\n\\end{center}\n\\caption{The Hurwitz diagram of $\\hat{\\cal L}$.}\n\\end{figure}\n\nThere is an automorphism $\\sigma$ of $\\hat{\\cal L}$ inherited from ${\\cal L}$\nwhich ensures $E_i^{\\sigma}=E_i$ and $F_i^{\\sigma}=F_i$. The orbit space\n${\\cal L}_H=\\hat{\\cal L}\/\\sigma$ is then, see~\\cite{algebro}, again a Riemann\nsurface, namely a hyperelliptic surface given by\n\\begin{equation}\\label{3.22a}\n\\mu_H^2=(K-{\\rm i}\\bar{z})(K+{\\rm i}z)\\prod_{i=1}^{g}(K-E_i)(K-F_i)\\enspace.\n\\end{equation}\nThus it is possible to construct components of the matrix $\\Phi$ on ${\\cal \nL}_H$ which makes it possible to use the powerful calculus of hyperelliptic \nRiemann surfaces. These functions may be lifted to $\\hat{{\\cal L}}$. \nAs we will show in the following, it is possible to construct a \nmatrix $\\Phi$ on ${\\cal L}$ in accordance with the conditions of Theorem 2.1 \nby projecting onto this surface. \n\n\n\\section{Hyperelliptic solutions of the Ernst equation}\n\\setcounter{equation}{0}\n\n\\subsection{Theta functions asscociated with a Riemann surface and the\nRiemann--Hilbert problem}\n\nIn this section, we want to give an explicit construction of the matrix \n$\\Phi$ in accordance with Theorem 2.1. Condition II can be used to \nconstruct solutions by \nprescribing the poles, essential singularities and cuts of $\\Phi$ \nwhich is equivalent to the solution of \na generalized Riemann--Hilbert problem for the matrix $\\Phi$. The \ninvestigation of such matrix Riemann--Hilbert problem turns out to be \nrather difficult and is not yet fully done (in general it can be merely \nreduced to the solution of a linear integral equation, see e.g.\\ \n\\cite{musk}). Therefore we will use here a different approach. The \noccurrence of the monodromy matrix suggests that it might be possible to \nconstruct a matrix $\\Phi$ on the Riemann surface $\\hat{{\\cal L}}$ of the \nprevious section. The additional freedom we thus gain is used to restrict \nthe problem to a scalar one, namely to a Riemann--Hilbert problem for one \ncomponent of $\\Phi$ on the hyperelliptic surface ${\\cal L}_H$ obtained \nfrom $\\hat{{\\cal L}}$ by factorizing with respect to the involution \n$\\sigma$. We impose the reality condition $E_i, F_i\\in \\bigBbb{R}$ or \n$E_i=\\bar{F}_i$ on the branch points in order to satisfy the reality \ncondition of Theorem 2.1. \nThen we construct the whole matrix $\\Phi$ in accordance with \nthis theorem. In fact it was shown in \\cite{prd} that all matrix \nRiemann--Hilbert problems with rational jump data are gauge equivalent \nto scalar problems on a suitably chosen hyperelliptic surface. Thus the \nlimitation to the scalar case is only a comparatively weak restriction \nwhich allows, as we will show below, for an explicit solution of the \nproblem in terms of theta functions. \n\nFor the moment, we fix the physical coordinates $z$ and \n$\\bar{z}$ in a way that $\\rho\\neq0$ and that $-{\\rm i}z$ and ${\\rm i}\\bar{z}$\ndo not coincide with the singular points of \n$\\Phi$ in order to ensure that the first condition of Theorem 2.1 is valid.\nIn the next section we study the dependence of the found solution on\n$z$ and $\\bar{z}$.\nIn order to give the solution to this special case of the generalized\nRiemann--Hilbert problem, we use the theory of theta functions associated to\na Riemann surface (see \\cite{dubrovin}) and the solution of the Riemann--Hilbert problem on a Riemann\nsurface, as given in~\\cite{zverovich1}.\nAs we will need only hyperelliptic Riemann surfaces\nof the form (\\ref{3.22a}), we restrict ourselves to this case.\n\nLet us denote by $(a_1,\\dots,a_g,b_1,\\dots,b_g)$ a basis of the\nfirst (integral) homology group $H_1({{\\cal L}_H})$ of ${{\\cal L}_H}$ (see \nthe picture below) where the\ncuts are either between real branch points (which are ordered \n$E_{k+1}From the properties of the theta function, we also find that $\\psi(P)$ has\n$g$ simple poles at the points $P_1,\\dots,P_g$ and $g$ simple \nzeros. Additional poles, zeros and essential singularities can be obtained by a\nsuitable choice of\nAbelian integrals of the second kind (essential singularities) \nand third kind (zeros\nand poles). We remark that the assumption $\\bar{\\Omega}(P)=\\Omega(\\bar{P})$ had\nto be introduced in order to satisfy the reality condition of Theorem 2.1. \n\\end{proof}\n\\begin{remark}\nWithout loss of generality we can choose $D$ to consist only of branch\npoints since $D$ gives the poles of $\\Psi$ due to the zeros of the theta\nfunction in the denominator. This can always\nbe compensated by a suitable choice of the zeros and poles of $\\Psi$\nwhich arise from the integrals of the third kind in\n$\\Omega$. All $P_i\\in D$ shall have multiplicity 1 and be chosen in a way\nthat $\\Theta\\left[\\alpha\\atop\\beta\\right](x)$ with\n$\\left[\\alpha\\atop\\beta\\right] =\\omega(D)+\\omega(\\bar{P}_0)+K_R$ \nhas the same reality\nproperties as the Riemann theta function $\\Theta(x)$.\n\\end{remark}\n\nOur next aim is to define a matrix\nvalued function $\\Phi(P)$ on ${\\cal L}$, satisfying the conditions of\ntheorem~\\ref{theorem2.1}, with the help of the above solution to the scalar\nRiemann--Hilbert problem on the hyperelliptic surface ${\\cal L}_H$. \nTo this end we define a further function on ${\\cal L}_H$ by\n\\begin{equation}\n\\chi(P_H)=\\chi_0\\frac{\\Theta(\\omega(P_H)+u-\\omega(\\bar{P}_0)\n-\\omega(D)-K_R)}{\\Theta(\\omega(P_H)-\\omega(D)-K_R)}\\exp\\left(\n\\frac{1}{2\\pi {\\rm i}}\\int_{\\Gamma}^{}\\ln G{\\rm d}\\omega_{P_H P_0}\\right)\n\t\\label{gauge17},\n\\end{equation}\nwhere $\\chi_0$ is again a normalization constant. \nIt can be easily seen that the analytic behaviour of $\\chi(P_H)$ is\nidentical to that of $\\psi(P_H)$, except that it changes the sign at every\n$a$-cut. $\\chi$ is thus not a single valued function on ${\\cal L}_H$.\nHowever, it is single valued on $\\hat{{\\cal L}}$ which can be\nviewed as two copies of ${\\cal L}_H$ cut along \n$\\left[P_0,\\bar{P}_0\\right]$ and glued together along this cut. We define \nthe vector $X$ on $\\hat{{\\cal L}}$ by fixing the sign in front of $\\chi$ in the \nvicinity of the points $P_0^{\\pm}=(K_0,0,\\pm\\hat{\\mu}(K_0))\\in \\hat{{\\cal L}}$,\n\\begin{equation}\n\tX(\\hat{P})=\\left(\n\t\\begin{array}{c}\n\t\t\\psi(\\hat{P}) \\\\\n\t\t\\pm \\chi(\\hat{P})\n\t\\end{array}\\right), \\quad \\hat{P}\\sim P_0^{\\pm}.\n\t\\label{gauge18}\n\\end{equation}\nWith the help of this vector, we can construct the matrix $\\Phi$ on ${\\cal \nL}$ via\n\\begin{equation}\n\t\\Phi(P)=(X(K,\\mu_0(K),+\\hat{\\mu}(K)),X(K,\\mu_0(K),-\\hat{\\mu}(K)))\n\t\\label{gauge19}\n\\end{equation}\nwhere the signs are again fixed in the vicinity of $P_0^{\\pm}$. Notice \nthat this matrix consists of eigenvectors of the monodromy matrix, \n$LX(K,\\mu_0(K),\\pm\\hat{\\mu}(K))=\\hat{\\mu} X(K,\\mu_0(K),\\pm\\hat{\\mu}(K))$ \nif $L$ is written as $L=\\hat{\\mu}\\Phi \\gamma \n\\Phi^{-1}$ where $\\gamma$ is the matrix from~\\ref{theorem2.1}.\n\nIt may be readily checked that this ansatz is in accordance with the reduction\ncondition (\\ref{lin7}) (this is in fact the reason why one has to define \nthe function $\\chi$ in the way (\\ref{gauge17})). The behaviour at the\nsingularities is as required in condition II: For the contour $\\Gamma$ and the\nsingularities of the Abelian integrals $\\Omega$, this is obvious. At the\nbranch points $E_i$ and $F_i$, one gets the following behaviour: at \npoints $P_i$ of the divisor $D$, the components of $\\Phi$ have a simple pole, and \nthe determinant diverges as $(K-P_i)^{-\\frac{1}{2}}$, if this branch point \nis not a singularity of an integral of the third kind in $\\Omega$ or lies \non the contour $\\Gamma$. If the same condition holds at \nthe remaining branch points, the components are regular there but the \ndeterminant vanishes as $(K-P_i)^{\\frac{1}{2}}$. If the branch points \ncoincide with one of the singularities of the integrals in the exponent in\n(\\ref{8.9}), this merely changes the singular behaviour of $\\Phi$ and its \ndeterminant there. Condition II of theorem~\\ref{theorem2.1} is however \nobviously satisfied.\n\nSince $\\Phi$ in (\\ref{gauge19}) is only a function of $P$, it will not be\nregular at the cuts $\\left[E_i,F_i\\right]$. At the $a$-cuts around non-real\nbranch points, we get $\\Phi^-=\\Phi^+\\sigma_1|_{a_i}$, whereas we have\n$\\Phi^-|_{a_i}=-\\Phi^+|_{a_i}\\sigma_2$ at\nthe $a$-cuts around real branch points. The logarithmic derivatives of $\\Phi$\nwith respect to $z$ and \n$\\bar{z}$ are however holomorphic at all these points. One can recognize that \nthe behaviour at the non-real branch\npoints is related to a gauge transformation of the form (\\ref{c}). This \nmeans that one can find a gauge transformed matrix $\\Phi'$ that is \ncompletely regular at these points if the integrals in the exponent are \nregular there. With \n\\begin{equation}\n\t\\alpha_1=\\frac{1}{2}(1+\\lambda), \\quad \\alpha_2=\\frac{1}{2}(1-\\lambda)\n\t\\label{cc}\n\\end{equation} \nand $\\lambda=\\prod_{i=1}^{g}\\sqrt{\\frac{K-\\bar{P}_i}{K-P_i}}$ where\n$D=\\sum_{i=1}^{g}P_i$, this may be checked by direct calculation. The real\nbranch points, however, cannot be related to gauge transformations.\n\nNormalizing $\\psi$ and $\\chi$ (if possible)\nin a way that $\\psi(\\infty^-_H)=1$ and $\\chi(\\infty^-_H)=-1$, one can see \nthat $\\Phi$ is then in accordance with all conditions of Theorem 1 since \nthe reality condition follows from the reality properties of the theta \nfunctions and the Riemann--Hilbert problem. The fact that $\\Phi$ is at least\ndifferentiable with respect to $z$ and $\\bar{z}$ at points where $P_0$ \ndoes not coincide with the singularities of the integrals in the exponent \nor the remaining branch points of ${\\cal L}_H$ follows from the modular \nproperties of the theta function. Let the paths between \n$\\left[P_0,\\infty^-\\right]$ and \n$\\left[P_0,\\infty^+\\right]$ be the same in \nall integrals and let them have the same projection into the complex plane \n(i.e.\\ one is the involuted of the other).\nThen the results may be summarized in\n\\begin{theorem}\\label{theorem3.2}\nLet $\\Theta\\left[\\alpha\\atop\\beta\\right](\\omega(\\infty^-)+u)\\neq0$. Then the function\t\n\t\\begin{equation}\\label{8.16}\n\tf(z,\\bar{z})=\\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](\\omega(\\infty^{+})+u+b)}{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right]\n\t(\\omega(\\infty^{-})+u+b)}\n\t\\exp\\left\\{\\Omega(\\infty^{+})-\\Omega(\\infty^{-})+\n\t\\frac{1}{2\\pi{\\rm i}}\\int\\limits_\\Gamma\n\t\\ln G(\\tau){\\rm d}\\omega_{\\infty^{+}\\infty^-}(\\tau)\n\t\\right\\}\n\t\\end{equation}\n\tis a solution to the Ernst equation. \n\\end{theorem}\n\\begin{remarks}\n\\item In the case $g=0$ the Ernst potential (\\ref{8.16}) is real, $f={\\rm\ne}^{2U}$. This means\nthat $U$ is a solution to the axisymmetric Laplace equation and belongs \ntherefore to the\nWeyl-class. For $g>0$, there are no real solutions other than $f=1$ which\ndescribes Minkowski space.\n\\item The multi-black-hole solutions which can be obtained via B\\\"acklund\ntransformations (see e.g.\\ \\cite{hkx}) are contained in the class \n(\\ref{8.16}) as the limiting case that the branch points $E_i$ and $F_i$ \ncoincide pairwise. In this limit, all branch points become double points \nand the theta functions break down to purely algebraic functions. Notice \nthat the analysis of $f$ at the branch points in the following section \nalways assumes a regular surface. The obtained results for the regularity \nof $f$ do not hold in this limit. \n\\end{remarks}\nThe above explicit construction of the solutions makes it possible to\nderive useful formulae for the metric function $a$ and the derivatives of the\nErnst potential. Let $\\int_{P_H}^{P_H+\\delta}{\\rm d}\\omega_i= \ng_i\\delta+o(\\delta)$ where\n$\\delta$ is the local parameter in the vicinity of $P_H\\in {\\cal L}_H$. We\ndefine the derivative\n\\begin{equation}\n\tD_{P_H}\\Theta(x)=\\sum_{i=1}^{g} g_i\\partial_{x_i}\\Theta(x).\n\t\\label{local}\n\\end{equation}\nUsing (\\ref{a}) and (\\ref{8.9}), (\\ref{gauge17}), we get \n\\begin{equation}\n\t(a-a_0) {\\rm e}^{2U}={\\rm i}D_{\\infty^-}\\ln \n\t\\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right]\\left(\\int_{\\bar{P}_0}^{\\infty^- }d\\omega+u+b\\right)}{\n\t\\Theta\\left[\\alpha\n\t\\atop\\beta\\right]\\left(\\int_{P_0}^{\\infty^-}d\\omega+u+b\\right)}\n\t\\label{fay6}.\n\\end{equation}\n>From the linear system (\\ref{lin1}) and (\\ref{lin2}), we obtain with\n(\\ref{8.9}) and (\\ref{gauge17})\n\\begin{eqnarray}\n\t\\frac{\\bar{f}_z}{f+\\bar{f}}&=&\\frac{{\\rm i}}{2\\sqrt{P_0-\\bar{P}_0}}\n\t\\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b-\\omega(\\bar{P}_0))\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-))}{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-)-\\omega(\\bar{P}_0))}\\times\\nonumber\\\\\n\t&&\\left(D_{P_0}\\ln \\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u-\\omega(\\bar{P}_0))+I_{P_0}\\right)\n\t\\label{fay7}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\t\\frac{f_z}{f+\\bar{f}}&=&\\frac{{\\rm i}}{2\\sqrt{P_0-\\bar{P}_0}}\n \\frac{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-)-\\omega(\\bar{P}_0))}{\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b-\\omega(\\bar{P}_0))\\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u+b+\n\t\\omega(\\infty^-))}\\times\\nonumber\\\\\n\t&&\\left(D_{P_0}\\ln \\Theta\\left[\\alpha\n\t\\atop\\beta\\right](u-\\omega(\\bar{P}_0))+I_{P_0}\\right)\n\t\\label{fay8},\n\\end{eqnarray}\nwhere $I_{P_0}$ is the linear term of the expansion of the integrals \nin the exponent of (\\ref{8.9}) in the local parameter around $P_0$.\n\n\n\\subsection{Finite gap solutions and Picard--Fuchs equations}\n\nThe original finite gap solutions of~\\cite{korot1} are those among\n(\\ref{8.16}) without the contour integral (in our notation\nonly an arbitrary linear combination of Abelian integrals of the second and\nthird kind $\\Omega$). They just correspond to the so called\nBaker--Akhiezer function (see \\cite{algebro}) for the Ernst system. This\nfunction that has essential singularities and poles gives the periodic or\nquasiperiodic solutions to the integrable nonlinear evolution equations. \nThere the essential singularity is uniquely determined by the structure of \nthe differential equation. In contrast to these equations, the solutions \n(\\ref{8.16}) are in general neither periodic nor quasiperiodic, and the \nessential singularity can be nearly arbitrarily chosen. The form of the \nsolution to the Riemann--Hilbert problem shows that one might even think\nof ``putting the singularities densely on a line and integrate over the\nintegrals with some measure\": an Abelian integral $\\Omega_p$ of the second\nkind with a pole of first order at $p$ can be used as an analogue to the Cauchy\nkernel. A contour integral over this kernel with some measure,\n$\\int_{\\Gamma}^{}\\ln G \\Omega_p{\\rm d} p$, is thus\njust another way to write down the solution to a Riemann--Hilbert problem \non a Riemann surface. \n\nIn studying the boundary value problem for the rigidly rotating disk of dust,\nMeinel and Neugebauer~\\cite{meinelneugebauer} observed that it is possible to\nobtain solutions to the Ernst equations via\n\\begin{equation}\nf=\\exp\\left(\\sum_{m=1}^{g}\\int_{E_m}^{C_m}\\frac{K^g{\\rm d} K}{\\mu_H}-I_g\\right)\n\\label{mn1}\n\\end{equation}\nwhere the divisor $C=\\sum_{m=1}^{g}C_m$ is determined by\n\\begin{equation}\n\t\\sum_{m=1}^{g}\\int_{E_m}^{C_m}\\frac{K^i {\\rm d} K}{\\mu_H}=I_i\n\t\\label{mn2}\n\\end{equation}\n($i=0,1,\\dots,g-1$), i.e.\\ as the solution of a Jacobi inversion problem. The\n$I_i$ are (in the absence of real branch points)\nreal solutions to the axisymmetric Laplace equation which satisfy the \nrecursive condition,\n\\begin{equation}\n {\\rm i} I_{n+1,z}=zI_{n,z}+\\frac{1}{2}I_n\\enspace.\n \\label{8.18}\n\\end{equation}\nThe relation to the class obtained in theorem~\\ref{theorem3.2} is the following:\nThe integral of the third kind in\n(\\ref{mn1}) can be expressed by the help of a formula in~\\cite{stahl}\nvia theta functions. Equation (\\ref{mn2}) ensures that the resulting \nexpression is independent of the chosen integration path which is shown in \nthe proof of the theorem. Thus the $u_i$ (obtained from the $I_i$ by \nnormalization) are as in our case the $b$-periods\nof the integral $I_g$ in the exponent. In fact, it was shown in\n\\cite{meinelneugebauer} that one of these periods, say $I_1$, can be chosen \nas an arbitrary solution to the axisymmetric Laplace equation. The other \nperiods as well as the integral in the exponent then follow from \ndifferential identities plus boundary conditions.\n\nThe underlying reason for this fact is that the \nErnst potential $f$ is studied on a family of Riemann surfaces parametrized\nby the moving branch points $-{\\rm i}z$ and ${\\rm i}\\bar{z}$. The periods \non this surface (i.e.~integrals along closed curves) are\nsubject to differential identities, the so called Picard--Fuchs equations. \nIt is a general feature of the periods of rational functions \n\\cite{griffiths,morrisson,foucault} that \nthey satisfy a differential system of finite order with Fuchsian \nsingularities. An elegant way to find the Picard--Fuchs system explicitly is\nvia the notion of the Manin connection in the bundle $H^1_{\\rm\nDR}(\\Sigma_g)\\to\\Sigma_g$, see~\\cite{manin1}. The investigation turns out to\nbe particularly simple if one uses the following standard form of the\n(hyperelliptic) Riemann surface $\\Sigma_g$ (all hyperelliptic surfaces of \ngenus $g$ are conformally equivalent to this standard form)\n\\begin{equation\ny^2=(x-z)\\prod\\limits_{i=1}^{2g} (x-E_i)\\doteq (x-z)P(x)\n=(x-z)\\sum_{j=0}^{2g}a_jx^j\\enspace,\n\\end{equation}\nwhere the $E_i$ do not depend on $z$. Using $j_0={\\rm d} x\/y$, \n$j_1=xj_0,\\dots,j_{2g-1}=x^{2g-1}j_0$ as the basis for the de Rham \ncohomology $H^1_{\\rm DR}(\\Sigma_g)$ we\nobtain for the matrix $M_n^m$ ($m,n=0,\\dots,2g-1$) of the Manin connection\n(defined by $\\frac{\\partial j_n}{\\partial z}=M^m_nj_m$)\n\\begin{equation\nM_n^m=\n\\left\\{\n\\renewcommand{\\arraystretch}{1.4}\n\\begin{array}{cl}\n\\displaystyle \\frac{z^n}{2P(z)}\\left((m+1)a_{m+1}+\nz^{-m-1}\\,\\sum_{j=0}^ma_j z^j\\right)& \\mbox{for $0\\leq m1$ with the cut system in use and $P_0=E_g+x$, \n\twhere $x$ is chosen as in the \n\tcase of the real branch points, the differentials of the \n\tfirst kind have a smooth expansion in $x$ and $\\bar{x}$. In contrast to \n\tthe case of real branch points, the coefficients in the expansion are \n\t$\\phi$--independent. The differentials\n\t${\\rm d}\\omega_{i}$ become in leading order the differentials of the first kind \n\t${\\rm d}\\omega''_i$ on $\\Sigma''$. The differential \n\t${\\rm d}\\omega_{g-1}$ becomes in the limit the differential \n\t$-{\\rm d}\\omega''_{E_g^+E_g^-}$, and similar for ${\\rm d}\\omega_g$ at $F_g$. \n\tThe differential of the third kind becomes \n\t${\\rm d}\\omega_{\\infty^+\\infty^-}=\n\t{\\rm d}\\omega_{\\infty^+\\infty^-}''$. All these differentials have \n\tcoefficients in the $x$ and $\\bar{x}$ expansion that contain Abelian \n\tintegrals of the second kind with poles in $E_g^{\\pm}$ and $F_g^{\\pm}$ as \n\tmay be checked by direct calculation. This implies for the \n\t$b$--periods that $\\pi_{ij}=\\pi''_{ij}$ for $i,j=1,...,g-2$ and \n\t\\begin{eqnarray}\n\t\t\\pi_{(g-1)(g-1)} & = & \\pi_{gg}=2\\ln\\delta +...,\n\t\t\\nonumber \\\\\n\t\t\\pi_{i(g-1)} & = & -2\\omega''(E_g^+),\n\t\t\\nonumber \\\\\n\t\t\\pi_{ig} & = & -2\\omega''(F_g^+),\n\t\t\\label{com1}\n\t\\end{eqnarray} \n\twhereas $\\pi_{(g-1)g}$ is finite in the limit $\\delta\\to0$.\n\tIf $E_g\\notin \\Gamma$, the $u_i$ as well as the Cauchy integral in the \n\texponent have a smooth expansion in $x$ and $\\bar{x}$ with finite \n\tcoefficients. The theorem of \\cite{mzh} then guarantees regularity if the\n\tlimiting value that may be calculated as on the axis exists. \n\tThe theta function \n\ton ${\\cal L}_H$ breaks down to a sum of four theta functions on \n\t$\\Sigma''$ times a multiplicative factor. If $E_g\\in\\Gamma$, \n\tthe coefficients in the expansion of $f$ in $x$ and $\\bar{x}$ will diverge \n\twhich implies that $f$ is possibly \n\tnot differentiable there though the limiting \n\tvalue exists if (\\ref{reg}) holds.\n\\end{proof}\n\n\n\\section{Metric functions and ergospheres}\n\nIn the previous sections we have made extensive use of the \ncomplete integrability of the Ernst equation to \nconstruct a large class of solutions. To discuss physical \nfeatures of the resulting spacetimes however, it would be helpful to have \nexpressions in closed form not only for the Ernst potential but for the \nmetric functions, at least for the functions ${\\rm e}^{2U}$ and $a$ that \ncan be expressed invariantly via the Killing vectors. It is a remarkable \nfact already noticed by Korotkin \\cite{korot1} that the metric function \n$a$ can be related to derivatives of the matrix $\\Phi$ without solving \nthe differential equation (\\ref{vac9}). In the following we will show that \na theta identity of Fay \\cite{fay} can be used to go one step \nfurther to obtain a formula for $a$ that is free of derivatives. The same \nidentity leads to a simplified expression for the metric function ${\\rm e}^{2U}$ \nthat can be directly used to identify ergospheres in the spacetime.\n\nFay's trisecant identity establishes a relation between four points\n$A_1,\\dots,A_4$ on a\nRiemann surface, in our case ${\\cal L}_H$, in arbitrary position (see e.g.\\ \n\\cite{mumford}, \\cite{taimanov}). Let $x$ be an arbitrary $g$-dimensional\nvector. Then the following identity holds,\n\\begin{eqnarray}\n & &\\Theta(x)\\Theta\\left(x+\\int_{A_1}^{A_3}{\\rm d}\\omega+\n \\int_{A_2}^{A_4}{\\rm d}\\omega\\right)\n -\\exp \\left(\\Omega_{A_1A_4}|^{A_3}_{A_2}\\right)\n \\Theta\\left(x+\\int_{A_2}^{A_3}{\\rm d}\\omega\\right)\n \\Theta\\left(x+\\int_{A_1}^{A_4}{\\rm d}\\omega\\right)\n\t\\nonumber \\\\\n &&- \\exp\\left(\\Omega_{A_2A_4}|^{A_3}_{A_1}\\right)\n \\Theta\\left(x+\\int_{A_1}^{A_3}{\\rm d}\\omega\\right)\n \\Theta\\left(x+\\int_{A_2}^{A_4}{\\rm d}\\omega\\right) = 0\\enspace,\n\t\\label{fay1a}\n\\end{eqnarray}\nwhere e.g.~$\\Omega_{A_1A_4}|^{A_3}_{A_2}$ denotes the integral of a normalized\ndifferential of the third kind with simple poles at $A_1$ respectively $A_4$\nwith residues $+1$ respectively $-1$ along a path from $A_2$ to $A_3$. For a\ngeometric interpretation of this identity in terms of the Kummer variety see\n\\cite{mumford}, \\cite{taimanov}, for an interpretation via generalized cross\nratio functions see \\cite{farkas}. The strength of the above identity arises\nfrom the fact that it holds for points $A_i$ in general position. By a \nsuitable choice of these points, we obtain for the metric function ${\\rm \ne}^{2U}$, the real part of the Ernst potential, \n\\begin{equation}\n\te^{2U}=\\frac{1}{2}\\exp \\left(\\Omega_{\\bar{P}_0\\infty^-}|^{P_0}_{\\infty^+}\n\t\\right)\n\t\\frac{\\Theta\\left[\\alpha\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\\atop\\beta\\right]\n\t(u+b+\\omega(\\bar{P}_0))}{\\Theta\\left[\\alpha\\atop\\beta\\right]\n\t(u+b+\\omega(\\infty^-)+\\omega(\\bar{P}_0))\n\t\\Theta\\left[\\alpha\\atop\\beta\\right](u+b+\\omega(\\infty^-))} e^{I}\n\t\\label{fay14}\n\\end{equation}\nwhere $I$ denotes the integral in the exponent of (\\ref{8.16}).\nThis formula makes it possible to identify directly the zeros of ${\\rm \ne}^{2U}$ which give the ergospheres, the limiting surfaces of stationarity \n(inside these surfaces there can be no observer at rest with respect to \nspatial infinity). Since the exponent of \nthe integral of the third kind in (\\ref{fay14}) in front of the fraction \ncannot vanish, the necessary condition for ergospheres is \n\\begin{equation}\n\t\\Theta\\left[\\alpha\\atop\\beta\\right](u+b)\\Theta\\left[\\alpha\\atop\\beta\\right]\n\t(u+b+\\omega(\\bar{P}_0))=0\n\t\\label{ergo}.\n\\end{equation}\nDefining the divisor $A$ as the solution of the Jacobi inversion problem\n\\begin{equation}\n\t\\omega(A)-\\omega(D)=u+b\n\t\\label{ergo1},\n\\end{equation}\nwe find that an ergosphere can occur if $P_0$ or $\\bar{P}_0$ are in $A$. It\nis however possible that the denominator of (\\ref{fay14}) vanishes at the \nsame time which would imply a violation of (\\ref{reg}) (and thus a \nsingularity of the spacetime). Summing up we get\n\\begin{proposition}\n\tI. Let $P_0$ or $\\bar{P}_0$ and $\\infty^-$ be in $A$ for some $P_0$, \n\tthen condition \n\t(\\ref{reg}) is violated and the Ernst potential is singular\n\tat these points.\\\\\n\tII. Let $P_0$ or $\\bar{P}_0$ but not $\\infty^-$ be in $A$ for some $P_0$, \n\tthen the real part of the Ernst potential vanishes at these points which \n\t describe an ergosphere.\n\\end{proposition}\n\nThe same formula can be used on the axis where we obtain for the metric \nfunction ${\\rm e}^{2U}$ in the notation of (\\ref{sing7})\n\\begin{eqnarray}\n\t{\\rm \n e}^{2U}&=&\\frac{1}{2}\\exp\\left(\n \\Omega_{\\infty^-\\zeta^+}|^{\\infty^+}_{\\zeta^-}\n\t\\right)\\nonumber\\\\\n &&\\frac{\\Theta'\\left[\\alpha' \\atop \\beta'\\right]^2(u'+b')}{\\Theta'\n \\left\n [\\alpha' \\atop \\beta'\\right]^2(\\omega'|^{\\infty^-}_{\\zeta^-}+u'+b')-\n \\exp(2(\\omega_g(\\infty^-)+u_g+b_g))\\Theta'\\left[\\alpha' \\atop\n \\beta'\\right]^2\n (\\omega'|^{\\infty^-}_{\\zeta^+}+u'+b')}\\nonumber\\\\\n\t\\label{ergo2}.\n\\end{eqnarray}\nThe condition for an ergosphere to hit the axis is then\n\\begin{equation}\n\\Theta'\\left[\\alpha'\\atop\\beta'\\right](u'+b')=0\\enspace,\n\t\\label{ergo3}\n\\end{equation}\nsince the integral of the third kind in the exponent in front of the fraction\ncannot diverge for finite values of $\\zeta$. The interesting feature of \nthis relation is that it is completely independent of the physical \ncoordinates. This implies that if an ergosphere extends to the axis, this \nwill be only possible if the metric function ${\\rm e}^{2U}$ vanishes on the\nwhole axis. In the case of the Kerr solution, the ergosphere touches the \naxis at the horizon. An interpretation of the fact \nthat the whole axis would be singular in the present case is given in the \nnext section where the above case is related to the ultrarelativistic limit \nin which the source of the gravitational field becomes so strong that it\nvanishes behind the horizon of the extreme Kerr metric.\n\nThe metric function $a$ can be calculated from (\\ref{a}) if one uses the \ntrisecant identity (\\ref{fay1a}) in the limit that two points coincide. \nUsing the trisecant identity several times, we get\n\\begin{eqnarray}\n (a-a_0){\\rm e}^{2U}&=&-\\rho\\left(\n \\frac{\\Theta\\left[\\alpha\\atop\\beta\\right](0)\n \\Theta\\left[\\alpha\\atop\\beta\\right]\n \\left(\\int_{\\bar{P}_0}^{P_0}{\\rm d}\\omega\\right)}{\n \\Theta\\left[\\alpha\\atop\\beta\\right]\\left(\n \\int_{\\infty^-}^{P_0}{\\rm d}\\omega\\right)\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n \\left(\\int_{\\bar{P}_0}^{\\infty^-}{\\rm d}\\omega\\right)}\\times\n \\right.\\nonumber\\\\\n\t&&\\left.\\frac{\\Theta\\left[\\alpha\\atop\\beta\\right](u+b)\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n \\left(u+b+\\int_{P_0}^{\\infty^-}{\\rm d}\\omega+\n \\int_{\\bar{P}_0}^{\\infty^-}{\\rm d}\\omega\\right)}{\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n \\left(u+b+\\int_{\\bar{P}_0}^{\\infty^-}{\\rm d}\\omega\\right)\n\t\\Theta\\left[\\alpha\\atop\\beta\\right]\n \\left(u+b+\\int_{P_0}^{\\infty^-}{\\rm d}\\omega\\right)}-1\\right)\n \\label{fay7a}\\enspace.\n\\end{eqnarray}\nThe constant $a_0$ can be obtained in a similar way from the condition \nthat $a=0$ on the regular part of the axis (we assume here that the \nsingularities in the exponent of (\\ref{8.16}) are situated in a compact \nregion of the $(\\rho,\\zeta)$--plane). Care has to be taken in the above \nformula that some of the terms in brackets explode as $1\/\\rho$ in the \nlimit $\\rho\\to0$. We get\n\\begin{equation}\n\ta_0=\\frac{{\\rm i}}{2}(D-\\bar{D})\\frac{\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n\t(u'+b'+\\int_{\\infty^+}^{\\infty^-}d\\omega')\n\t\\Theta'\\left[\\alpha'\\atop\\beta'\\right](0)}{\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n\t(\\int_{\\infty^+}^{\\infty^-}d\\omega')\n\t\\Theta'\\left[\\alpha'\\atop\\beta'\\right](u'+b')}e^{-I'}\n\t\\label{fay18b}.\n\\end{equation}\nIt can be seen from this formula that $a_0$ does not vanish if there \nare no singularities in the exponent ($I=u=b=0$) in which case $f=1$ which \ndescribes Minkowski spacetime. This reflects, as already noted, the fact \nthat $a_0$ is a gauge dependent quantity. The metric function $a$ however \nis gauge independent. In the above example of Minkowski spacetime, it will \nof course vanish in the used asymptotically non-rotating coordinates.\n\n\n\\section{Asymptotic behaviour and equatorial symmetry}\n\nSince we are mainly interested in solutions to the Ernst equation that could\ndescribe the gravitational field outside a compact matter source, we will study\nthe asymptotic behaviour (near spatial infinity) of the solutions (\\ref{8.16}).\nIt is generally believed\nthat the Ernst potentials of the corresponding spacetimes are regular except\nat the contour $\\Gamma_z$ in \nthe $(\\rho,\\zeta)$--plane which corresponds to the surface of the body, \nasymptotically flat and equatorially symmetric. We will investigate in the \nfollowing whether it is possible to identify solutions with these properties in \nthe class (\\ref{8.16}).\n\n\n\\paragraph{Asymptotic behaviour}\n\nAsymptotic flatness implies that the Ernst potential is of the form\n$f=1-2m\/|z|+o(1\/|z|)$ for $|z| \\to \\infty$ where $m$ is a positive real\nconstant. A complex $m$ is related to a so called NUT-parameter that is\ncomparable to a magnetic monopole.\n\nThe asymptotic properties of the solutions (\\ref{8.16}) can be read off at the \naxis. Notice that the ${\\rm d}\\omega_i'$ are independent of $\\zeta$. \nFor ${\\rm d}\\omega_g$, we get\n\\begin{equation}\n\t{\\rm d}\\omega_g={\\rm d}\\omega'_{\\infty^+\\infty^-}\\left(1-\\frac{1}{2\\zeta}\n\t\\sum_{i=1}^{g}(E_i+F_i)\\right)+\\frac{1}{\\zeta}{\\rm d}\\omega'_{\\infty^+,1}\n\t+o(1\/\\zeta)\n\t\\label{sing9}\n\\end{equation}\nwhere ${\\rm d}\\omega'_{\\infty^+,1}$ is the differential of the second kind \nwith a pole of second order at $\\infty^+$. Furthermore it can be seen \nthat $\\exp(-\\omega_g(\\infty^+))$ is proportional to $1\/\\zeta$ for $\\zeta\\to\n\\infty$. Thus we get\n\\begin{proposition}\n Let $\\lim_{\\tau\\to \\infty}\\tau\\ln G(\\tau)=0$ \n on all contours that go through\n\t$\\infty^+$ or $\\infty^-$ and let $\\Theta'\\left[\\alpha'\\atop\\beta'\\right]\n \\left(u'+b'\\right)\\neq0$. Then $f$ has the form $f=1-2m\/\\zeta$ for \n\t$\\zeta\\to\\infty$ where $m$ is a complex constant. \n\\end{proposition}\nThe proof of this proposition follows from (\\ref{sing9}) and (\\ref{sing7}).\n\n\n\\paragraph{Equatorial symmetry}\n\nThe fact that the mass is in general complex implies that the class we are \nconsidering here is too large if one wants to study only solutions that are\nasymptotically flat in the strong sense ($m$ real). There is the belief that\nstationary axisymmetric spactimes describing isolated bodies in thermodynamical\nequilibrium are equatorially symmetric. This implies for the Ernst potential\n$f(-\\zeta)=\\bar{f}(\\zeta)$. Solutions with this property always have a real\nmass due to the symmetry. It is therefore of special interest to single out\nequatorially symmetric solutions among those in (\\ref{8.16}). We get\n\\begin{theorem}\nLet ${\\cal L}_H$ be a hyperelliptic surface of the form (\\ref{3.22a}) with \neven genus $g=2s$ and the property $\\mu(-K,-\\zeta)=\\mu(K,\\zeta)$.\t\nLet $\\Gamma$ be a piecewise smooth contour on ${\\cal L}_H$ such that \nwith $P=(K,\\mu(K))\\in \\Gamma$ also $\\bar{P}\\in\n\\Gamma$ and $(-K,\\mu(K))\\in \\Gamma$. Let there be \ngiven a finite nonzero function $G$ on \n$\\Gamma$ subject to $G(\\bar{P})=\\bar{G}(P)=G((-K,\\mu(K)))$. If $(p,\\mu(p))$\nis a singularity of $\\Omega$, the same should hold for $(-p,\\mu(-p))$. \nChoose a cut sytem in a way that the cuts $a_i^1$ ($i=1,\\ldots ,s$) encircle \n$\\left[-F_i,-E_i\\right]$ and $a_i^2$ encircle\n$\\left[E_i,F_i\\right]$ in the $+$--sheet (in the case of real branch \npoints, the points are ordered in the way $E_i