diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjbom" "b/data_all_eng_slimpj/shuffled/split2/finalzzjbom" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjbom" @@ -0,0 +1,5 @@ +{"text":"\\section{\\label{sec:intro}INTRODUCTION}\n\nIndividual qubits with the help of which we attempt to \nbuild our would-be quantum computers are fragile because \nwe must have full control over the states of each of them \nseparately and of all of them collectively within any time window. \nIn many proposals qubits are held in traps---electrical, magnetic, \nor optical. Controlling their states in traps over time and \ngetting information from them during calculations can \ndecohere and unstabilize them. The less energy \ntransferred by probes, usually flying qubits, to a qubit \nsitting in a trap, the better. It would be the best to have no \nenergy transfer at all. Quantum mechanics provides a way to \ndo this. Whenever we {\\em fail} to detect a quantum system in \none of the paths it would take to interfere with itself, we \n``erase'' the interference fringes we would have got, if we \nhad not carried out any detection or \nmeasurement at all \\cite{renninger,dicke,my-PhD-int-f-c-osa}.\nDicke called such a measurement an {\\em interaction-free \nmeasurement\\\/} \\cite{dicke} and Elitzur and Vaidman realized that such \nmeasurements might be useful whenever we do not want to \ntransfer energy to a measured object \\cite{elitzur-vaidman}.\nThe measurements are also called \n{\\em energy-exchange-free\\\/} \\cite{pav-pla-96}, \n{\\em absorption-free\\\/} \\cite{mitch-mass-01}, and \n{\\em counterfactual\\\/} \\cite{mitch-jozsa-01} measurements, as well as \n{\\em quantum interrogations\\\/} \\cite{gilc02}. \n\nTwo implementations of interaction-free measurements in \nquantum computation have recently been proposed. One, proposed \nby Richard Jozsa \\cite{jozsa-99}, considers atoms as quantum \ncomputers (or parts of them) and flying qubits \n(photons) as their switches. Thus one of the two possible \nstates of an atom resulting from a ``computation'' can be read \n``for free,'' with no transfer of energy to the atom, i.e., \nwithout the ``computer'' actually running. Unfortunately, it \nturns out that the other state cannot be obtained for \nfree \\cite{mitch-jozsa-01}. \nThe controlled-NOT (CNOT) gate used in this approach has \nphoton states as its control qubits and output register \nstates as its target qubits. \n\nThe other proposal considers interaction-free measurements\nthat are used to implement essential parts of \nquantum computations, notably CNOT gates and \nentanglement \\cite{azuma03,azuma04,gilc02,horod01,methot01}.\nCNOT gates are essential for quantum computing because they \nenable us to set up any quantum gate and therefore to implement \nany available algorithm. Entanglement appears in almost \nall blueprints of quantum computer candidates. \n\nIn constructing an interaction-free CNOT gate, Hiroo \nAzuma \\cite{azuma03} uses a positron as a straight moving \ncontrol qubit that blocks ($N$ times, where $N\\to\\infty$) \nthe paths of a zig-zagging electron---a \ntarget qubit. The CNOT gate is constructed not directly \nbut using Bell-basis measurements by the method of\nGottesman-Chuang \\cite{gottesman-99}. \nIn the M\\'{e}thot-Wicker method, the qubits of the \nCNOT gate are two-level atoms, and photons only pass \ninformation between atoms \\cite{methot01}. \nGilchrist, White, and Munro \\cite{gilc02} do use atoms as \ncontrol qubits and photons as target qubits in a quantum \nZeno setup, but to obtain a gate which could be called a \n{\\em destructive} CNOT gate, \nsince one of the target qubits must be destroyed and \ntherefore one of the four outputs of the standard CNOT \ngate is not available. \n\nIn this paper we propose a {\\em nondestructive} CNOT gate in \nwhich a two-level atom in a trap is the control qubit and a \nphoton interrogating it---without transferring a single \nquantum of energy to it---is the target qubit. \n``Nondestructive'' means that all four modes of the gate \nare available.\\ \\cite{zhao-cnot05} The proposal is an \nelaboration of the CNOT-gate setup put forward in \nRef.\\ \\cite[p.\\ 166]{pavicic-book-05}. \nA particular feature of the gate is that the target qubit can also \n{\\it physically control\\\/} its control qubit by preventing the \nlatter from entering into a superposition of its two \navailable states. We carry out this {\\em interaction-free} \ninterrogation with the help of a photon resonator, and we \nprepare the atom by {\\em stimulated Raman adiabatic \npassage}, STIRAP. \n\nWe organize the paper as follows. In Sections\n\\ref{sec:resonator} and \\ref{sec:stirap} \nwe briefly present those details of interaction-free and \nSTIRAP experiments (respectively) that are indispensable \nfor understanding the construction of our CNOT gate. In \nSec.~\\ref{sec:i-f-cnot} we present the interaction-free \nCNOT gate itself, and in Sec.~\\ref{sec:suppr-sup} an \ninteraction-free control of superposition. The conclusion \nof the paper is given in Sec.~\\ref{sec:concl}. \n\n\\section{\\label{sec:resonator}THE RESONATOR}\nLet us consider the setup proposed by Paul and \nPavi\\v{c}i\\'{c} in 1996 \\cite{ppijtp96,pav-pla-96,p-p-josab97} \nand shown in Fig.~\\ref{fig:pellin}. The predictions were\nconfirmed in an actual experiment carried out by Tsegaye, Goobar, \nKarlsson, Bj\\\"{o}rk, Loh, and Lim in 1998 \\cite{bjork-karlsson-98}.\n\nWe make use of a resonator which consists of two perfect mirrors \nand two highly asymmetrical mirrors that determine photon \nround trips as shown in Fig.~\\ref{fig:pellin}. Other setups that \nminimize reflection losses are also possible \n\\cite{ppijtp96,p-p-josab97}. A laser beam enters the resonator \nthrough highly asymmetrical beam splitter ABS. \nWhen there is no object in the resonator, an incoming laser\nbeam is almost completely transmitted into detector \\it D\\rm$_t$. \nWhen there is an object, the beam is almost totally \nreflected into detector \\it D\\rm$_r$. \nTo increase efficiency, frustrated \ntotal reflection (which is an optical version of quantum \nmechanical tunnelling) can be used \ninstead \\cite{ppijtp96,p-p-josab97,pav-evanston}.\nThus the reflectivity $R$ can reach 0.9999 or higher. \nThe uniqueness of the reflectivity at the beam splitters and \nperfect mirrors is assured by choosing the orientation of the \npolarization of the incoming laser beam perpendicular to the \nplane of incidence. The source of the incoming beam should be a\ncontinuous wave (cw) laser (e.g., Nd:YAG), because\nof its coherence length (up to 300$\\>$km) and because of its \nvery narrow linewidth (down to 10$\\>$kHz in the visible range).\n\n\\begin{figure}\n\\includegraphics[width=0.99\\textwidth]{pavicic-pra-1-07-fig1.eps}\n\\caption{Schematic of an interaction-free device \naccording to Ref.~\\cite{pav-evanston}. A single photon \nenters the resonator. M's are perfect mirrors \n(total-reflection Pellin--Broca prisms can be substituted \nfor M's for higher efficiency \\cite{pav-evanston}). ABS's \nare highly asymmetrical mirrors with $R=0,999$ or higher \n(with frustrated total reflection, \ni.e., optical tunnelling \\cite{pav-evanston}); \n(a) When there is no object in its path\nthe photon exits into \\it D\\rm$_t$ (with a realistic \nefficiency of over 98\\%); (b) When there is an object in \nits path, the photon is reflected into \\it D\\rm$_r$.}\n\\label{fig:pellin}\n\\end{figure}\n\nEach subsequent round trip contributes to a geometric \nprogression whose infinite sum in the plane wave approach \nyields the total amplitude of the \nreflected beam: \n\\begin{eqnarray}\nB=-A\\sqrt{R}{1-e^{i\\psi}\\over1-R\\,e^{i\\psi}}\\,,\n\\label{eq:total}\n\\end{eqnarray}\nwhere $\\psi=(\\omega-\\omega_{res})T$ is the phase added by the\nround trip, $\\omega$ is the frequency of the incoming beam, \n$T$ is the round-trip time, and $\\omega_{res}$ is the resonance \nfrequency corresponding to a wavelength which satisfies \n$\\lambda\/2=L\/j$, where $L$ is the round-trip length of the cavity \nand $j$ is an integer. \nWe see that, in the long run, for any $R<1$ \nand $\\omega=\\omega_{res}$ we get no reflection at all---i.e., no \nresponse from $D_r$---if nothing obstructs the \nround trip, and almost a perfect reflection when the object blocks the \nround trip and $R$ is close to one. In terms of single photons \n(obtained by attenuating the intensity of the laser until \nthe chance of having more than one photon at a time becomes \nnegligible) the probability of detector $D_r$ reacting when \nthere is no object in the system is zero. A response from $D_r$ indicates \nan interaction-free detection of an object in the system. The \nprobability of the response is $R$, the probability of making the \nobject absorb the photon $R(1-R)$, and the probability of a photon \nexiting into $D_t$ detector $(1-R)^2$. \n\n \nDetailed wave packet calculations based on classical optical \ninterference (analogous to the calculations for laser resonators) \nare carried out in Refs.~\\cite{p-p-josab97,ppfphy98,pav-evanston} \nand they yield the following efficiencies of the suppression \nof the reflection ($r$) into $D_r$ and of the throughput ($t$)\ninto $D_t$ when there is no object in the resonator:\n\\begin{eqnarray}\nr=(1-R)(1-\\rho^2 R)\\,\\Phi, \\qquad\\qquad\nt=(1-R)^2\\,\\Phi\n\\,,\\label{eq:tau}\n\\end{eqnarray}\nwhere $\\rho\\le1$ is a measure of overall losses and \n\\begin{eqnarray}\n\\Phi=\n{\\displaystyle\\int_0^\\infty{\\displaystyle\\exp[-\n{\\cal T}^2(\\omega-\n\\omega_{res})^2\/2]d\\omega\\over\\displaystyle1-\n2\\rho R \\cos[(\\omega-\n\\omega_{res})T]+\\rho^2R^2}\\over\\displaystyle\\int_0^\\infty\n\\exp[-{\\cal T}^2(\\omega-\\omega_{res})^2]d\\omega}\n\\,,\\label{eq:phi}\n\\end{eqnarray}\nwhere ${\\cal T}$ is the coherence time and $T$\nthe round-trip time.\n\nIn effect, the resonator has to be ``charged'' to yield a \nsuperposition, i.e., we have to allow the beam a sufficient \nnumber of round-trips to build up a destructive or \nconstructive interference even when it contains just one photon. \nThis corresponds to the sum which we used to obtain \nEq.~(\\ref{eq:total}) and it is shown in Fig.~\\ref{fig:round-trip}.\n \n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{pavicic-pra-1-07-fig2.eps}\n\\caption{$r$ as a function of ${\\cal T}\/T$ \nfor $\\rho=0.99$ and \n$0.9\\le R \\le 1$: \n${\\cal T}\/T=500$ (top), 150, 50, 20, and 10 (bottom). \nThe differences in the shapes stem from the amount of \nlosses.}\n\\label{fig:round-trip}\n\\end{figure}\n\nNow the difference between the classical and the quantum \npicture of interference lies in the statistical behaviour\nof the flying quantum system---the photon. The classical \napproach does not permit an interaction-free detection, because \nthere is always an exchange of energy, e.g.~$\\hbar\\omega\/100$. \nIn the quantum approach, there can be no exchange of energy \nsmaller than a quantum of energy $\\hbar\\omega$ corresponding to a \nsingle photon, and therefore only in the long run and on average \nthe quantum energy transferred to an object does equal the \nclassical energy. \n\nTherefore we cannot narrow down the time window so as to make \nthe coherence time less than the time required for \ninterference to build up (at least 100 round trips). \nIf we did so, a photon could not enter the resonator whether or \nnot the object was in the photon path. As a consequence, \ndownconverted photons (the signal to enter the resonator \nand the idler to control the event) are not suitable sources \nof photons, because their coherence time is too short (in the \nrange of picoseconds). The use of a cw laser or a low emission \nLED and their inability to control the number of photons within \nthe time window do not, however, pose a problem to our interaction-free \nCNOT gate, because we have two distinct outgoing ports, and \nbecause the time required for a sufficient number of round-trips \nis a few nanoseconds, which is short enough for use \nin quantum computation, where the decoherence typically ranges \nbetween nanoseconds and seconds. \n\n\\section{\\label{sec:stirap}DARK STATES AND SUPERPOSITIONS}\n\nThe purpose of interaction-free detection of a macroscopic \nobject is to wipe out photon interference fringes so as \nto put the object in a photon path. In the case of atoms we \ndo not physically block the photon path but make them opaque \nor transparent by bringing them into states in which \nthey can or cannot absorb a photon of a chosen frequency. \nThis also means that if an atom can be in a superposition \nof such two states, its interaction-free interrogation by \na photon will prevent it from entering the superposition. \nIn this section we present a setup which can be used to \nmake an atom (in)visible to a photon in a resonator and \nto build up a CNOT gate.\n\nLet us consider the rubidium isotope $^{87}$Rb \\cite{yu-pra04}.\n(We can use many other atoms and ions that enable $\\Lambda$ \nscheme presented below; e.g., $^{40}$Ca$^+$ that we discuss in \nthe next section.) \nIt has the closed shells $nl=1s, 2s, 2p, 3s, 3p, 3d, 4s$, $4p$, \nand one electron in the $5s$ shell, which is pushed below the $4d$ \nand $4f$ shells by spin--orbit interaction. Thus $^{87}$Rb\nbehaves like a system with one electron in the $5s$ ground \nstate. The total angular momentum is given by {\\bf J=L+S}. \nFor the ground state $5s$, \nwe have $s=1\/2$ and $l=0$ and therefore $j=1\/2$. The first \nexcited states are the $5p$ states $5p_{1\/2}$ and \n$5p_{3\/2}$, corresponding to $s=1\/2$, $l=1$, $j=3\/2$, and \n$j=5\/2$, respectively. They are separated by the \nspin--orbit interaction ${\\mathbf L}\\cdot{\\mathbf S}$. \nWe will consider only $j=3\/2$. \n\nThe total nuclear angular momentum {\\bf K} combines with \n{\\bf J} to give the total angular momentum of the atom: \n${\\mathbf F}={\\mathbf J}+{\\mathbf K}$. \n$^{87}$Rb has $K=3\/2$, and its $j=1\/2$ \nground states are split by hyperfine interaction \ninto doublets with $F=K\\pm j=3\/2\\pm 1\/2=2,1$. \nNow we apply an external \nmagnetic field {\\bf B} to the atom to split the levels \ninto magnetic Zeeman sublevels \nwith magnetic quantum numbers $m=-F,-F+1,\\dots,F$. \nThe levels are given in Fig.~\\ref{fig:rubidium} \n(cf.~Ref.~\\cite{87rb}).\n\n\\begin{figure}\n\\includegraphics[width=0.99\\textwidth]{pavicic-pra-1-07-fig3.eps}\n\\caption{(a) STIRAP $|g_1\\rangle\\leftrightarrow|g_2\\rangle$\nin which the population of $|e\\rangle$ is completely\navoided is obtained by subsequent application of\ntwo laser beams of frequencies $\\omega_2$ and $\\omega_1$,\ndetuned by amount $\\Delta$. The $\\omega_2$ one is called\nthe {\\em Stokes} beam and it corresponds to\nRabi frequency $\\Omega_2$. The $\\omega_1$ one is called\nthe {\\em pump} beam (with Rabi frequency $\\Omega_1$);\n(b) Two pump beams ($\\Omega_1$,\n$\\Omega_2$) and a cavity with atom--cavity coupling ($G$)\ninstead of the Stokes laser beams produce superposition\n $\\alpha|g_1\\rangle +\\beta|g_2\\rangle$.}\n\\label{fig:rubidium}\n\\end{figure}\n\nTo excite and deexcite electrons between \n$m=\\pm 1$ and $m=0$ we must use circularly polarized\nphotons with angular momentum $j_p=1$ and \ntwo additional degrees of freedom (eigenvalues of \n${\\mathbf k}\\cdot{\\bf j}_p\/k$) denoted $m_{j_p}=\\pm 1$ \n\\cite{messiah}. Linearly polarized photons \ncannot be used because the selection \nrules require $\\Delta m=0$ for them.\n\nWhen an atom absorbs a circularly polarized photon, it \nabsorbs its energy and receives its angular \nmomentum in its transition from the ground state to \nthe excited state, and therefore the following selection \nrules must be met: \n\\begin{eqnarray}\n\\Delta l=\\pm1, \\qquad \n\\Delta m=m_{j_p}=\\pm 1.\\quad\n\\label{eq:select-rules}\n\\end{eqnarray}\nWhen a photon is emitted, the same selection rules must \nbe observed. Thus for $\\Delta m=\\pm 1$ we get a circularly\npolarized photon and for $\\Delta m=0$ a linearly \npolarized photon. \n\n\nBy solving the Schr\\\"{o}dinger equation for our three-level system\n\\begin{eqnarray}\n\\hat H|\\Psi\\rangle=i\\hbar\\frac{\\partial|\\Psi\\rangle}{\\partial t}\\,,\n\\label{eq:sch}\n\\end{eqnarray} \nwe arrive (after starting with a more general Hamiltonian, doing some \napproximations, and re-introducing an intermediate form of \nthe wave function) at the following Hamiltonian\n\\begin{eqnarray}\n\\hat H=\\frac{\\hbar}{2}\\left[ \\begin{array}{ccc}\n 0 & \\Omega_1(t) & 0 \\\\\n\\Omega_1(t) & 2\\Delta & \\Omega_2(t) \\\\ \n 0 & \\Omega_2(t) & 0 \\\\ \n \\end{array}\n \\right],\\quad\n\\label{eq:3lev-Ham}\n\\end{eqnarray}\nwhere Rabi frequencies $\\Omega_1$ and $\\Omega_2$ \n(coefficients of the general solution to \nEq.~(\\ref{eq:sch})) \ncorrespond to two pump laser \nbeams of frequencies $\\omega_1$ and $\\omega_2$ that are \ndetuned from resonance for $\\Delta=\\omega_{eg_1}-\\omega_1=\n\\omega_{eg_2}-\\omega_2$. Hamiltonian (\\ref{eq:3lev-Ham})\nhas three eigenstates that are linear combinations of \n$|g_1\\rangle$, $|g_2\\rangle$, and \n$|e\\rangle$. One of them is \\cite{kuk-hioe-bergm-89}: \n\\begin{eqnarray}\n|\\Psi^0\\rangle=\\frac{1}{\\sqrt{\\Omega_1^2(t)+\\Omega_2^2(t)}}\n\\Big(\\Omega_2(t)|g_1\\rangle-\\Omega_1(t)|g_2\\rangle\\Big)\\,.\n\\label{eq:psi-0}\n\\end{eqnarray}\nWe see that this state is completely independent of the \nintermediate state $|e\\rangle$ and that its \neigenvalue---being zero---is independent of the Rabi \nfrequencies $\\Omega_1$ and $\\Omega_2$. We call states \n$|g_1\\rangle$ and $|g_2\\rangle$ {\\em dark states}.\nExperimentally, we would obtain complete population \ntransfer: \n \\begin{eqnarray}\n\\left|\\langle g_1|\\Psi^0\\rangle\\right|^2=1\\quad{\\rm for}\\quad \nt\\to -\\infty,\\qquad\\ \n\\left|\\langle g_2|\\Psi^0\\rangle\\right|^2=1\\quad{\\rm for}\\quad \nt\\to +\\infty,\\quad\n\\label{eq:popul-transfer-prob}\n\\end{eqnarray}\nif we assumed $\\left.\\frac{\\Omega_1(t)}{\\Omega_2(t)}\n\\right|_{t\\to -\\infty}\\to 0 \\ {\\rm and} \\ \n\\left.\\frac{\\Omega_2(t)}{\\Omega_1(t)}\\right|_{t\\to +\\infty}\\to 0$, \nand this corresponds to switching on and off the \nsecond laser before switching on and off the first one. When the \ntransfer $|g_1\\rangle\\to|g_2\\rangle$ is {\\em adiabatic} \n(the laser beams are gradually switched on and off; for the \nadiabaticity criteria see Ref.~\\cite{berg-rmp98}), \nthe system prepared in $|\\Psi^0\\rangle$ remains in this state \nat all times and the process is called STIRAP (Stimulated Raman \nAdiabatic Passage) \\cite{berg-rmp98}. The first laser beam \n(historically called the {\\em pump beam}) is right-hand circularly \npolarized, denoted $\\sigma^+$ \n(because the transition $|g_1\\rangle\\to|e\\rangle$ requires it) \nand the second beam (historically called the {\\em Stokes beam})\nis left-hand circularly polarized, \ndenoted $\\sigma^-$ (for $|e\\rangle\\to|g_2\\rangle$).\n\nIn a quantum computer, control of single states $|g_1\\rangle$ \nand $|g_2\\rangle$ is less important than control of their \nsuperposition $\\alpha|g_1\\rangle+\\beta|g_2\\rangle$. We can \nobtain a superposition by carrying out two simultaneous \nSTIRAPs to $|g_1\\rangle$ and $|g_2\\rangle$ from a common \nthird one $|g_3\\rangle$ as shown in Fig.~\\ref{fig:rubidium}. \nMany such designs for controlling and transferring\nsuperpositions have been proposed and implemented \nrecently \\cite{pellizzari-97,bose-vedral-prl99,yu-pra04}.\nCavities are often used instead of the second (Stokes) laser \nbeams in each STIRAP \\cite{yu-pra04} and we consider \nsuch a design. \n\nIn Fig.~\\ref{fig:rubidium} (b) a schematic is given of a \nstrongly coupled atom-cavity system where the cavity is \ntuned (by shifting the mirrors) to the same frequency the \nStokes beam would have for each transition. The cavity stimulates \nthe population of levels $g_1$ and $g_2$ in the same way the \nStokes laser fields would, and therefore the whole process \nis characterized by the following Hamiltonian: \n\\begin{eqnarray}\n\\hat H=\\frac{\\hbar}{2}\\left[ \\begin{array}{ccc}\n 0 & \\Omega_i & 0 \\\\\n\\Omega_i & 2\\Delta & 2G \\\\ \n 0 & 2G & 0 \\\\ \n \\end{array}\n \\right],\\quad\n\\label{eq:cav-Ham}\n\\end{eqnarray}\nwhere $i=1,2$ and \n$G=\\sqrt{\\hbar\\omega\/(2\\varepsilon_0 V_{\\rm cavity})}$ \nis the {\\em atom-cavity coupling constant} ($V_{\\rm cavity}$ \nis the cavity mode volume). The photon which supports \nthe cavity modes and the population of $g_1$ and $g_1$ \nlevels eventually leaks from the cavity. \n\nThe state of the atom coupled to the cavity photon state is: \n\\begin{eqnarray} \n|\\Psi(t)\\rangle&=&\\frac{\\alpha}{\\sqrt{4G^2+\\Omega_1(t)}}\n(2G|g_3,\\emptyset\\rangle-\\Omega_1(t)|g_1,R\\rangle)\\nonumber\\\\ \n&&+\\ \\ \\frac{\\beta}{\\sqrt{4G^2+\\Omega_2(t)}}\n(2G|g_3,\\emptyset\\rangle-\\Omega_2(t)|g_2,L\\rangle).\\qquad\\qquad\n\\label{eq:two-dark-states-b}\n\\end{eqnarray} \nThus at the beginning of the STIRAP process, the system \nis in state $|g_3,\\emptyset\\rangle$, where $|\\emptyset\\rangle$ \nmeans that there is no cavity photon coupled to $|g_s\\rangle$. \nAs $\\Omega_1$ and $\\Omega_2$ gradually increase, the system \nadiabatically evolves to state\n\\begin{eqnarray} \n|\\Psi(t)\\rangle=\\alpha|g_1,R\\rangle+\n\\beta|g_2,L\\rangle\\,,\n\\label{eq:stirap-state-e-p}\n\\end{eqnarray}\nand when the photon in the state $|R\\rangle+|L\\rangle$\nleaves the cavity, the atom state {\\em jumps} \n\\cite{plenio-knight-rmp98} into the required superposition:\n\\begin{eqnarray} \n|\\Psi\\rangle=\\alpha|g_1\\rangle+\\beta|g_2\\rangle\\,.\n\\label{eq:stirap-state-e}\n\\end{eqnarray}\nThe jump is probabilistic and has a success probability \nof 50\\%.\n\n\\section{\\label{sec:i-f-cnot}INTERACTION-FREE CNOT GATE}\n\nIn this section we show how the resonator described in \nSec.~\\ref{sec:resonator} can be used to construct an \ninteraction-free CNOT gate and then in Sec.~\\ref{sec:suppr-sup}\nwe discuss how to control and suppress an atom superposition \nobtained during quantum computation. \n\nTo construct an interaction-free CNOT gate, we substitute an \natom, for example $^{87}$Rb of Sec.~\\ref{sec:stirap}, \nfor the object in our resonator in Fig.~\\ref{fig:pellin}. \nThe $^{87}$Rb atom will be transparent to properly \npolarized photons of specific frequency when there is \nno electron in the ground level that a photon \ncould excite to a higher level (a photon will not ``see''\nthe atom), and nontransparent when there is an electron \npopulating the ground level. \n\nIn Sec.~\\ref{sec:stirap} we saw that a left-hand \ncircularly polarized photon can excite an atom from \nits ground state $|g_1\\rangle$ ($5s_{1\/2},\\ F=1,\\ m=-1$) to \nits excited state $|e\\rangle$ ($5p_{1\/2},\\ F=2,\\ m=0$), and that \nthe right-hand circularly polarized photon can excite the atom \nfrom $|g_2\\rangle$ ($5s_{1\/2},\\ F=1,\\ m=+1$) to $|e\\rangle$ \n($5p_{1\/2},\\ F=2,\\ m=0$). So an $L$-photon will ``see'' \nthe atom in $|g_1\\rangle$ but will not ``see'' it when it is \nin $|g_2\\rangle$. With an $R$-photon, the opposite is true. \nThe energy differences to the detuned excited level \nare the same, so both photons have the same frequency \n(as the ``G part'' of Fig.\\ \\ref{fig:rubidium}$\\>$(b): \n$|g_1\\rangle\\to\\Delta$ and $|g_2\\rangle\\to\\Delta$). \nWe can induce a change of the atom from $|g_1\\rangle$ to \n$|g_2\\rangle$ and back by a STIRAP process, with two additional \nexternal laser beams, as explained in the previous section. \n\nTo build our CNOT gate we use the resonator we introduced \nin Sec.\\ \\ref{sec:resonator}. When a photon does not ``see'' \nthe atom, its laser beam will interfere with itself in a resonator \nso that classical and quantum descriptions of photons give the same \nresult \\cite{carmichael-book,mandel-wolf-book} as we already stressed\nin Sec.\\ \\ref{sec:resonator}. On the other hand, when we say \nthat a photon ``sees'' an atom, that means that the atom would \nhave absorbed the photon if it had come to it---in reality it \ncannot come to it because it is being reflected from the entrance \nto the resonator (see Fig.~\\ref{fig:pellin}). To show that \nthe atom in $|g_1\\rangle$ ($|g_1\\rangle$) is realistically \nopaque for $L$ ($R$) circularly polarized photons, we must resort \nto quantum theory and we shall come back to this point in the \nsecond half of this section. \n\nThis feature of our resonator approach that there are no \nphoton loops in it when the atom is opaque is yet another \nadvantage over a Zeno-like setup. We have to carry out quantum \ncalculations only of a single absorption of a photon by the atom \nwhich reduces to a single strong atom-photon interaction while in \na Zeno setup each photon loop inside a cavity with an opaque \natom includes a strong photon-atom interaction. \nCf.\\ quantum calculations for a Zeno-like atom-photon \ninteraction-free setup carried out by Luis and \nS{\\'a}nchez-Soto.\\ \\cite{luis-98, luis-99,luis-99a}\n \nWe feed our resonator with $+45^\\circ$ and $-45^\\circ$ linearly \npolarized photons to achieve the same conditions for round trips \nof both kinds of photons within the resonator. To the right \nof the atom we place a quarter-wave plate (QWP) to turn \na $45^\\circ$-photon into an $R$-photon and a $-45^\\circ$-photon \ninto an $L$-photon. To the left of the atom we place a half-wave \nplate (HWP) to change the direction of the circular polarization \nand then another QWP to transform the polarization back into \nthe original linear polarization. We denote the atom states as \nfollows:\n\\begin{eqnarray} \n|0\\rangle=|g_1\\rangle, \\qquad |1\\rangle=|g_2\\rangle,\n\\label{eq:a-states-int-f-a}\n\\end{eqnarray} \nand take these atom states as control states \nand the atom itself to be our control qubit. \nWe denote the photon states as follows:\n\\begin{eqnarray} \n|0\\rangle=|45^\\circ\\rangle, \\qquad |1\\rangle=|-45^\\circ\\rangle,\n\\label{eq:a-states-int-f-f}\n\\end{eqnarray} \nand we take these photon states as the target states and the photons\nas target qubits. For example, $|01\\rangle$ means that the atom is \nin state $|g_1\\rangle$ and the photon is polarized along $-45^\\circ$. \n\n\\begin{figure}\n\\includegraphics[width=0.99\\textwidth]{pavicic-pra-1-07-fig4.eps}\n\\caption{Interaction-free CNOT. (a) The atom is in state \n$|g_1\\rangle$ and can absorb a $-45^\\circ$ polarized photon. \nTherefore a photon in state $|1\\rangle$ cannot enter the cavity\nand thus $|0\\rangle\\to|0\\rangle$ and $|1\\rangle\\to|1\\rangle$. \n(b) The atom is in state $|g_2\\rangle$ and can absorb a \n$+45^\\circ$ polarized photon. Therefore photon $|0\\rangle$ \ncannot enter the cavity and thus $|0\\rangle\\to|1\\rangle$ \nand $|1\\rangle\\to|0\\rangle$. ABS are highly asymmetrical beam \nsplitters with $R=0.999$; SBS is a symmetric 50:50 beam splitter; \nM are perfect mirrors; PBS is a polarizing beam \nsplitter which lets $|0\\rangle$ photons through and reflects \n$|1\\rangle$ photons; HWP and QWP are half- and quarter-wave plates, \nrespectively---the plates in the resonator turn linear polarization \ninto circular and back into linear and HWP after PBS turns \n$|0\\rangle$ ($|1\\rangle$) photon into $|1\\rangle$ ($|0\\rangle$) \nphoton; D is a detector---ideally, when it does not click, \nthe target qubit exits at the other side of SMS.} \n\\label{fig:int-f-cnot} \n\\end{figure}\n\nNow consider Fig.~\\ref{fig:int-f-cnot}$\\>$(a). A photon in \nstate $|0\\rangle$ does not ``see'' the atom in state \n$|g_1\\rangle$ and will therefore exit the resonator through the \nright port and will pass through the polarizing beam \nsplitter PBS. A photon in state $|1\\rangle$ ``sees'' the atom \nin the state $|g_1\\rangle$ and therefore does not enter the resonator \nbut goes down to the PBS and is reflected by it. \nFig.~\\ref{fig:int-f-cnot}$\\>$(b) refers to the atom in \nstate $|1\\rangle$. A photon in state $|0\\rangle$ sees it, \ngoes down and passes through it, then goes through the \nhalf wave plate (HWP) which changes its state to $|1\\rangle$. \nA photon in state $|1\\rangle$ does not see the atom and exits \nthrough the right port, reflects at PBS and changes to\nstate $|0\\rangle$ when passing through HWP. This would \n(before the 50:50 beam splitter shown in Figure \n\\ref{fig:int-f-cnot}) give us a classical reversible CNOT \ngate and a nondestructive method of detecting the \nstates of an atom.\n\nHowever if we wanted to integrate the obtained CNOT gate \ninto the circuits of a would-be quantum computer, then we \nshould make the target operation unitary, i.e., we have \nto erase the which-path information the photons carry. \nWe do so with the help of a symmetrical 50:50 beam splitter\nshown as SBS in Figure \\ref{fig:int-f-cnot}. Ideally, when \ndetector D does not response, the target qubit will exit \nat the opposite side of the beam splitter so as to yield \nthe following CNOT qubit values\n\\begin{eqnarray} \n|00\\rangle\\to|00\\rangle,\\qquad|01\\rangle\\to|01\\rangle,\\qquad\n|10\\rangle\\to|11\\rangle,\\qquad|11\\rangle\\to|10\\rangle.\\qquad\n\\label{eq:int-f-cnot-ver}\n\\end{eqnarray} \n\nBeam splitter SBS makes our CNOT probabilistic. Detector \nwill ideally {\\em not\\\/} response in half of the cases and this \nmeans that we would be able to forward on average every second \ntarget qubit to subsequent computation stages. Realistically, \nsingle photon detectors have recently reached the efficiency \nof 50\\%\\ \\cite{s-phot-det06} and photon-number-resolving detectors \nthe efficiency of 90\\%\\ \\cite{s-phot-det05}, so, the efficiency \nof the CNOT gate would be less than 50\\%. As a \npartial remedy, we could use two resonators simultaneously to \nobtain two CNOT gates consisting of one control atom qubit and \ntwo photon target qubits. Although statistically independent, \ntheir outcomes could support each other for getting more \nreliable final results, for obtaining middle stage results, \nand perhaps even for obtaining an error correction scheme for \nthe target qubits (an algorithm for the purpose should be \ndevised). This would enable us to compare our CNOT \ngate with the recent feed-forwardable all-optical CNOT gates.\\ \n\\cite{pit-frans03,pan-zeil-cnot-prl04,pit-frans05}\n\nTo estimate how efficiently an $L$-photon will ``see'' \nan atom in $|g_1\\rangle$ state (and an $R$-photon an \natom in $|g_2\\rangle$ state) we have to calculate the \nprobability with which an atom in $|g_1\\rangle$ ($|g_2\\rangle$) \nstate would absorb a photon in $L$ ($R$) state. \nThe total Hamiltonian for the atom-photon coupled system \ncan be decomposed into three \nparts \\cite{carmichael-book,mandel-wolf-book}:\n\\begin{eqnarray}\n\\hat H=\\hat H_a+\\hat H_p+\\hat H_i,\n\\end{eqnarray}\nwhere $\\hat H_a$ is the Hamiltonian of the two-level atom, \n$\\hat H_p$ of the photon field, and $\\hat H_i$ of their \ninteraction. We assume that both the atom and the field \nare quantized so as to have\n\\begin{eqnarray}\n\\hat H_i={\\rm i}\\omega\\left(\\langle g|\\hat{\\pmb{\\mu}}|e\\rangle\n|g\\rangle\\langle e|-\n\\langle g|\\hat{\\pmb{\\mu}}^*|e\\rangle\n|e\\rangle\\langle g|\\right)\\cdot \n\\hat{\\pmb{\\rm{A}}}(r_0,t),\n\\label{eq:tot-ham}\n\\end{eqnarray}\nwhere $\\pmb{\\mu}$ is the atomic dipole moment, \n$\\hat{\\bf{A}}$ is the vector potential of the electric field\nof the laser beam and $\\pmb{\\rm r}_0$ is the position of the \natom, which we take to be fixed. The latter assumption is \nmade on the following grounds.\n\nWithin an ion trap, ionized atoms can be confined to a region \nmuch smaller than the optical wavelength and their position can \nbe controlled with a precision of under 10 nm.\\ \\cite{mundt-blatt-02} \nThe cavity (Sec.\\ \\ref{sec:stirap}) and the resonator \ncan be put around the ion trap following a proposal recently \nelaborated theoretically by Maurer, Becher, Russo, Eschner, and Blatt \n\\cite{blatt04} and Keller, Lange, Hayasaka, Lange, and \nWalther \\cite{kell-walther} and confirmed experimentally by \nboth groups \\cite{mundt-blatt-02,kell-walther-apb03} for \n$^{40}$Ca$^+$. Our $^{87}$Rb cannot be easily ionized because \n$^{87}$Rb$^+$ would lack its $5s$ electron which forms our ground \nstates and the first option for $^{87}$Rb$^-$ is $5p$ which builds \nour excited states. Higher $^{87}$Rb$^-$ are difficult to obtain \nand are unstable.\\ \\cite{rb-ion-94} But other stable ions, as e.g. \n$^{43}$Ca$^+$ have abundant available states ($^{43}$Ca$^+$ has \nnuclear spin 7\/2) with a structure that enables STIRAPs of the \nkind we analysed in Sec.\\ \\ref{sec:stirap} for $^{87}$Rb and can \nbe used instead of $^{87}$Rb$^-$. We however leave details of such \na reformulation to a more realistic experimental future proposal \nsimply because $^{87}$Rb structure and its Zeeman splitting has \nalready been given a detailed experimentally tested model.\\ \n\\cite{zhu-pra99,87rb} Zeeman splitting of other \ncandidates, including $^{43}$Ca$^+$, has not been sufficiently \nexplored as of yet.\n\nIn the interaction picture we use the interaction Hamiltonian $H_i$ \nfrom Eq.~(\\ref{eq:tot-ham}) to obtain the following probability for \nthe photon absorption by the atom in time $\\Delta t$ \n\\cite{mandel-wolf-book}: \n\\begin{eqnarray}\n\\frac{\\omega_0^2}{2\\hbar\\omega\\varepsilon_0 V}\n|\\langle g|\\pmb{\\mu}|e\\rangle\\cdot\\pmb{\\varepsilon}|^2\n\\cos^2\\frac{1}{2}\\Theta\\left[\\frac{\\sin\\frac{1}{2}\n(\\omega-\\omega_0)\\Delta t}{\\frac{1}{2}(\\omega-\\omega_0)}\\right]^2,\n\\label{eq:abs-prob}\n\\end{eqnarray}\nwhere $\\omega_0$ is the atomic frequency, $\\omega$ is the \nlaser field frequency, $\\Theta$ is the polar angle of the atomic\nBloch vector, $\\pmb{\\varepsilon}$ is the polarization vector, \nand $V$ the quantization volume. \n\nThis means that several conditions have to be satisfied \nto get a high absorption probability. First we have to tune the\nlaser frequency $\\omega$ close enough to the atomic frequency \n$\\omega_0$ and this is achieved by the level of precision already \nexperimentally reached in targeting individual ions trapped in \nPaul traps within an optical cavity as we mentioned above. \nThis has also been achieved experimentally in cavity quantum \nelectrodynamics (CQED) by Brune, Schmidt-Kaler, Maali, Dreyer, \nHagley, Raimond, and Haroche \\cite{haroche-prl96} already ten \nyears ago. They obtained a coherent exchange of photons between \nthe cavity field and individual atoms (vacuum Rabi oscillations). \nOn the other hand, lasers can have linewidths that are less than \n1 Hz and the precision of determining frequency of atomic \ntransitions also approaches 1 Hz.\\ \n\\cite{hertz-precision-05} \n\nNext, for an atomic transition $m=0\\to m=-$1 (when \n$\\langle g|\\pmb{\\mu}|e\\rangle=\n|\\mu|(\\mathbf{x}+i\\mathbf{y})\/\\sqrt{2}$, \nwhere $|\\mu|=|\\langle g|\\pmb{\\mu}|e\\rangle|$) \nthe probability (\\ref{eq:abs-prob}) is greatest for right \ncircularly polarized photons \n$\\pmb{\\varepsilon}=(\\mathbf{x}+i\\mathbf{y})\/\\sqrt{2}$ and \nvanishes for left circularly polarized ones \n$\\pmb{\\varepsilon}=(i\\mathbf{x}+\\mathbf{y})\/\\sqrt{2}$. \nThus we have to orient the external magnetic field \n$\\mathbf B$ along the direction of the beam that hits the \natom. As for $\\cos^2\\frac{1}{2}\\Theta$ term, $\\Theta=0$ \nmeans that the atom is in pure $|g \\rangle$ state \nand $\\Theta=\\pi$ that its $|g \\rangle$ state is not \npopulated at all. \n\nWhat remains to be evaluated to estimate the level \nof coupling of the atom to the photon field is the quantization \nvolume $V$ and the terms containing $\\omega$ in the probability \n(\\ref{eq:abs-prob}). \nThe square root of the first two terms \n\\begin{eqnarray}\ng=\\sqrt{\\frac{|\\mu|^2\\omega_0^2}{2\\hbar\\omega\\varepsilon_0 V}}, \n\\end{eqnarray} \nis called the {\\em dipole coupling constant} \\cite{carmichael-book}, \nor the {\\em rate of coupling} of an atom to a single cavity mode \n\\cite{kell-walther,grangier-fp00} (usually with the assumption \nthat $\\omega\\approx\\omega_0$). \nIn CQED and STIRAP cavities, $g$ is compared with spontaneous \nemission from the atom (transverse damping rate $\\gamma$) and with leaking \nout of the cavity (damping rate $\\kappa$). The calculations for these \ncavities differ from the one carried above, but the overall results are \ncomparable. For the former cavities a {\\em strong coupling} is achieved \nwhen $g$ is much larger than both $\\gamma$ and $\\kappa$.\\ \n\\cite{carmichael-book,kell-walther,grangier-fp00} One can achieve \nthis by making a cavity as small as possible \nthereby decreasing $V$ and making $g$ larger, by choosing \natoms with a narrower atomic linewidth $\\gamma$, and by decreasing\nthe linewidth of the cavity mode $\\kappa$. With our resonator \ncavity, however, we do not have to care about spontaneous emission \nand\/or leaking from the cavity because the interaction-free \nabsorption (almost) never really excites the atom and the photon \n(almost) never really reaches the atom. \nSo, our resonator cavity does not have to be small. In Sec.\\\n\\ref{sec:resonator} we have seen that $\\lambda_{\\rm res}=2L\/j$, \nwhere $L$ is the round-trip length of the cavity and $j$ is an integer. \nBy picking a larger $j$ we get a larger cavity in which the \nbirefringent optics would fit. \n\nAfter integrating the terms containing $\\omega$ in \n(\\ref{eq:abs-prob}) over $\\omega$ we get the rate of absorption: \n$q^2\\Delta t$ (the probability turns out to be a function of \n$(\\Delta t)^2$). This rate is higher than the one we get in CQED \nexperiments \\cite{haroche-prl96} where the atoms move through \na cavity with a velocity of over 100$\\>$m\/s and nevertheless \ncouple strongly to the cavity field within less than \n100$\\>\\mu$s. In our resonator we can achieve much \nlonger and also shorter times ($1\\>{\\mu\\rm s}<\\Delta t<1\\>{\\rm ms}$; \nthe shortest time is limited by the resonance build-up time when a \ntransparent object is in the resonator and the longest one by the \ncoherence time of cw lasers). So, we can conclude that under \nrealistic conditions the atom will be strongly coupled to our \nresonator cavity field. The shortest required times for the \ncoupling have to be determined for a chosen atomic system. \n\nWe should add that the Stokes and pump beams for STIRAP \nswitching $|g_1\\rangle\\leftrightarrow|g_2\\rangle$ can be used \ntogether with the resonator cavity inclined at a small angle to \nits beam because that beams are much stronger than the resonator \nbeam and a small misalignment with the magnetic field $\\mathbf B$ \nwill not significantly influence the transitions.\\ \n\\cite{arimondo,berg-rmp98} \nWe cannot carry out STIRAP transitions and run interaction-free\nCNOT simultaneously because of the electromagnetically induced \ntransparency (EIT) by the transitions.\\ \n\\cite{arimondo,lukin03,fleischhauer05} If we wanted to carry \nout interaction-free interrogations of STIRAP transitions, \nwe should employ transitions that are not used by the transitions, \ne.g., $m=-1,F=1\\ \\to\\ m=-2,F=2$ and $m=1,F=1\\ \\to\\ m=2,F=2$ as \nshown in Fig.\\ \\ref{fig:superp}. \n\n\\section{\\label{sec:suppr-sup}INTERACTION-FREE CONTROL OF \nSUPERPOSITION}\n\nOur design can also be used to control the STIRAP \nsuperposition of the states discussed in Sec.\\\n\\ref{sec:stirap}, Eq.~(\\ref{eq:stirap-state-e}), in two \nways. The first one is by determining a time span within which an \natomic superposition can be built up and the second one is by \nsuppressing the formation of atomic superposition using the \ninteraction-free erasure of interference fringes obtained in \nRef.~\\cite{pav-pla-96}. \n\nIn the STIRAP process presented in Sec.~\\ref{sec:stirap} \nthe electron is moved from the state $|g_3\\rangle$ \nshown in Fig.~\\ref{fig:rubidium}$\\>$(b) into the \nsuperposition of the states $|g_1\\rangle$ and $|g_2\\rangle$\ngiven by Eq.~(\\ref{eq:stirap-state-e}). Interaction-free\ndetection can be used to determine, within a few microseconds, \nwhen the electron leaves state $|g_3\\rangle$, as follows. \n\nWe use our interaction-free resonator with the interrogating \npaths perpendicular to the axis of the cavity and wave plates \nremoved in order to probe the atom with a linearly polarized \nphoton $\\hbar\\omega_5$, denoted $\\pi$ in Fig.~\\ref{fig:superp} \n(a circularly polarized photon corresponding to the transition \n$|g_3\\rangle\\to|e_3\\rangle$ would also work). \nWe tune the resonator and photons to the frequency $\\omega_5$ \nthat corresponds to a possible transition $|g_3\\rangle\\to|e_5\\rangle$.\nWhen our resonator is applied to an atom in which a STIRAP transition\n$|g_3\\rangle\\to(\\alpha|g_1\\rangle+\\beta|g_2\\rangle)$ is induced, \nits repeated interrogation will pinpoint the time of this transition. \nThis is because the photon ``sees'' the electron if \ntransition $|g_3\\rangle\\to|e_5\\rangle$ is possible and does not \n``see'' it when it leaves $|g_3\\rangle$. In the former case it \nexits at one port of the resonator and in the latter at the other. \nThe photon cannot of course really excite the atom, because it \ncannot transfer any energy to it.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{pavicic-pra-1-07-fig5.eps}\n\\end{center}\n\\caption{Transitions which make an atom opaque for photons in \nthe interaction-free resonators of Fig.~\\ref{fig:int-f-cnot}. \nIf an electron is in one of the $|g\\rangle$ states, then a photon\nwill exit through one port of the gate, and if not, through \nthe other.} \n\\label{fig:superp}\n\\end{figure}\n\nAssuming that a realistic round trip of our resonator can be reduced \nto a few cm and calculating \\cite{ppijtp96} that 200 \nto 300 round trips (under 100$\\>$ns) are sufficient to establish \ninterference within the resonator, we calculates that \na laser has to be {\\em on} for several $\\mu$s in \norder to detect whether or not an electron is in state \n$|g_3\\rangle$, i.e., whether the atom is opaque or transparent. \nWe must therefore reduce the intensity of a cw laser so as to \nobtain on average one photon within this time window.\n\nThe other way of controlling our STIRAP is to prevent \nbuilding of superposition $\\alpha|g_1\\rangle+\\beta|g_2\\rangle$ and \nforcing the electron into either state $|g_1\\rangle$ or \nstate $|g_2\\rangle$. For this purpose, we use two interaction-free \nresonators with interrogating paths oriented perpendicularly \nto the axis of the cavity and tuned to frequencies \n$\\omega_3$ and $\\omega_4$ with circularly polarized photons, as shown \nin Fig.~\\ref{fig:superp}. Linearly polarized photons corresponding \nto transitions to $m=-1$ and $m=1$, respectively, would also work. \nNote that the scheme with photons tuned to transitions \n$|g_1\\rangle\\to|e_5\\rangle$ and $|g_2\\rangle\\to|e_5\\rangle$ \nwould not work, because, as shown in Sec.~\\ref{sec:stirap}, \nthe atom is transparent to such photons in both \ncases---when there is no electron in states $|g_1\\rangle$ and \n$|g_2\\rangle$ and when STIRAPs are in progress. In the latter \ncase we speak of electromagnetically induced transparency (EIT) \n\\cite{arimondo,lukin03,fleischhauer05}.\n\n\n\\section{\\label{sec:concl}CONCLUSION}\n\nIn conclusion, we have obtained a probabilistic interaction-free \nCNOT gate in which two atom Zeeman states represent the control \nqubit and two photon polarization states represent the target qubit. \nThe gate, which is a photon ring resonator, is robust because \nit does not transfer any energy to atoms in over 95\\%\\ \nof tests. Unlike the previous atom-photon CNOT gate \n\\cite{gilc02}, this gate has all four modes available, \nbecause it has two exit ports for photons, \neach of which can let photons in both polarization \nstates out, depending on the state the atom is in. \n\nIn Sec. \\ref{sec:i-f-cnot} we carried out quantum calculations \nand made realistic estimations for a possible experiment. \nIf we confine ions (e.g., $^{40}$Ca$^+$) \nto under 10$\\>$nm by using a Paul trap and mount the \nresonator around it, we arrive at the rate of absorption \nwhich is higher than in other experimentally tested cavities. \nThis amounts to a strong coupling between the atoms in our \nring cavity and its field modes. \n\nThe interaction-free resonator can also be used to \ncontrol a stimulated Raman adiabatic passage (STIRAP) \nfrom an individual state to a state of superposition. \nWe can control the time when an atom changes its \nstate. Such detection takes under 1$\\>\\mu$s. \n\nAnother control we can exert over qubits is to suppress \natom-state superposition. The quantum system is altered\nwithout any energy transfer to the system in observation of \nthe quantum indistiguishability principle, which states that \nno information can be acquired about the population of\nparticular states which take part in a superposition. \n\nThus the interaction-free resonator can be used in \nquantum computation to manipulate qubits and to \nobtain information on them during computation \nwithout decohering their states. It is suitable for \nsystems that can be scaled up because it is non-destructive, \ni.e., it does not destroy the output states, and because it \nprovides information on the success of the CNOT-gate \noperation that can be used for subsequent manipulation\nof the same photon qubits. More specifically, we can \namplify the null detections of detector D in Fig.\\ \n\\ref{fig:int-f-cnot} by combining two resonators, \nwhere one of them would give information on \nthe output of the other. \n\n\\begin{acknowledgments}\nThis work was supported by the Ministry of Science, \nEducation, and Sport of Croatia, Project No.~0082222. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLet $\\mathbb{K}$ be a field. One of the fundamental problems in affine algebraic geometry is to try to describe the structure of ${\\rm GA}_n(\\mathbb{K})$, the group of polynomial automorphisms of $\\mathbb{A}^n$. There are a few natural subgroups:\n\\begin{itemize}\n\\item The general linear group ${\\rm GL}_n(\\mathbb{K})$;\n\\item The affine group ${\\rm Aff}_n(\\mathbb{K})$ consisting of automorphisms of degree one;\n\\item The triangular subgroup ${\\rm BA}_n(\\mathbb{K})$;\n\\item The subgroup ${\\rm EA}_n(\\mathbb{K})$ generated by elementary automorphisms, i.e. those with unital Jacobian determinant fixing $n-1$ variables;\n\\item The tame subgroup ${\\rm TA}_n(\\mathbb{K})$ generated by the triangular and affine automorphisms;\n\\item The special automorphism group ${\\rm SA}_n(\\mathbb{K})$, consisting of automorphisms with unital Jacobian determinant.\n\\end{itemize}\n\nIt is a classical result of Jung and van der Kulk \\cite{Jung,vanderKulk} that in dimension two, the tame subgroup is the entire automorphism group, while Shestakov and Umirbaev \\cite{Shestakov-Umirbaev} famously showed that this does not hold in dimension three (in characteristic zero); this question, known as the tame generators problem, remains open in higher dimensions. \n\nA natural area of inquiry is to describe subgroups lying between the affine and the tame subgroup. In dimension two, there are many such subgroups due to the classical result that ${\\rm TA}_2(\\mathbb{K})$ is an amalgamated free product of ${\\rm Aff}_2(\\mathbb{K})$ and ${\\rm BA}_2(\\mathbb{K})$ over their intersection, but in higher dimensions (and characteristic zero; see \\cite{Edo-Kuroda} for the positive characteristic case) this is a surprisingly delicate question. It was not until recently that Edo and the author \\cite{Edo-Lewis} gave the first example of such an intermediate subgroup in characteristic zero. The idea there was to study {\\em co-tame automorphisms}, defined by Edo \\cite{Edo} as those that together with the affine group generate the entire tame subgroup; the example of \\cite{Edo-Lewis} is an automorphism that is tame but not co-tame, which therefore generates a proper intermediate subgroup between ${\\rm Aff}_n(\\mathbb{K})$ and ${\\rm TA}_n(\\mathbb{K})$. Interestingly, Edo \\cite{Edo} showed that certain wild maps, including the Nagata map, are co-tame.\n\nOne key difficulty in describing this subgroup lattice between the affine and tame subgroups arises from the fact that many simply constructed automorphisms are co-tame. To describe this difficulty further, let us make a precise definition.\n\n\\begin{definition}\nA tame automorphism $\\phi$ is called {\\em $m$-triangular} if it can be written in the form $\\phi = \\alpha _0 \\tau _1 \\alpha _1 \\cdots \\tau _m \\alpha _{m}$ for some $\\tau _i \\in {\\rm BA}_n(\\mathbb{K})$ and $\\alpha _i \\in {\\rm Aff}_n(\\mathbb{K})$.\n\\end{definition}\n\nThe author and Edo \\cite{Edo-Lewis17} recently showed that, for $n \\geq 3$, all 3-triangular automorphisms are co-tame, while in the $n=3$ case, for all $m \\geq 4$ there exist $m$-triangular automorphisms that are not co-tame (and thus generate proper intermediate subgroups between the affine and tame subgroups). \n\nThis phenomenon of single automorphisms generating large subgroups also appears in the work of Furter and Lamy \\cite{Furter-Lamy}, who were studying normal subgroups in dimension two with an eye towards establishing the non-simplicity of the two-dimensional Cremona group (later proved over an algebraically closed field by Cantat and Lamy \\cite{CantatLamy}). To be more precise, let us quickly fix some notations.\n\n\\begin{itemize}\n\\item If $H \\subset {\\rm SA}_n(\\mathbb{K})$, we use $\\langle H \\rangle^S$ to denote the normal subgroup generated by $H$ in ${\\rm SA}_n(\\mathbb{K})$.\n\\item If $H \\subset {\\rm GA}_n(\\mathbb{K})$, we use $\\langle H \\rangle^G$ to denote the normal subgroup generated by $H$ in ${\\rm GA}_n(\\mathbb{K})$.\n\\item The group ${\\rm SLIN}_n(\\mathbb{K}):=\\langle {\\rm SL}_n (\\mathbb{K}) \\rangle ^S$ is the smallest normal subgroup of ${\\rm SA}_n(\\mathbb{K})$ that contains ${\\rm SL}_n(\\mathbb{K})$.\n\\item The group ${\\rm GLIN}_n(\\mathbb{K}):= \\langle {\\rm GL}_n(\\mathbb{K}) \\rangle ^G$ is the smallest normal subgroup of ${\\rm GA}_n(\\mathbb{K})$ that contains ${\\rm GL}_n(\\mathbb{K})$.\n\\end{itemize}\n\nDanilov \\cite{Danilov} showed that ${\\rm SA}_2(\\mathbb{K})$ (for a field of characteristic zero) is not simple by constructing a $13$-triangular map that generates a proper normal subgroup. Furter and Lamy \\cite{Furter-Lamy} showed that the normal subgroup generated by any single nontrivial $4$-triangular automorphism in ${\\rm SA}_2(\\mathbb{K})$ is the entire group ${\\rm SA}_2(\\mathbb{K})$. Moreover, by taking advantage of the amalgamated free product structure of ${\\rm GA}_2(\\mathbb{K})$, they showed that for $m \\geq 7$, generic $m$-triangular automorphisms generate proper normal subgroups. More recently, the non-simplicity of ${\\rm SA}_2(\\mathbb{K})$ was shown for all fields by Minasyan and Osin \\cite{MinasyanOsin}.\n\nIn dimension 3 (and characteristic zero), while ${\\rm TA}_3(\\mathbb{K})$ is a proper subgroup of ${\\rm GA}_3(\\mathbb{K})$ \\cite{Shestakov-Umirbaev}, the tame subgroup is still an amalgamated free product \\cite{Wright} (of three subgroups along their pairwise intersections). Recently Lamy and Przytycki \\cite{LamyPrzytycki} took advantage of this to give a class of examples of $m$-triangular automorphisms $$\\phi _m = (x_2,x_1+x_2x_3,x_3)^m(x_3,x_1,x_2)$$ such that $\\langle \\phi _m \\rangle ^{{\\rm SA}_3(\\mathbb{K}) \\cap {\\rm TA}_3(\\mathbb{K})}$ is a proper subgroup of ${\\rm SA}_3(\\mathbb{K}) \\cap {\\rm TA}_3(\\mathbb{K})$ for every even $m \\geq 12$; moreover, they showed that ${\\rm TA}_3(\\mathbb{K})$ is acylindrically hyperbolic. However, it remains to our knowledge an open question whether $\\langle \\phi _m \\rangle ^S ={\\rm SLIN}_3(\\mathbb{K})$.\n\n\nThe group ${\\rm GLIN}_n(\\mathbb{K})$ was introduced by Maubach and Poloni \\cite{Maubach-Poloni}, who were investigating a weaker form of Meister's Linearization problem\\footnotemark:\n\\begin{problem}\\label{prob:Meister}\nFor which $\\phi \\in {\\rm GA}_n(\\mathbb{C})$ do there exist some $s \\in \\mathbb{C}^*$ such that $(sx_1,\\ldots,sx_n)\\phi$ is conjugate to an element of ${\\rm GL}_n(\\mathbb{C})$?\n\\end{problem}\n\n\\footnotetext{We feel obliged to point the reader to Section 8.3 of \\cite{ArnoBook}, in which van den Essen gives a delightful accounting of the story of the construction of counterexamples to Meister's original Linearization Conjecture and the related Markus-Yamabe Conjecture.}\n\nWhile van den Essen \\cite{vandenEssen} gave an example of an automorphism that does not have this property (see Example \\ref{ex:vdE}), Maubach and Poloni showed that the (wild) Nagata map does have this property, and thus lies in ${\\rm GLIN}_n(\\mathbb{C})$. This led them to make the following conjecture.\n\n\\begin{conjecture}\\label{con:g}\nIf $\\mathbb{K} \\neq \\mathbb{F}_2$, then ${\\rm GLIN}_n(\\mathbb{K}) = {\\rm GA}_n(\\mathbb{K})$.\n\\end{conjecture}\n\nThis is trivial for $n=1$, and a consequence of the Jung-van der Kulk theorem for $n=2$ (see Theorem \\ref{thm:GLIN}), but remains open for $n \\geq 3$. We remark that Maubach and Willems showed the necessity of the $\\mathbb{K}\\neq \\mathbb{F}_2$ hypothesis in \\cite{Maubach-Willems}. Here, we add the following, slightly stronger conjecture:\n\\begin{conjecture}\\label{con:s}\nIf $\\mathbb{K} \\neq \\mathbb{F}_2$, then ${\\rm SLIN}_n(\\mathbb{K}) = {\\rm SA}_n(\\mathbb{K})$.\n\\end{conjecture}\n\nIn section \\ref{secSLIN}, we study the group ${\\rm SLIN}_n(\\mathbb{K})$ in any characteristic, and show that ${\\rm SLIN}_n(\\mathbb{K}) = \\langle {\\rm EA}_n(\\mathbb{K})\\rangle ^S$ for all fields other than $\\mathbb{F}_p$ for a prime $p$. Since ${\\rm TA}_n(\\mathbb{K}) \\cap {\\rm SA}_n(\\mathbb{K})={\\rm EA}_n(\\mathbb{K})$, this motivates us to make the following definition.\n\n\\begin{definition}\nA special automorphism $\\theta \\in {\\rm SA}_n(\\mathbb{K})$ is called {\\em normally co-tame} if $\\langle \\theta \\rangle ^S \\geq {\\rm SLIN}_n(\\mathbb{K})$. \n\\end{definition}\n\nNote that if Conjecture \\ref{con:s} is true, then an automorphism $\\theta \\in {\\rm SA}_n(\\mathbb{K})$ is normally co-tame if and only if $\\langle \\theta \\rangle ^S = {\\rm SA}_n(\\mathbb{K})$. Thus, in this paper we turn our attention to describing classes of maps that are normally co-tame. In particular, we generalize a result of Furter and Lamy to all dimensions, and show\n\n\\begin{theorem}[Main Theorem]\nOver an field of characteristic zero, every nontrivial $4$-triangular automorphism is normally co-tame.\n\\end{theorem}\n\nWe also quickly show that a class of exponential maps, including the (wild) Nagata map, are all normally co-tame (cf. \\cite{Maubach-Poloni}). Finally, we show that a related class consisting of triangular maps composed with exponential maps are all normally co-tame. In particular, this shows that the example of van den Essen \\cite{vandenEssen} that does not satisfy Problem 1 does in fact lie in ${\\rm SLIN}_n(\\mathbb{K})$, lending some more support to Conjectures \\ref{con:g} and \\ref{con:s}. \n\n\n\\section{Preliminaries}\nWe begin by recalling some standard definitions; see \\cite{vandenEssen} for a general reference on polynomial automorphisms. We use $\\mathbb{K}^{[n]}=\\mathbb{K}[x_1,\\ldots,x_n]$ to denote the $n$-variable polynomial ring.\n\n\\begin{itemize}\n\\item ${\\rm GA}_n(\\mathbb{K})$ is the group of automorphisms of $\\Spec \\mathbb{K}^{[n]}$ over $\\Spec \\mathbb{K}$. It is anti-isomorphic to the group of $\\mathbb{K}$-automorphisms of $\\mathbb{K}^{[n]}$. We abuse this correspondence freely, and for $\\phi \\in {\\rm GA}_n(\\mathbb{K})$ and $P \\in \\mathbb{K}^{[n]}$ will write $(P)\\phi$ for the image of $P$ under the corresponding automorphism of $\\mathbb{K}^{[n]}$. By writing the automorphism on the right, the usual composition holds, namely if $\\psi \\in {\\rm GA}_n(\\mathbb{K})$ as well, then $(P)\\phi \\psi = ((P)\\phi)\\psi$.\n\\item ${\\rm Tr}_n(\\mathbb{K})$ denotes the group of translations. \n\\item ${\\rm EA}_n(\\mathbb{K})$ denotes the subgroup generated by elementary automorphisms, i.e. those of the form $$(x_1,\\ldots,x_{i-1},x_i+P(x_1,\\ldots,x_{i-1},x_{i+1},\\ldots,x_n),x_{i+1},\\ldots,x_n)$$\nfor some $P \\in \\mathbb{K}[x_1,\\ldots,x_{i-1},x_{i+1},\\ldots,x_n]$.\n\\item ${\\rm BA}_n(\\mathbb{K})$ denotes the subgroup of (lower) triangular automorphisms, i.e. those of the form $$\\left(a_1x_1+P_1,a_2x_2+P_2(x_1),\\ldots,a_nx_n+P_n(x_1,\\ldots,x_{n-1})\\right)$$\nfor some $a_i \\in \\mathbb{K}^*$ and $P_i \\in \\mathbb{K}[x_1,\\ldots,x_{i-1}].$\n\\item The tame subgroup is ${\\rm TA}_n(\\mathbb{K}) = \\langle {\\rm EA}_n(\\mathbb{K}), {\\rm GL}_n(\\mathbb{K}) \\rangle = \\langle {\\rm BA}_n(\\mathbb{K}), {\\rm Aff}_n(\\mathbb{K}) \\rangle$. \n\\item We use ${\\rm D}_n(\\mathbb{K})$ to denote the diagonal subgroup of ${\\rm GL}_n(\\mathbb{K})$, and define ${\\rm Df}_n(\\mathbb{K}) = {\\rm D}_n(\\mathbb{K}) \\ltimes {\\rm Tr}_n(\\mathbb{K})$. This group consists of all automorphisms of the form \n$$(a_1x_1+b_1,\\ldots,a_nx_n+b_n)$$ for some $a_i \\in \\mathbb{K}^*$ and $b_i \\in \\mathbb{K}$.\n\\item ${\\rm PA}_n(\\mathbb{K})$ is the group of parabolic automorphisms, i.e. those of the form \n$$\\left(H_1,\\ldots,H_{n-1}, a_nx_n+P_n(x_1,\\ldots,x_{n-1})\\right)$$\nfor some $H_i \\in \\mathbb{K} ^{[n-1]}$, $a_n \\in \\mathbb{K}^*$, and $P_n \\in \\mathbb{K}[x_1,\\ldots,x_{n-1}]$. \n\\end{itemize}\n\n\n\\begin{definition}\nWe define the {\\em vector degree} $\\vd: {\\rm BA}_n (\\mathbb{K}) \\rightarrow \\mathbb{N}^n$ by writing $\\tau = \\left(a_1x_1+P_1,a_2x_2+P_2(x_1),\\ldots,a_nx_n+P_n(x_1,\\ldots,x_{n-1})\\right)$ for some $a_i \\in \\mathbb{K}$, $P_i \\in \\mathbb{K}[x_1,\\ldots,x_{i-1}]$\nand setting \n\n$$\\vd(\\tau) = \\left( \\deg \\left( P_1 \\right), \\ldots, \\deg \\left(P_n\\right) \\right).$$\nWe will, somewhat unusually, adopt the convention that $\\deg(0)=0$ for convenience.\n\\end{definition}\n\n\n\\begin{example}\n$\\vd\\left( (x_1+2,x_2+x_1^2,x_3-x_1^2+x_1x_2^4) \\right) = (0,2,5)$.\n\\end{example}\n\nIt will be convenient to order $\\mathbb{N}^n$ lexicographically; we denote this partial order by $<_{\\rm lex}$ and write, for example, $(0,2,5) <_{\\rm lex} (0,3,3)$.\nThe utility of the vector degree is made clear by the following lemma.\n\n\\begin{lemma}\\label{lem:vdreduce}\nLet $\\tau \\in {\\rm BA}_n (\\mathbb{K})$.\n\\begin{enumerate}\n\\item We have $\\tau \\in {\\rm Df}_n(\\mathbb{K})$ if and only if $\\vd(\\tau)=(0,\\ldots,0)$.\n\\item If $\\gamma \\in {\\rm Tr}_n(\\mathbb{K})$ and $\\tau \\notin {\\rm Df}_n(\\mathbb{K})$, then $\\vd \\left(\\tau ^{-1} \\gamma \\tau\\right) <_{\\rm lex} \\vd \\left(\\tau\\right)$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThe first statement is immediate from our definition of ${\\rm Df}_n(\\mathbb{K})$. For the second, write $\\tau = (a_1x_1+P_1,a_2x_2+P_2(x_1),\\ldots,a_nx_n+P_n(x_1,\\ldots,x_{n-1}))$ for some $a_i \\in \\mathbb{K}^*$, $P_i \\in \\mathbb{K}[x_1,\\ldots,x_{i-1}]$, and write $\\gamma =(x_1+b_1,\\ldots,x_n+b_n)$ for some $b_i \\in \\mathbb{K}$. Since $\\tau \\notin {\\rm Df}_n(\\mathbb{K})$, we have $(0,\\ldots,0) <_{\\rm lex} \\vd(\\tau)$. Therefore we let $r>1$ be minimal with $\\deg P_r >0$, so that $P_1,\\ldots,P_{r-1} \\in \\mathbb{K}$. Then it is easy to see that for $i1$. Note that $p \\nmid d_k$, and thus ${p+d_k \\choose p} \\neq 0$; so since $\\mathbb{K}$ is finite we can choose $b \\in \\mathbb{K}$ such that $b^p{p+d_k \\choose p} = a$. \nThen setting $f = \\left(x_k^pM\\right)\\epsilon _{k,b}-x_k^pM$, we have\n$$\\epsilon _{1,f} = \\epsilon _{k,-b} \\left(\\epsilon _{1,x_k^pM} \\epsilon _{k,b} \\epsilon _{1,x_k^pM}^{-1}\\right) \\in {\\rm SLIN}_n(\\mathbb{K}).$$\n\n\\begin{claim}\nLet $S$ be the set of tuples $(r_2,\\ldots,r_n) \\in \\mathbb{N}^{n-1}$ satisfying either\n\\begin{enumerate}\n\\item $r_{2}+\\cdots+r_{n} < \\deg M$, or\n\\item $(q-1) \\nmid (r_k+1)$.\n\\end{enumerate}\nThen $$f=aM + \\sum _{(r_2,\\ldots,r_n) \\in S} c_{r_2,\\ldots,r_n} x_2^{r_2}\\cdots x_n^{r_n}$$ for some $c_{r_2,\\ldots,r_n} \\in \\mathbb{K}$.\n\\end{claim}\n\\begin{proof}\nIt is straightforward to compute that\n$$f=\\left(\\frac{M}{x_k^{d_k}}\\right) \\sum _{i=1} ^{d_k+p} {d_k+p \\choose i} b^i x_k ^{d_k+p-i}.$$\nNote that $\\frac{M}{x_k^{d_k}} \\in \\mathbb{K}[x_2,\\ldots,\\hat{x}_k,\\ldots,x_n]$ and that $\\deg _{x_j} \\left( \\frac{M}{x_k^{d_k}}\\right)=d_j$ for $j \\neq 1,k$. \nBy assumption $d_k+1 \\equiv 0 \\pmod {(q-1)}$; since $q>p$ (by hypothesis), we thus have $r+1 \\not \\equiv 0 \\pmod{(q-1)}$ for any $d_k 0$, then $\\deg _{x_n}(F-(F)\\epsilon _{n,1})=\\deg _{x_n}F - 1$, so inducting downwards on $\\deg _{x_n} F$, we are left to deal with the case that $F \\in \\mathbb{K}[x_1,\\ldots,x_{n-1}]$. But in this case, either $\\exp(FD)$ is triangular, or there exists $1 \\leq i \\leq n-1$ with $(x_i)\\exp(FD) = x_i+Q$ for some nonzero $Q \\in \\mathbb{K}[x_1,\\ldots,x_{n-1}]$. Then, letting $$\\epsilon _{n,x_i}^{-1}\\exp(-FD)\\epsilon _{n,x_i} \\exp(FD)=\\epsilon _{n,Q}$$\nwe see that $\\exp(FD)$ is normally co-tame since $\\epsilon _{n,Q}$ is elementary.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:triangularexponential}\nLet $\\mathbb{K}$ be a field of characteristic zero.\nLet $D$ be a nonzero triangular derivation and $F \\in \\ker D$; let $\\tau \\in {\\rm BA}_n(\\mathbb{K})$ and $\\alpha \\in {\\rm GL}_n(\\mathbb{K})$. Then $\\tau \\alpha \\exp(FD)$ is normally co-tame.\n\\end{theorem}\n\\begin{proof}\nLet $\\phi = \\tau \\alpha \\exp(FD)$. By Lemma \\ref{reductionlemma} it suffices to show that $\\phi _0 = \\epsilon _{n,1} ^{-1} \\phi^{-1} \\epsilon _{n,1} \\phi$ is normally co-tame. Applying Lemma \\ref{reductionlemma} once more, it suffices to show that $\\phi _1 = \\exp(FD) \\phi _0 \\exp(-FD)$ is normally co-tame. So we compute $\\phi _1$, letting $\\gamma = \\alpha ^{-1} \\epsilon _1 \\alpha \\in {\\rm Tr}_n(\\mathbb{K})$:\n\\begin{align*}\n\\phi _ 1 &= \\exp(FD) \\left( \\epsilon _{n,1} ^{-1} \\exp(-FD) \\alpha ^{-1} \\tau ^{-1} \\epsilon _{n,1} \\tau \\alpha \\exp(FD) \\right) \\exp(-FD) \\\\\n&=\\exp(FD) \\epsilon _{n,1} ^{-1} \\exp(-FD) \\gamma.\n\\end{align*}\nNow, letting $G=(F)\\epsilon _{n,1}^{-1}-F$ as in the proof of Theorem \\ref{thm:exponential}, we have\n$$\\phi _1 = \\epsilon _{n,1} ^{-1} \\exp(GD) \\gamma.$$\nBut then, applying Lemma \\ref{reductionlemma} once more, it suffices to show $\\phi _2 = \\epsilon _{n,1} \\phi _1 \\epsilon _{n,1}^{-1} \\phi _1 ^{-1}$ is normally co-tame, so we compute \n$$\\phi _2 = \\exp(GD) \\gamma \\epsilon _{n,1}^{-1} \\gamma ^{-1} \\exp(-GD) \\epsilon _{n,1} \n= \\exp(GD) \\epsilon _{n,1}^{-1} \\exp(-GD) \\epsilon _{n,1}.$$\nBut letting $H= G-(G)\\epsilon _{n,1}$, we have $\\phi _2 = \\exp(HD)$ which is normally co-tame by Theorem \\ref{thm:exponential}.\n\\end{proof}\n\n\\begin{example}\\label{ex:vdE}\nThe automorphism $$ (x_1,x_2+x_1^3,x_3-x_2(x_1x_3+x_2x_4),x_4+x_1(x_1x_3+x_2x_4)) \\in {\\rm SA}_4(\\mathbb{C})$$ is normally co-tame by Theorem \\ref{thm:triangularexponential}, as it can be written as $(x_1,x_2+x_1^3,x_3,x_4) \\exp(FD)$ where $F=x_1x_3+x_2x_4$ and $D= -x_2\\frac{\\partial}{\\partial x_3}+x_1 \\frac{\\partial}{\\partial x_4}$. This automorphism is conjugate (by a permutation) to van den Essen's counterexample \\cite{vandenEssen} to Problem \\ref{prob:Meister}. \n\\end{example}\n\n\n\\subsubsection*{Acknowledgements}\nThe author would like to thank the referees for a number of helpful comments, and for pointing out the recent paper \\cite{LamyPrzytycki}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWe start with reminding the standard def\\\/inition for a complex manifold.\n Suppose a manifold of dimension $D = 2d$ is\ncovered by several overlapping $D$-dimensional disks. Suppose that in\neach such map complex coordinates $w^{j = 1,\\ldots,d}$,\n$\\bar w^{\\bar j = 1,\\ldots,d} $ are introduced such that\nthe metric has a Hermitian form\n \\begin{gather*}\nds^2 \\ =\\ h_{j\\bar k}(w, \\bar w) dw^j d \\bar w^{\\bar k} , \\qquad\nh_{j\\bar k}^* = h_{k\\bar j} .\n \\end{gather*}\nIn the region where\na couple of the maps with coordinates $w$, $\\bar w$ and $\\tilde w$,\n$\\tilde {\\bar w}$ overlap the latter are expressed into one another.\nThe manifold is called {\\it complex} if this relationship\ncan be made holomorphic, $\\tilde w^j = f^j(w^k)$.\n\nFor example, $S^2$ (actually, any 2-dimensional manifold) is complex.\nTo see that, introduce the stereographic complex coordinates\n\\[\nw = \\frac {x + iy} {\\sqrt{2}} , \\qquad \\bar w = \\frac {x - iy} {\\sqrt{2}}\n\\]\n such that\n \\begin{gather*}\nds^2 = \\frac {2 dw d \\bar w}{(1 + \\bar w w)^2} .\n \\end{gather*}\nThis map covers the whole sphere except its north pole (corresponding to $w = \\infty$).\nIntroduce now another stereographic map that covers the whole sphere but its south pole.\nThe metric is again\n \\begin{gather*}\nds^2 = \\frac {2 d \\tilde w d {\\tilde {\\bar w}}}{(1 + {\\tilde {\\bar w}} \\tilde w)^2} .\n \\end{gather*}\n In the region where the maps overlap (the whole sphere but two points), the holomorphic\nrelation $\\tilde w = 1\/w$ holds.\n\nLet us try to do the same for $S^4$. Again, we can cover it by two stereographic maps with the coordinates $w_j$ and\n$\\tilde w_j$\\,\\footnote{$j = 1,2$ and, when going down to $S^4$ with its conformally\nf\\\/lat metric, we will not bother to distingush between\ncovariant and contravariant indices. Neither will we distiguish in this case\nthe indices $j$ and $\\bar j$. The summation over the repeated indices in equations (\\ref{metrS4}), (\\ref{metrS4tilde})\n and in all the formulas in Sections~\\ref{section2}--\\ref{section4} is assumed, as usual.}\nsuch that the metric is, on one hand,\n \\begin{gather}\n\\label{metrS4}\nds^2 = \\frac {2 d w_j d \\bar w_j}{(1 + \\bar w w)^2} .\n \\end{gather}\n($\\bar w w \\equiv \\bar w_j w_j = (x^2 + y^2 + z^2 + t^2)\/2$) and, on the other hand,\n\\begin{gather}\n\\label{metrS4tilde}\nds^2 = \\frac {2 d \\tilde w_j d {\\tilde {\\bar w}}_j}{(1 + {\\tilde {\\bar w}} \\tilde w)^2} .\n \\end{gather}\nBut the relationship $\\tilde w_j = \\bar w_j\/(\\bar w w)$ {\\it is} not holomorphic anymore meaning that $S^4$ is not\ncomplex.\n\nFor complex compact manifolds, one can consider a set of holomorphic $(p,0)$-forms, introduce the operator of exterior holomorphic\nderivative $\\partial$, its Hermitian conjugate $\\partial^\\dagger$ and def\\\/ine thereby the {\\it Dolbeault} complex~(see e.g.~\\cite{EGH}).\nThe operators $\\partial$ and $\\partial^\\dagger$ are nilpotent and the Hermitian\nDolbeault Laplacian $\\partial \\partial^\\dagger +\n\\partial^\\dagger \\partial$ commutes with both $\\partial$ and $\\partial^\\dagger$.\nThis algebra is isomorphic to the simplest supersymmetry algebra,\n \\[\nQ^2 = \\bar Q^2 = 0, \\qquad \\{Q, \\bar Q\\} = H .\n \\]\n The supersymmetric description of the Dolbeault complex for any (not necessarily K\\\"ahler) complex manifold\nhas been constructed in recent~\\cite{IvSm}. The superf\\\/ield action\n(f\\\/irst written in~\\cite{Hull})\n is expressed in terms of $d+d$ chiral and antichiral\nsuperf\\\/ields\n\\begin{gather*}\nW^j = w^j + \\sqrt{2} \\theta \\psi^j - i\\theta \\bar \\theta \\dot{w}^j, \\qquad\n\\bar W^{\\bar j} = \\bar w^{\\bar j} - \\sqrt{2} \\bar\\theta \\bar\\psi^{\\bar j} + i\\theta \\bar \\theta \\dot{\\bar w}^{\\bar j} ,\n\\\\\nS = \\int dt d^2\\theta \\left[ - \\frac 14 h_{j\\bar k} \\big(W^l, \\bar W^{\\bar l}\\big)\n {DW^j \\bar D \\bar W^{\\bar k}} + G(\\bar W, W) \\right] .\n \\end{gather*}\nDeriving with this action the component Lagrangian,\nthen classical and quantum Hamiltonian, using the N\\\"other theorem, and accurately\nresolving the ordering ambiguities~\\cite{howto}, we arrive at the expressions for the quantum supercharges\n \\begin{gather}\n Q = \\psi^c e^k_c\\left[\\Pi_k -\\frac{i}{4} \\partial_k (\\ln \\det h)\n+ i \\psi{\\,}^b \\bar\\psi{\\,}^{\\bar a} \\Omega_{k, \\bar a b}\\right], \\nonumber\\\\\n \\bar Q = \\bar\\psi{\\,}^{\\bar c} e^{\\bar k}_{\\bar c}\n\\left[\\bar\\Pi_{\\bar k} -\\frac{i}{4} \\partial_{\\bar k} (\\ln \\det h)\n+ i \\bar\\psi{\\,}^{\\bar b} \\psi{\\,}^{a} \\bar{\\Omega}_{\\bar k, a \\bar b}\\right] , \\label{Qcovgen}\n\\end{gather}\nwhere $e_j^c$ are the vielbeins, $e_j^c \\bar e_{\\bar k}^{\\bar c} = h_{j\\bar k}$,\nchosen such that $\\det e = \\det \\bar e = \\sqrt{\\det h}$,\n\\[\n\\Omega_{j, \\bar b a} \\equiv \\Omega_{j, \\ a}^{\\ \\ b} = e^b_p \\big(\\partial_j e^p_a + \\Gamma^p_{jk} e^k_a\\big)\n\\]\n(and the complex conjugate $\\bar \\Omega_{\\bar j, b \\bar a} \\equiv \\Omega_{\\bar j, \\ \\bar a}^{\\ \\ \\bar b} $)\nare the holomorphic and antiholomorphic components of the standard Levi-Civita spin connections\\footnote{In the K\\\"ahler case, nonholomorphic components like $\\Omega_{j \\ \\bar b}^{\\ a}$ vanish. For generic\ncomplex manifold, they do not vanish (though they vanish again for some special torsionful connections (\\ref{hatOm})\nbelow) but {\\it do} not enter the supercharges~(\\ref{Qcovgen}). We refer the reader to~\\cite{IvSm} and\nto recent~\\cite{HRR} for further pedagogical explanations.},\n$\\bar \\psi^{\\bar a} = \\partial\/\\partial \\psi^a$, and\n \\begin{gather*}\n\\Pi_k = -i\\left(\\frac{\\partial}{\\partial z^k} - \\partial_k G\\right), \\qquad\n\\bar\\Pi_{\\bar k} = -i \\left(\\frac{\\partial}{\\partial \\bar z{\\,}^{\\bar k}} + \\partial_{\\bar k} G\\right).\n\\end{gather*}\nare the covariant derivatives involving the gauge f\\\/ield\n\\begin{gather}\n\\label{A}\nA_{j,\\bar k} = (-i\\partial_j G, i\\bar \\partial_{\\bar k} G) .\n \\end{gather}\n\n The quantum supercharges (\\ref{Qcovgen}) act on the wave\nfunctions\n \\begin{gather*}\n\\Psi\\big(w^j, \\bar w^{\\bar k}; \\psi^a\\big) = A^{(0)}\\big(w^j, \\bar w^{\\bar k}\\big) + \\psi^a A^{(1)}_a\\big(w^j, \\bar w^{\\bar k}\\big) + \\cdots\n+ \\psi^{a_1}\\cdots \\psi^{a_d}A^{(d)}_{[a_1 \\cdots a_d]}\\big(w^j, \\bar w^{\\bar k}\\big).\n \\end{gather*}\n The components of this wave function $A^{(0)}$, $A^{(1)}_a $, etc. can be mapped onto the space of\nthe holomorphic forms $A^{(0)}$, $e^a_j A^{(1)}_a dw^j$, etc. A $(p,0)$-form corresponds to the wave\nfunction with the eigenvalue $p \\equiv F$ of the fermion charge operator, $\\bar F = \\psi^a \\bar \\psi^{a}$.\n Each component is normalized with the covariant measure,\n \\begin{gather}\n\\label{measure}\n \\mu d^Dx = \\sqrt {\\det g} d^Dx = \\det h d^d w d^d \\bar w .\n \\end{gather}\nThe supercharges (\\ref{Qcovgen}) are conjugate to each other with respect to this measure,\n$\\bar Q = \\mu^{-1} Q^\\dagger \\mu$, where $Q^\\dagger$ is a ``naive'' Hermitian conjugate.\n\n\nIt was shown in \\cite{IvSm} that the supercharge $Q$ in (\\ref{Qcovgen}) is isomorphic in this setting\nto the exterior derivative operator $\\partial$ and the Dolbeault complex is reproduced,\nif choosing the function $G$ in a~special way,\n \\begin{gather}\n\\label{G}\nG = \\frac 14 \\ln \\det h .\n \\end{gather}\n Another distinguished choice is $G = -(1\/4) \\ln \\det h $ when the operator\n$\\bar Q$ is isomorphic to the antiholomorphic exterior derivative $\\bar \\partial$, and we arrive at the anti-Dolbeault\ncomplex. For an arbitrary $G$, we are dealing with a {\\it twisted} Dolbeault complex.\n\nThe Hamiltonian is given by the expression\n \\begin{gather}\nH = - \\frac 12 \\triangle^{\\rm cov} + \\ \\frac 18 \\left (R - \\frac 12 h^{\\bar k j}h^{\\bar l t}h^{\\bar i n}\n C_{j\\,t\\, \\bar i}\\,C_{\\bar k\\,\\bar l\\, n}\\right) \\nonumber\\\\\n\\phantom{H =}{} - 2 \\langle \\psi^a \\bar \\psi^{\\bar b} \\rangle\\,e^k_a e^{\\bar l}_{\\bar b}\\partial_k\\partial_{\\bar l} G\n- \\langle \\psi^a \\psi^c \\bar \\psi^{\\bar b} \\bar \\psi^{\\bar d} \\rangle\n e^t_a e^j_c e^{\\bar l}_{\\bar b} e^{\\bar k}_{\\bar d} (\\partial_t\\partial_{\\bar l}{\\,}h_{j\\bar k}) . \\label{kvantH}\n\\end{gather}\n Here, $\\langle \\ldots \\rangle$ denotes the Weyl-ordered products of fermions, $\\langle \\psi^a \\bar \\psi^{\\bar b} \\rangle =\n (\\psi^a \\bar \\psi^{\\bar b} - \\bar\\psi^{\\bar b} \\psi^a)\/2 $, etc.\n$R$~is the standard scalar curvature\nof the metric $h_{j\\bar k}$, while\n\\begin{gather}\n\\label{Cikl}\nC_{j k\\bar l} = \\partial_{k} h_{j\\bar l} - \\partial_{j} h_{k\\bar l} , \\qquad\nC_{\\bar j \\bar k l} = (C_{j k \\bar l})^* = \\partial_{\\bar k} h_{l \\bar j} - \\partial_{\\bar j} h_{l \\bar k}\n\\end{gather}\nis the metric-dependent torsion tensor. The covariant Laplacian\n $\\triangle^{\\rm cov}$ is def\\\/ined with taking into account the torsion,\n\\begin{gather*}\n-\\triangle^{\\rm cov} = h^{\\bar k j} \\big( {\\cal P}_j {\\bar {\\cal P}}_{\\bar k} +\ni \\hat \\Gamma^{\\bar q}_{j \\bar k} {\\bar {\\cal P}}_{\\bar q}\n+ {\\bar {\\cal P}}_{\\bar k} {\\cal P}_j + i \\hat \\Gamma^{s}_{\\bar k j} {{\\cal P}}_s \\big),\n \\end{gather*}\nwhere ${\\cal P}_j = \\Pi_j + i \\hat \\Omega_{j, \\bar b a} \\langle\\psi^a \\bar \\psi^{\\bar b} \\rangle $ and\n ${\\bar {\\cal P}}_{\\bar k} = \\bar \\Pi_{\\bar k} -\ni \\hat {\\bar \\Omega}_{\\bar k, a \\bar b} \\langle \\psi^a \\bar \\psi^{\\bar b} \\rangle $\n with some particular torsionfull af\\\/f\\\/ine and spin connections\n(the so called Bismut connections\n\\cite{Bismut}),\n \\begin{gather}\n\\hat{\\Gamma}^M_{NK} = \\Gamma^M_{NK} + \\frac{1}{2}g^{ML}C_{LNK}, \\label{genGamma}\n\\\\\n\\hat \\Omega_{M, AB} = \\Omega_{M, AB} + \\frac{1}{2}e_A^K e_B^L C_{KML}, \\qquad M \\equiv \\{m, \\bar m\\}. \\label{hatOm}\n\\end{gather}\n\nNote that this rather complicated expression for the Hamiltonian is greatly simplif\\\/ied in the K\\\"ahler\ncase. Then the torsion (\\ref{Cikl}) vanishes, the 4-fermion term in (\\ref{kvantH}) vanishes too, and{\\samepage\n \\begin{gather*}\nH_{\\rm K\\ddot{a}hl} = - \\frac 12 \\triangle^{\\rm cov} + \\frac R8\n- 2 \\langle \\psi^a \\bar \\psi^{\\bar b} \\rangle e^k_a e^{\\bar l}_{\\bar b}\\partial_k\\partial_{\\bar l} G ,\n\\end{gather*}\n where now $-\\triangle^{\\rm cov} = h^{\\bar k j} \\left( {\\cal P}_j {\\bar {\\cal P}}_{\\bar k} +\n{\\bar {\\cal P}}_{\\bar k} {\\cal P}_j \\right)$\nwith $\\hat \\Omega_{j, \\bar b a} = \\Omega_{j, \\bar b a} =\n e^{\\bar k}_{\\bar b} \\partial_j e^{\\bar a}_{\\bar k}$.}\n\nIn this paper, we are interested, however, with $S^4$, which is not K\\\"ahler and, as was mentioned,\nnot even globally complex.\nThis notwithstanding, one can write the supercharges~(\\ref{Qcovgen})\nand the Hamiltonian~(\\ref{kvantH}) with the metric~(\\ref{metrS4}), which is\nwell def\\\/ined everywhere on $S^4$ except the north pole and study the spectrum.\n\n\n\nFor sure, to determine the spectrum, we have to def\\\/ine f\\\/irst the {\\it spectral problem} and to specify\nthe boundary conditions for the wave functions. There are two dif\\\/ferent reasonable choices:\n$(i)$~We can consider the functions that are regular on $S^4$. $(ii)$~We can allow the singularity at the pole, but\nrequire that the functions are square integrable with the measure~(\\ref{measure}),\n \\begin{gather*}\n \\int \\frac {|\\Psi|^2 d^2 w d^2 \\bar w}{(1+\\bar w w)^4} < \\infty .\n \\end{gather*}\nIt turns out that, for the {\\it first} spectral problem, the Hamiltonian is well def\\\/ined and Hermitian. However, the Hilbert\nspace of all nonsingular on $S^4$ functions does not constitute the domain of the supercharges: there exist nonsingular functions\n$\\Psi$ such that $ Q\\Psi$ are singular. In physical language, this means that the supersymmetry is broken~-- some states do not\nhave superpartners. In mathematical language, this means that the Dolbeault complex is not well def\\\/ined on the manifolds\nthat are not complex, of which $S^4$ is an example.\n\nWhat is, however, rather nontrivial and somewhat surprising is that,\nin the Hilbert space of square integrable functions, everything works f\\\/ine. In the main body of the paper,\nwe will show that all excited square integrable states of the Hamiltonian are doubly degenerate (i.e.\\ supersymmetry {\\it is} there)\nand that there are 3 bosonic zero modes such that the Witten index of this system is $I_W = 3$. In other words, even though\n the Dolbeault complex is not well def\\\/ined on $S^4$, there is a nontrivial\nself-consistent way to def\\\/ine it on $S^4\\backslash \\{\\cdot\\}$.\n\nThere is a kinship between the problem under consideration and a problem of the Dirac complex on $S^2$ with noninteger\nmagnetic f\\\/lux~\\cite{flux}. In both cases, the requirement for the spectrum to be supersymmetric brings about\nrestrictions on the Hilbert space (see also~\\cite{SSV}). However, for a noninteger f\\\/lux,\nthese restrictions are extremely stringent: they simply leave the Hilbert space empty,\na Dirac complex with noninteger f\\\/lux is not def\\\/ined. And, for $S^4$, the Hilbert\nspace of regular wave functions is not supersymmetric, while\nits {\\it extension}~-- the space of square integrable functions is.\n\n\n\\section{The Dolbeault Hamiltonian and its spectrum}\\label{section2}\n\nOn $S^4$ with the metric (\\ref{metrS4}) and with $G(\\bar W, W)$ given by~(\\ref{G}),\n the supercharges (\\ref{Qcovgen}) acquire the following\nsimple form\n\\begin{gather}\n Q =i(1 + \\bar w w) \\psi_j \\partial_j + i \\psi_j \\psi_k\n\\bar\\psi_j \\bar w_k , \\nonumber \\\\\n\\bar Q = i \\bar \\psi_j \\left[ (1 + \\bar w w) \\bar \\partial_j -\n2w_j \\right] + i \\bar \\psi_j \\bar \\psi_k \\psi_j w_k .\\label{QS4}\n \\end{gather}\nThe complicated expression (\\ref{kvantH}) for the Hamiltonian also simplif\\\/ies a lot. There are three sectors:\n$F=0$, $F=1$, and $F=2$.\nConsider f\\\/irst the sector $F=0$. We obtain\n \\begin{gather}\n\\label{HF0}\nH^{F=0} = -(1 + \\bar w w)^2 \\partial_j \\bar\\partial_{j} + 2(1 + \\bar w w) w_j \\partial_j .\n\\end{gather}\n It is instructive to compare this Dolbeault Laplacian with the standard covariant Laplacian on~$S^4$,\n\\begin{gather}\n\\label{lapl}\n-\\triangle_{S^4} = -(1 + \\bar w w)^2 \\bar \\partial_j \\partial_{j} +\n(1 + \\bar w w)\n\\big(w_j \\partial_j + \\bar w_j \\bar \\partial_j \\big) .\n\\end{gather}\nNote that both (\\ref{HF0}) and (\\ref{lapl}) commute with the angular momentum operator\n$m = w_j \\partial_j - \\bar w_j \\bar \\partial_j $.\\footnote{The Hamiltonian (\\ref{HF0}) commutes\nalso with two other generators of $SU(2)$ such that the states represent $SU(2)$-multiplets.\nThe standard Laplacian (\\ref{lapl}) has $O(5)$ symmetry.} The eigenvalues of $ m $ are integer.\n\nThe supercharges (\\ref{QS4}) admit 3 normalizable zero modes satisfying\n $Q\\Psi^{(0)} = \\bar Q\\Psi^{(0)} = 0$ in the sector\n$F=0$,\n \\begin{gather}\n\\label{zero}\n \\Psi^{(0)} = 1, \\bar w_1, \\bar w_2 .\n \\end{gather}\nThey represent the ground states of the Hamiltonian (\\ref{HF0}).\n\nConsider now excited states.\n The eigenfunctions of the Hamiltonian can be sought for in the form\n \\begin{gather}\n\\label{Ansatz}\n \\Psi_{ms} = S_{ms} F_{ms}(\\bar w w ) ,\n \\end{gather}\nwhere $S_{ms}$ ($ m = 0, \\pm 1, \\ldots$; $ s = 0,1,\\ldots$) are mutually orthogonal tensor structures\nthat vanish under the action of the ``naive Laplacian''\n$\\bar \\partial_{j} \\partial_j$. Each structure $S_{ms}$ has $2s + |m| + 1$ independent components\n(and the corresponding energy level has degeneracy $2s + |m| + 1$). The explicit form of f\\\/irst few such structures is\n \\begin{gather*}\n S_{00} = 1 , \\qquad S_{01} = w_j \\bar w_k - \\frac {\\bar w w}2 \\delta_{jk} ,\n\\nonumber \\\\\nS_{02} = w_i w_j \\bar w_k \\bar w_l - \\frac {\\bar w w}4 \\left( w_i \\bar w_k \\delta_{jl} +\n w_i \\bar w_l \\delta_{jk} + w_j \\bar w_k \\delta_{il} + w_j \\bar w_l \\delta_{ik} \\right)\n+ \\frac {(\\bar w w)^2}{12} (\\delta_{ik} \\delta_{jl} + \\delta_{il} \\delta_{jk} ) , \\nonumber \\\\\n S_{10} = w_j, \\qquad S_{11} = w_i w_j \\bar w_k - \\frac {\\bar w w}3 ( w_i \\delta_{jk} +\n w_j \\delta_{ik} ) , \\qquad\n S_{-1,0} = \\bar w_j .\n \\end{gather*}\n It is straightforward to see that the action of the Hamiltonian on the Ansatz~(\\ref{Ansatz}) preserves its\ntensor form. The radial dependence is then determined from the solution of scalar spectral equations for $F_{ms}$.\nIt is convenient to introduce the variable\n\\begin{gather*}\nz = \\frac {1-\\bar w w}{1+\\bar w w}\n \\end{gather*}\n(it is nothing but $\\cos \\theta$, $\\theta$ being the polar angle on $S^4$).\n\nThe spectral equations acquire then the form\n \\begin{gather*}\n(z^2-1) F''(z) + 2(2z + m + 2s) F'(z) + \\frac {4(m + s)}{1+z} F(z) = \\lambda F(z) , \\qquad m \\geq 0 ,\\nonumber \\\\\n (z^2-1) F''(z) + 2(2z + |m| + 2s) F'(z) + \\frac {4s}{1+z} F(z) = \\lambda F(z) , \\qquad m \\leq 0 \n \\end{gather*}\n Their formal solutions are\n\\begin{gather}\nF(z) = (1 + z)^{\\gamma_{ms}} P_n^{|m|+ 2s +1, \\pm \\Delta_{ms}}(z) ,\n\\nonumber \\\\\n\\lambda_{msn} = \\gamma_{ms}^2 + 3\\gamma_{ms} + n(n + |m| + 2s\n+ 2 \\pm \\Delta_{ms} ) ,\\label{solFz}\n \\end{gather}\nwith\n \\begin{gather}\n\\label{gam}\n \\gamma_{ms} = \\frac {|m| + 2s -1 \\pm \\Delta_{ms} }2\n \\end{gather}\n and\n \\begin{gather*}\n \\Delta_{ms} = \\sqrt{(1-m - 2s)^2 + 8(m + s)} , \\qquad m \\geq 0 , \\nonumber \\\\\n\\Delta_{ms} = \\sqrt{(1-|m| - 2s)^2 + 8s} , \\qquad m \\leq 0 .\n \\end{gather*}\n$P_n^{\\alpha, \\beta}$ ($n = 0,1,\\ldots$) are the Jacobi polynomials,\n\\begin{gather*}\nP^{\\alpha,\\beta}_n(z) = \\frac 1{2^n} \\sum_{k=0}^n \\begin{pmatrix} n+\\alpha \\\\ k \\end{pmatrix}\n \\begin{pmatrix} n+\\beta \\\\ n- k \\end{pmatrix} (1 + z)^k (z-1)^{n-k} .\n \\end{gather*}\nFor $\\alpha > -1$, $\\beta > -1$, the Jacobi polynomials are mutually orthogonal\non the interval $z \\in (-1,1)$ with the weight\n$\\mu = (1-z)^\\alpha (1+z)^\\beta$.\n\nNot all the solutions in (\\ref{solFz}) are admissible, however. One can observe the following:\n \\begin{itemize}\\itemsep=0pt\n\\item First of all, all the solutions with $s > 0$ and\/or $m > 0$ and the negative\nsign of $\\Delta_{ms}$ in~(\\ref{solFz}),~(\\ref{gam}) are not square integrable and should not be included in\nthe spectrum.\nIf $s=0$ and $m \\leq 0$, the solution with negative sign of $\\Delta_{ms}$ are not independent being expressed into\nthe solutions with positive sign in virtue of the identity\n \\begin{gather}\n\\label{relJacobi}\nP^{\\alpha, -\\beta}_{n+\\beta}(z) = 2^{-\\beta} (z+1)^\\beta \\frac {n!\n(n+\\alpha + \\beta)!}{(n+\\alpha)! (n+\\beta)!} P_n^{\\alpha, \\beta}(z) ,\n \\end{gather}\nwhich holds for integer $\\alpha$~\\cite{WuYang}.\n\\item On the other hand, the solutions with positive sign of $\\Delta_{ms}$\nand with nonnegative $m$ are all not only square integrable,\nbut also nonsingular on $S^4$. In addition, they belong to the domain of $Q$:\n$ Q \\Psi_{m \\geq 0, s}$ is never singular.\n \\item Most of the solutions with $m < 0 $ also have this property. However, there are three distinguished families of\nsolutions: the solutions\n \\begin{gather}\n\\label{m=-1}\n \\Psi_{-1,0,n} = \\bar w_j P_n^{2,0}(z) ,\n \\end{gather}\n the solutions\n \\begin{gather}\n\\label{m=-2}\n \\Psi_{-2,0,n} = \\frac {\\bar w_j \\bar w_k}{1 + \\bar w w} P_n^{3,1}(z) ,\n \\end{gather}\nand the solutions\n\\begin{gather}\n\\label{m=-3}\n \\Psi_{-3,0,n} = \\frac {\\bar w_j \\bar w_k \\bar w_l}{(1 + \\bar w w)^2 }\n P_n^{4,2}(z) .\n \\end{gather}\n\n$(i)$ The functions (\\ref{m=-1}) are all singular at inf\\\/inity, but integrable. Two lowest such functions $\\Psi = \\bar w_j$\nare zero modes of the Hamiltonian (\\ref{HF0}). The functions ${ { Q}} \\Psi_{-1,0,n}$ are less singular: they do not grow at inf\\\/inity\n(though do not have a def\\\/inite value there when $n > 0$).\n\n$(ii)$ The functions (\\ref{m=-2}) are bounded at inf\\\/inity. The supercharge action produces growing functions,\n${{ Q}} \\Psi_{-2,0,n} (w=\\infty) = \\infty$. Still, ${{ Q}} \\Psi_{-2,0,n}$ is square integrable.\n\n$(iii)$ The functions (\\ref{m=-3}) are regular at inf\\\/inity. The supercharge action produces singular bounded functions.\n\n\n\n\\item Note that if $\\Psi$ is a\n{\\it non-normalizable} eigenfunction in the sector $F=0$, the function $Q \\Psi$ is also not\nnormalizable. Indeed, the action of $ Q$ brings about\ngenerically an extra power of $|w|$, which makes the divergence still stronger.\n An exception would only be provided by the\nfunctions with the asymptotics $\\propto \\bar w_{j_1} \\cdots \\bar w_{j_k}$ at inf\\\/inity. But the only {\\it eigenfunctions}\n with\nsuch asymptotics are written in equation~(\\ref{m=-1}). They are normalizable.\n \\end{itemize}\n\n\n\nConsider now the sector $F=2$. The Hamiltonian is\n\\begin{gather*}\nH^{F=2} \\ =\\ -(1 + \\bar w w)^2 \\bar \\partial_j \\partial_j + 2(1 + \\bar w w) w_j \\partial_j\n+ 2(2 + \\bar w w) .\n\\end{gather*}\n The eigenfunctions have the same form as in (\\ref{Ansatz}), (\\ref{solFz}), (\\ref{gam}) with\n modif\\\/ied\n\\begin{gather*}\n \\Delta_{ms}^{F=2} = \\sqrt{(1- m - 2s)^2 + 8(m+s+1)} , \\qquad m \\geq 0 , \\nonumber \\\\\n \\Delta_{ms}^{F=2} = \\sqrt{(1-|m| - 2s)^2 + 8( s+1)} , \\qquad m \\leq 0 \n \\end{gather*}\nAgain, almost all functions with negative sign in (\\ref{gam}) are not normalizable. The exceptions are the sectors\n$s=0$, $m = 0, -2$, where the functions with the negative sign are expressed into the functions with positive sign.\n Speaking of the latter, they are not only normalizable, but also nonsingular in this case.\n All these functions are annihilated by $Q$ and belong\nto the domain of~${ {\\bar Q}}$,\n${{\\bar Q}} \\Psi^{F=2}$ being regular on~$S^4$.\n\nIn the sector $F=1$, the wave functions have two components,\n$\\Psi^{F=1} = \\psi_j C_j(\\bar w_k, w_k)$. No new zero modes appear. Indeed,\nif the Hamiltonian\n{\\it had} zero modes in this sector, they would\nsatisfy the conditions $Q \\Psi = { {\\bar Q}} \\Psi = 0$ giving\n \\begin{gather*}\n (1 + \\bar w w) \\partial_{[j} C_{k]} - \\bar w_{[j} C_{k]} = 0 , \\nonumber \\\\\n (1 + \\bar w w) \\bar \\partial_k C_k - 3 w_k C_k = 0 \n \\end{gather*}\nThe f\\\/irst equation can be rewritten as\n\\[\n \\partial_{[j} \\left[ \\frac {C_{k]}}{1+\\bar w w} \\right] = 0\n \\]\nwith a generic solution $C_k = (1+ \\bar w w) \\partial_k \\Phi$. Then the second equation gives\n$H^{F=0} \\Phi = 0$. If $C_k$ is normalizable, $\\Phi$ must also be normalizable (modulo a pure antiholomorphic\npart). But we have seen that the only normalizable zero modes of $H^{F=0}$ are 1 and $\\bar w_j$\nannihilated by holomorphic derivatives and giving $C_k = 0$.\\footnote{Note that if one lifts the normalizability condition, a nontrivial solution of the equation\n$H^{F=0} \\Phi = 0$ exists: $\\Phi = \\bar w w + 2 \\ln (\\bar w w) - \\frac 1{\\bar w w}$ giving\n $C_k = \\bar w_k (1 + \\bar w w)^3\/(\\bar w w)^2$.}\n\nTo f\\\/ind the nonzero modes in the sector $F=1$, one needs not to solve the Schr\\\"odinger equation again. All such normalizable\nfunctions are obtained by the action of $Q$ or ${ {\\bar Q}}$ onto the normalizable functions in the sectors $F=0$\nor $F=2$, correspondingly. This follows from the last itemized statement above, which is valid also in the sector $F=2$.\n By construction, these functions are annihilated by $Q$ or $\\bar Q$ and belong to the domain\nof $\\bar Q$ or $Q$, correspondingly.\n\n\\subsection{Twisted Dolbeault complex}\n\nThe Hamiltonian (\\ref{kvantH}) is supersymmetric not only under the condition\n(\\ref{G}) that distinguishes the pure Dolbeault complex, but also with other choices of $G$ describing\n twisted Dolbeault complexes.\n\n First of all, we can set $G=0$. As was shown in \\cite{IvSm},\nthe Hamiltonian (\\ref{kvantH}) with $G=0$ coincides with the extended $N=4$ supersymmetric Hamiltonian\nwritten in~\\cite{Konush},\n \\begin{gather}\n\\label{HamKon}\nH = - \\frac 12 f^3 \\partial_M^2 \\frac 1f - \\frac 12\\, \\psi\n\\sigma_{[M}^\\dagger \\sigma_{N]} \\bar \\psi f (\\partial_M f) \\partial_N + f (\\partial^2 f)\n\\left ( \\psi \\bar \\psi - \\frac 12 (\\psi \\bar \\psi)^2 \\right) ,\n \\end{gather}\nwith $f = 1 + x_M^2\/2$.\n\nThis model belongs to the class of the so called ``hyperk\\\"ahler with torsion'' (HKT) mo\\-dels~\\cite{Howe},\nwhich were classif\\\/ied using the harmonic superspace formalism in recent \\cite{Delduc}.\nThe Hamiltonian~(\\ref{HamKon}) does not admit normalizable zero-energy solutions and its index is zero.\n\n\nConsider now a model with\n \\begin{gather*}\n G = \\frac q4 \\ln \\det h = -q \\ln(1 + \\bar w w)\n \\end{gather*}\nwith an integer $q >1$. The supercharges are then\n \\begin{gather*}\n Q = i(1 + \\bar w w) \\psi_j \\partial_j + i (q-1) \\bar w w \\psi_j \\bar w_j\n+\ni \\psi_j \\psi_k \\bar\\psi_j \\bar w_k , \\nonumber \\\\\n\\bar Q = i \\bar \\psi_j \\left[ (1 + \\bar w w) \\bar \\partial_j -\n(q+1) w_j \\right] + i \\bar \\psi_j \\bar \\psi_k \\psi_j w_k \n \\end{gather*}\n The zero modes all dwell in the sector $F=0$. They have the form\n \\begin{gather*}\n\\Psi^{(0)} = \\frac {P(\\bar w)}{(1+\\bar w w)^{q-1}} ,\n \\end{gather*}\nwhere $P(\\bar w)$ is an antiholomorphic polynomial of degree $2q-1$.\nIt has $2q^2 + q$ independent coef\\\/f\\\/icients,\nwhich gives $2q^2 + q$ independent zero modes.\n\nWhen $q$ is negative, the analysis is similar.\nIt gives $2q^2-q$ zero modes in the sector $F=2$. The same consideration\n as in the\npure Dolbeault case displays the absence of the normalized\nzero modes in the sector $F=1$.\n The f\\\/inal result for the index of the twisted Dolbeault complex~is\n \\begin{gather}\n\\label{Indexq}\n I(q) = 2q^2 + |q| .\n \\end{gather}\n\n\n\n\\section{The index and the functional integral}\\label{section3}\n\n\nAs was mentioned, for pure Dolbeault complex, there are 3 bosonic zero modes (\\ref{zero})\nin the sector $F=0$ and no zero modes in the\nother sectors. This means\nthat the Witten index of this system,\n\\begin{gather*}\nI_W = {\\rm Tr} \\big\\{(-1)^F e^{-\\beta H} \\big\\}\n \\end{gather*}\n is equal to 3.\n\nFor {\\it compact} complex manifolds, the Witten index of the supersymmetric Hamiltonian (\\ref{kvantH}) under\nthe condition (\\ref{G}) is known to mathematicians by the name of {\\it arithmetic genus} of the manifold. This invariant\nadmits an integral representation known as the Hirzebruch--Riemann--Roch theorem \\cite{Hirz},\\footnote{Following the ideas of \\cite{A-GFW}, it has been recently derived also in physical way\nby studying the path integral for the supersymmetric partition function~\\cite{HRR}.}\n \\begin{gather*}\nI \\ =\\ \\int \\, {\\rm Td}(TM) ,\n \\end{gather*}\nwhere the symbol Td$(TM)$ ({\\it Todd class of a complex tangent bundle} associated with the manifold~$M$) is spelled out as\n \\begin{gather}\n\\label{Todd}\n {\\rm Td}(TM) =\n \\prod_{\\alpha = 1}^n \\frac {\\lambda_\\alpha\/2\\pi}{1 - e^{-\\lambda_\\alpha\/2\\pi} } ,\n \\end{gather}\nwhere $\\lambda_\\alpha$ are eigenvalues of the curvature matrix corresponding to this bundle\\footnote{For (\\ref{Todd}) to be correct and simply to make sense, the connection and its curvature should respect the complex\nstructure. For example, the Bismut connection (\\ref{genGamma}) is appropriate for this purpose, while the usual\ntorsionless Levi-Civita connection is not, if the manifold is not K\\\"ahler.}.\n\nThe representation (\\ref{Todd}) can be derived by using the fact that the sum of the supercharges~(\\ref{QS4}) can be interpreted\nas a Dirac operator involving an Abelian gauge f\\\/ield and torsions. (The presence of torsions is a complication\nthat distinguishes the HRR theorem from a version of the Atiyah--Singer theorem\ndiscussed usually by physicists.) It is important in this derivation that the gauge f\\\/ield\nrepresents a regular f\\\/iber bundle on the manifold, while torsions are regular tensors.\n\nIn our $ S^4$ case, however, these\nconditions are not fulf\\\/illed: the gauge f\\\/ield and the torsion are singular at $w = \\infty$.\nIndeed, the torsion (\\ref{Cikl}) with the metric (\\ref{metrS4}) behaves at inf\\\/inity as $\\sim |x|^{-5}$.\nThen $g^{MN} g^{PQ} g^{ST} C_{MPS} C_{NQT} \\sim (x^{4})^3 \\cdot (x^{-5})^2 \\sim x^2$ and diverges.\nThe gauge f\\\/ield (\\ref{A}), (\\ref{G}) is rather peculiar. It is {\\it disguised} as a benign f\\\/iber bundle having\n an integer Chern class\n \\begin{gather}\n\\label{Chern}\n {\\rm Ch}_2 = \\frac 1{8\\pi^2} \\int F \\wedge F = 2 .\n \\end{gather}\nHowever, a topologically nontrivial $U(1)$ bundle on $S^4$ {\\it does} not exist because $\\pi_3[U(1)] = 1$.\\footnote{On the other hand, topologically\nnontrivial bundles on $S^4$ with non-Abelian gauge groups ({\\it instantons}), of course, exist.} Indeed, the f\\\/ield strength tensor\nis singular in this case, which manifests itself in the fact that the ``action integral'' $\\sim\n\\int d^4 x \\sqrt{g} F_{MN} F^{MN}$ diverges logarithmically.\n\nFor such a singular Dirac operator, one cannot get rid of torsions by a smooth deformation\n(a key step in the derivation of~(\\ref{Todd})), because the index integral may in this case acquire contributions from total\nderivatives of singular expressions. Moreover, with all probability, the Dirac operator on $S^4$ with the singular\nf\\\/ield~(\\ref{A}) and\n{\\it without} torsions does not describe a benign supersymmetric system~-- the situation must be the same as for\nthe gauge f\\\/ield on~$S^2$ with non-integer magnetic f\\\/lux~\\cite{flux}.\n\nHaving no further mathematical methods at our disposal (at least, we are not aware of such methods), we can try to\ncalculate the Witten index in a physical way by evaluating directly\n the corresponding path integral,\n \\begin{gather*}\n I = \\int d\\mu \\exp \\left\\{ - \\int_0^\\beta L_E(\\tau) d\\tau \\right \\} ,\n\\end{gather*}\n where $L_E$ is the Euclidean Lagrangian of our supersymmetric quantum system, $d\\mu$ is the appropriate\nfunctional integral measure, and the periodic boundary conditions are imposed onto all variables.\n\n For most supersymmetric systems, this integral is reduced for small $\\beta$ to an ordinary phase space integral \\cite{Cecotti}.\nThis is true e.g.\\ for a supersymmetric Hamiltonian describing the de Rham complex on a compact manifold,\nwhere the Witten\nindex is given by its Euler characteristics. For the Dirac complex on compact\nmanifolds, the situation is more complicated, a naive semiclassical reduction is not\njustif\\\/ied and one has to perform\na honest calculation of the path integral in the one-loop approximation~\\cite{A-GFW}, which is not so trivial\n (see~\\cite{IvSm} for detailed pedagogical explanations). For the Dolbeault complex on compact non-K\\\"ahler\ncomplex manifolds, the life is still more dif\\\/f\\\/icult.\nGenerically, one has to perform a~two-loop calculation for $4d$ and $6d$ manifolds, a~three-loop\ncalculation for $8d$ and $10d$ manifolds, etc. This complication is due to the appearance of the new 4-fermionic term\nin the Lagrangian,\n \\begin{gather}\nL_E = \\frac{1}{2}\\left[ g_{MN} \\dot x{\\,}^M \\dot x{\\,}^N + g_{MN}\\,\\psi^M \\hat{\\nabla} \\psi^N\n+ \\frac{1}6 \\partial_P C_{MNT} \\psi^P\\psi^M\\psi^N\\psi^T \\right] \\nonumber\\\\\n\\phantom{L_E =}{} - i A_M \\dot{x}^M + \\frac i2 F_{MN} \\psi^M \\psi^N \\label{LE} \n\\end{gather}\n($\\hat{\\nabla} \\psi^M = \\dot{\\psi}^M + \\hat \\Gamma^M_{NK} \\dot{x}^N \\psi^K$ is the Bismut covariant derivative).\nFor example, for a 4-dimensional manifold, the leading (at small $\\beta$) contribution to the index is\n \\begin{gather}\n\\label{Ising}\nI \\sim \\frac 1\\beta \\int d^4 x\\, \\epsilon^{MNPQ} \\partial_M C_{NPQ} .\n \\end{gather}\nFor sure, the integrand is a total derivative here and the integral vanishes, but the appearance of the large factor $1\/\\beta$\ndoes not allow one to ignore 2-loop corrections anymore. They are essential. For $8d$ manifolds, the leading contribution in\nthe integrand is of order $\\sim 1 \/\\beta^2$, and this makes essential 3 loop contributions, etc.\n\nAs was mentioned above, for compact manifolds, one needs not actually to come to grips with these complicated\nmultiloop contributions. One can, instead, perform a smooth deformation that kills the torsion and makes the problem and the\ncorresponding path integral tractable. For a particular class of manifolds where the fermion term\nin~(\\ref{LE}) vanishes (the so called SKT manifolds), this program was in fact carried out in~\\cite{Bismut}. (In this mathematical\npaper, path integrals and\nsupersymmetry were not mentioned and the author described the results in the language of {\\it heat kernel} technique,\n which is equivalent, however, to the path integral approach.) The generic case is discussed in~\\cite{HRR}.\n\nFor $S^4$, we have no other choice than to try to evaluate the path integral directly. In 4~dimensions, this is dif\\\/f\\\/icult,\nbut feasible and, in the case when the Dirac operator involves only torsions, but no extra gauge f\\\/ield ($G=0$ in our language),\nhas been performed in \\cite{Waldron}. The index integral has been represented in these papers as\n \\begin{gather}\nI = - \\frac 1{4\\pi^2} \\int d^4x \\, \\sqrt{g} \\bigg\\{ \\frac 1{2\\beta} \\nabla_M B^M \\nonumber \\\\\n\\phantom{I =}{}+\\frac 1{192} \\frac 1{\\sqrt{g}}\n\\epsilon^{RSKL} \\left[ R_{MNRS} R^{MN}_{\\ \\ \\ \\ KL} + \\frac 12 B_{RS} B_{KL} \\right]\n + \\frac 1{24} \\nabla_M {\\cal K}^M\n\\bigg\\} + {\\cal O} (\\beta)\\label{IWald}\n \\end{gather}\n with\n \\begin{gather}\n\\label{KB}\n{\\cal K}^M = \\left( \\nabla^N \\nabla_N + \\frac 14 B^N B_N +\n\\frac 12 R \\right) B^M, \\qquad B_{MN} = \\nabla_M B_N - \\nabla_N B_M .\n \\end{gather}\nHere $\\nabla_M$ is the standard Levi-Civita covariant derivative and $R_{MNRS}$ is the standard Riemann tensor. $B_M$ is the\naxial vector dual to the torsion tensor,\n \\begin{gather*}\nB^M = \\frac 1 {6\\sqrt{g}} \\epsilon^{MNPQ} C_{NPQ} = 2x^M \\big(1 + x^2\/2\\big) .\n \\end{gather*}\nThe f\\\/irst term in equation~(\\ref{IWald}) is a singular ($\\sim 1\/\\beta$ )\n but vanishing\nintegral of a total derivative. It was discussed before. The last term also represents a total derivative and also vanishes\\footnote{This vanishing is due to a certain cancellation. One can easily check that {\\it individual} terms\n$\\sim \\nabla^N \\nabla_N$ and $\\sim B^N B_N $ in (\\ref{KB})\nare expressed into the integral $\\sim \\int d^4x \\, \\partial_M (x^M\/x^4)$ that does not vanish.\nThese contributions cancel, however, in the sum.}. The ``f\\\/ield strength'' $B_{MN}$ is zero in our case. The quadratic in the\nRiemann tensor integral is proportional to a certain topological invariant called Hirzebruch signature. For $S^4$, it vanishes.\n\nAs a result, the index of the corresponding supersymmetric Hamiltonian vanishes.\nThis agrees with the direct analysis of its spectrum (see the remark after equation~(\\ref{HamKon})).\n\nWhen $G \\neq 0$, the Lagrangian and the Hamiltonian involve an extra gauge f\\\/ield. The functional integral\nfor the index acquires the (tree-level) contribution~(\\ref{Chern}).\n\nOn top of this, there might have been a 1-loop contribution associated with the 4-fermion term.\nTo evaluate it, it is convenient to expand the periodic f\\\/ields $x^M(\\tau)$ and $\\psi^M(\\tau)$\ninto the Fourrier modes,\n \\begin{gather*}\nx^M (\\tau) = x^{M}_0 + \\sum_{m \\neq 0} x^{M}_m e^{2\\pi i m \\tau\/\\beta},\\qquad\n\\psi^M (\\tau) = \\psi^{M}_0 + \\sum_{m \\neq 0} \\psi^{M}_m e^{2\\pi i m \\tau\/\\beta} ,\n \\end{gather*}\nwith integer $m$ ($\\bar x_m^M = x_{-m}^M$, $\\bar \\psi_m^M = \\psi_{-m}^M$). We obtain instead of (\\ref{Ising})\n \\begin{gather}\nI \\sim \\frac 1\\beta \\int d^4 x_0\\, \\epsilon^{MNPQ} \\partial_M C_{NPQ} \\mu\\nonumber\\\\\n\\phantom{I \\sim }{}\n\\times \\prod_{M, m \\neq 0} dx^M_m d\\psi^M_m\n\\exp \\Bigg\\{ {-} \\frac 1{2\\beta} \\sum_m (2\\pi m)^2 x^M_m x^N_{-m}\n \\left( g_{MN} - \\frac{\\beta F_{MN}} {2\\pi m} \\right)\\nonumber \\\\\n\\phantom{I \\sim }{}\n +i \\sum_m (2\\pi m) \\psi^M_m \\psi^N_{-m} \\left( g_{MN} - \\frac{\\beta F_{MN}} {2\\pi m} \\right) \\Bigg\\} ,\\label{Isinggauge}\n \\end{gather}\nwhere $\\partial C$, $g$, $F$, and the inf\\\/inite factor $\\mu$\\,\\footnote{It can be made f\\\/inite when imposing an ultraviolet cutof\\\/f,\nsee equation~(6.27) of~\\cite{IvSm}. When $F=0$,\n\\[\n\\mu \\int \\prod_{M, m \\neq 0} dx^M_m d\\psi^M_m\n\\exp \\{ \\cdots \\} = 1 .\n\\]}\ndepend on the zero coordinate modes $x_0^M$.\n\nThe individual contributions due to bosonic and fermionic Gaussian integrals are nontrivial.\nFor example, the fermionic integral gives\n \\begin{gather*}\n {\\rm fermion\\ factor} = \\prod_{ m = 1}^\\infty \\det \\left\\| \\delta_M^{\\ N} - \\frac {\\beta F_M^{\\ N}}{2\\pi m} \\right \\|\n= \\det\\nolimits^{ 1\/2} \\frac {2\\sin (\\beta F\/2 ) }{\\beta F } .\n \\end{gather*}\n Formally, the correction to unity is proportional to $\\beta^2$, but it multiplies\n${\\rm Tr}\\{ F^2\\} = - F_{MN} F^{MN}$, which grows at inf\\\/inity $\\sim x^4$. The integral is then\nsaturated by large $x$ values, and, as a result, the correction\nis of order $\\beta$. When multiplied by the overall factor~$1\/\\beta$ in front of the integral, this gives a\ncorrection of order 1 to the index.\n\nAnyway, as is clear from (\\ref{Isinggauge}), this fermionic correction exactly cancels the bosonic one\n(obviously, this cancelation is due to supersymmetry), and we have to conclude that the functional\nintegral calculation gives the value (\\ref{Chern}) for the index, which contradicts the direct analysis above\ngiving $I = 3$. In addition, for the twisted Dolbeault complex, we obtain\n \\begin{gather*}\n\nI_{\\rm funct.\\ int.} = 2q^2 ,\n \\end{gather*}\nwhich contradicts the estimate (\\ref{Indexq}) above.\n\nCertainly, this mismatch is disappointing and paradoxical. We want to emphasize, however, that there is\nno {\\it logical} contradiction here. We calculated the functional integral by semiclassical methods expanding\nit in $\\beta$. In particular, we studied only one-loop corrections to the index associated with gauge f\\\/ield, because\ntwo- and higher-loop corrections are suppressed by the naive~$\\beta$ counting.\n We have seen, however, that this expansion breaks down near the singularity where~$\\beta$ is multiplied\nby a large factor~$\\sim x^2$. In this situation, one cannot reliably justify ignoring higher-loop contributions.\nThey can give something (though we do not see at the moment how this can come about).\n\nNote that there are some other examples where the presence of singularities invalidates the\n semiclassical calculation of the path integral. In particular, in~\\cite{Blok}, we constructed SQM systems\nassociated with chiral supersymmetric gauge theories in f\\\/inite volume. The Hamiltonian\nof these systems is singular near the origin, $H \\sim 1\/x^2$. And, though this singularity is of repulsive benign\nnature, unitarity is not broken, and the spectrum of the Hamiltonian is discrete, the semiclassical approximation\nfor the path integral breaks down near the origin. This manifests itself in the senseless\nfractional values of the path integral for the index evaluated at the leading order.\n\nDef\\\/initely, more studies of this very interesting question are necessary.\n\n\n\n\\section[$S^6$]{$\\boldsymbol{S^6}$}\\label{section4}\n\nA similar analysis can be done for $S^6$ and also for higher even-dimensional spheres.\nThe metric of $S^6$ is still given by equation~(\\ref{metrS4}) where now $j = 1,2,3$. The supercharges\nof the SQM system describing\nthe pure Dolbeault complex have almost\nthe same form as for $S^4$,\n\\begin{gather*}\n Q = i(1 + \\bar w w) \\psi_j \\partial_j + i \\psi_j \\psi_k\n\\bar\\psi_j \\bar w_k , \\nonumber \\\\\n\\bar Q = i \\bar \\psi_j \\left[ (1 + \\bar w w) \\bar \\partial_j -\n3w_j \\right] + i \\bar \\psi_j \\bar \\psi_k \\psi_j w_k \n \\end{gather*}\nThe Hamiltonian in the sector $F=0$ is\n\\begin{gather*}\nH^{F=0} = -(1 + \\bar w w)^2 \\partial_j \\bar\\partial_{j} + 4(1 + \\bar w w) w_j\n\\partial_j ,\\qquad j=1,2,3 ,\n\\end{gather*}\nto be compared with the standard covariant Laplacian on $S^6$,\n\\begin{gather*}\n-\\triangle_{S^6} = -(1 + \\bar w w)^2 \\bar \\partial_j \\partial_{j} +\n2(1 + \\bar w w)\n(w_j \\partial_j + \\bar w_j \\bar \\partial_j ) .\n\\end{gather*}\nWe choose the basis\n \\begin{gather*}\n\\label{AnsatzS6}\n \\Psi_{pq} = T_{pq} F(\\bar w w ) ,\n \\end{gather*}\nwhere a tensor structure $T_{pq}$ having $p$ factors $w$ and $q$ factors\n $\\bar w$ and annihilated by $\\partial_j \\bar \\partial_j $\nrepresents a $\\begin{pmatrix} p \\\\ q \\end{pmatrix}$ multiplet of $SU(3)$\nand has $(p+1)(q+1)(p+q+2)\/2$ independent components. For example,\n \\begin{gather*}\nT_{11} = w_j \\bar w_k - \\frac {\\bar w w}3 \\delta_{jk}\n \\end{gather*}\nis an octet\\footnote{The $(p,q)$ notation can also be used for $S^4$, in which case $m = p-q$ and $s = \\min\\{ p,q\\}$.}.\n The spectral equations for the coef\\\/f\\\/icients $F(z)$ are\n \\begin{gather*}\n\\big(z^2-1\\big) F''(z) + 2(3z + p + q) F'(z) + \\frac {4p}{1+z} F(z) = \\lambda F(z) .\n \\end{gather*}\nTheir solutions are\n \\begin{gather}\n\\label{solFz3}\nF(z) = (1 + z)^{\\gamma_{pq}} P_n^{p+q+2, \\pm \\Delta_{pq}}(z) ,\n\\qquad \\lambda_{pqn} = \\gamma_{pq}^2 + 5\\gamma_{pq} + n(n + p+q\n+ 3 \\pm \\Delta_{pq} ) ,\n \\end{gather}\nwith\n\\begin{gather}\n\\label{gam3}\n \\gamma_{pq} = \\frac {p+q -2 \\pm \\Delta_{pq} }2\n \\end{gather}\n and\n \\begin{gather*}\n \\Delta_{pq} = \\sqrt{(2-p-q)^2 + 16p} .\n \\end{gather*}\n The observations to be made are exactly parallel to the observations in the $S^4$ case.\nIn particular,\n\\begin{itemize}\\itemsep=0pt\n\\item The solutions with $p > 0$ and the negative\nsign of $\\Delta_{pq}$ are not square integrable and should not be included in\nthe spectrum.\nIf $p=0$ or $p = q = 1$, the solutions with negative sign of $\\Delta_{ms}$ are not independent being expressed into\nthe solutions with positive sign in virtue of~(\\ref{relJacobi}).\n\\item On the other hand, the solutions with positive sign of $\\Delta_{pq}$\nand $p > 0$ are all not only square integrable,\nbut also nonsingular on $S^6$. In addition, they belong to the domain of~$Q$\n(meaning here that $ Q \\Psi$ are all normalizable).\n \\item The solutions with $p= 0$ and $q > 4$ also have this property.\n\\item\nThe normalizable\\footnote{This means here that\n\\[\n \\int \\frac {|\\Psi|^2 \\, d^3 w d^3 \\bar w}{(1+\\bar w w)^6} < \\infty ,\n\\]\nsuch that the singularity $\\Psi \\sim |w|^2$ is still allowed, while $\\Psi \\sim |w|^3$ is already not.}\n families of solutions with $p=0$ and $q = 0,1,2,3,4,5$ are\n \\begin{gather*}\n \\Psi_{00n} = P_n^{22}(z), \\qquad \\Psi_{01n} = \\bar w_j P_n^{31}(z) , \\nonumber \\\\\n \\Psi_{02n} = \\bar w_j \\bar w_k \\, P_n^{40}(z) , \\qquad\n\\Psi_{03n} = \\frac {\\bar w_j \\bar w_k \\bar w_l}{1 + \\bar w w} \\, P_n^{51}(z) , \\nonumber \\\\\n\\Psi_{04n} = \\frac {\\bar w_j \\bar w_k \\bar w_l \\bar w_p}{(1 + \\bar w w)^2} P_n^{62}(z) , \\qquad\n\\Psi_{05n} = \\frac {\\bar w_j \\bar w_k \\bar w_l \\bar w_p \\bar w_s} {(1 + \\bar w w)^3} P_n^{73}(z) .\n \\end{gather*}\nThe families with $q=1,2,3$ grow at inf\\\/inity. The functions $\\Psi_{04n}$ are bounded, but still singular\n($\\Psi(\\infty)$ is not def\\\/ined). However,\none cannot restrict oneself with the regular functions. The last family in the list above\nis regular on $S^6$, but would not belong to the domain of $Q$ in this case: $Q \\Psi_{05n}$ is not regular\nat inf\\\/inity.\nIn addition, by the same token as for $S^4$,\nthe family\n$ Q \\Psi_{01n}$ is regular on $S^6$, but does not belong to the domain of $\\bar Q$ (because\n$\\Psi_{01n}$ are singular).\n \\end{itemize}\nFor normalizable functions, we probably have a nice complex.\nAs we have just shown, all normalizable functions in the sector $F=0$ have normalizable superpartners.\n\nThe Hamiltonian in the sector $F=3$ is\n\\begin{gather*}\nH = -(1 + \\bar w w)^2 \\partial_j \\bar \\partial_j + (1+\\bar w w)\\left(\n\\bar w_j \\bar \\partial_j + 3w_j \\partial_j \\right) + 3(3+\\bar w w) .\n \\end{gather*}\n The spectral equations for the coef\\\/f\\\/icients $F(z)$ of the structures\n$T_{pq}$ are\n \\begin{gather*}\n\\big(z^2-1\\big) F''(z) + 2(3z + p + q) F'(z) + \\frac {3(p+1) + q}{1+z} F(z) = (\\lambda - 6) F(z) ,\n \\end{gather*}\nand the solutions are also given by (\\ref{solFz3}), (\\ref{gam3}) with\n\\begin{gather*}\n \\Delta_{pq}^{F=3} = \\sqrt{(2-p-q)^2 + 4[3(p + 1) + q] } .\n \\end{gather*}\nThe eigenfunctions have better convergence here than in the sector $F=0$. Actually, all normalizable\neigenfunctions as well as their superpartners (they have fermion charge $F=2$) are regular on $S^6$.\n\n To {\\it prove} that the Dolbeault complex is well def\\\/ined in this case in the space of square integrable\nfunctions, we have also to solve the Schr\\\"odinger equation\n in the sectors $F=1$ and $F=2$. In this case, it is more dif\\\/f\\\/icult than for $S^4$ because {\\it some} states\nin the sector $F=1$ are annihilated by $\\bar Q$ and cannot be found as superpartners of the states\nin the sector $F=0$. Likewise, there are states in the sector $F=2$ that are not superpartners to the states\nwith $F=3$. (These new states are superpartners to each other.) A special analysis of the matrix\nSchr\\\"odinger equation is thus required. We do not think,\nhowever, that such an analysis would unravel unpleasant surprises and believe that the Dolbeault\ncomplex is well def\\\/ined on~$S^6\\backslash \\{\\cdot\\} $.\n\n\nThe Witten index of this system is equal to\n \\begin{gather*}\nI^{S^6\\backslash \\{\\cdot\\}} = 1 + 3 + 6 = 10 .\n \\end{gather*}\n (There is one zero mode, $\\Psi = 1$,\nin the sector $F=0$, $p=q=0$, three zero modes, $\\Psi = \\bar w_j$, in the sector $F=0$, $p = 0$, $q = 1$ and six\nzero modes $\\Psi = \\bar w_j \\bar w_k $, in the sector $F=0$, $p = 0$, $q = 2$.\nNo zero modes in the other sectors are present.)\nGeneralizing this analysis to higher spheres, we obtain the result\n \\begin{gather*}\n\n I^{S^{2d}\\backslash \\{\\cdot \\}} = C^{d-1}_{2d-1}\n\\end{gather*}\n for the index of the pure Dolbeault complex.\n\nAgain, we can try to make contact of this result with functional integral calculations.\nUnfortunately, this does not work better here than in the $S^4$ case. At the tree level, one obtains\n a~{\\it fractional} contribution to the path integral,\n\\begin{gather*}\n \\frac 1{48\\pi^3} \\int F \\wedge F \\wedge F = \\frac 92 .\n \\end{gather*}\n The one-loop contribution associated with the gauge f\\\/ield\nvanishes by the same token as for~$S^4$ (see equation~(\\ref{Isinggauge})\nand the discussion thereabout).\nHigher loops seem to be suppressed for small~$\\beta$, but the presence of singularity does not allow one to make\na def\\\/inite statement \\dots.\n\n\n\n\n\n \\subsection*{Acknowledgements}\n\nI am indebted to G.~Carron, E.~Ivanov, and V.~Roubtsov for useful discussions.\n\n\n\\pdfbookmark[1]{References}{ref}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\\label{sec:intro}\n\nSince its explosion as the first optical SN observed in the year 1987,\nSN\\,1987A\\ has been the subject of intense study at every possible\nwavelength. Its location in the Large Magellanic Cloud (LMC), \nat a relatively close distance of 51.4\\,$\\pm$1.2~kpc \\citep{Panagia99},\nhas enabled us to\nstudy details regarding the aftermath of a SN explosion\nnot previously possible; we have learned and will\ncontinue to learn more about the physics of SNe and the formation\nof supernova remnants (SNRs).\n\nThe X-ray emission from SN\\,1987A\\ has likewise been studied with every\navailable satellite. However, SN\\,1987A\\ is by far the intrinsically dimmest X-ray SN to be\nobserved, at least in its first decade, and is only now approaching\nthe luminosity of other comparable-age X-ray SNe \\citep{Dwarkadas12}.\nIt is clear that the only reason it was detected in early\nobservations at high X-ray energies (6--28~keV) with the Ginga satellite \\citep{Inoue91}\nwas because of its proximity. The emission appeared to decrease\nover the next couple of years, and it appeared as though SN\\,1987A\\ would\nbehave like every other X-ray SN then known, with an X-ray emission\ndecreasing with time. \n\nJust over 3 years after its explosion however, the X-ray\n\\citep[0.5--2~keV ROSAT observations]{Hasinger96} and radio\n\\citep[1.4--8.6~GHz with the Australian Telescope Compact Array]{Gaensler97}\nemission from SN\\,1987A\\ began instead to increase with time. The\nincrease in radio emission was initially interpreted by\n\\citet{Chevalier92} as arising from the interaction of the SN forward\nshock (FS)\nwith the wind termination shock of the blue supergiant (BSG) wind\nof the progenitor. However this would not result in a continued\nincrease but would instead be limited in time. The observations\nindicated otherwise, and the X-ray emission continued to\nincrease.\nThe subsequent increases have suggested that the interaction is\nwith a much higher density and more extended region than expected from the wind\ntermination shock. This was interpreted by \\citet{Chevalier95} as\nionized red-supergiant (RSG) wind emitted during a pre-BSG phase,\nand subsequently swept-up by the fast BSG wind.\nThe ejecta interaction with the resulting H{\\footnotesize \\,II}\\ region\nproduces the usual two-shock structure with the FS moving\ninto the H{\\footnotesize \\,II}\\ region and a reverse shock (RS) moving back\ninto the ejecta; a contact-discontinuity (CD) defines the ejecta-H{\\footnotesize \\,II}\\\nboundary. Such a model was used to explain the early X-ray emission\n\\citep{BBMcC97let} and predicted line widths of 5000~{km\\,s$^{-1}$}\\ or more.\n\nThe X-ray emission since then has continued to rise, as shown in\nFigure 1 for the 0.5--2~keV and 3--10~keV bands.\nA review of the X-ray emission over the first 20 years is\ngiven in \\citet{McCray07}. The hard X-ray and radio emission have\ncontinued to rise together, but after about 6000 days the soft X-ray\nlight curve has steepened still further, indicating a source of\nemission that leads predominantly to soft X-rays. This has been\ninterpreted by \\citet{McCray07} as due to the FS interacting with\ndense fingers of emission arising from the inner edge of the\nequatorial ring (ER) surrounding SN\\,1987A. \nFrom days 6500 to 8000 the soft X-ray flux continued to grow\nat a roughly linear rate\\footnote{This was first pointed out by D.N. Burrows in\ndiscussions during the preparation of \\citet{Park11}.},\n that is $dF\/dt\\approx$ constant.\nMost recently, \\citet{Park11} have reported a flattening of the X-ray light curve\nsince day $\\sim$\\,8000,\nindicated in Figure~\\ref{fig:lcoverview}, and have suggested\nthe possibility that the FS has now propagated beyond the majority\nof the dense inner ring material. This could lead to further\nchanges in the light curve and, hence, future measurements are eagerly\nanticipated.\n\nEven considering its low luminosity, the proximity of SN\\,1987A\\ has\nallowed for high resolution grating observations to be taken.\nThese observations, including recent ones from 2011, are\nsummarized and analyzed in \\S\\,\\ref{sec:gratobs}.\nThere is much to learn from this set of high-resolution X-ray spectra;\nhowever, here we focus on the existence and evolution of a ``very-broad''\n({v-b}) emission line component in the data, \\S\\,\\ref{sec:vb}.\nThis {v-b}\\ component is persistent over the last decade \nand it is likely the continuation of the broad lines expected by \\citet{BBMcC97let} and \nmeasured in the {\\it Chandra}\\ HETG data of \\citet{Michael02}.\nIn \\S\\,\\ref{sec:hydro} we review and present our hydrodynamic modeling\nof SN\\,1987A\\ which consists of two contributions:\nthe H{\\footnotesize \\,II}\\ material above and below the equatorial plane\nand the dense equatorial ring material.\nWe compare the results of the model with observed quantities \nin \\S\\,\\ref{sec:results},\nshowing that the {v-b}\\ component naturally arises from the shocked \nH{\\footnotesize \\,II}\\ material and is the main source of the 3--10~keV flux.\nAlso, our very simplified hydrodynamics of the ring\ninteraction is used to show how the 0.5--2~keV light curve\nwould behave if the FS has indeed gone beyond the densest ring material.\nIn the final section we summarize our conclusions.\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.55]{dataonly_lightcurve.ps} \\\\\n\\end{center}\n\\caption{SN\\,1987A\\ X-ray light curves with grating observation epochs indicated.\nThe 0.5--2~keV data (black, red) show an initial exponential behavior whose\nslope increases after day $\\sim$\\,4500 and then decreases\nfrom day $\\sim$\\,6300, possibly leveling off at the\nrecent epoch $\\sim$\\,8800 days.\nIn constrast, the 3--10~keV flux (green) has more closely followed a\nconsistent exponential behavior to the present.\nFor reference, the sum of an exponential (beginning at day 2000)\nand a linear ramp increase (from day\n6000 to day 8200), is included (magenta, blue) to highlight the general phases of the\n0.5--2~keV light curve. The ROSAT fluxes are taken from \\citet{Haberl06}\nand the ACIS values are from \\citet{Park11}.\n\\label{fig:lcoverview}}\n\\end{figure*}\n\n\n\\clearpage\n\\section{GRATING OBSERVATIONS AND DATA REDUCTION}\n\\label{sec:gratobs}\n\n\n\nThe grating observations of SN\\,1987A\\ are listed in Table~\\ref{tab:gratobs},\nincluding those from instruments on {\\it Chandra}~(High- and Low- Energy Transmission\nGratings: HETG, LETG) and {\\it XMM-Newton}~(Reflection Grating Spectrometer, RGS).\nThe observation epochs are marked on the SN\\,1987A\\ light curve of\nFigure~\\ref{fig:lcoverview}, showing that only the\nHETG-99 observation \\citep{Michael02} was taken before day 5500\nafter which the flux from the shock--protrusion collisions became significant.\nAlong with HETG-99, the deep non-grating {\\it Chandra}\\ ACIS-00 observation\nat 5036 days \\citep[ObsId\\,1967]{Park02} gives the best pre-collision X-ray spectra (albeit at\nonly CCD resolution) and is included in the table as well.\nWith the collision getting underway, the RGS-03 \\citep{Haberl06} and LETG-04\n\\citep{Zhekov05,Zhekov06} observations cover a transition period.\nSince 2007, with most of the X-ray flux due to the shock-ring\ninteraction, there have been regular grating observations\nmade with both {\\it Chandra}\\ \\citep{Dewey08,Zhekov09} and {\\it XMM-Newton}\\ \\citep{Sturm10}.\n\nFor our data analysis, we use standard spectral extractions from the\nobservations as described in \\S\\,\\ref{sec:data} \\&\n\\S\\,\\ref{sec:rgsdata}.\nThe various analyses,\ndescribed in \\S\\,\\ref{sec:3shock} \\& \\S\\,\\ref{sec:vb},\nare then carried out in ISIS \\citep{Houck00}\nusing a common set of scripts that include some simple tailoring\nof the energy ranges and spatial-spectral parameters (see\nAppendix~\\ref{sec:gsmooth})\nbased on the type of data, e.g., HETG vs.\\ RGS.\n\n\n\\subsection{{\\it Chandra}\\ Data}\n\\label{sec:data}\n\nPublic {\\it Chandra}\\ grating data and their spectral extractions are\navailable and were obtained from the {\\it TGCat}\\ archive\\,\\footnote{\\,\n{\\tt http:\/\/tgcat.mit.edu\/}}\n\\citep{TGCat11}.\nAll of these had been re-processed in 2010 or later and therefore the\ndownloaded {\\tt pha}, {\\tt arf}\\ and {\\tt rmf}\\ files\ncould be used without any further re-processing.\nMultiple ObsId's at the same epoch were combined in\nISIS by summing the {\\tt pha}\\ counts, creating\nan exposure-weighted {\\tt arf}, and summing the exposure times.\nTo reduce background from the less sensitive front-illuminated CCDs,\nthe {\\tt pha}\\ and {\\tt arf}\\ for wavelengths longward of 17.974\\,\\AA\\ were set\nto zero in the plus [minus] order of the MEG [LEG] data.\nThe plus and minus orders are then combined, even though\nthey do have small but significant\ndifferences in their line-widths because of\nthe resolved nature of SN\\,1987A\\ \\citep{Zhekov05,Dewey08}.\nThe composite line-shapes are accounted for in the model by a smoothing\nfunction that has parameters based on the {\\it observed}\nline widths, \\S\\,\\ref{sec:3shock} and Appendix~\\ref{sec:gsmooth}.\nIn this way we reasonably approximate the combined-orders' line shapes.\nThus, further analysis is\ncarried out on a single spectrum in the case of LETG data\nand on two spectra (MEG and HEG) for HETG observations.\n\nSome HETG observations are indicated with a lower-case ``hetg-YY''\ndesignation in the table.\nThese spectra are from the combined ``monitoring observations'' (PI -\nDavid Burrows) in the year 20YY. Starting in 2008 the HETG was\ninserted for SN\\,1987A\\ observations in order to reduce the\neffects of pileup \\citep{Park11}; as a by-product, additional HETG high-resolution data\nare taken.\nHowever, to obtain the shortest exposure times only a subframe of the ACIS\nis read out; because of this the outer portions of the\nHETG dispersed ``X'' pattern \\citep{Canizares00,Canizares05}\nfall outside of the subframe, removing response at longer wavelengths.\nInstead of centering SN\\,1987A\\ in the subframe, it is offset\nin cross-dispersion to get more of the \nsensitive MEG minus order in the subframe and this extends the\nresponse to just include the O{\\footnotesize \\,VIII}\\ line at $\\sim$\\,0.65~keV.\n\nThe most recent HETG observations of SN\\,1987A, HETG-11 in\nTable~\\ref{tab:gratobs}, were carried out during 2011 March 1--13\nas part of the GTO program (PI - Claude Canizares).\nThe proprietary data were retrieved from the\n{\\it Chandra}\\ archive and scripts from {\\it TGCat}\\,\\footnote{\\,See\n Help$\\Rightarrow$Software on the {\\it TGCat}\\ web page:\n{\\tt http:\/\/tgcat.mit.edu\/}} were used to carry out the spectral extraction\nusing standard CIAO tools; specifically\nCALDB 4.4.5 and CIAO 4.3 were used to process the data. \n\nThe non-grating ACIS-00 observation, ObsId\\,1967, was downloaded from the\narchive and processed through level 1 event filtering with CIAO tools.\nThe {\\tt psextract} routine was then used to generate {\\tt pha}, {\\tt arf},\nand {\\tt rmf}\\ files for a $\\sim$\\,6 arc second diameter region around SN\\,1987A;\na background extraction was made but not used because it had very\nfew counts.\n\n\n\\subsection{{\\it XMM-Newton}\\ RGS Data}\n\\label{sec:rgsdata}\n\nThe European observatory, {\\it XMM-Newton}, has\nbeen used to study SN\\,1987A\\ at high spectral resolution via\nits reflection-grating spectrometer, RGS \\citep{denHerder01}. The data and\nprocessing steps for the four epochs of\nRGS data taken through January 2009 have been presented in \\citet{Sturm10}.\nTwo subsequent observations, RGS-10 and RGS-11,\nhave been carried out and were similarly processed;\n{\\it XMM-Newton}\\ SAS 10.0.0 was used in (re-)processing all RGS data.\nWhen read into ISIS\nthe RGS-1 and RGS-2 counts (and backgrounds) are kept separate and jointly fit\nin further analysis.\n\n\\input{tab_gratobs.tex}\n\n\n\\clearpage\n\\subsection{Fluxes and Three-Shock Model Fitting}\n\\label{sec:3shock}\n\nThe {\\it Chandra}\\ grating observations can robustly determine the observed\nflux (and statistical error) in energy bands directly\nfrom the data without using a model.\nInstead of the the usual coarse ``flux in the 0.5--2~keV band,''\nfluxes in Table~\\ref{tab:gratobs} are given for \nthree bands covering this range. The band boundaries are chosen to\navoid strong lines and are dominated by lines of N\\,\\&\\,O,\nFe\\,\\&\\,Ne, and Mg\\,\\&\\,Si, respectively. Because of their reduced\nwavelength coverage the ``hetg-YY'' data sets do not allow an accurate flux\nmeasurement in the lowest-energy band. For the RGS data sets\nthe fluxes given in the table are from 3-shock model fits (described\nbelow) because the RGS spectral gaps and background counts complicate\na direct-from-the-data approach.\n\nComparing HETG-11 fluxes to the previous HETG-07 ones shows\nthat the main deviation from the trend of continued flux increase\nhas happened primarily in the lowest energy range:\nwhile the 0.79--1.1 and 1.1--2.1 keV fluxes have increased by factors\nof 1.78 and 2.17, respectively, the 0.47--0.79 keV flux has\nincreased by a smaller factor of only 1.18.\\footnote{\\,The apparent\nlow-energy decrease may have a contribution due to calibration issues,\nin particular with the ACIS contamination modeling.\nAn HETG observation of the SNR E0102 was taken\non 2011 February 11, just a\nmonth before the SN\\,1987A\\ data; its initial analysis shows\ncalibration changes in the oxygen lines, but only of order 20\\,\\%.}\nLikewise, similar spectral changes are seen\nin the recent RGS data of 2010 December (``RGS-11''),\ncontinuing trends seen earlier; for example, \nwhereas the Ne{\\footnotesize \\,X}\\ flux increased by 36\\,\\% from RGS-08 to RGS-09\nthe lower-energy N{\\footnotesize \\,VII}\\ flux increased by only 10\\,\\% \nin the same period \\citep{Sturm10}.\n\nA nominal non-equilibrium ionization (NEI) model was fit independently to each\nepoch's data with the goal of capturing all of the relevant\nlines in that epoch's spectrum. In order to allow for the wide range of shock\ntemperatures seen in SN\\,1987A\\ \\citep[Figure~2]{Zhekov09},\nthe model consists of the sum of three plane-parallel shock components \nwith common (``tied'') abundances. Physically, a single plane-parallel shock\nmodel represents the emission from a uniform slab of material into which a\nshock wave has propagated for some finite time, $t_s$ \\citep{Borkowski01}.\nThe shocked portion of the slab has uniform electron temperature, $kT_e$,\nand density, $n_e$, but has a linear variation in the time since shock passage, \ngoing from $\\sim$\\,0 immediately postshock to a maximum of $t_s$ for the\nearliest-shocked material. Thus, the plane-parallel shock emission can be\nparameterized by a normalization ($\\propto n_e n_H V$), the abundances, \nthe postshock electron temperature, and the maximum ionization\nage\\,\\footnote{\\,Specifically, the maximum ionization age is specified by the {\\tt\nTau\\_u} parameter of the {\\tt vpshock} model in XSPEC; there is also\na {\\tt Tau\\_l} parameter which in keeping with our physical description is set to 0.}, \n$\\tau=n_e t_s$.\n\nThe sum of the three shock components is then smoothed with a Gaussian function to account for\nline-broadening and the finite size of SN\\,1987A.\nFinally, there is an overall photo-electric absorption term to account\nfor Galactic and LMC absorption.\nThe model is implemented in ISIS, making use of the XSPEC spectral\nmodels library; the model specification (Expression~\\ref{eq:3shock}) and\na description of the smoothing parameters\nare given in Appendix~A.\nMany of the model parameters are fixed as in previous\nwork \\citep{Zhekov09}: the column density\\,\\footnote{\\,The\ncolumn density could have a measureable local component\nthat would change as the SN\\,1987A\\ system evolves: taking a\ndensity of 1\\tttt{4}~H\\,cm$^{-3}$\nwith a 0.01~pc path length sets an $N_{H\\,{\\rm local}}$ scale of $\\sim$\\,0.3\\tttt{21}.\nClumping, the ionization state, and the geometry of SN\\,1987A\\ have to be taken into account as well.\nIn terms of data analysis, the $N_H$ is very degenerate with the low-energy\nline abundances (N, O) and\nwith the continuum teperatures and norms.\nFor our purposes we fixed $N_H$\nat the two-shock-model value determined in \\citet{Zhekov09},\nroughly half of which is due to the Galactic contribution of $\\sim$\\,0.6\\tttt{21}\\,cm$^{-2}$.\n}\n($N_H=$\\,1.3\\tttt{21}\\,cm$^{-2}$),\nthe redshifts (286.5~{km\\,s$^{-1}$}), and the abundances of elements\nwithout strong, visible lines (H=1, He=2.57, C=0.09, N=0.56, Ar=0.54,\nCa=0.34, Ni=0.62); as previously the solar photospheric abundances\nof \\citet[Table 2]{AG89} are used as the reference set.\nThere remain the free parameters\nconsisting of 3 normalizations, 3 temperatures, 3 ionization ages,\nand the abundances of O, Ne, Mg, Si, S, and Fe.\n\nThe parameter set was reduced further by fixing the middle temperature at a value \nnear the (logarithmic) center of the emission measure distribution of SN\\,1987A,\n$kT_{\\rm \\,mid}=$\\,1.15~keV \\citep[Figure~2]{Zhekov09}. \nSince the $\\sim$\\,1~keV emission had grown from\n2004 to 2007, the expectation, borne out in\nTables~\\ref{tab:3shock}\\,\\&\\,\\ref{tab:3shVB}, is that this component\nwould have a growing contribution to the model, especially at later epochs.\nTo test the (in)sensitivity of the fit values to the exact value\nof $kT_{\\rm \\,mid}$, the HETG-11 data were fit with additional\n$kT_{\\rm \\,mid}$ values of 1.0\\,\\&\\,1.3~keV. All of the HETG-11 fit parameters\nshowed small variations over the test set of mid-temperature values,\n\\{1.0, 1.15, 1.3\\} keV. Some examples are: \n$kT_{\\rm \\,lo}$ takes values \\{0.50, 0.54, 0.56\\},\nthe $kT_{\\rm hi}$ norm varies as \\{2.4, 2.0, 1.5\\},\nthe silicon abundance varies as \\{0.41, 0.39, 0.38\\}, and\nthe $\\chi^2$ changed slightly as \\{1069, 1088, 1111\\}\n(for 560 degrees of freedom.)\nThese small, gradual variations demonstrate that the exact choice \nfor the middle temperature is somewhat arbitrary and nearly degenerate\nwith other parameters.\n\nTwo more parameters are removed by constraining the ionization ages\nto cover a factor of 2, e.g., as seen in \\citet{Zhekov09},\nspecifically setting:\n$\\tau_{\\rm \\,lo}=\\sqrt{2}\\,\\tau_{\\rm \\,mid}$ and\n$\\tau_{\\rm \\,hi}=\\tau_{\\rm \\,mid}\\,\/\\sqrt{2}$.\nHence, in addition to the abundances there are 6 free parameters:\n$kT_{\\rm \\,lo}$, $\\tau_{\\rm \\,mid}$, $kT_{\\rm \\,hi}$, and\nthe 3 normalizations, $N_{\\rm lo}, N_{1.15}, N_{\\rm hi}$. This is the same\nnumber of parameters as for a general 2-shock model (2 norms, 2 $kT$'s, 2 $\\tau$'s) and gives\nslightly better fits.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.65]{HETG-11_3shplt.ps} \\\\\n\\end{center}\n\\caption{HETG-11 data and model. The data\n(black) are globally fit by a 3-{\\tt vpshock} model (red).\nRemoving the high-temperature component gives the blue curve and\njust the low-temperature component is shown in green. \n\\label{fig:3shock}}\n\\end{figure*}\n\nThe 3-shock model was fit to the HETG and LETG data in the 0.6 to 5 keV\nrange, giving good overall fits, Figure~\\ref{fig:3shock}.\nFor fitting, the spectra were binned to a minumum of 20--30 counts per bin and a\nminimum bin width of $\\sim$\\,0.05~\\AA.\nThe HEG data are ignored below 0.78~keV ($>$15.9\\,\\AA) and ignored\ncompletely for the HETG-99 data. \nBecause of their reduced wavelength coverage,\nseveral changes were made for the ``hetg-YY'' data sets:\nthey were ``noticed'' in the range 0.69--5~keV for the MEG (1.25--5~keV for\nthe HEG), the O abundance was fixed, and the $kT_{\\rm \\,lo}$\nvalue was fixed (these latter two were set to values based on the HETG-07\nand HETG-11 fits.)\nThe RGS data sets do a good job of covering the N line $\\sim$\\,0.5~keV\nbut have reduced sensitivity above the Si{\\footnotesize \\,XIV}\\ line, \nso these data were fit in the range 0.47 to 2.20~keV with the N\nabundance free, the S abundance frozen at a nominal value (0.36), and\nthe $kT_{\\rm \\,hi}$ value fixed.\nThe ACIS-00 spectrum was fit over the 0.4 to 8.1~keV range with N free.\n\nThe values for the 3-shock fit parameters and their\n1-$\\sigma$ confidence ranges are given in Table~\\ref{tab:3shock}.\nAs expected from their low fluxes, the two pre-collision data sets (HETG-99,\nACIS-00) show comparatively low normalizations.\nNote that the RGS values and the HETG\/LETG values differ significantly at similar\nepochs; this is probably due to {\\it near-degeneracies in the model}\nwhich amplify the differences between the spectrometers in terms of their\nenergy-range coverage, statistical weighting (counts vs.\\ energy), and\nline-spread functions. However they do show the same general trends,\ne.g., toward larger $\\tau_{\\rm \\,mid}$ values at more recent epochs.\nIn the next section we use these 3-shock fits as good starting points\nto look for and measure the presence of the {v-b}\\ component.\n\n\n\\clearpage\n\\input{tab_3shock.tex}\n\n\n\\clearpage\n\\subsection{The Very-Broad Component and Its Evolution}\n\\label{sec:vb}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.48]{HETG-07_vbresid.ps} \\\\\n\\vspace{0.1in}\n\\includegraphics[angle=270,scale=0.48]{HETG-11_vbresid.ps} \\\\\n\\end{center}\n\\caption{Deep HETG spectra and their very-broad ({v-b}) models.\nThe HETG-07 (top pair) and HETG-11 (bottom pair) data are shown (black)\nwith the best-fit {v-b}\\ 3-shock model (red). The non-broad\nmodel component is shown as well (blue). The lower half plots\nshow the difference (black) between the data and the non-broad component\nwith the {v-b}\\ component over plotted (red).\n\\label{fig:vbresid}}\n\\end{figure*}\n\nTo look for and quantify a {v-b}\\ line component in the SN\\,1987A\\ spectra\nwe start with the data and 3-shock model fits (above) and extend the\nmodel to include a {v-b}\\ component that is a broadened\nversion of the 3-shock model itself; the model definition is explicitly\ngiven in Appendix~A, Expression~\\ref{eq:vb}. The {v-b}\\ extension\nadds two additional parameters: the fraction of the total flux that is in the {v-b}\\\ncomponent ($f_{\\mathrm{v}\\textrm{\\footnotesize -}\\mathrm{b}}$) and the width of the {v-b}\\ component expressed\nas an equivalent FWHM in velocity ($v_{\\mathrm{v}\\textrm{\\footnotesize -}\\mathrm{b}}$). \n\nThe choice of broadening the whole spectrum is the simplest given the\nlow signal-to-noise ratio of the {v-b}\\ component in any given line, provided\nthere is some expectation that the bright lines of the {v-b}\\ component\nare similar to those of the narrow emission. At face value this may\nseem to be {\\it unlikely}, with the {v-b}\\ lines produced by a plasma\nwith very different temperatures and ionization ages from the plasma\nproducing the narrow lines.\nHowever, our hydrodynamic modeling shows that the approximation is a\nsurprisingly good one as seen in Figure~\\ref{fig:vbhydro} and\ndiscussed in \\S\\,\\ref{sec:vbspectrum}.\nHowever, based on this comparison\nwe have chosen to ignore the Fe{\\footnotesize \\,XVII}\\ 0.70--0.75~keV range\nwhen doing {v-b}\\ fitting (below) as it is the only\nbright line-complex that is not predicted to have a {v-b}\\ component.\n\nThe {v-b}--3-shock model is fit to the data using $\\sim$\\,60\\%\nfiner binning and a somewhat smaller energy range that\nfocusses on the brightest, high-resolution lines in the particular spectra,\ngenerally those of O, Ne, Mg, Si, and Fe-L.\nFor HETG, LETG and RGS data these ranges are, respectively: 0.62--2.1~keV, 0.6--1.6~keV\n(no Si),\nand 0.47--1.1~keV (including N but not Mg and Si).\nFor ``hetg-YY'' data sets because of their limited low-energy range,\nthe O line is not included and the O abundance is frozen\nat the average of the HETG-07 and HETG-11 O {v-b}\\ values.\nBecause of the reduced energy range of the {v-b}\\ fitting, the $kT_{\\rm \\,lo}$, \n$\\tau_{\\rm \\,mid}$ and $kT_{\\rm \\,hi}$ parameters were\nfixed at their 3-shock values; this leaves the 3 normalizations and the\nabundances to be fit along with the {v-b}\\ fraction and width.\n\nAs examples, the {v-b}\\ fits to the HETG-07 and HTEG-11 data are shown in\nFigure~\\ref{fig:vbresid}; note that the y-axes of the lower, difference\nplots are the same, showing that the {v-b}\\ flux is {\\it increasing}\nin absolute terms.\nThe statistics of this {v-b}\\ fitting are given in the left-half of\nTable~\\ref{tab:vbstats}; in particular the values of $F_2$ are large\nenough to indicate that the addition of the two {v-b}\\ parameters has\nimproved the model fit significantly.\n\n\n\\clearpage\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.50]{HETG_fv_contours.ps} \\\\\n\\end{center}\n\\caption{Confidence contours for the very-broad component\nparameters. Solid curves give the\n68\\,\\% ($\\Delta\\chi^2=2.3$) contours for fits to \neach grating data set; the best-fit values are indicated with ``$+$'''s.\nThe two deep HETG observations, HETG-07 \\& HETG-11,\nshow a similar, well-determined {v-b}\\ width of $\\sim$\\,9300~{km\\,s$^{-1}$}~FWHM.\nSee the text for a discussion of the spread seen among the contours.\n\\label{fig:vbcontours}}\n\\end{figure*}\n\nInstead of tabulating the best-fit {v-b}\\ parameters and their ranges, it is more\nuseful to generate two-dimensional confidence contours\nin ``{v-b}\\ fraction vs.\\ {v-b}\\ width'' space.\nThese contours and their best-fit values are shown in\nFigure~\\ref{fig:vbcontours}; for HETG-99 the data point with errors\nis based on the 2-Gaussian ``stacked fitting'' result given in \\citet{Michael02}.\nThe earliest data sets, HETG-99 and\nRGS-03, show large values of {v-b}\\ fraction ($>$\\,0.6) with widths of\norder 10\\tttt{3}~{km\\,s$^{-1}$}. In contrast the later\nobservations show {v-b}\\ fractions within the range 0.15 to 0.30 and \ncover a large spread in widths, with the HETG values below\n15\\tttt{3}~{km\\,s$^{-1}$}\\ and RGS-determined widths above this value.\n\nThis separation suggests that the measurement of the {v-b}\\ component depends on the\nparticular spectrometer used. Figure~\\ref{fig:vbLSFs} shows how\nthe spectrometers ``see'' a monochromatic line consisting of a narrow\nand a {v-b}-component; these plots include SN\\,1987A's spatial-spectral\nblur descibed in Appendix~\\ref{sec:gsmooth}.\nThe HETG, even at the higher Mg{\\footnotesize \\,XII}\\ energy, clearly separates\nthe narrow and {v-b}\\ components, although it's also clear that the\n{v-b}\\ presence is subtle.\nFor the LETG the narrow component is significantly broadened by\nthe spatial extent of SN\\,1987A\\ and at high energies the {v-b}\\ component is\nalmost completely covered by the narrow component.\nThe RGS line-spread function is more peaked than the LETG\nat O{\\footnotesize \\,VIII}, however it includes extended wings so that even the\nnarrow component produces counts far from the line center.\nHence the RGS {v-b}\\ measurement will be particularly sensitive\nto the accuracy of its LSF calibration.\n\nWith these considerations in mind, we see that the two deep\nHETG observations, HETG-07 and HETG-11,\nshould give the most accurate measure of the {v-b}\\ component.\nComparing their small confidence contours,\nthere is a clear change in the {v-b}\\ fraction from 2007 to 2011.\nIn terms of the {v-b}\\ width, these two HETG contours show very\nsimilar values of $\\approx$ 9300~{km\\,s$^{-1}$}~FWHM, with an estimated \n90\\,\\% confidence range of $\\pm$\\,2000~{km\\,s$^{-1}$}.\nThis value is also in rough agreement with the (lower\nstatistics) HETG-99 and RGS-03 widths and so it is reasonable to\npostulate a constant {v-b}\\ width with a fraction that changes\nover time as shown by the\nblue path in Figure~\\ref{fig:vbcontours}.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,scale=0.6]{plots_comp_oviii.ps}\n\\includegraphics[angle=0,scale=0.6]{plots_comp_mgxii.ps} \\\\\n\\end{center}\n\\caption{Comparing the grating spectrometer line responses when observing SN\\,1987A.\nThe HETG(MEG), LETG, and RGS views of the same combination of a\nnarrow line plus a very-broad\ncomponent are shown for the case of $f_{\\mathrm{v}\\textrm{\\footnotesize -}\\mathrm{b}} =25$\\% and\n$v_{\\mathrm{v}\\textrm{\\footnotesize -}\\mathrm{b}} =9300$~{km\\,s$^{-1}$}\\ FWHM.\nBecause the resolution varies with energy, responses are shown\nfor O{\\footnotesize \\,VIII}\\,(left colum) and for Mg{\\footnotesize \\,XII}\\,(right column).\nOn each plot the orange curve shows a continuum level, the blue\ncurve is the continuum plus the {v-b}\\ component, the magenta curve is the\ncontinuum plus the narrow component, and the red curve is the sum of \nall three. Vertical gray lines indicate the line center and\n$\\pm$\\,1500~{km\\,s$^{-1}$}\\ from it.\nFor reference, example data are shown in the background by the light gray \nhistograms; the O{\\footnotesize \\,VIII}\\ region includes the He-like L-$\\beta$ line at\n$\\sim$\\,0.665~keV. The HETG provides the best separation of the {v-b}\\ and narrow\ncomponents especially at the higher energies.\n\\label{fig:vbLSFs}}\n\\end{figure*}\n\n\n\\input{tab_vb9300.tex}\n\n\nGiven the above, we fix the width of the {v-b}\\\ncomponent at $v_{\\mathrm{v}\\textrm{\\footnotesize -}\\mathrm{b}}=9300$~{km\\,s$^{-1}$}~FWHM and fit for just the {v-b}\\ fraction.\nThe results of this ``constrained'' fitting are given in the right-half of\nTable~\\ref{tab:vbstats}, including the\n{v-b}\\ fractions and their 68\\,\\% confidence ranges.\nNote that for most data sets the $\\chi^2$ change when doing this\nconstrained fitting, $\\Delta\\chi^2_1$, is a substantial portion of\nthe $\\chi^2$ change when the {v-b}\\ width is also being fit, $\\Delta\\chi^2_2$.\nThis suggests that the data do not strongly constrain the width to\ndiffer from our assigned value;\nthe elongated contours of Figure~\\ref{fig:vbcontours} suggest this as well.\nConversely, the {v-b}\\ fractions given by the constrained\nfitting have only a small dependence on the assigned width.\nAs examples, when the data sets LETG-07, RGS-11, HETG-11, and hetg-11 were\nfit with the width set to 6500\\,\\&\\,13000\\,~{km\\,s$^{-1}$}~FWHM, the best-fit fractions\nfound are: 0.132\\,\\&\\,0.160, 0.216\\,\\&\\,0.210, 0.163\\,\\&\\,0.178, and\n0.200\\,\\&\\,0.188, respectively.\nThese variations are actually within the statistical\n1-$\\sigma$ ranges given for the $f_{\\mathrm{v}\\textrm{\\footnotesize -}\\mathrm{b}}$ values in\nTable~\\ref{tab:vbstats}, and so the choice of the {v-b}\\ width\nwill not significantly change the {v-b}\\ fractions and light curve.\n\nHaving determined the {v-b}\\ fractions for the data sets we can now\nre-do the 3-shock fitting of \\S\\,\\ref{sec:3shock} with\nthe appropriate {v-b}\\ component fixed and included in the model.\nThis gives the results in Table~\\ref{tab:3shVB}.\nNote that the abundances have increased compared to the non--{v-b}\\ values\nin Table~\\ref{tab:3shock}.\nThis is expected since a larger flux in a given line is now needed in\norder to match the height of the narrow-line component in the high-resolution spectrum.\nThe change is especially pronounced\nfor HETG-99 where the {v-b}\\ fraction is 0.78 and the abundances\nare now in better agreement with those of ACIS-00 (although there is a large error range.)\nAlso as expected, the ACIS-00 values hardly change when\nthe {v-b}\\ component is included because the lower-resolution CCD spectra\nare primarily determined by the total line flux with little\nsensitivity to any underlying velocity structure.\n\n\n\\input{tab_3shVB.tex}\n\n\n\\clearpage\n\\section{HYDRODYNAMIC MODELING OF SN\\,1987A}\n\\label{sec:hydro}\n\nBesides being an extremely well-studied object, SN\\,1987A\\ has been\nfrequently modelled in almost every wavelength regime. The\nearliest calculations, which delineated the circumstellar material\n(CSM) into a low-density\ninner and high-density outer region, were made by \\citet{Itoh87}. The\nX-ray emission was modelled by \\citet{Masai91} and\n\\citet{Luo91a} using 1-dimensional (1D) models. Two-dimensional (2D) models were\nexplored by \\citet{Luo91b} for the case of an hour-glass equatorial\nring (ER) structure.\n\\citet{Suzuki93} carried out 2D axisymmetric smoothed particle\nhydrodynamics (SPH) simulations with the ER modeled as a torus;\ntheir Figures 4--8 demonstrate the hydrodynamic complexity by showing\nthe different components' spatial distributions.\n\\citet{Masai94} went back to the 1D models but took into\naccount the light-travel times from different parts of the remnant.\n\\citet{Luo94} carried out 2D calculations approximating the ER\nas a rigid boundary; they also introduced the ejecta density distribution that\nhas been subsequently used by most authors, and forms the basis for\nthe ejecta density profile used in this paper.\n\nThe increasing X-ray and radio emission led \\citet{Chevalier95} to\npropose the existence of an ionized H{\\footnotesize \\,II}\\ region interior to the ER. The\nFS had collided with this region around day 1200, and was\nexpanding in this moderately-high density medium,\n$\\rho\\approx 100$~{amu\\,cm$^{-3}$}, giving rise to increasing\nX-ray and radio emission. The slowing down of the shock wave observed\nat radio wavelengths was consistent with this conjecture. The X-ray\nflux from the H{\\footnotesize \\,II}\\ region was modelled by \\citet{BBMcC97let}, whereas a\ndetailed prediction of the expected X-ray flux and spectrum when the\nshock collided with the dense ER was given in\n\\citet{BBMcC97}.\n\nFor the next decade or so, investigations of the X-ray emission were\nmainly made via simpler, non-hydrodynamical models that attempted to\ntie in the observed X-ray emission to the CSM properties\n\\citep{Park04,Park05, Park06, Haberl06, Zhekov10}.\n\\citet{Dwarkadas07RevMex} revised the older\nsimulations made (but not presented) in \\citet{Chevalier95} using\nmore recent data, and managed to create spherically symmetric\nhydrodynamical models that successfully reproduced the FS\nradius and velocity as measured at radio frequencies. Using\nthese simulations he was also able to make a crude estimate of the\nX-ray light curve. This model was updated in \\citet{Dwarkadas07AIP}, and forms\nthe basis of the hydrodynamical simulations that are presented in the\nfollowing.\n\n\n\\subsection{Our Hydrodynamic Model of SN\\,1987A}\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=0,scale=0.33]{z10_compare_t0.eps}\n\\hspace{-0.1in}\n\\includegraphics[angle=0,scale=0.33]{z10_compare_t7300.eps} \\\\\n\\end{center}\n\\caption{Schematic diagrams of the CSM as modeled in\n\\citet{Zhekov10}, top row, and in this work, bottom row. The left diagrams show the\nCSM configuration assumed at the time of the SN explosion.\nThe coresponding diagrams at the right show the configuration\n$\\sim$\\,7300~days after explosion when the forward shock, driven by\nthe expanding ejecta, has progressed into the dense, clumpy ER.\nThe Zhekov model is based on a single 1D hydrodynamics whereas our\nmodel consists of two 1D models, one that includes the dense ER and\none that accounts for the H{\\footnotesize \\,II}\\ material above and below the\nequatorial plane. It is this latter component that gives rise to\nthe very-broad X-ray emission.\n\\label{fig:rsscompare}}\n\\end{figure*}\n\n\nThe structure of the CSM around SN\\,1987A\\ is inherently 3-dimensional, as is\nvisible in the amazing images from the {\\it Hubble Space Telescope},\nmost recently in \\citet{Larsson11}.\nModelling the ejecta-CSM interaction in 3-dimensions\nis a highly challenging and computationally demanding problem,\ndoes not encourage iterative parameter exploration, and requires\nmulti-dimensional non-equilibrium ionization (NEI) X-ray\nemission calculations.\nHowever, given the substantial symmetries in the overall geometry,\nit is possible to partition the domain into one or more\nregions whose emission can then be approximated by a\nspherically symmetric 1D simulation.\nThese simulations can then be summed together, weighted\nby the fraction of the full $4\\pi$ solid angle that they each subtend.\n\nAs examples of this approach we show schematics in\nFigure~\\ref{fig:rsscompare} for the modeling of \\citet{Zhekov10} and\nour model. The former consists of a single 1D solution with a smooth,\nincreasing-density CSM component. To match the observed spectra, especially at\nlate times, dense ``clumps'' are included in the CSM with the clump\nX-ray emission based on transmitted and reflected shock equations \\citep{Zhekov09}.\nOur model in the lower portion of Figure~\\ref{fig:rsscompare}\nis motivated by the on-going existence of the {v-b}\\ component which\nrequires a combination of two 1D solutions:\n\\begin{itemize}\n\\item[i)] A ``with ER'' simulation that consists of an H{\\footnotesize \\,II}\\ region\nfollowed at larger radii by a density jump to high values\nat the ER location; the density profile is shown in\nFigure~\\ref{fig:initprofile}. This component subtends a small solid\nangle, $\\Omega_{\\rm ER}$, and the high-density ER includes\nadditional ``clumping''\n(\\S~\\,\\ref{sec:lcs}\\,\\&\\,\\ref{sec:erabunds}).\nThis component is similar to the 1D model of \\citet{Zhekov10}.\n\\item[ii)] A ``no ER'' simulation in which the H{\\footnotesize \\,II}\\ region continues\nto large radii and there is no ER density jump.\nThis simulation produces the {v-b}\\ emission and\ncorresponds to material above and below the\nequatorial plane; it subtends a moderate solid angle,\n$\\Omega_\\mathrm{H\\,II}$.\n\\end{itemize}\n\\noindent This multi-1D technique does have its\nlimitations, especially for the ``with ER'' case,\nand we will keep these in mind.\n\nThe simulations here were carried out using the VH-1 code, a\n1, 2, and 3-dimensional finite-difference hydrodynamic code based on\nthe Piecewise Parabolic Method \\citep{Colella84}. The code solves the\nhydrodynamical equations in a Lagrangian framework, followed by a\nremap to the original Eulerian grid. Radiative cooling is included in\nthe form of the X-ray cooling function scaled by a factor of 3.5 to\naccount for IR dust emission, $L_\\mathrm{IR}\/L_\\mathrm{X}\\approx 2.5$. \n\nFor each of a set of time steps, the code outputs a snapshot file giving\nthe hydrodynamic quantities density, velocity, and pressure at each\nradial zone; two examples are shown in Figure~\\ref{fig:modprofile}.\nThe set of snapshot files is then processed by custom routines coded\nin S-Lang\\,\\footnote{\\,S-Lang web page: {\\tt http:\/\/www.jedsoft.org\/slang\/}}\nand run within the ISIS software environment. The\nradial grid is rebinned into mass cuts, so that the time\nevolution of each parcel of gas in a given mass shell can be\naccurately followed; in this way the NEI state is tracked and\nthe X-ray emission can be calculated \\citep{Dwarkadas10}.\n\nOf course we did not simply pick parameters, run the simulation, and\nhave agreement with observations. An iterative process of trial\\,\\&\\,error\nwas carried out to converge on model parameters that reproduced the\nvarious measured properties of SN\\,1987A, especially in the X-ray.\nRather than take the reader through these steps (and mis-steps), we\nsummarize in the following sections the final model properties and\nthen compare the model results to data in \\S\\,\\ref{sec:results}.\n\n\n\n\\subsection{Ejecta Parameters}\nThe ejecta are modelled using the \\citet{Chevalier82selfsim}\nprescription where the ejecta are in homologous expansion with a density \nthat decreases as a power-law with radius, i.e.,\n\\begin{equation}\\label{eq:ejecta}\n\\rho_{\\rm ej} = C~ t^{-3}\\,(r\/t)^{-n}~.\n\\end{equation}\n\\noindent This is written to show that one can also consider the density profile as\na function of $v=r\/t$. Since a power-law extending all the way back to the origin\nis unphysical, below a\ncertain velocity, $v_t$, the density is assumed to be a constant,\nsee the ``Ejecta'' portion of the density in Figure~\\ref{fig:initprofile}.\nThus the profile can be specified by the three parameters: $C$, $n$, and\n$v_t$ . The ejecta's total kinetic energy and mass\nare functions of these parameters \\citep[equ.\\,2.1]{Chevalier89}\nand so we can equivalently choose other 3-parameter sets, e.g.:\n$n$, $E_{\\rm ej}$, and $M_{\\rm ej}$ .\n\nIn our 1D hydrodynamic\nsimulations, the RS moves into the ejecta and at most shocks\nonly the outer $\\sim$\\,0.5\nsolar masses of the ($4\\pi$) ejecta profile,\nat which point the ejecta velocity is\n$\\sim$\\,5700~{km\\,s$^{-1}$}. Provided that $v_t$ is less than this value,\nwhich is generally the case, we\nare insensitive to the plateau transition location.\nOur simulations then only depend on the outer profile,\nthat is the two parameters of Equation~\\ref{eq:ejecta}:\nthe normalization of the power-law profile\n($C=$\\,2.03\\tttt{85}~g\\,cm$^6$\\,s$^{-6}$)\\,\\footnote{\\,The\nvalue and units of $C$ do not mean much at face value; following\nmany authors we can instead specify the value of $\\rho_{\\rm ej}$ at\nsome appropriate reference values of $v$ and $t$.\nFor $v=$\\,1\\tttt{4}~{km\\,s$^{-1}$}\\ and\n$t=$\\,10~years we get $\\rho_{\\rm ej}\\approx$~390~{amu\\,cm$^{-3}$}, \nvery similar to the value of 360~{amu\\,cm$^{-3}$}\\ used by \\citet{BBMcC97let}.}\nand the exponent ($n=9$ from \\citet{Luo91b}). \nImplicitly fixing these two, we are then left with a degenerate\ndegree of freedom, our insensitivity to $v_t$, in choosing the ejecta parameters.\nHence there are pairs of ($E_{\\rm ej}$,\\,$M_{\\rm ej}$)\nthat will equivalently give the hydrodynamics presented here:\n(2.4,\\,16), (1.5,\\,7.9), or (1.0,\\,4.3), where the energy is in units\nof 10$^{51}$~erg and the mass is in solar masses. It's important\nto note that we do not make any determination of the \\textit{actual}\nglobal values of these parameters, they are only used to define the outer ejecta\nproperties relevant to our hydrodynamics.\n\n\n\\subsection{H{\\footnotesize \\,II}\\ Region Parameters}\n\nEstimates of the parameters of the H{\\footnotesize \\,II}\\ region are determined mainly\nfrom observations. This was first done by \\citet{Lundqvist99} by\nanalyzing the ultraviolet line emission from\nSN\\,1987A. \\citet{Dwarkadas07RevMex, Dwarkadas07AIP} attempted to refine these\nparameters by calculating the radius and velocity of the expanding SN\nshock wave in a spherically symmetric case, comparing the results to radio observations,\nand iterating until a good fit was obtained. They also calculated a\nreasonable fit to the hard X-ray emission under collisional-ionization\nequilibrium (CIE) conditions.\n\nIn the present work we further develop these earlier\ncalculations. The H{\\footnotesize \\,II}\\ region radius and density profile are adjusted\nso that our simulated emission\nagrees with the early X-ray light curves (up to day $\\sim$\\,5000);\nin particular, producing a reasonably sharp ``turn-on'' at $\\sim$\\,1400 days\nand agreeing with the HETG-99 and ACIS-00 measured spectra at later times.\nA further constraint on the H{\\footnotesize \\,II}\\ region, especially near the equatorial\nplane, comes from the requirement that the FS reaches the\nER (and\/or protrusions) at a time {\\it and} radius in agreement with\nthe X-ray imaging observations \\citep{Racusin09}.\nThe parameters we have found appropriate for the H{\\footnotesize \\,II}\\ region,\nshown in Figures~\\ref{fig:initprofile},\nhave it beginning at 3.61\\tttt{17}\\,cm (0.117~pc) with a density of\n$\\sim$\\,130~{amu\\,cm$^{-3}$}\\ \nand gradually increasing to $\\sim$\\,250~{amu\\,cm$^{-3}$}\\ at 6.17\\tttt{17}\\,cm (0.20~pc)\nand remaining at this density to larger radii.\nThese X-ray-based values have an inner radius closer to the expectations of\n\\citet{Chevalier95} and differ from the previous H{\\footnotesize \\,II}\\ values\n\\citep[beginning at 4.3\\tttt{17}\\,cm with\n$\\rho\\approx 6$~{amu\\,cm$^{-3}$}, increasing to $\\rho\\approx 200$~{amu\\,cm$^{-3}$}\\ at\n6.2\\tttt{17}\\,cm]{Dwarkadas07AIP} which were based primarilly on the radio\nsize at early time;\nwe discuss the reconciliation of these further in \\S\\,\\ref{sec:locs} and\n\\S\\,\\ref{sec:img}.\n\n\n\\subsection{Equatorial Ring Parameters}\n\\label{sec:er}\n\nUnlike the H{\\footnotesize \\,II}\\ region where the simple 1D approximation may suffice, it is\nclear that only a far cruder approximation to the X-ray emitting ER can be made\nwith simple 1D constructs. Note that in this paper we often use the term ``ER'' as a\ngeneric term to refer to the dense ($\\sim$\\,10$^4$~{amu\\,cm$^{-3}$}) equatorial material\nwith which the SN shock interacts.\nThe ER therefore includes the finger-like extensions that\n\\citet{McCray07} has postulated as extending inwards from the ring.\nConsequently, the ER is not located at any single radius; this is\nclearly demonstrated\nby the gradually increasing number of optical hot spots\n\\citep{Sugerman02}.\nA superposition of ER collisions in time is thus expected to make\nup the ER contribution to the X-ray light curve.\n\nFor modeling purposes we choose a single location of the ER starting at\na radius of 5.4\\tttt{17}\\,cm (with a density of $\\sim$\\,9000~{amu\\,cm$^{-3}$}).\nAs we'll see, this choice reproduces the ``kink'' in the measured X-ray radius\nvs time (\\S\\,\\ref{sec:locs}) \nand the bulk of the dramatic soft X-ray increase after day 6000\n(\\S\\,\\ref{sec:lcs}); hence this model component is\nrepresentative of the majority of shocked ER material.\nSince there is an indication that the X-ray flux\nmay be leveling off \\citep{Park11} we also consider the case of a\nfinite or ``thin'' ER with a density that then drops from\n$\\sim$\\,10800~{amu\\,cm$^{-3}$}\\ to (a lower, arbitrary\nvalue of) $\\sim$\\,1200~{amu\\,cm$^{-3}$}\\ at a radius of 5.77\\tttt{17}\\,cm,\nas shown in Figure~\\ref{fig:initprofile}.\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.55]{vmov_locs_1009.ps} \\\\\n\\end{center}\n\\caption{Initial 1D model radial profiles.\nThis early-time ($\\sim$\\,900 d) configuration shows the \nejecta, the H{\\footnotesize \\,II}\\ region, and the options of a ``thick'' or ``thin'' ER.\nThe ejecta (black) and CSM (blue)\ndensities are separated at the contact discontinuity (CD); the\nmean plasma temperature (red) and velocity (green) are also plotted.\nThe gray vertical lines indicate the boundaries of the mass-shells that\nare tracked and used to compute the NEI X-ray emission; note that they\ndo not need to extend into the plateau region of the ejecta profile.\n\\label{fig:initprofile}}\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.48]{vmov_locs_1060.ps} \\\\\n\\vspace{0.1in}\n\\includegraphics[angle=270,scale=0.48]{vmov_locs_1103.ps} \\\\\n\\end{center}\n\\caption{Model profiles at later times.\n{\\it Top}: The simulation at the ACIS-00 epoch when the FS\nhas moved well into the H{\\footnotesize \\,II}\\ region; note that the early-time\ndensity profile is shown in gray for reference.\n{\\it Bottom}: At the HETG-11 epoch, the FS has encountered\nthe ``thin ER'', slowed, and is just about to exit the ER.\nThe color coding is the same as for Figure~\\ref{fig:initprofile}.\n\\label{fig:modprofile}}\n\\end{figure*}\n\n\n\\clearpage\n\\section{MODEL RESULTS AND COMPARISONS}\n\\label{sec:results}\n\nThe previous section summarized the ejecta-CSM properties of our\nhydrodynamic simulations. In the following we compare several results derived\nfrom the simulations with the observed properties of SN\\,1987A.\n\n\\subsection{Model Locations}\n\\label{sec:locs}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.60]{sn87a_wER_110802_locs485.eps} \\\\\n\\vspace{0.2in}\n\\includegraphics[angle=270,scale=0.30]{sn87a_noER_110718_locs485.eps} \n\\includegraphics[angle=270,scale=0.30]{sn87a_noER_090410_locs485.eps} \\\\\n\\end{center}\n\\caption{Model radii compared with X-ray and radio measurements. \n {\\it Top}: The ``with ER'' simulation has most X-ray emission between the FS (blue)\n and CD (red) radii and their locations\n agree very well with the ACIS-measured average radii (black).\n Note how the kink due to the ER collision occurs first in the FS curve\n (around day 6000) and then at later times in the CD and RS (green) curves.\n {\\it Lower-left}: As expected, the ``no ER'' simulation shows\n continued expansion; this applies to the case of the out-of-plane H{\\footnotesize \\,II}\\ region\n giving rise to the very-broad X-ray component.\n {\\it Lower-right}: The 8.6~GHz average radii (orange) are better\n matched with a different CSM profile, here we use the one given in\n \\citet{Dwarkadas07AIP}.\n\\label{fig:modradii}}\n\\end{figure*}\n\nThe radio and X-ray measured radii are used to constrain the\nhydrodynamic models. In Figure~\\ref{fig:modradii} we show these radii\nalong with the locations of the simulations' FS, CD, and RS.\nThe measured radii are given in arc seconds and are based on\nde-projected model fitting for both the radio\n\\citep{Ng08} and X-ray \\citep{Racusin09} values.\nTo convert from the simulation spatial units (cm) to observed angular units\na distance of 48.5~kpc has been used to get the agreement shown;\nthis is very close to the accepted distance to SN\\,1987A\\ \n(\\S\\,\\ref{sec:intro}) and further ``fine tuning'' of the hydrodynamics\nis not warranted for these simple models.\n\nFor the ``with ER'' simulation, Figure~\\ref{fig:modradii} top panel,\nthe locations of the model emitting region (between the FS and\nCD) are in good agreement with the measured ACIS radii and reproduce\nthe ``kink'' that occurs at about day 6000\nat a radius of $\\sim$\\,0.74 arc seconds \\citep{Racusin09}.\nThis radius is set by the start of our ER at 5.4\\tttt{17}~cm, or 0.744 arc\nseconds, and it\nalso agrees well with the inner radii of the majority of the optical ring\nmaterial that is seen in the very useful Figure\\,7(b) of \\citet{Sugerman02}.\nHence, our ER model location is a good representation of the \nlocation of the start of the bulk of the ER.\nRegarding a possible end to the ER, our ``thin ER'' case\nends at a radius of 5.77\\tttt{17}~cm, or 0.795 arc seconds.\nThis gives our thin ER a radial extent of $\\sim$\\,0.05 arc seconds, and,\nlooking again at Sugerman's Figure 7(b), we see that this is\na reasonable radial extent for the central portions\nof {\\it individual} bright optical features.\nAt late times,\nthe locations shown in Figure~\\ref{fig:modradii} are\nbased on the ``thin ER'' profile (labeled in Figure\\,\\ref{fig:initprofile})\nwhich has the FS leaving the\ndense ER at $\\sim$\\,9000 days; this\nproduces a slight upward kink in the FS location curve at this time.\n\nThe ``no ER'' simulation, representing the out-of-plane H{\\footnotesize \\,II}\\ region,\ndoes a good job of matching the ACIS radii before day 6000\n(since it is the same as the ``with ER'' simulation at those times)\nand continues to expand at a roughly constant rate.\nBy 25 years, the FS in the H{\\footnotesize \\,II}\\ region simulation has progressed to\na distance of $\\sim$\\,6.8\\tttt{17}\\,cm (0.22~pc),\nequivalent to $\\sim$\\,0.94 arc seconds for the model-scaling distance of 48.5~kpc.\nComparing this to the mean radius and extent of the optical ER,\n0.83 and 0.65--1.0 arc seconds, respectively\n\\citep[Figure\\,7]{Sugerman02},\nsuggests that the FS in the out-of-plane H{\\footnotesize \\,II}\\ region is now beyond\nmost of the ER.\n\nComparing the model locations with radio measurements on these plots,\nthe 8.6~GHz data \\citep{Ng08}\nappear to be different from both the ``with ER'' and\n``no ER'' X-ray-based models; in particular showing a $>0.6$~arc second\nradius by day 2000.\nHowever, using the previous CSM profile of \\citet{Dwarkadas07AIP}\ndoes give a CD--FS location curve that is in good agreement \nwith the measured average radii for the 8.6\\,GHz radiation,\nFigure~\\ref{fig:modradii} lower-right panel.\nThe single 36\\,GHz radius measurement \\citep{Potter09} is located\nbetween the X-ray and the 8.6~GHz values, although it has an\nuncertainty range including each of these.\nOne way to accomodate the larger 8.6~GHz radio radii is to have that\nemission come from material further from the equatorial plane as\nschematically shown in Figure~\\ref{fig:hydrogeom}.\n\n\\subsection{Model Light Curves}\n\\label{sec:lcs}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.65]{hydro_wdata_lc.ps} \\\\\n\\end{center}\n\\caption{ Comparing X-ray light curves with the hydrodynamic model fluxes.\nThe H{\\footnotesize \\,II}\\ region emission from the ``no ER'' model\nis shown by the curves beginning before day 3000 (magenta, green).\nThe model does a reasonable job of matching the initial and late-time 3--10~keV flux\n(green diamonds)\nas well as modeling the initial 0.5--2~keV turn-on and growth (ROSAT points). \nAt late-times the 0.5--2~keV flux from the H{\\footnotesize \\,II}\\ region \nis in reasonable agreement with the measured very-broad flux (magenta\ndiamonds), strengthening the identification of the {v-b}\\ emission\nwith the H{\\footnotesize \\,II}\\ region.\nThe emission from the shock collision with the dense ER \nrises steeply around day 6000 (orange, light blue).\nThe light curve here (orange) is for the\nthe ``thin (or finite) ER'' case\nof Figure\\,\\ref{fig:initprofile}; the gray line indicates the\n0.5--2~keV flux when the ER continues in the ``thick ER'' case.\n\\label{fig:modlc}}\n\\end{figure*}\n\n\nWe use the hydrodynamic simulations as input to calculate the X-ray emission from each\nshocked mass-cut shell at each time step, employing the same scheme\nto track the NEI plasma state and carry out spectral calculations\nas outlined in \\citet{Dwarkadas10}.\nOnce the hydrodynamic solution is determined -- fixing temperatures,\nionization ages, and normalizations for the mass-cuts -- the only free parameters are the\nabundances,\nan optional clumping set by a clumping fraction and its\nover-density, and the overall normalization set by the solid angle\nwhich each 1D model actually fills, $\\Omega_\\mathrm{H\\,II}$ and $\\Omega_{\\rm ER}$.\nThe abundances can be determined by fitting to the measured spectra,\nas described further in \\S\\,\\ref{sec:hiiabunds}\\,\\&\\,\\ref{sec:erabunds}.\nIn terms of line broadening, the hydrodynamic velocities of the shells\nare used to include Doppler broadening in the synthesized spectra;\nin \\S\\,\\ref{sec:vbspectrum}\nthis ``real'' broadening is compared with the simple smoothing\napproximation that was used to quantify the {v-b}\\ fraction\n(\\S\\,\\ref{sec:vb}).\nFinally, for high shock velocities and low $\\tau$'s, the simple $\\propto\\,v^{-2}$\nbehavior of $\\beta=T_e\/T_i$ \\citep{Ghavamian07} produces $T_e$ values that are very low.\nAs in \\citet{Dwarkadas10} we have used a\nmodified $\\beta(v)$ relation, suggested by the results of\n\\citet{vanAdelsberg08}, to\nset a minimum value of $\\beta_{\\rm min}\\approx 0.06$.\n\nIntegrating the simulated spectra over the standard energy bands,\n0.5--2~keV and 3--10~keV, we created model light curves, shown in\nFigure~\\ref{fig:modlc}. The light curves for the two hydrodynamic simulations\nhave been plotted separately and their (imagined) sum can reasonably describe\nthe observed SN\\,1987A\\ light curves over the full time span.\n\n\n\\clearpage\n\n\nA filling fraction of $\\Omega_\\mathrm{H\\,II}\/4\\pi=$~0.26 is\nused to match the overall flux level for the H{\\footnotesize \\,II}\\ (``no ER'')\nsimulation, corresponding to emission from\n$\\pm$\\,15 degrees of the ring plane. This is roughly similar to \nother estimates of the H{\\footnotesize \\,II}\\ region's out-of-plane extent, e.g., \nby \\citet[$\\pm$\\,30 degrees]{Michael03},\n\\citet[$\\pm$\\,10 degrees]{Zhekov10}, and \\citet[$\\pm$\\,7.2 degrees]{Mattila10}.\nThe abundances for the H{\\footnotesize \\,II}\\ light curve are set from fits\nto the early ACIS observation, described in \\S\\,\\ref{sec:hiiabunds}.\nOne characteristic of this early-time H{\\footnotesize \\,II}\\ emission is\nits relative hardness, e.g., for the deep ACIS-00 observation\nat day 5036 the ratio $F_{3-10} \/ F_{0.5-2}$ is 0.35\nwhereas at later times (day 7997, January 2009) the ratio has\ndropped to 0.12. As the light curves show, this is caused by the\nincrease in the 0.5--2~keV flux due to the collision with\nthe ER, rather than any decrease in the 3--10~keV emission.\n\nIn contrast to the H{\\footnotesize \\,II}\\ region, the ER needs to produce\nemission primarily in the 0.5--2~keV range while keeping\nthe 3--10~keV flux below the measured levels.\nEven though the ER is dense, our simulation still gives\nelectron temperatures at\/above 3~keV and\nthus relatively too much 3--10~keV flux.\nWe could further increase the density of the ER, however given the\nresonable agreement of the slope of the ``with ER'' simulation\n(top panel of Figure~\\ref{fig:modradii}) and the expectations of an inhomogeneous\nER (e.g., the optical spots), we consider another, approximate approach.\nWe can match the observed spectra\nby including emission from ``clumps'' in the ER\nwith a density enhancement $\\eta_{\\rm clump}\\approx \\times$5.5 that\nfills $f_{\\rm clump}\\approx$ 30\\% of the CSM volume.\nThe clumping is implemented at the spectral synthesis stage and\nintroduces $kT_e\\approx$ 0.5--0.8~keV emission components\ngreatly enhancing the 0.5--2~keV emission (further details of the\nclumping implementation are given in \\S\\,\\ref{sec:erabunds}.)\nWe acknowledge that for these clumping parameters most of the mass is in the very dense\nclumps (i.e., 0.30$\\times$5.5\\,\/\\,(0.70$\\times$1+0.30$\\times$5.5)\n~$\\approx$ 70\\%)\nand so the clumping is not a small perturbation on the\nER hydrodynamics; therefore our 1D clumped ER simulation is just a placeholder\nfor a realistic multi-dimensional simulation.\n\nThe single set of ``with ER'' light curves plotted in Figure~\\ref{fig:modlc}\nrepresents the bulk of the 0.5--2~keV flux increase; the fractional area used to\nscale the 1D model's emission is\nvery small: $\\Omega_{\\rm ER}\/4\\pi =$~0.0016 .\nTo put this into perspective,\nthe fraction subtended by a uniform region with the dimensions of\nthe optical ring, i.e., a region extending\n$\\pm$\\,0.05 arc seconds out of the plane at a radius of 0.83 arc seconds,\nis almost 40$\\times$ larger, $\\sim$\\,0.0600.\nHence, the small value indicates that the the ring is not uniformly\ndense and clumped, but rather it has ``X-ray hot spots'' analogous to\nthe (still) discrete optical spots around the ring.\nSpecifically, the 0.0016 value would arise from $\\sim$\\,20 regions\neach subtending a diameter of $\\sim$\\,2 degrees around the ring.\n\nIt is seen in Figure~\\ref{fig:modlc} that there is \nearlier-time flux growth above the H{\\footnotesize \\,II}\\ emission that our plotted ER\ncurve does not include. As mentioned in \\S\\,\\ref{sec:er}, a range of inner radii is\nexpected for the ``ER'' with some material at radii smaller than the\n0.74~arc seconds where our modeled ER starts.\nMentally ``translating'' our ER curve, we would expect to get\nflux beginning $\\sim$\\,5500 days\nby having a small fraction of ER material (perhaps\n0.1\\,$\\times$\\,$\\Omega_{\\rm ER}$) that is impacted\nat $\\sim$\\,4600 days. From our hydrodynamics this time\ncorresponds to a radius of 4.8\\,\\tttt{17}\\,cm (0.155~pc)\nor 0.67 arc seconds. This radius is\nin reasonable agreement with the locations of the early optical spots\n\\citep[Figure\\,7]{Sugerman02}.\n\n\n\n\\subsection{H{\\footnotesize \\,II}\\ Abundances from the Hydrodynamics}\n\\label{sec:hiiabunds}\n\nUsing the set of shocked mass-cut shells at a given time-step of the \nhydrodynamics, we can create a spectral model in ISIS which is the sum\nof emission from each of the shells. The relative normalizations, temperatures,\nand ionizations ages for each model component (mass-cut shell) are set from the\nhydrodynamics. Tying the abundances of all components together,\nwe then have a model which can be fit to measured spectra\nby adjusting the abundances and an overall normalization.\nBy using the hydrodynamics to set the plasma properties we reduce the\ndegeneracy in fitting that arises when the temperature(s) and ionization\nage(s) of the plasma are unconstrained and therefore we get accurate abundance\nvalues.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.30]{110718_Ttaudist_1060.ps}\n\\hspace{0.1in}\n\\includegraphics[angle=270,scale=0.30]{obs1967_hydro-based.ps} \n\\end{center}\n\\caption{{\\it Left}: The distribution of $kT_e$ and $\\tau$ values in\nthe H{\\footnotesize \\,II}\\ hydrodynamics at the ACIS-00 epoch.\nPoints of $(\\tau, kT_e)$ are plotted for all shocked\nmass shells in the model;\nsymbols for the shocked ejecta (black)\nand CSM (blue) are larger\/bold for brighter shells.\n{\\it Right}: The ACIS-00 data (black) are fit with a model (red) that\nis the sum of the emission from the 14 shocked shells of the left panel;\na single norm and the abundances of N, O, Ne, Mg, Si, S, and Fe were the\nonly free parameters in the fit. The emission from the CSM shells (blue)\nis much larger than that of the shocked ejecta (green) at this early epoch.\n\\label{fig:TTauHII}\\label{fig:obs1967}}\n\\end{figure*}\n\n\\input{tab_obs1967}\n\nThe early deep {\\it Chandra}\\ observation, ACIS-00, is the best data set to use\nto determine the abundances of the H{\\footnotesize \\,II}\\ region (as opposed to the\ndense ER abundances, which may differ).\nThese data also provide a useful example of how the fit abundances and\ntheir confidence ranges vary\ndepending on the $kT$--$\\tau$ model assumptions. The results of fitting four\ndifferent cases are shown in Table~\\ref{tab:obs1967}.\nIn all cases the redshifts and $N_H$ were frozen\nand the normalization(s) and abundances (N, O, Ne, Mg, Si, S, and Fe) were fit.\nIn the first case\na single {\\tt vpshock} component is fit giving $kT\\approx 3.0$~keV\nand $\\tau\\approx$ 1.0\\tttt{11}~{s\\,cm$^{-3}$}; these are similar to the values in \\citet{Park02}.\nAs a second case, we fit a 2-shock model fixing the temperatures,\nionization ages, and $N_H$ based on values from \\citet{Zhekov10};\nthis has a high temperature component similar to the 1-shock value\nalong with a very low temperature component.\nIn the third case, our model of Table~\\ref{tab:3shock} is shown, and\nbecause we have set $N_{\\rm mid}$ to 0 for this early data it is\nalso a 2-shock model with a different set of $kT$'s and $\\tau$'s.\nFinally, we fit the data with the temperatures and ionization ages\nfixed based on the \nvalues from the ``no ER'' hydrodynamic model at the ACIS-00 epoch;\nthese values are plotted in the\nleft panel of Figure~\\ref{fig:TTauHII} and correspond to the model\nat the epoch shown in the top panel of Figure~\\ref{fig:modprofile}.\n\nThe reduced $\\chi^2$ values are near 1.0 for all four of the fits, so they\nare equally acceptable from a data-fitting perspective. However, as the table\nshows, the 1-$\\sigma$ ranges of the fit abundances depend very much on the\n$kT$--$\\tau$ values used in the model. \nIf it is possible to constrain the $kT$--$\\tau$ values through\nother means or assumptions, as in the fourth case here, then\nthe fit abundances will be both more realistic and better constrained.\nHence, we adopt the hydrodynamics-based model as the most physical and\ninclude its abundances in the H{\\footnotesize \\,II}\\ column of Table~\\ref{tab:abundsdisc}.\n\n\n\\input{tab_abundsdisc}\n\n\n\n\\clearpage\n\\subsection{ER Abundances and Clumping}\\label{sec:erabunds}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.30]{110718_Ttaudist_1101.ps}\n\\hspace{0.1in}\n\\includegraphics[angle=270,scale=0.30]{110719_Ttaudist_1103clmp.ps}\n\\end{center}\n\\caption{Distributions of $kT_e$--$\\tau$ for the hydrodynamics at the HETG-11 epoch.\nPoints of $(\\tau, kT_e)$ are plotted for all shocked\nmass shells in the H{\\footnotesize \\,II}\\ (no-ER, left panel) and ``with ER'' (right\npanel) simulations.\nSymbols for the shocked ejecta (black)\nand CSM (blue) are larger\/bold for brighter shells.\nThe H{\\footnotesize \\,II}\\ distribution is similar to earlier times\n(Figure~\\ref{fig:TTauHII},\nleft panel) although shifted to somewhat higher $\\tau$ and $kT_e$ values.\nThe ``with ER'' distribution (right panel) is dominated by components at\nmuch higher $\\tau$ values;\nthe addition of a clumped component introduces additional\n$kT_e$--$\\tau$ values (red) that are scaled from the\nnon-clumped CSM values.\n\\label{fig:TTauHETG11}}\n\\end{figure*}\n\n\nWe can similarly fit hydrodynamics-based models to the later\nobservations where the emission is dominated by the shocked ER material\nand in this way determine the ER abundances.\nAt these times there will still be a contribution from the H{\\footnotesize \\,II}\\\nregion, i.e., the {v-b}\\ component. This contribution is removed\nfrom the data before fitting by evaluating our ``no ER'' model at the\nappropriate epoch and subtracting the predicted counts from the data\nbefore proceeding with ER-model fitting.\n\nThe $kT_e$--$\\tau$ values of the shocked mass-cuts in our H{\\footnotesize \\,II}\\ and\nER models at the HETG-11 epoch are shown in Figure~\\ref{fig:TTauHETG11}.\nAs the blue points in the ``with ER'' (right) panel show, the $kT_e$\nvalues in the shocked dense ring are mostly at\/above 3~keV. Given\nthese temperatures, simply adjusting the abundances is insufficient to\nfit the data, there is simply too little low-$kT_e$, high-$\\tau$ plasma. \nWe have therefore included ``clumps'' in the ER material\nfollowing the scheme in \\citet{Dwarkadas10}.\n\nFor each of the shocked CSM mass-cuts (blue circles in the right panel\nof Figure~\\ref{fig:TTauHETG11})\nwe add an additional model component \nwith its temperature reduced by the clump's density enhancement,\n$kT_e^\\prime = kT_e\/\\eta_{\\rm clump}$, and its\nionization age increased to $\\tau^\\prime = \\eta_{\\rm clump}\\,\\tau$.\nThe additional clumped components are shown as\nred circles in the right panel of Figure~\\ref{fig:TTauHETG11}\nfor $\\eta_{\\rm clump}\\approx$ 5.5.\nThe original and clumped components have additional\nfactors in their norms of $(1-f_{\\rm clump})$ and\n$f_{\\rm clump}\\,\\eta_{\\rm clump}^2$, respectively,\nwhere $f_{\\rm clump}\\approx$ 30\\% is the volume fraction\nof clumped material.\nThe resulting ISIS fit function at the HETG-11 epoch consists of 45\nNEI model components but with only 9 free parameters: an overall \nnormalization, the 2 clumping parameters, and the ER abundances of O, Ne,\nMg, Si, S, and Fe.\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.60]{HETG-07_data_hydro_MEG.ps} \\\\\n\\vspace{0.1in}\n\\includegraphics[angle=270,scale=0.60]{HETG-11_data_hydro_MEG.ps}\n\\end{center}\n\\caption{HETG(MEG) spectra and full hydrodynamics-based models.\nThe HETG-07 (top) and HETG-11 (bottom) data (black) are reasonably\nfit by the total (H{\\footnotesize \\,II}\\ plus nominal clumped-ER) model spectra (red).\nThe H{\\footnotesize \\,II}-shocked-CSM (blue) and the H{\\footnotesize \\,II}-shocked-ejecta (green)\ncomponents are also shown individually.\nNote the relative growth of the shocked ejecta (green) between the epochs;\nhence at late times the ejecta and its abundances will dominate the\nH{\\footnotesize \\,II}\\ (i.e., {v-b}) component.\n\\label{fig:hydrospect}}\n\\end{figure*}\n\nWe fit both the deep HETG-07 and HETG-11 data sets with their appropriate-epoch\nhydrodynamics-based models, determining the abundances and clumping\nparameters for each. There are only small differences\nbetween the 2007 and 2011 fits,\nso we have adopted nominal ER parameters that are the average values given in the\n``ER'' column of Table~\\ref{tab:abundsdisc}.\nNote that the N abundance was set based on similar fitting\nto the RGS-11 spectrum.\nThese values were then used\nto generate our light curve (\\S\\,\\ref{sec:lcs})\nand to fit the hydrodynamics-based models to all of the grating spectra\nwith only a single normalization adjustment. As examples,\nFigure~\\ref{fig:hydrospect} compares the two deep HETG spectra\nwith the sum of the hydrodynamics-based emission from the H{\\footnotesize \\,II}\\ region and\nthe nominal-parameter ER emission. \n\nGiven our crude clumping implementation and the system's physical\ncomplexity, we have\nless confidence in the accuracy, i.e., agreement with reality, of our ER abundances\nthan those we determined for the H{\\footnotesize \\,II}\\ region at the ACIS-00 epoch.\nHowever since the ER $kT$s and $\\tau$s are based on a hydrodynamic\nsolution there is some expectation that they will be an improvement over\nsimpler multi-shock models.\nFor comparison, we've also included in Table~\\ref{tab:abundsdisc}\nthe abundances given by\n\\citet[multi-shock fits to grating data]{Zhekov09},\n\\citet[RSS fits to the ACIS data]{Zhekov10}, and\n\\citet[optical \\& NIR data before the ER collision]{Mattila10}.\nOur ER abundances for N, O, Ne, and S do appear to be in someshat\nbetter agreement with the \\citet{Mattila10} values\nthan are the other X-ray-determined abundances.\n\nThis is a good place to compare the free parameters of the\n\\citet{Zhekov10} model (Z10 in the following)\nwith ours, in terms of their number and their values. \nThe two model geometries are similar at later times, both are dominated by the\nER emission, Figure~\\ref{fig:rsscompare}, and both make approximations\nto include clumped ER material.\nEach model fixes time-independent values for the CSM's $\\rho(r)$,\nthe $N_H$, and the abundances.\nIn addition to these, Z10 fit six parameters\n{\\it for each observation}: the values\nof the temperature, ionization age, and emission measure for both the \nblast wave (BW) and the transmitted shock (TS).\nFor example, for their latest data point which is near the HETG-07\nepoch, the values for these are: $kT_{\\rm BW}\\approx$\\,1.7~keV, \n$kT_{\\rm TS}\\approx$\\,0.33 keV,\n$\\tau_{\\rm BW}\\approx$\\,1.4\\tttt{11}~{s\\,cm$^{-3}$},\n$\\tau_{\\rm BW}\\approx$\\,30\\tttt{11}~{s\\,cm$^{-3}$}, and the respective \nemission measures are 1.05 and 9.5 in units of 10$^{59}$\\,cm$^{-3}$.\nIn contrast, all of our mass-shell $kT$ and $\\tau$\nvalues are fixed by the hydrodynamics, e.g., as for HETG-11\nin Figure~\\ref{fig:TTauHETG11}, and our ER modeling adds only three\nfurther time-independent parameters: the overall normalization,\n$\\Omega_{\\rm ER}$, and the two clumping parameters.\nIn our approximation the clump over-density, $\\eta_{\\rm clump}\\approx\n5.5$, sets the ratio of ER-to-clump temperature.\nFor the Z10 values above, this\nratio is $T_{\\rm BW}\/T_{\\rm TS}\\approx 5$, close to ours.\nLikewise, the Z10 emission measure ratio, ${\\rm EM}_{\\rm TS}\/{\\rm EM}_{\\rm BW}\\approx 9$,\nis in the ball park\nof our value which is given by $f_{\\rm clump}\\eta_{\\rm clump}^2\/(1-f_{\\rm\nclump})\\approx13$, using our clump volume fraction $f_{\\rm clump}\\approx 30$\\,\\%.\nIn constrast, the clump volume fraction for the Z10 model is of order 5\\,\\% because\nthe clump-to-ambient density ratio is large: \n$\\rho_{\\rm TS}\/\\rho_{\\rm BW}\\approx 14$; this latter value is\na function of the temperature ratio as shown in a\nplot in the Appendix of \\citet{Zhekov09}. \nThese comparisons suggest that there may be some degeneracy between over-density\nand volume fraction for the clumps. In anycase, it is clear that both\napproaches here are only approximating the complex 2D\/3D density structures\nof the real ER.\n\n\n\n\\clearpage\n\\subsection{Very-Broad Spectrum from the Simulation}\n\\label{sec:vbspectrum}\n\nWe can check our decision to use\na smoothed version of the spectrum as the model for the\n{v-b}\\ component by directly comparing the two of them, as shown in\nFigure~\\ref{fig:vbhydro}. Here, the broad lines in the ``real''\n(from the hydrodynamics) {v-b}\\ component are similar to those\nin the smoothed version of the 3-shock fit to the data.\nThis similarity can be roughly explained using the $kT_e$--$\\tau$\ndistributions shown in Figure~\\ref{fig:TTauHETG11}: the H{\\footnotesize \\,II}\\ ({v-b})\nemission is dominated by components with $kT_e \\approx$ 2.5~keV\nand $\\tau \\approx$ 6\\tttt{10}~{s\\,cm$^{-3}$}, whereas the main ER emission\nis from clumped material with $kT_e \\approx$ 0.7~keV\nand $\\tau \\approx $ 2\\tttt{13}~{s\\,cm$^{-3}$}. Putting these values into a {\\tt vnei}\nmodel gives similar line structures for the two cases, hence,\nthe high-$kT$-low-$\\tau$ plasma\nproduces ionization states and emission lines that are similar\nto those of a low-$kT$-high-$\\tau$ plasma.\nBased on this comparison we did, however, decide to ignore the\nFe{\\footnotesize \\,XVII}\\ range 0.70--0.75~keV when doing the {v-b}\\ fitting as these\nlines were clearly not present in the hydrodynamic-based {v-b}\\ spectrum.\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=270,scale=0.60]{compare_HETG-07_3shnoER.ps} \\\\\n\\end{center}\n\\caption{Comparing the heuristic very-broad spectrum\nwith the hydrodynamics-based H{\\footnotesize \\,II}\\ {v-b}\\ spectrum.\nThe HETG-07 data (black) are shown with the best-fit 3-shock\nmodel (red) that includes a {v-b}\\ component (blue).\nThis ``smoothed version of the model'' form of the {v-b}\\ component is compared\nwith the multi-shells model spectrum at the HETG-07 epoch\n(magenta). Ignoring the overall continuum shape,\nthey are in reasonable line-broadening agreement except for \nthe lack of significant Fe{\\footnotesize \\,XVII}\\ lines between 0.70--0.75~keV in\nthe hydro-based model.\n\\label{fig:vbhydro}}\n\\end{figure*}\n\n\nIn terms of the assumption of a constant width of the {v-b}\\ emission\nin time (the evolution path shown in Figure~\\ref{fig:vbcontours}),\nthe average bulk motion of the shocked H{\\footnotesize \\,II}\\ material\nin the ``no ER'' simulation varies (only) from 4200~{km\\,s$^{-1}$}\\ to\n3850~{km\\,s$^{-1}$}\\ from the HETG-99 epoch to an age of over 25 years (9000+ days).\nThis is also seen in\nthe locations of the ``no ER'' CD and FS locations, lower-left panel \nof Figure~\\ref{fig:modradii}, which\nfollow nearly straight lines with overall slopes of\n4300~{km\\,s$^{-1}$}\\ and 5100~{km\\,s$^{-1}$}, respectively.\n\n\n\n\n\\subsection{The Mass of the X-Ray Emitting Material}\\label{sec:mass}\n\nWe can calculate the shocked mass for different components and epochs of the\nmodel. For the H{\\footnotesize \\,II}\\ region, at the ACIS-00 epoch the\nFS is at $\\sim$\\,0.162~pc (top panel of Figure~\\ref{fig:modprofile})\nand the mass of shocked CSM in the simulation, correcting for the opening angle\nnormalization $\\Omega_\\mathrm{H\\,II}\/4\\pi\\approx$ 0.26 (\\S\\,\\ref{sec:lcs}),\nis 0.012 solar masses. At this time our model shows $\\sim$\\,0.005 solar masses of ejecta\nare shocked within the same opening angle.\nLater, at the HETG-11 epoch, \nthe shocked H{\\footnotesize \\,II}\\ material that produces the {v-b}\\ emission in our model\nhas grown to 0.042 solar masses along\nwith an additional 0.024 solar masses of shocked ejecta.\nOur H{\\footnotesize \\,II}\\ mass values compare in order of magnitude\nwith the mass of 0.018 solar masses that \\citet{Mattila10}\nrequire in their low-density (10$^2$ atoms\\,cm$^{-3}$) H{\\footnotesize \\,II}\\ component.\n\nThe total ``4$\\pi$'' mass of the thin ER is 1.18 solar masses;\nhowever, correcting for the small solid-angle fraction,\n$\\Omega_{\\rm ER}\/4\\pi\\,\\approx\\,0.0016$,\nwe get only $\\sim$\\,0.0019 solar masses of ER that are shocked at the\nHETG-11 epoch (when the FS is exiting the thin ER).\nIncluding the extra mass due to clumping (multiplying by a factor of 2.35, \\S\\,\\ref{sec:lcs}),\nwe have 0.0045 solar masses as the total shocked-ER mass that produces the observed\nnon-{v-b}\\ X-rays in our model at the HETG-11 epoch.\nThis is to be compared with an estimate of the total mass of the UV-ionized ER\nof 0.058 solar masses distributed among three densities of 1.4\\tttt{3},\n4.2\\tttt{3}, and 4.2\\tttt{4}~{amu\\,cm$^{-3}$}\\ \\citep[using\n$\\sim$\\,1.4~amu per atom]{Mattila10};\nof this total, 0.0120 solar masses are in the highest-density component,\nalmost 3 times our X-ray inferred mass.\nHowever, it could well be that our ER X-ray emission is a factor of a\nfew overly efficient per mass and including this as an error bar puts us\nin a gray area in trying to use this mass comparison\nto decide if all of the ionization-visible dense ER has been shocked at the\ncurrent $\\sim$\\,25 year age of SN\\,1987A.\n\n\n\\subsection{Image Implications}\n\\label{sec:img}\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[angle=0,scale=1.0]{sn87a_hymod_Pole_20045.eps}\n\\hspace{0.15in}\n\\includegraphics[angle=0,scale=1.00]{sn87a_hymod_Pole_20115.eps} \\\\\n\\vspace{0.05in}\n\\includegraphics[angle=0,scale=1.0]{sn87a_hymod_Side_20045.eps}\n\\hspace{0.15in}\n\\includegraphics[angle=0,scale=1.00]{sn87a_hymod_Side_20115.eps} \\\\\n\\vspace{0.05in}\n\\includegraphics[angle=0,scale=1.0]{sn87a_hymod_Sky_20045.eps}\n\\hspace{0.15in}\n\\includegraphics[angle=0,scale=1.00]{sn87a_hymod_Sky_20115.eps} \\\\\n\\end{center}\n\\caption{Hydrodynamics-based geometric emission models of SN\\,1987A\\ at epochs\n2004.5 (left column) and 2011.5 (right column). Three views are shown: a ``polar''\nview (top; looking perpendicular to the equatorial plane),\na ``side'' view (middle; north is up and the Earth is to the left),\nand the usual sky view (bottom; north up, east to left).\nFour emission components are color-coded: the shocked H{\\footnotesize \\,II}\\ material (blue),\nthe reverse-shocked ejecta (green), the emission from the shocked\nER (red spots), and higher-lattitude 8.6~GHz radio emission (gray).\n\\label{fig:hydrogeom}}\n\\end{figure*}\n\nWe can construct schematic geometric visualizations of SN\\,1987A\\ using\nour simple hydrodynamic models to set the locations and fluxes \nof the components at a given epoch.\nThe main emission components at two different epochs\nare shown in multiple views in Figure~\\ref{fig:hydrogeom}\nusing a color-coding similar to that for the shocked plasma in the schematic of\nFigure~\\ref{fig:rsscompare}.\nThese images were created using\nsimple software extensions to ISIS \\citep{Dewey09}\\,\\footnote{\\,See\nalso: {\\tt http:\/\/space.mit.edu\/home\/dd\/Event2D\/}}.\n\nThe H{\\footnotesize \\,II}\\ (``no ER'') simulation gives rise to emission from\ni) shocked H{\\footnotesize \\,II}\\ material and ii) reverse-shocked ejecta; these\nregions extend out of the ER plane as shown in the images.\nThe ``with ER'' simulation is shown using a schematic geometry of\n21 discrete spots to emphasize our ``X-ray hot spots'' conclusion that\nis based on the small value of $\\Omega_{\\rm ER}$.\nThe ``with ER'' emission components include: iii)\nthe shocked dense ER and its clumps, iv) the shocked H{\\footnotesize \\,II}\\ material that is\nin the plane of the ER, and v) reverse-shocked ejecta also in\nthe ER plane.\nBecause the effective solid angle associated with the ER simulation is so small\n($\\Omega_{\\rm ER}\/4\\pi=$~0.0016) these last two components make only\nminor flux contributions which are exceeded by the first\nthree. We note that component ``iv)'' is the plasma that\nis ``shocked again'' in the reflected shock structure (RSS)\nparadigm \\citep{Zhekov09,Zhekov10};\nour ``with ER'' hydrodynamics does show\nthe reflected shock(s) having velocity effects on the shocked H{\\footnotesize \\,II}\\ region\ninterior to the shocked ER for\nabout 1.5 years, after which the region's velocity recovers and is similar to\nother regions, e.g., see the velocity curve in\nFigure~\\ref{fig:modprofile}, bottom panel.\n\nThese visualizations suggest the need for fitting the X-ray\nimaging data with multiple spatial components\nand provide some guidance on the component properties.\nRecent work of \\citet{Ng09} has explored a\ntwo-component spatial model in fitting SN\\,1987A's image, and\nas they emphasize, further observations are needed to come\nto a firm conclusion.\n\n\n\\clearpage\n\\section{CONCLUSIONS}\n\nIn this paper we have modelled the X-ray emission from SN\\,1987A,\nfocussing on the very-broad ({v-b}) component seen with the\nhigh-resolution grating instruments. \nAlthough the CSM of SN\\,1987A\\ is known to have a complex\nshape from optical observations, we have chosen to model it using 1D\nsimulations in and out of the equatorial plane, our H{\\footnotesize \\,II}\\ and ER\nhydrodynamics. Even with these simplifying\nassumptions the models yield a good fit to the X-ray-measured radii\nand growth rates, the multi-band X-ray light curves, and\nthe high-resolution X-ray spectra.\n\nOur results show that better fits to the X-ray spectra over the last\ndecade are obtained if a very-broad component of emission, with a FWHM of\nabout 9300 km s$^{-1}$, is included. Such a {v-b}\\ component is present\nin the data over at least the last decade.\nOur hydrodynamic simulations provide\na natural explanation for this component: it arises from the shocked H{\\footnotesize \\,II}\\\nmaterial and it is the dominant X-ray component\nuntil $\\sim$\\,5500~days after the SN explosion.\nSince then the total 0.5--2~keV flux dramatically\nincreased due to the ``ER'' collision, yet\nthe inferred 0.5--2~keV flux of the {v-b}\\ component itself\ncontinued to grow in agreement with our hydrodynamics expectations.\nIdentifying the 3--10~keV flux as originating primarilly from the\n{v-b}\\ component provides a natural explanation for the difference in the\n0.5--2 and 3--10~keV light curves.\nAt the present epoch, the {v-b}\\ contribution is $\\sim$\\,20\\% of the\ntotal 0.5--2~keV flux, and of this the hydrodynamics suggests that\nroughly half is coming from reverse-shocked {\\it ejecta}.\nThis ejecta contribution should grow and may cause\nthe {v-b}\\ component to become selectively\nenhanced in lines that reflect the (outer) ejecta composition.\n\nOur ER hydrodynamics adquately reproduces the observed X-ray image radii\nand the bulk of the late-time 0.5--2~keV flux increase.\nWe do require clumps in the ER with density enhancements of\n$5.5\\times$ the $\\sim$\\,10$^4$~{amu\\,cm$^{-3}$}\\ ambient ER values, not\nunlike the clumps that \\citet{Zhekov10} have added within their smooth\nCSM density profile.\nThese are approximations to the actual 2D\/3D density structure of the\nshock-ER interaction; \nit would not be surprising if the effective clumping parameters might change\nin time, e.g., as clumps evaporate and\/or turbulent structures evolve.\n\nOur approach allows us to make predictions of what the image size, X-ray spectrum\nand light curve will look like in the next few years, in particular\nfor the two cases of i) a continuing on-going interaction with dense ER material\n(our ``thick ER'' case) and ii) the drop in flux if the ER is\n``thin''. In this latter case, the FS in the\nequatorial plane has generally exited the densest parts of the ER\nand has begun moving into lower-density material as suggested by\n\\citet{Park11}. For this case our hydrodynamics shows a decrease in\nthe 0.5--2~keV flux of $\\sim$\\,17\\,\\% per year.\nNote that a third option that cannot be completely ruled out\nis that the FS will encounter {\\it larger amounts} \nof dense ER material, e.g., the main body of the ER; this would\nresult in even greater increases in flux than we have shown here.\nComparing these and more realistic multi-dimensional\npredictions with future observations will provide significant\ninformation regarding the thickness, density, and structure of the equatorial ring\nand the H{\\footnotesize \\,II}\\ region within and outside the equatorial plane\nas SN\\,1987A\\ moves into a new phase.\n\n\n\\acknowledgments\n\nWe thank the anonymous referee for useful suggestions and for holding our statistical feet\nto the fire regarding the significance of the {v-b}\\ component measurements.\nWe thank David Burrows for access to the ``hetg-11'' data in advance\nof their public release. R.S.\\ acknowledges support from the German\nBundesministerium f\\\"ur Wirtschaft und Technologie~\/~Deutsches Zentrum\nf\\\"ur Luft- und Raumfahrt (BMWI\/DLR) grant FKZ 50 OR 0907. Support for\nthis work was provided by the National Aeronautics and Space\nAdministration (NASA) through Chandra Award Number TM9-0004X to V.V.D.\\\nat the University of Chicago issued by the {\\it Chandra}\\ X-ray Center (CXC),\nwhich is operated by the Smithsonian Astrophysical Observatory (SAO) for and\non behalf of NASA under contract NAS8-03060.\nNASA also provided support through the SAO contract\nSV3-73016 to MIT for support of the CXC and Science Instruments.\n\n{\\it Facilities:} \\facility{CXO (HETG)}.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:level1}Introduction}\n\nAllene (propadiene, $\\mathrm{C_3H_4}$, CH2:C:CH2) is the prototype and first member of the group of molecules called cumulenes. It is thought to be present in the interstellar medium \\cite{kaiser1997neutral, jones2011formation} at least as an intermediate, but as it lacks a dipole moment it has not yet been detected there directly. It is closely related to the important transient molecule ketene (CH2:C:CO) which has been detected in the interstellar medium \\cite{ruiterkamp2007organic}. This paper will report on single-photon double and triple ionisation of allene, as well as on allene's dissociation pathways.\n\nThe single ionisation of allene and its consequent dissociations have been studied using mass spectrometry \\cite{stockbauer1979ionization}, photoelectron spectroscopy \\cite{baker1969photoelectron, thomas1974photoelectron,bieri1977valence} and photoelectron-photoion coincidence spectroscopy \\cite{dannacher1978unimolecular}. The electronic structure of the singly charged molecule has recently attracted theoretical interest, focusing on \"M\u00f6bius\" and \"helical\" frontier orbitals \\cite{garner2018coarctate, hendon2013helical,soriano2014allenes} peculiar to cumulenes. \n\nMuch less is known about multiple ionisation of allene, but double and triple ionisation of the molecule have both been examined in experiments using the electron capture charge exchange technique, \\cite{andrews1992allene,andrews1995allene}, where the lowest vertical double ionization energy to the expected ground-state triplet was determined as 28.2 $\\pm 0.3$ eV. Allene double ionisation and the rates of its dissociation reactions have also been examined using fs IR multi-photon laser pulses \\cite{hoshina2011metastable,xu2013experimental}. In these and other previous studies of allene double ionization, dissociation producing H$_3^+$ ions has been found, surprisingly, to be more abundant than from molecules where three hydrogen atoms are initially contiguous in \u2013 CH$_3$ groups. The mechanism was interpreted in the specific case of H$_3^+$ formation from a series of hydrocarbon molecules with no methyl group including allene as involving a \"roaming\" mechanism where a neutral H$_2$ is detached from the residual C$_3$H$_2^{2+}$ fragment, after which a proton is transferred from the doubly charged parent to form the H$_3^+$ ion \\cite{hoshina2008publisher}, in line with the theoretical work of Mebel and Bandrauk \\cite{mebel2008theoretical}. The creation of H$_3^+$ is found to be a significant dissociation mechanisms for doubly charged allene, and must include at least migration of one hydrogen atom \\cite{hoshina2011metastable, mebel2008theoretical} and possibly more extensive rearrangement. A number of experiments probing this mechanism, mainly by IR multi-photon ionisation in different molecules (mostly methanol) followed \\cite{livshits2020time}. Calculations on methanol indicate that the formation of H$_3^+$ competes with that of H$_2^+$ on a sub-100 fs time scale \\cite{livshits2020time}. \nIn the present work we examine both H$_2^+$ and H$_3^+$ detection in correlation with the electrons involved in the creation of the nascent allene dications which dissociate to form them.\n\nOur study was carried out using electron-electron (ee) coincidence measurements at photon energies above and below the C1s inner shell (approximately 291 eV), and electron-electron-ion (eei) as well as electron-electron-ion-ion (eeii) coincidence measurements at 40.8 eV. These measurements determine the spectra of states of nascent doubly ionised allene, and the spectra coincident with undissociated parent dications as well as each set of dissociation products. Photoionization mass spectra were also measured at all photon energies and ion-ion coincidence maps were taken to identify the decay pathways and clarify the mechanisms. The spectra and dissociation pathways are discussed and interpreted with the help of quantum chemical calculations carried out by ourselves and are also compared with calculations available in the literature \\cite{mebel2008theoretical}. \n\n\\section{Experimental Methods}\n\nThe experiments were carried out in our laboratory at the University of Gothenburg and at the synchrotron radiation facility BESSY-II of the Helmholtz Zentrum f\u00fcr Materialien in Berlin. In both cases, the target gas was let into the spectrometer using a hollow needle to create an effusive gas beam in the interaction region. The basic configuration of our system, which builds on a more compact version of the original instrument of this type \\cite{eland2003}, comprises a strong conical magnet with a divergent field of approximately 1 T at the light-matter interaction point and a 2 m flight tube surrounded by a weak homogeneous solenoid field (few mT). At the end of the flight tube, electrons are registered by a multi-channel plate (MCP) detector, making the overall collection-detection efficiency of this magnetic bottle electron spectrometer (MBES) for low energy (< ca. 400 eV) electrons about 50-60 \\%. The electron energy resolution of the set-up is about $\\mathrm{E_{kin}\/\\Delta E_{kin} \\sim 50}$ when collecting only electrons. For collection of both electrons and ions the strong conical magnet is replaced by a hollow ring magnet with a weaker magnetic field in the interaction zone, which limits the electron energy resolution to about $\\mathrm{E_{kin}\/\\Delta E_{kin} \\sim 20}$, but with the benefit that ions can be extracted in the opposite direction to the electrons \\cite{eland2006,feifel2006}. In this latter configuration, electron-ion coincidence data are obtained by first letting the electrons leave the interaction region, before accelerating the ions in the opposite direction, through the ring magnet and into a two-field time of flight mass spectrometer with a 0.12 m long flight tube. The fields are optimized to achieve time focus conditions, giving a numerical resolution for thermal ions of about 25. Under these conditions peaks in the parent ion group are only partially resolved. Complementary non-pulsed ion-only measurements were carried out with the magnetic bottle in its basic configuration by using the 2 m long flight tube for ions instead of electrons, and similar measurements were made using the shorter flight tube to capture processes on shorter time scales. With the longer flight path the numerical mass resolution was about 50, sufficient to resolve all the main ion peaks. \n\nIn the Gothenburg laboratory, a pulsed helium gas discharge lamp with a repetition rate of approximately 4 kHz was used as ionisation source where the atomic emission lines of HeI$\\alpha$ and HeII$\\alpha$ provided photon energies of 21.2 and 40.8 eV, respectively. The discrete energies were selected using a monochromator based on an ellipsoidal grating of 550 lines\/mm groove density which also focuses the radiation of a selected wavelength into the interaction region. \n\nThe flight times of the electrons were converted to kinetic energies on the basis of a calibration derived from measurements of known photoelectron and Auger electron spectra. For calibration in the low energy region we used the spectra of molecular oxygen at 21.2 and 40.8 eV photon energy, and for higher energies the spectra of atomic argon and krypton at photon energies of 100 eV and above. \n\nAt BESSY-II, the experimental set-up was mounted on undulator beamline UE52\/SGM where the photon energy can be tuned in the soft X-ray energy region. In order to doubly ionize allene through valence-valence, core-Auger and core-valence electron removal photon energies of 100 eV, 110 eV, 300.5 eV and 350.4 eV were used. The set up was essentially the same as in Gothenburg but with the addition of a mechanical chopper synchronized to the radio frequency signal of the storage ring operating in single bunch mode. The chopper was set to increase the time interval between photon bunches passed to the experiment (otherwise 805.5 ns) to between 10 and 100 $\\mu$s, to allow detection of electrons and ions without interference from subsequent radiation pulses during their expected maximum flight times.\nThe stated purity of the sample was 99.9 \\%. This was verified by on-line valence and core level photoelectron spectroscopy and mass spectroscopy which showed no impurity lines. \n\n\n\\section{Theoretical Methods}\n\n \nAll calculations were carried out using the ORCA program (v 4.1.2) \\cite{neese18}. Optimisations were carried out at the CASSCF(4,4) and CASSCF(2,4) levels with the def2-TZVP basis set \\cite{weigend05} for neutral and doubly-ionised states respectively, reflecting ionisation from the 1e and 2e ($\\pi$) orbitals of allene. Ionisation energies were obtained at the NEVPT2 level \\cite{angeli01}, based on CASSCF(4,4) and CASSCF(2,4) wavefunctions for neutral and doubly-ionised states respectively. NEVPT2 calculations used the cc-pVQZ basis set \\cite{dunning89}.\nStatic correlation arising from (near)-degeneracy, that is often a challenge for single-reference methods \\cite{andrews1992allene}, is explicitly accounted for using the current procedure, allowing the accurate calculation of the double ionisation energy to the S$_0$ state. There is good agreement on the vertical double ionisation energy with the extensive calculations of Mebel and Bandrauk\\cite{mebel2008theoretical}, who also found that after geometry optimization i.e. relaxation, the lowest singlet state becomes much lower in energy than the triplet.\n\n\n\n\\section{Results and Discussion}\n\n\\subsection{Double ionisation of Allene}\n\n\\subsubsection{Valence double ionisation}\n\n\nFigure \\ref{fig:DIP} shows double ionization spectra from electron-electron coincidence measurements at photon energies of 40.8 and 100 eV. Accidental coincidences have been subtracted, but a constant background remains. \nThe two spectra agree well with each other, reflecting the onset of double ionization at about 27 eV as part of the first band with a vertical double ionisation energy of about 28.5 eV, the latter in good agreement with the theoretical value of 28.05 eV \\cite{mebel2008theoretical}. Additional structures appear above 30 eV double ionisation energy, which are centred near 33.5, 36, 40, 44 and 48 eV and which involve removing more tightly bound electrons in allene. The bands associated with electron removal from the degenerate orbitals are expected to be the most intense because of the larger number of electrons in them, but they probably conceal underlying bands from combinations with the other ($\\sigma$-type) orbitals.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure1.png}\n\\caption{\\label{fig:DIP} Double ionisation electron spectra of allene measured at the photon energies of 40.8 eV (upper panel) and 100 eV (lower panel). The 40.8 eV data were taken in Gothenburg with a helium gas discharge lamp and the 100 eV data were taken at BESSY-II in Berlin. Both spectra are based on electron pair measurements. Assignments in terms of leading configuration (lower panel), and for the lowest structure in the 40.8 eV spectrum in terms of different electronic states (upper panel) are included.}\n\\end{figure}\n\n\nThe first band is especially interesting, with substructures at 27.7, 28.6 and 29.5 eV, which are interpreted with the aid of our calculations of vertical ionisation energies to the lowest triplet T$_0$ (\\supp{3}A\\subb{2} in D\\subb{2d}) state, the lowest singlet S\\subb{0} state, which undergoes Jahn-Teller distortion, possibly to \\supp{1}A\\subb{g} in D\\subb{2h} or to a lower symmetry structure, and the second singlet S\\subb{1} state which retains D\\subb{2d} geometry, as illustrated in Fig. \\ref{fig:DIP_structure}. \n\nOur theoretical values for the double ionisation energy of the different states as well as the experimental values taken from Fig. \\ref{fig:DIP}, are summarized in Table \\ref{tab:DIP}. The theoretical and experimental values are very close to each other, but adiabatic ionisation to the rearranged ground state is not observed. Our calculations are in excellent agreement with the double ionisation calculations by Andrews et al. \\cite{andrews1992allene, andrews1995allene} at the MP2 and MP4 levels of theory. In the double ionisation of allene, the molecule goes from being described by the point group D$_{2d}$ to D$_{2h}$ through a reorganisation of the orbitals and a reduction of degenerate states. This process, where the 3D structure of neutral allene changes to a planar form upon double ionisation is schematically illustrated in Fig. \\ref{fig:DIP_structure}. Although their calculations are otherwise very extensive, Mebel and Bandrauk \\cite{mebel2008theoretical} did not report calculated energies or structures for S$_1$ or any higher states of doubly ionised allene.\n\n\n The lowest (adiabatic) double ionisation energy is calculated as 26.1 eV, well below the range seen to be accessed by single-photon ionisation in our spectra. The triplet state also relaxes, but much less, ending up at an adiabatic double ionisation energy of 27.6 eV, which is just in the range seen in our spectrum.\n\n\\begin{table}[H]\n \\centering\n \\caption{Theoretical values for the vertical and 0-0 double ionisation of allene from calculations carried out here using CASSCF(4,4) and CASSCF(2,4) for neutral and doubly-ionised states, respectively. For comparison, experimental values extracted from Fig. \\ref{fig:DIP} are also included.}\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n \\textbf{\\makecell{Electronic \\\\ state}} & \\textbf{\\makecell{Vertical double \\\\ ionisation (eV)}} & \\textbf{\\makecell{0-0 double \\\\ ionisation (eV)}} & \\textbf{\\makecell{Experiment \\\\ (eV)}} \\\\\n \\hline\n S$_0$ & 28.6 & 26.1 & 28.6 \\\\\n \\hline\n S$_1$ & 29.5 & 29.2 & 29.5 \\\\\n \\hline\n T$_0$ & 28.2 & 28.0 & 27.7 \\\\\n \\hline\n \\end{tabular}\n \\label{tab:DIP}\n\\end{table}\n\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure2.png}\n\\caption{\\label{fig:DIP_structure} Changes in the orbital structure of neutral allene in transitions to the three states that make up the first peak in the double ionisation spectrum in Fig. \\ref{fig:DIP}. Neutral allene is stripped of two electrons in three different ways, ending up in three different states. The S$_0$ state rearranges by Jahn-Teller distortion, resolving the degeneracy of two equivalent forms of the state in D$_{2d}$ symmetry, resulting in a planar molecule described by the D$_{2\\mathrm{h}}$-point group instead.} \n\\end{figure}\n\n\n\\subsubsection{Double ionisation by Auger decay}\n\nElectronic states with two vacancies in valence orbitals can also be formed by initial inner shell (C1s) ionization followed by emission of a secondary Auger electron upon filling of the short-lived core hole. In the case of allene, single ionisation from the C1s orbitals was reported by Travnikova et al. \\cite{travnikova2008structure} to occur at 290.6 and 290.8 eV, giving rise to photoelectron lines that were only partially resolved in our electron pair measurements carried out at 300.5 eV photon energy. In analyzing the coincidence data, partially separate selection is possible by choosing the extreme sides of the composite photoelectron line feature. Double ionisation electron spectra (incorporating the energy of the selected photoelectrons) obtained in this way are shown in Fig. \\ref{fig:DIP_Auger}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure3.png}\n\\caption{\\label{fig:DIP_Auger} Double ionisation spectra of allene measured with 300.5 eV photon energy. The spectra have been selected on outer and inner C1s core electron, respectively. The inner C1s electron has a binding energy of 290.8 eV and the outer carbon 1s electron has 290.6 eV \\cite{travnikova2008structure}, and the selections comprise binding energy ranges of $\\pm 0.1$ eV.\nThe arrows indicate structures in the ionization spectra at energies of about 33, 40, 44 and 48 eV.}\n\\end{figure}\n\nAs can be seen, there is a clear difference between the two selectively extracted spectra. Also, even though the experimental resolution is low (ca. 5 eV) because of the high electron kinetic energies, substructures that correspond well with the distinct bands seen in the 100 eV spectra (cf. Fig. \\ref{fig:DIP}) are discernible as indicated by arrows, at about 33, 40, 44 and 48 eV.\n\n\n\\subsection{Triple ionisation of Allene}\n\nTriple ionization of allene can occur by different main pathways, according to the photon energy. At 100 eV there is only triple valence electron removal. At all energies above about 300 eV core ionization and subsequent double Auger decay is dominant, while at and above 330 eV there is also core-valence double ionization followed by Auger decay. Spectra corresponding to the three different pathways are shown in Fig. \\ref{fig:TIP}. \n\nThe 100 eV spectrum (panel A) is essentially structureless, showing only a smoothly rising signal with a possible start at about 50 eV (and an uncertainty of nearly 2 eV). To remove spurious low energy electrons, the lowest energy included in the events was selected to 3$\\pm$0.5 eV. \n\nThe triple ionization spectra displayed in panels B and C were extracted from measurements at 300.5 and 350.4 eV photon energy by selecting events with one core electron and two other electrons. These selections were made by visual inspection of the electron-electron coincidence maps \\textcolor{blue}{(?)} where electrons that were not part of the desired energy sharing have been removed. As can be seen, the spectra reflect a broad, featureless bump with onset at about 51 \u00b1 2.3 eV and maximum intensity in the 79 \u2013 80 eV energy range. We note that the ratio of double to single Auger is experimentally found to be 13\\% at both 300.5 eV and 350.4 eV photon energy which is slightly higher than reported ratios in other C\\subb{3} carbon compounds from the work of Hult Roos et al. \\cite{hult2019multi}. \n\nAt 350.4 eV we could also extract the triple ionization spectrum shown in panel D, based on selection of initial core-valence double ionization, for which full spectra will be presented in a forthcoming publication, followed by an additional Auger electron emission. The spectrum locates the lowest triple ionization energy at about 50 $\\pm$ 5.4 eV. All these onset values are in reasonable agreement with the calculated adiabatic triple ionisation values given by Mebel and Bandrauk \\cite{mebel2008theoretical} as 52.29 eV, and 53.05 eV from different forms of theory. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure4.png}\n\\caption{\\label{fig:TIP} Triple ionization at 100, 300.5 and 350.4 eV photon energies. In panel A, the triple ionization spectrum has been extracted by limiting the data to include events with 100 eV photons where the electron with the lowest energy is 3$\\pm$0.5 eV. In panels B and C, events with one core electron emission and two other electron emissions have been selected. The triple ionization spectrum in panel D is derived by identifying events where a core-valence doubly ionised state is formed before an Auger electron is emitted.}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Fate of Allene above lowest double ionisation threshold}\n\nTo investigate the fate of allene exposed to photon energies above the lowest double ionisation threshold ion detection in multiplex was employed.\n\n\\subsubsection{Ion time-of-flight mass spectra}\n\nFigure \\ref{fig:mass_spectrum_100_301eV} shows mass spectra of allene obtained at the photon energies of 100 eV (upper panel) and 300.5 eV (lower panel). Both spectra were obtained using the 2 m flight tube for ion detection (instead of electron detection). The 100 eV spectrum is very similar to the mass spectrum at 40.8 eV which is not shown separately.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure5.png}\n\\caption{\\label{fig:mass_spectrum_100_301eV} Mass spectra of allene at photon energies 100 eV (upper panel) and 300.5 eV (lower panel). The mass spectrum at 40.8 eV (not shown) is very similar to the 100 eV spectrum. \n}\n\\end{figure}\n\n\nAs can be seen, the 100 eV spectrum is dominated by the parent ion group (m\/e = 40, 39, 38, 37, 36). The weak and otherwise unexpected peak(s) at 32 (and 28) suggest(s) that a small amount of air was present during this recording. More interestingly, there are relatively strong peaks for doubly charged ions containing the C$_3$ species at m\/e = 20, 19.5 and 19, all three of roughly equal intensity. The very weak peak at m\/e = 16 is probably due to O$^+$, while the peak at m\/e = 15 of similar intensity is attributed to the CH$_3^+$ species. Its intensity is about 1\/4 that of the m\/e = 14 peak associated with CH$_2^+$. Very interestingly, there are distinct peaks for m\/e = 2 and 3, i.e. for H\\subb{2}\\supp{+} and H\\subb{3}\\supp{+}.\n\nThe most striking points in comparing the mass spectrum at 300.5 eV with the 100 eV data are the greatly increased dissociation at 300.5 eV, where double ionisation by the Auger effect dominates, and the stability of the doubly-charged ions C$_3$H$_4^{2+}$, C$_3$H$_3^{2+}$ and C$_3$H$_2^{2+}$ (but not C$_3$H$^{2+}$ or C$_3^{2+}$). We note that the structure and energetics of the parent doubly charged ion were calculated by Mebel and Bandrauk \\cite{mebel2008theoretical}, but those of C$_3$H$_2^{2+}$ and C$_3$H$^{2+}$ were not. They, and the apparently less stable or unstable C$_3$H$^{2+}$ and C$_3^{2+}$ species are fundamental entities, likely to be of relevance in astrophysical contexts. From a theoretical point of view, not much is known about their structure, except for the C$_3^{2+}$ species for which a linear configuration has been suggested \\cite{hogreve1995ab}. However, for C$_3^+$ there is experimental evidence from Coulomb explosion imaging \\cite{faibis1987geometrical} and theory \\cite{taylor1991ab,diaz2006ionization} that it is non-linear.\n\n\n\\subsubsection{Ion-ion coincidences}\n\nAn ion-ion coincidence map associated with the 100 eV ion time-of-flight spectrum from Fig. \\ref{fig:mass_spectrum_100_301eV} and after subtraction of accidental coincidences is shown in Fig. \\ref{fig:ion_map_100}. The detectable ion pairs seen in this map are summarized in Table \\ref{tab:ion_coincidences}.\n\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure6.png}\n\\caption{\\label{fig:ion_map_100}Ion-ion coincidence map at 100 eV photon energy using the 2 m flight tube. Accidental coincidences have been removed and the mass over charge information is included in the plot.}\n\\end{figure}\n\n\n\n\n\\begin{table}[H]\n \\centering\n \\caption{\\label{tab:ion_coincidences}Ion pairs detected in the 100 eV ion-ion coincidence map shown in Fig. \\ref{fig:ion_map_100}. Note: the H\\supp{+} + CH\\supp{+} or C\\supp{+} channels are not included in the figure.}\n \\begin{tabular}{|l|l|}\n \\hline\n \\textbf{Ion pair} & \\textbf{\\makecell{Relative abundance \\\\ ($\\Sigma = 1000$)} } \\\\\n \n \\hline\n H\\supp{+} + C\\subb{3}H\\subb{3}\\supp{+} or C\\subb{3}H\\subb{2}\\supp{+} or C\\subb{3}H\\supp{+} or C\\subb{3}\\supp{+} & 562 \\\\\n \\hline\n H\\supp{+} + C\\subb{2}H\\supp{+} or C\\subb{2}\\supp{+} & 92 \\\\\n \\hline\n H\\supp{+} + CH\\supp{+} or C\\supp{+}\\supp{*} & 72 \\\\\n \\hline\n H\\subb{2}\\supp{+} + C\\subb{3}H\\subb{2}\\supp{+} or C\\subb{3}H\\supp{+} or C\\subb{3}\\supp{+} & 33 \\\\\n \\hline\n H\\subb{3}\\supp{+} + C\\subb{3}H\\supp{+} or C\\subb{3}\\supp{+} & 12 \\\\\n \\hline\n C\\supp{+} + C\\subb{2}H\\subb{3}\\supp{+} or C\\subb{2}H\\subb{2}\\supp{+} or C\\subb{2}H\\supp{+} or C\\subb{2}\\supp{+} & 49 \\\\\n \\hline\n CH\\supp{+} + C\\subb{2}H\\subb{3}\\supp{+} or C\\subb{2}H\\subb{2}\\supp{+} or C\\subb{2}H\\supp{+} or C\\subb{2}\\supp{+} & 76 \\\\\n \\hline\n CH\\subb{2}\\supp{+} + C\\subb{2}H\\subb{3}\\supp{+} or C\\subb{2}H\\subb{2}\\supp{+} or C\\subb{2}H\\supp{+} or C\\subb{2}\\supp{+} & 104 \\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\n\nApart from the numerous ion pairs observed at 100 eV, some of which may come from triple ionisation and involve an unobserved third ion, four double-charge-retaining dissociation channels producing C\\subb{3}H\\subb{3}\\supp{2+}, C\\subb{3}H\\subb{2}\\supp{2+}, C\\subb{3}H\\supp{2+} and C\\subb{3}\\supp{2+} as seen in Fig. 5, arise entirely from double ionisation. From these and other data we identify the H\\supp{+} + C\\subb{3}H\\supp{+} + H\\subb{2} double ionisation channel as the most probable charge separation channel \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure7.png}\n\\caption{\\label{fig:metastable} Panel A displays an ion-ion coincidence map at 40.8 eV photon energy using the shorter (0.12 m) flight tube. Mass\/charge is denoted by the numbers in the figure. The sloping feature which starts at the correlation island of mass 1 and 39 and extends towards the lower right corner, indicates the presence of a metastable doubly charged state with a lifetime on the order of the flight times of the ions in this shorter tube. This metastable state dissociates into the two singly charged ions seen in the map, H\\supp{+} and C\\subb{3}H\\subb{3}\\supp{+}, as illustrated in panel B by the pink markings. The markings have been obtained by simulating the decay in SIMION with a lifetime of the metastable state of 129$\\pm$9.9 ns.}\n\\end{figure}\n\nFig. \\ref{fig:metastable} displays an ion-ion coincidence map of allene obtained at the photon energy of 40.8 eV in DC mode using a much shorter (about 0.12 m long) ion time-of-flight spectrometer mounted in tandem configuration to the electron flight tube. This map confirms, though at lower resolution, several of the ion pair channels involving the H\\supp{+}, H\\subb{2}\\supp{+}, and H\\subb{3}\\supp{+} species. The significant appearance of H\\subb{3}\\supp{+} ions in coincidence is in agreement with the findings of Hoshina et al. \\cite{hoshina2011metastable} and related theory \\cite{mebel2008theoretical}. While the mechanism of formation of the H\\subb{3}\\supp{+} ions is discussed in the literature in terms of a \"roaming\" mechanism, which involves an initial H\\subb{2} species that \"orbits\" within a doubly-charged precursor until it succeeds extracting the third proton, the identity of the electronic state from which it happens in allene has not been determined experimentally. \n\\textcolor{magenta}{According to the energy range it could be any of the three states in the first double ionisation band (cf. Fig. \\ref{fig:DIP}). If Mebel and Bandrauk's RRKM rate calculation \\cite{mebel2008theoretical} is taken at face value, it means that H$^+$ + C$_3$H$_3^+$ come from the relaxed ground state S$_0$. Further support for that can be obtained from figure 3 of Mebel and Bandrauk\\cite{mebel2008theoretical} according to which the ground state can make hiving off \"roaming H$_2$\" more likely by the involvement of a methyl acetylene transition state which posses favourable structural characteristics.}\n\n\n\nThe most interesting aspect in the 40.8 eV map is the presence of an unambiguous, though weak metastable tail, which is essentially invisible at 100 eV and higher photon energies. The map identifies the most intense part of the tail as belonging to the H\\supp{+} + C\\subb{3}H\\subb{3}\\supp{+} reaction channel but its comparatively broad spread may include a contribution from the H\\supp{+} + C\\subb{3}H\\subb{2}\\supp{+}, too. The H\\supp{+} + c-C\\subb{3}H\\subb{3}\\supp{+} (cyclic cyclo-propenyl) ion pair has the lowest thermodynamic formation limit at 25 eV, while almost all the other observed pairs have limits near 26 eV for formation with no kinetic or internal energy release. Since there is no detectable \"V\" shape from the metastable centred on the doubly charged parent ion's position as apex on the diagonal (t$_1$ = t$_2$) \\cite{field1993lifetimes}, no longer-lived group of ions dissociates in the flight tube. This implies that essentially all the metastable decay takes place in the source field of our spectrometer. \n\nThe existence of this metastable decay was already noted by Barber and Jennings in 1969 \\cite{barber1969kinetic} in electron impact mass spectrometry. To investigate the metastable lifetime, we plotted the coincidence data as a map of t$_1$+t$_2$ vs. t$_2$-t$_1$, where the metastable tail gets concentrated as a strip almost parallel to the (t$_2$-t$_1$)-axis. The intensity as a function of the time difference was extracted and plotted, as is done in Fig. \\ref{fig:lifetime}, for comparison with the theory given by Field and Eland \\cite{field1993lifetimes}. The fit to the metastable state in the figure has a 95\\% confidence interval of 229.2 - 266.8 ns which translates to a lifetime of 130.5 $\\pm$ 9.9 ns. The observation that the experimental points fit well to a single exponential decay with a mean lifetime of 130.5 ns is in contrast to most other cases, where metastable lifetimes decays curves investigated in this way represent a mixture or wide distribution of lifetimes \\cite{field1993lifetimes}. This may imply that only one single vibration level or a very close group lies just above the barrier to this reaction channel. \n\nBecause significant approximations are involved in the theory behind the analysis of Fig. \\ref{fig:lifetime}, the metastable decay was also investigated by a Monte-Carlo type simulation using a realistic model of the apparatus implemented in the software package SIMION \\cite{simion}, the results of which are presented in panel B of Fig. \\ref{fig:metastable}. The decay was modelled to have a 3 eV kinetic energy release (KER) in line with the observed difference in energy between the thermochemical threshold and the observed appearance energy, reported below. The pink contour shows the simulation results on top of the experimental coincidence map. This simulation fits the experimental observations well with a mean lifetime of 130\u00b110 ns, in good agreement with the simple exponential fit. We note that this lifetime is within the range predicted by Mebel and Bandrauk on the basis of RRKM theory, that is, assuming free flow of internal energy within the molecular ion allowing statistical energy redistribution. \n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure8.png}\n\\caption{\\label{fig:lifetime} Metastable tail intensity as a function of the inverted flight time difference with a fit to the exponential decay in yellow. The coincidences are the same as in the tail in Fig. \\ref{fig:metastable} plotted against t$_2$-t$_1$ where t$_1$ and t$_2$ are the flight times of the first and second ion to be detected, respectively. The location of the mass peak for H$^+$ ions on the same scale is included below the decay curve for comparison.}\n\\end{figure}\n\n\n\\subsubsection{Multiple-electron-ion coincidences and action spectra}\n\nTo determine the appearance energies of the different sets of products and to obtain \"action spectra\" for the different channels as a function of double ionisation energy, threefold (eei) and fourfold (eeii) coincidences are indispensable. In view of the achievable electron energy resolution, this was primarily done at the photon energy of 40.8 eV. If a particular ion is formed by only one decay channel, an eei spectrum is sufficient to define the action spectrum. This is true for all three doubly charged ions C\\subb{3}H\\subb{4}\\supp{2+}, C\\subb{3}H\\subb{3}\\supp{2+} and C\\subb{3}H\\subb{2}\\supp{2+} for which the spectra are shown in Fig. \\ref{fig:DIP_chargeretain}. It is also true for ions formed only in two-body decays of the parent including the metastable decay channel shown in Fig. \\ref{fig:metastable}. Analysis of the ion-ion coincidence maps suggests that it applies as a good approximation to H\\subb{3}\\supp{+} formation (action spectrum in the upper panel of Fig. \\ref{fig:DIP_ions}) and to the ionization pathways leading to C\\subb{2}H\\subb{2}\\supp{+} or CH\\subb{2}\\supp{+} and to C\\subb{2}H\\subb{3}\\supp{+} or CH\\supp{+} whose spectra are shown in Fig. \\ref{fig:DIP_pi}. The other ions, H\\supp{+}, H\\subb{2}\\supp{+} and C\\subb{3}H\\subb{n}\\supp{+}, are all products of more than one decay channel, so eeii coincidences are formally required to get unambiguous action spectra. Unfortunately, the ion collection efficiency of the apparatus is so low that spectra with useful numbers of counts cannot be obtained for most such fourfold coincidences so eei events have been used to generate the spectra shown in Fig. \\ref{fig:DIP_ions}.\n\nFrom these spectra, we can get an overview of the fates of allene dications in the range of the excitation energies covered by the double photoionization spectrum at 40.8 eV. As can be seen in Fig. \\ref{fig:DIP_chargeretain}, the first peak of the total double ionization spectrum is primarily associated with undissociated electronic states of the doubly ionized parent molecule at up to 3 eV above the calculated adiabatic double ionization threshold of 26 eV. Because of an overlap in the mass spectrum of the peaks reflecting the parent ion and the parent ion minus one and minus two hydrogen atoms, it was impossible to separate signals for these species completely. We believe that the peak in the double ionization spectrum of C\\subb{3}H\\subb{3}\\supp{2+} between 26 and 30 eV actually originates from C\\subb{3}H\\subb{4}\\supp{2+} instead, because otherwise its appearance energy would be impossibly low. Also, the high energy part of the C\\subb{3}H\\subb{4}\\supp{2+} spectrum may be affected in a similar way \nIn order to demonstrate more realistic spectra for the doubly-charged ions we have carried out an iterative subtraction process, based on the estimated extent of peak overlap. The resulting spectra are shown in red together with the raw spectra in black. \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure9.png}\n\\caption{\\label{fig:DIP_chargeretain} Double ionisation spectra based on electrons that were detected in coincidence with the charge retaining parent species or with charge retaining fragments which lost one or two hydrogen atoms. Because the detected species is doubly ionised and triple ionisation is impossible at this photon energy, no other ions can be involved in these events. \n}\n\\end{figure}\n\nOver the first 2 eV above the onset of vertical double ionisation at about 27 eV, the parent C$_3$H$_3^{2+}$ ion remains stable on the mass spectrometer time scale. The lowest energy dissociation by charge separation is the slow metastable decay by H\\supp{+} ejection, with onset at 28.5 \u00b1 0.3 eV and peak intensity at 29 eV $\\pm$ 0.3 eV shown in Fig. \\ref{fig:DIP_comb}. The metastable signal appears only in a narrow range, presumably just above threshold, \\textcolor{blue}{with a width of ...}. in line with the expected electron energy resolution at this ionisation energy of ca 0.6 eV. This observation strengthens the view that a single vibrational level of the parent dication is responsible for the slow dissociation. Once the ionisation energy exceeds 29 eV formation of H$^+$ + C$_3$H$_3^+$ occurs rapidly and becomes the most probable dissociation pathway. But at the same energy both H$_2^+$ and H$_3^+$ are formed with low intensity (cf. Fig. \\ref{fig:DIP_ions}). The three lowest panels in Fig. \\ref{fig:DIP_comb} all represent the process of H$^+$ + C$_3$H$_3^+$ formation detected in different ways, and the differences between them illustrate the problems of poor statistics in fourfold coincidences and background subtraction of overlapping mass peaks.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure10.png}\n\\caption{\\label{fig:DIP_comb} Double ionisation spectra based on electrons that were detected in coincidence with ions as specified in the figure. The uppermost spectrum represents decay events detected in the metastable tail shown in Fig. \\ref{fig:metastable}. The next lower spectra are derived from selection of events with two ions such as, $\\mathrm{H^+ \\;and\\; C_3H_3^+}$, and two electrons, eeii. The lower spectra are from events with one ion $\\mathrm{H^+ \\;or\\; C_3H_3^+}$, respectively, and two electrons, eei. One coincident count is equivalent to two electrons being detected in the same event.}\n\\end{figure}\n\nFig. \\ref{fig:DIP_ions} shows the yields of selected single ions in coincidence with electron pairs. For all the C$_3$H$_n^+$ ions there is an overlap problem similar to that encountered for the doubly-charged ions (cf. Fig. \\ref{fig:DIP_chargeretain}) and we have compensated for it in a similar way. In this case, fourfold coincidence measurements give clear guidance on the extent of overlap and the necessary subtractions. Because of the threefold coincidence selection, the contributing ions may be formed by three-body as well as two-body dissociations, but the lowest energy pathways must be the two-body reactions forming H$^+$, H$_2^+$ and H$_3^+$ with observed onset energies in the region of 28.5 to 29.5 eV. Where two-body reactions dominate, the spectra for H$_3^+$ and C$_3$H$^+$ and those of H$_2^+$ and C$_3$H$_2^+$ should be the same. This is borne out by comparison of the spectra up to 35 or 36 eV, but at higher energy the different intensities indicate that three-body reaction releasing additional neutral fragments take over. The calculations of Mebel and Bandrauk\\cite{mebel2008theoretical} indicate that barriers to the three hydrogen ion loss pathways should lie in the range of 28.84 to 29.03 eV, and their RRKM calculations predict that H\\subb{3}\\supp{+} peak formation should occur at 29.53 eV. The calculated onsets and the peak production energy agree well with the observations in Fig. \\ref{fig:DIP_ions}. However, further predictions based on the RRKM model assumption that all channels are in competition at all energies are not in line with our experimental data since the intensities of the channels leading to H$^+$ and H$_3^+$ are not comparable at any energy. An alternative interpretation, that chimes with the \"roaming\" mechanism is that some of the initial population becomes isolated as a $[\\mathrm{C_3H_2-H_2}]^{++}$ complex, where the $\\mathrm{H_2}$ can either escape or capture a proton. Such a mechanism would explain why $\\mathrm{H_3^+}$ formation does not take over from $\\mathrm{H^+}$ production and why the intensities and spectra for H$^+$ and H$_3^+$ production are similar.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure11.png}\n\\caption{\\label{fig:DIP_ions} Double ionisation spectra based on two electrons detected in coincidence with a singly charged ion as specified for each spectrum. Both two-body and three-body fragmentations can contribute to these spectra, but two-body decays are probably dominant. Modify as necessary }\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.5\\textwidth]{Figure12.png}\n\\caption{\\label{fig:DIP_pi} Double ionisation spectra for channels where a C-C bond is broken, based on two electrons detected in coincidence with a singly charged ion as specified for each spectrum.}\n\\end{figure}\n\n \nIn the higher double ionization energy range, one of the main observation from our data is that the three-body products $\\mathrm{H^+ + C_3H_2^+ + H}$ take over completely from $\\mathrm{H^+ + C_3H_3^+}$ at energies above 34 eV, as demonstrated by the spectra in Fig. \\ref{fig:DIP_comb}. The most natural explanation for that is that this is a sequential decay; $\\mathrm{C_3H_3^+}$ ions with sufficient internal energy (8 eV initially) to dissociate further by H atom loss. Another very interesting channel is the production of $\\mathrm{H_2^+ + C_3H_2^+}$, which according to the eei data given in Fig. \\ref{fig:DIP_ions} occurs weakly between 28 and 30 eV, then more strongly above 32 eV. Formation near 29 eV is possible only for the cyclic form of C$_3$H$_2^+$ (thermochemical threshold 25.6 eV) with a sensible kinetic energy release. Both the cyclic and perhaps more naturally produced linear C$_3$H$_2^+$ ion (threshold 27.8 eV) can be formed at 32 eV, which is the energy where the second main double ionization band begins (cf. Fig. \\ref{fig:DIP}). According to a recent fs laser pump-probe study of the roaming mechanism for $\\mathrm{H_3^+}$ formation from doubly ionised methanol \\cite{livshits2020time}, the two exit channels giving $\\mathrm{H_3^+}$ and $\\mathrm{H_2^+}$ are both ultrafast and in competition. If that is also true here, the channel forming H$_3^+$ must have entropic or related factors in its favour, as it takes over completely from H$_2^+$ formation in the second part of the first main double ionisation band (cf. Fig. \\ref{fig:DIP_ions}). It is also striking that the charge retaining channel $\\mathrm{H_2 + C_3H_2^{2+}}$ appears at 29.6 \u00b1 0.3 eV (cf. Fig. \\ref{fig:DIP_chargeretain}), the energy where H$_2^+$ ceases to be produced and close to its calculated asymptote. This suggests that charge separation and charge retention by the heavier fragment are also in competition at this point.\n\nThere is an apparent reappearance of intensity for H$^+$ + C$_3$H$_3^+$ peaking at around 34 eV (cf. Fig. \\ref{fig:DIP_comb}) in the second double ionisation band. In this range both the cyclic and linear isomers of the ion may be formed and charge-retaining formation of both C$_3$H$_3^{2+}$ and C$_3$H$_2^{2+}$ appear to compete strongly in the same energy range. As can be seen from Fig. \\ref{fig:DIP_pi}, this is also the energy range where the charge-separating C-C bond-breaking reaction giving rise to $\\mathrm{CH_2^+ + C_2H_2^+}$ becomes intense. It is slightly surprising that so many very different dissociation routes can all compete in the same energy range with comparable intensities in all the channels.\n\n\\begin{table}[H]\n \\centering\n \\caption{Double ionisation of allene at 40.8 eV photon energy in comparison to thermodynamic thresholds and theoretical predictions. The uncertainties in the observed values are a consequence of the kinetic energy resolution of the electron spectrometer used.}\n\\begin{tabular}{|c|cc|cc|}\n \\hline\n \\multirow{2}{*}{Channel} & \\multicolumn{2}{c|}{0 K threshold (eV)} & \\multicolumn{2}{c|}{Appearance energy (eV)} \\\\\n & Thermo. & Calc (M\\&B) & Calc (M\\&B) & Obs. (us) \\\\\n \\hline\n \\multirow{2}{*}{H\\supp{+} + 39\\supp{+}} & cyclic 25.0 & 24.95 & 28.8 & 28.9 $\\pm$0.3 \\\\\n & linear 26.1 & 26.2 & & \\\\\n \\hline\n \\multirow{2}{*}{H\\supp{+} + 38\\supp{+} + H }& cyclic 28.3 & & & 32$\\pm$0.3 \\\\\n & linear 30.4 & & & \\\\\n \\hline\n \\multirow{2}{*}{H\\supp{+} + 37\\supp{+} + H\\subb{2}} & 30.4 & & & \\\\\n & & & &\\\\\n \\hline\n \\multirow{2}{*}{H\\subb{2}\\supp{+} + 38\\supp{+} }& cyclic 25.6 & 27.7 & & 32.8$\\pm$0.2\\\\\n & linear 27.8 & 27.9 & 29.44 & \\\\\n \\hline\n \\multirow{2}{*}{H\\subb{2}\\supp{+} + 37\\supp{+} + H} & 32.2 & & & \\\\\n & & & &\\\\\n \\hline\n \\multirow{2}{*}{H\\subb{3}\\supp{+} + 37\\supp{+}} &26.0 & 26.0 & 29.03 & 29$\\pm$0.3\\\\\n & & & &\\\\\n \\hline\n \\multirow{2}{*}{CH\\subb{2}\\supp{+} + C\\subb{2}H\\subb{2}\\supp{+}} & 26.1 & 26.2 & 30.8 (vinylidene) & 31$\\pm$0.3\\\\\n & & & &\\\\\n \\hline\n \\multirow{2}{*}{CH\\supp{+} + C\\subb{2}H\\subb{3}\\supp{+} }& & &30.36 & 32.1$\\pm$0.2 \\\\\n & & & &\\\\\n \\hline\n C\\subb{3}H\\subb{4}\\supp{2+} & & \\makecell{adb. 25.84 \\\\ vert. 28.05} & & \\makecell{ \\\\ 27.5$\\pm$0.3 }\\\\\n \\hline\n \\multirow{2}{*}{C\\subb{3}H\\subb{3}\\supp{2+} + H }& & c-30.55 & & 32.1$\\pm$0.2\\\\\n & & & &\\\\\n \\hline\n \\multirow{2}{*}{C\\subb{3}H\\subb{2}\\supp{2+} + H\\subb{2}} & & 29.92, 29.39 & &29.6$\\pm$0.3\\\\\n & & & &\\\\\n \\hline\n\\end{tabular}\n \\label{tab:app_energy}\n\\end{table}\n\n\nThe appearance energies of the different sets of products determined from the experimental spectra in Figs. \\ref{fig:DIP_chargeretain}-\\ref{fig:DIP_pi}, are listed in Table \\ref{tab:app_energy} together with thermodynamic thresholds and theoretical predictions. For all the thermodynamic 0K thresholds for ions formed with no internal energy or kinetic energy, heats of formation are taken from the NIST database or are estimated by combining known thermodynamic data with theoretical calculations (e.g. from Mebel and Bandrauk (M\\&B)\\cite{mebel2008theoretical}). We note that the ions of mass 39, 38 and possibly 37 can have either cyclic or linear structures, with significantly different heats of formation (the heats of formation for mass 37 is uncertain). The Mebel and Bandrauk-calculated thresholds are taken from their figure 2, with some degree of ambiguity.\n\n\n\\section{Conclusions}\n\nUsing multi-particle coincidence experiments we have obtained single-photon double and triple ionization spectra of allene, and have determined how dissociation of the doubly charged ions depends on the ionisation energy. New high-level calculations confirm that adiabatic double ionisation would require isomerization to a different structure, not accessible to vertical ionisation processes. The calculations allow substructure in the first double ionisation band to be assigned to different electronic states of the dications. Double ionisation of allene by Auger decay of C1s vacancy states is found to populate different spectra of dication states according to location of the vacancy on either the central or an outer C atom.\n\nTriple ionization of allene by three routes, valence ionization at 100 eV, double Auger decay of C1s vacancies and Auger decay of a C1s core-valence doubly ionised intermediate state yield different spectra, but all exhibit an onset of triple ionization at approximately 50 eV, in line with predictions available in the literature. \n\nEight significant decay pathways for dissociative decay of nascent allene dications have been identified and the energy dependence of their relative intensities is reported. At the lowest energies there is evidence that formation of H$_2^+$ + C$_3$H$_2^+$ and H$_3^+$ + C$_3$H$^+$ are in competition with each other and possibly with the charge-retaining H + C$_3$H$_3^{2+}$ channel, but not with charge separation to H$^+$ + c-C$_3$H$_3^+$. This finding supports the roaming mechanism, already proposed, where an H$_2$ molecule becomes partially detached from the heavy residual molecular dication. \n\tFor the decay to H$^+$ + c-C$_3$H$_3^+$, we have confirmed the existence of a slow metastable decay happening in a narrow energy range just above threshold. This decay is well characterised as a single exponential with a mean lifetime determined as 130 \u00b110 ns. We suspect that a single vibrational level of the parent dication may be involved and might be identified by future calculations.\n\n\n\\begin{acknowledgments}\nThis work has been financially supported by the Swedish Research Council (VR) and the Knut and Alice Wallenberg Foundation, Sweden. We thank the Helmholtz Zentrum Berlin for the allocation of synchrotron radiation beam time and the staff of BESSY-II for smooth running of the storage ring during the single-bunch runtime. The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 730872.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}