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[
"The overtone level spacing of a black hole quasinormal frequencies: a\n fingerprint of a local $SL(2,\\mathbb{R})$ symmetry"
],
[
"Abstract The imaginary part of the quasinormal frequencies spectrum for a static and spherically symmetric black hole is analytically known to be equally spaced, both for the highly damped and the weakly damped families of quasinormal modes.",
"Some interesting attempts have been made in the last twenty years to understand in simple ways this level spacing for the only case of highly damped quasinormal frequencies.",
"Here, we show that the overtone level spacing, for both the highly damped and weakly damped families of quasinormal modes, can simply be understood as a fingerprint of a hidden local $SL(2,\\mathbb{R})$ symmetry, near different regions of the black hole spacetime, i.e.",
"the near-horizon and the near-photon sphere regions."
],
[
"Introduction",
"In the context of small perturbations of a black hole (BH) spacetime, the quasinormal modes (QNM), which are damped modes of perturbation which propagate towards spatial infinity, are playing a fundamental role both in our experimental and theoretical understanding of gravity and of BH in particular.",
"The QNM are related to complex quasinormal frequencies (QNF) at which they occur.",
"The real part of the QNF corresponds to the frequency of oscillation and the imaginary part to the damping rate of the QNM.",
"Analytically, for example in the case of static and spherically symmetric (asymptotically flat) BH perturbed by a massless scalar field, two families of QNM are known the highly damped QNM with the associated QNF (at first order in $n$ ) $\\omega _n = \\frac{\\kappa }{2\\pi }\\ln \\left[1+2\\cos \\left(\\frac{\\pi (qD-2)}{2}\\right)\\right] - i\\kappa \\left(n+\\frac{1}{2}\\right)$ where $\\kappa $ is the surface gravity of the BH, $D=d-2$ for $d$ -dimensional spacetimes and $q$ is a purely real power law exponent that characterizes the BH metric near its singularity [1], [2], and the weakly damped QNM with the associated QNF (at first order in $\\ell $ and $n$ ) $\\omega _{\\ell ,n}=\\Omega _c\\left(\\ell +\\frac{1}{2}\\right)-i|\\Lambda _c|\\left(n+\\frac{1}{2}\\right)$ where $\\Omega _c$ is the angular velocity of massless particles on unstable circular orbits around the BH, and $|\\Lambda _c|$ is the Lyapunov exponent associated with these unstable circular orbits [3], [4].",
"The imaginary part of the QNF, in both cases, is interesting.",
"It contains the overtone number $n$ and is equally spaced.",
"Some remarks about its amplitude will also be given in this paper.",
"Then, it would be natural to ask whether the seemingly simple behavior of the imaginary part of the QNF in both cases may, or may not, hide a “universal” feature of BH spacetimes.",
"For static and spherically symmetric spacetimes, such an imaginary part for the BH QNF, in the only case of the highly damped QNM, has been discussed in [5], where it has been obtained as poles of the scattering amplitude in the Born approximation, and in [6] where it has been extracted via a Coulomb-like phase shift.",
"In this paper, we would like to show that such a behavior for the imaginary part of a BH QNF is in fact more “universal” and can simply be understood, in both the highly and weakly damped regimes, as a fingerprint of a hidden local $SL(2,\\mathbb {R})$ symmetry.",
"We will focus on static and spherically symmetric BH endowed with a photon sphere (also known as “lightrings”, or “photon rings”).",
"The strategy is made simple as long as one can show that certain particular regions of a BH spacetime admit a Rindler metric as an approximation.",
"Indeed, on the one hand, the resonant scattering of a massless scalar field in a Rindler spacetime is shown to be deeply linked to the resonant scattering problem of a scalar field by an inverted harmonic oscillator (IHO) potential of the form $V(x)=-(\\alpha ^2/4) x^2$ , where $\\alpha $ is the potential curvature, or potential strength.",
"On the other hand, the resonant scattering problem by an IHO potential can be algebraically solved using the $SL(2,\\mathbb {R})$ algebra for which the representations and the eigenvalues spectrum are discrete and indexed by a non negative integer.",
"The general claim of this paper follows: in any BH geometry, if there exist some regions where the BH metric can be locally approximated as a Rindler one, in some set of coordinates, then the underlying algebra is $SL(2,\\mathbb {R})$ , and the imaginary part of the BH QNF will be indexed by a non negative integer, i.e.",
"the overtone number, and it will be equally spaced.",
"The amplitude of the overtone level spacing is shown to be related to the IHO potential curvature.",
"The paper is organized as follows.",
"In section , we give an $SL(2,\\mathbb {R})$ approach to the resonant scattering problem of a scalar field by an IHO potential.",
"This allows to highlight that the eigenvalues spectrum of the corresponding Hamiltonian is discrete, purely imaginary and indexed by a non negative integer.",
"In section , we focus first on the motions of massless particles in Rindler spacetime, and then on the resonant scattering problem of a massless scalar field in such a spacetime.",
"We show that massless dynamics in Rindler spacetime is deeply linked to the problem treated in section , and we apply the results we have obtained to the Rindler case.",
"Finally, in section , we apply the $SL(2,\\mathbb {R})$ algebra to the case of static and spherically symmetric BH endowed with a photon sphere, and show that the imaginary part of the BH QNF in both the highly an weakly damped regimes are equally spaced.",
"In sections and we discuss the amplitude of the overtone level spacing and gives somes conclusions about the results obtained in this paper."
],
[
"An $SL(2,\\mathbb {R})$ approach to the resonant scattering by an inverted harmonic oscillator potential",
"With a particular choice of “position” operator $X$ and conjugate “linear momentum” operator $P$ , acting both on a same Hilbert space, the Hamiltonian of a particle in an IHO potential can always be written as $H=P^2 - \\frac{\\alpha ^2}{4}X^2,$ where $\\alpha $ is the curvature of the IHO potential, and the operators, which satisfy $[X,P]=i\\mathbb {I}$ , are defined, as usual, as $P=-i\\frac{d}{dx}$ and $X=x$ , in an “$x$ -representation”.",
"In a system of units such that $h=c=1$ , it should be noted that if one wants $H$ to have dimension $L^{-1}$ , $P$ and $X$ have to be defined here with dimensions $L^{-1/2}$ and $L^{1/2}$ respectively.",
"The IHO potential curvature $\\alpha $ has dimension $L^{-1}$ .",
"From the point of view of a scalar field theory, the Hamiltonian gives the following equation for its eigenstates $\\psi $ in the $x$ -representation $H \\psi (x)=\\left(-\\frac{d^2}{d x^2}-\\frac{\\alpha ^2}{4}x^2\\right)\\psi (x)=h \\psi (x).$ Here $h$ is the Hamiltonian eigenvalue associated with the eigenstate $\\psi $ .",
"This equation is a Weber differential equation whose solutions are related to parabolic cylinder functions [7].",
"They describe the scattering states of the scalar field by an IHO potential.",
"Amongst the scattering states, the system presents a discrete set of eigenstates, i.e.",
"the resonant modes, or QNM, which can be obtained, for example, by looking at the poles of the corresponding $S$ -matrix in the complex energy plane.",
"An expression of $H$ in the form $(\\ref {Hamiltonian})$ (or $(\\ref {ScalarFieldEq})$ ) plays an “universal” role for the resonant scattering problem of a scalar field by the top of any potential barrier.",
"Indeed, in such case, the QNM are associated with energies close to the top of the potential.",
"For such energies, one can always approximate any potential barrier close to its maximum by a second order Taylor series expansion which locally gives an inversed parabola.",
"With the right choice of $X$ and $P$ operators, the corresponding Hamiltonian $H$ can then always be reduced to the form $(\\ref {Hamiltonian})$ (or $(\\ref {ScalarFieldEq})$ ).",
"This form of $H$ is not only “universal” for a problem of resonant scattering by a potential barrier, but it also permits to reveal a hidden (dynamical) conformal symmetry near the top of the potential through an $SO(2,1) \\sim SL(2,\\mathbb {R})$ algebra which allows to obtain the associated set of QNM from the algebra representations and, especially for the aim of this paper, to obtain the overtone level spacing in the imaginary part of the quasinormal frequencies.",
"We briefly recall below the outline of one of the possible proofs that mainly lies on the introduction of “creation and annihilation operators” analogues in the factorization of $H$ , which allows to highlight the underlying $SL(2,\\mathbb {R})$ algebra (see [8] for more details).",
"We have seen that the time-independent, one-dimensional Hamiltonian of a quantum particle scattered by an IHO potential reads in the $x$ -representation $H=-\\frac{d^2}{dx^2} - \\frac{\\alpha ^2}{4}x^2.$ As one usually proceeds in the context of the well-known harmonic oscillator problem, let us begin by factorizing $H$ .",
"From the operators $P=-i\\frac{d}{dx}$ and $X=x$ (mentioned in the previous section), one can introduce “creation and annihilation operators” analogues $U_{\\pm }=\\pm \\frac{1}{\\sqrt{\\alpha }}P+\\frac{\\sqrt{\\alpha }}{2}X=\\mp i\\frac{1}{\\sqrt{\\alpha }}\\frac{d}{dx}+\\frac{\\sqrt{\\alpha }}{2}x,$ with $(U_{\\pm })^{\\dagger } = U_{\\pm } \\ne U_{\\mp }$ (with respect to the usual scalar product) and $[U_{-},U_{+}]=i\\mathbb {I}$ .",
"Let us note that in our system of units, $U_{\\pm }$ are dimensionless operators.",
"The Hamiltonian $H$ can then be written in terms of $U_{\\pm }$ as $H=P^2-\\frac{\\alpha ^2}{4}X^2=-\\frac{\\alpha }{2}\\left(U_{+}U_{-}+U_{-}U_{+}\\right).$ It is also possible to introduce the dilation operator $D$ and the operator $S$ , which is associated with special conformal transformations generator $K=X^2$ , as follows $&D&=\\frac{1}{2}\\left(PX+XP\\right)=\\frac{1}{2}\\left(U_{+}^2-U_{-}^2\\right)=-i\\left(x\\frac{d}{dx} + \\frac{1}{2}\\right),\\nonumber \\\\&S&=P^2+\\frac{\\alpha ^2}{4}X^2=\\frac{\\alpha }{2}\\left(U_{+}^2+U_{-}^2\\right)=-\\frac{d^2}{dx^2}+\\frac{\\alpha }{4}x^2.$ Up to dimensional factors, it is important to note that the expressions of $H$ and $D$ are switched whether one is looking at their expressions in terms of $X$ and $P$ operators, or in terms of $U_{\\pm }$ operators.",
"In other words, for what concerns the IHO problem, $H$ can be seen as a dilation process or as a scattering process by an IHO potential, depending on the choice of “coordinates” ($x$ or $u_{\\pm }$ ) and the associated conjugate momenta ($p$ or $u_{\\mp }$ ).",
"It is interesting to note that the operator $S$ remains unchanged in both “coordinates” systems, as a sum of squared operators, and that it would play the role of the Hamiltonian in the usual harmonic oscillator problem.",
"It becomes then quite trivial to write the expressions of $H$ , $D$ and $S$ as differential operators in “$u_{\\pm }$ ”-representations, from their expressions in the $x$ -representation (see [9] and references therein, for a nice exhaustive analysis of the physics of the IHO).",
"For example, $H$ in the $u_{\\pm }$ -representation reads simply $H = \\mp i \\alpha \\left(u_{\\pm }\\frac{d}{du_{\\pm }}+\\frac{1}{2}\\right).$ In order to simplify the underlying algebra, one can finally define the following three (dimensionless) operators $J_1&=&\\frac{1}{2}D=-\\frac{i}{2}\\left(x\\frac{d}{dx}+\\frac{1}{2}\\right),\\\\J_2&=&-\\frac{i}{2\\alpha }S=\\frac{i}{2\\alpha }\\left(\\frac{d^2}{dx^2}-\\frac{\\alpha ^2}{4}x^2\\right),\\\\J_3&=&\\frac{i}{2\\alpha }H=-\\frac{i}{2\\alpha }\\left(\\frac{d^2}{dx^2}+\\frac{\\alpha ^2}{4}x^2\\right).$ Those operators satisfy the commutation relations of an $SO(2,1)$ algebra $[J_1,J_2]=-iJ_3; \\quad [J_2,J_3]=iJ_1; \\quad [J_3,J_1]=iJ_2.$ Finally, from $J_1$ and $J_2$ , one can also introduce “ladder operators”.",
"For example $J_{\\pm }=\\pm iJ_1-J_2=\\frac{i}{2}U_{\\pm }^2,$ which, together with $J_3$ , satisfy an $SL(2,\\mathbb {R})$ algebra $[J_{+},J_{-}]=-2J_3; \\quad [J_3,J_{\\pm }]=\\pm J_{\\pm }.$ It should be noted first that one could have obviously constructed $J_3$ and $J_{\\pm }$ , which satisfy commutation relations $(\\ref {LadderOp_CommutationRelation})$ , more directly from $H$ and $U_{\\pm }^2$ , i.e.",
"without any references to the operators $D$ and $S$ .",
"Finally, let us also note that that the ladder operators $J_{\\pm }$ are not self-adjoint conjugate to each other.",
"In our example here, one has instead $\\left(J_\\pm \\right)^\\dagger =-J_\\pm $ , i.e.",
"anti-self-adjoint operators.",
"Once the algebra is found, the computation of the eigenstates and eigenvalues of $H$ follows.",
"Indeed, as usually done in the harmonic oscillator case, one can begin by looking for the ground states of the ladder operators $J_\\pm $ .",
"Then, by an application of $(J_\\pm )^n$ to the associated ground state (for every non negative integer $n$ ), it is an easy task to construct the set of eigenstates which belong to the associated rigged Hilbert space, that allows, in a few words, states to be defined as distributions (see [10], [11] for more details about the rigged Hilbert space formalism).",
"The interested reader may found a detailed construction of the set of eigenstates and eigenvalues of $(J_\\pm )^n$ and $H$ in [8], in the context of a hidden $SL(2,\\mathbb {R})$ symmetry on a BH photon sphere.",
"Finally, one shows that the associated spectrum of $H$ is discrete, purely imaginary, and indexed by an integer.",
"The eigenvalues $h_n^{(\\pm )}$ of $H$ , for every non negative integer $n$ , can finally be written as $h_n^{(\\pm )} = \\mp i \\alpha \\left(n+\\frac{1}{2}\\right).$ These discrete and purely imaginary eigenvalues are shown to be related to the frequencies of the resonant modes in the resonant scattering problem of a scalar field by an IHO potential [9], [8].",
"The non negative integer index $n$ will be the key to understand the overtone level spacing of a BH quasinormal frequencies.",
"It should be noted that the curvature $\\alpha $ of the IHO potential appears to be the amplitude of the eigenvalues spacing of $H$ .",
"In the following, we will first show that the resonant scattering problem of a massless scalar field in a Rindler spacetime can precisely be reduced to the study of a resonant scattering problem by an IHO potential with the corresponding Hamiltonian $H$ .",
"From the above, this obviously allows its algebraic treatment through the $SL(2,\\mathbb {R})$ algebra with all the related results.",
"By a natural extension, it should be noted that every regions of a black hole spacetime that locally admit the Rindler metric as an approximation will be places where a similar treatment can be applied.",
"In particular, we will show in the following sections that the horizon (which is the best known case) but also the photon sphere are both such regions."
],
[
" Massless particles in Rindler spacetime",
"We consider the line element of a $(1+1)$ -dimensional Rindler spacetime, in Rindler's coordinates $(t,x)$ : $ds^2=a^2x^2 dt^2-dx^2.$ It is known to describe a line element seen by an uniformly accelerated observer (with a constant acceleration $a$ ), in coordinates $(t,x)$ .",
"The study can be restricted, for example, to the right Rindler wedge, for which $t\\in ]-\\infty ,+\\infty [$ and $x\\in [0,+\\infty [$ .",
"In the following, we will be focusing on the motions of massless particles in those coordinates.",
"With this aim in mind, we introduce an affine parameter $\\lambda $ to describe the geodesics, and we decide to use a (quadratic) Lagrangian approach which is a minimalist but efficient way to highlight our point.",
"Let us consider the following quadratic lagrangian: $\\mathcal {L}=\\frac{1}{2}\\frac{ds^2}{d\\lambda ^2}=\\frac{1}{2}(a^2x^2\\dot{t}^2-\\dot{x}^2)$ where $\\dot{t}=dt/d\\lambda $ and $\\dot{x}=dx/d\\lambda $ .",
"In units such that $h=c=1$ , here $x$ and $t$ have dimension $L$ , $\\lambda $ has dimension $L^2$ , while $a$ has dimension $[x]^{-1}=L^{-1}$ .",
"From Euler-Lagrange equations and spacetime symmetries, the only integral of motion (equivalently associated with the Killing vector $\\partial /\\partial t$ ) is here the “energy” $E$ defined by $E=\\frac{\\partial \\mathcal {L}}{\\partial \\dot{t}}=a^2x^2\\dot{t}.$ The “linear momentum” is defined as $p=\\frac{\\partial \\mathcal {L}}{\\partial \\dot{x}}=-\\dot{x}.$ Inserting $(\\ref {E})$ into the line element $(\\ref {RindlerMetric})$ for a massless particle, i.e.",
"$ds^2=0$ , one finds $\\dot{x}^2 - \\frac{E^2}{a^2x^2}=0$ or $\\left(\\frac{dx}{dt}\\right)^2 - a^2x^2=0.$ This equation, whose solution grows exponentially with an associated Lyapunov exponent $\\Lambda =|a|$ (see [12] and references therein), describes the trajectory of a massless particle in Rindler's coordinates $(t,x)$ , i.e.",
"seen by an uniformly accelerated observer.",
"Equation $(\\ref {MasslessMotion2})$ clearly shows that this motion, in $(t,x)$ coordinates, is actually a motion in an IHO potential.",
"This equation can then also be interpreted as describing an instability near the (hyperbolic) point $x=0$ (the top of the IHO potential barrier) in the phase space $(dx/dt,x)$ .",
"In other words, the Rindler horizon corresponds to an unstable equilibrium point for massless particles, in the phase space $(dx/dt,x)$ of an accelerated observer.",
"Let us note that for massless geodesics ($ds^2=0$ ), dividing $(\\ref {RindlerMetric})$ by $dt^2$ would have given directly $(\\ref {MasslessMotion2})$ .",
"It should be noted that there exist a simple dispersion relation between the energy $E$ and the linear momentum $p$ of the particle, obtained by inserting $(\\ref {p})$ in $(\\ref {MasslessMotion1})$ $E=\\pm a x p.$ This dispersion relation is characteristic of a particle in a IHO potential.",
"Indeed, with a change of variables of the form $\\tilde{x}=x-\\frac{2}{a}p \\quad \\textrm {and}\\qquad \\tilde{p}=\\frac{a}{2}x + p,$ which is equivalent to $(\\ref {CreationAnnihilationOperators})$ , one can rewrite $(\\ref {RindlerDispersionRelation})$ as $E = \\pm \\left(\\tilde{p}^2-\\frac{a^2}{4}\\tilde{x}^2\\right).$ where the IHO potential appears explicitly."
],
[
" Scattering of a massless scalar field in a Rindler spacetime",
"The Klein Gordon equation for a massless scalar field $\\Phi (x^\\mu )$ in a spacetime with metric $g_{\\mu \\nu }$ reads $\\Box \\Phi = g^{\\mu \\nu } \\nabla _\\mu \\nabla _\\nu \\Phi = \\frac{1}{\\sqrt{-g}} \\partial _\\mu \\left(\\sqrt{-g} g^{\\mu \\nu }\\partial _\\nu \\Phi \\right)=0.$ In a Rindler spacetime with metric $(\\ref {RindlerMetric})$ , the Klein Gordon equation becomes $\\left[-\\partial _t^2 + a^2x\\partial _x\\left(x\\partial _x\\right)\\right]\\Phi (t,x)=0.$ Let us assume a stationary scalar field with harmonic time dependence of the form $\\Phi (t,x)=\\phi (x)e^{-i\\omega t}$ .",
"The Klein-Gordon equation becomes $\\left[(-i\\omega )^2 - ax\\partial _x\\left(x\\partial _x\\right)\\right]\\Phi (t,x)=0,$ which can be easily factorized in the form $\\left[(-i\\omega )+ax\\partial _x\\right]\\left[(-i\\omega )-ax\\partial _x\\right]\\Phi (t,x)=0.$ It should be noted that a massless scalar plane wave, e.g.",
"$\\Phi (t,x)=\\Phi _0 e^{-i(\\omega t \\mp kx)}$ , solution of each term of $(\\ref {KG_factorized})$ gives a dispersion relation $\\omega = \\pm a x k$ which is equivalent to the relation $(\\ref {RindlerDispersionRelation})$ .",
"In order to establish explicitly the link between $(\\ref {KG_factorized})$ and $(\\ref {H_Diff_Upm})$ , we will have to introduce a “trick” that will highlight an important aspect of thermality comparing the Rindler and the IHO case.",
"Let us begin to rewrite $(\\ref {KG_factorized})$ as $\\left[-i\\omega -\\frac{a}{2}+\\frac{a}{2}+ax\\partial _x\\right]\\left[-i\\omega +\\frac{a}{2}-\\frac{a}{2}-ax\\partial _x\\right]\\Phi (t,x)=0.$ Most of the solutions $\\Phi $ [13] can be reduce to a linear combination of $\\tilde{\\Phi }^{(\\pm )}$ , such as $\\left\\lbrace \\begin{array}{ll}\\left(ax\\partial _x + \\dfrac{a}{2}\\right)\\tilde{\\Phi }^{(+)}=\\left(i\\omega +\\dfrac{a}{2}\\right)\\tilde{\\Phi }^{(+)}\\\\-\\left(ax\\partial _x + \\dfrac{a}{2}\\right)\\tilde{\\Phi }^{(-)}=\\left(i\\omega -\\dfrac{a}{2}\\right)\\tilde{\\Phi }^{(-)}.\\end{array}\\right.$ In other words, $\\tilde{\\Phi }^{(\\pm )}$ both satify $\\pm ia\\left(x\\partial _x +\\frac{1}{2}\\right)\\tilde{\\Phi }^{(\\pm )} = - \\left(\\omega \\mp \\frac{ia}{2}\\right)\\tilde{\\Phi }^{(\\pm )},$ which reads $H \\tilde{\\Phi }^{(\\pm )} = \\left(\\omega \\mp \\frac{ia}{2}\\right)\\tilde{\\Phi }^{(\\pm )}$ where $H$ is the Hamiltonian of an IHO written in the form $(\\ref {H_Diff_Upm})$ , i.e.",
"as a dilation operator.",
"In other words, the equation of motion of a massless scalar field, of frequency $\\omega $ , seen by an uniformly accelerated observer (with acceleration $a$ ) in Rindler coordinates $(t,x)$ , can simply be expressed in term of an IHO Hamiltonian.",
"It should be noted that the naive “trick” used in $(\\ref {KG_factorized_2})$ that leads to $(\\ref {H_Rindler_IHO})$ , could be understood as being related to the thermal behavior of the number of quantized particles in the Unruh effect.",
"Indeed, on the one hand, the factorization of the massless Klein Gordon equation for a scalar field $\\Phi $ in a $(1+1)$ -dimensional Rindler spacetime with metric $(\\ref {RindlerMetric})$ gives a system of two massless scalar field $\\tilde{\\Phi }^{(\\pm )}$ equations, the same way the usual $(1+1)$ -dimensional massless scalar wave equation can be factorized into a system of two $(1+1)$ -dimensional uncoupled Weyl-like equations.",
"On the other hand, it is well-known that, for a scalar field, the thermal behavior of the number of quantized particles seen by an accelerated observer is given by a Bose-Einstein factor $(e^{2\\pi \\omega /T}-1)^{-1}$ with Unruh temperature $T$ , whereas for a spin-$1/2$ field the thermal behavior is given by a Fermi-Dirac factor $(e^{2\\pi \\omega /T}+1)^{-1}$ .",
"From this perspective, the “trick” in $(\\ref {KG_factorized_2})$ formally corresponds to the passage from one thermal behavior, that originates from equation $(\\ref {KleinGordon_Rindler})$ , to the other, that originates from equations $(\\ref {Weyl_like_eqs1})$ , with the change $\\omega \\rightarrow \\omega \\mp ia/2$ in the computation of the related Unruh effect [14].",
"We have explicitly shown in this section that the scattering by an IHO potential is actually hidden in the scattering problem of a massless scalar field in Rindler spacetime, in coordinates $(t,x)$ .",
"The resonant scattering in Rindler spacetime is then simply given by the $SL(2,\\mathbb {R})$ algebraic approach presented in section .",
"Indeed, by combining $(\\ref {HEigenvalues})$ and $(\\ref {H_Rindler_IHO})$ gives a set of purely imaginary frequencies, for every non negative integer $n$ $\\omega _{n}^{(\\pm )}=\\mp i a n.$ As anticipated previously, it should be noted that the discrete eigenvalues spectrum of the IHO Hamiltonian for the associated resonant modes can be understood as being at the origin of the regular level spacing in the QNF.",
"This level spacing appears clearly here, in the case of resonant scattering in Rindler spacetime.",
"Let us note that the amplitude of the level spacing is nothing less than the IHO potential curvature, which is interpreted as a constant acceleration in the Rindler case.",
"This will appear in a very similar manner in the QNF spectrum for BH, as long as the BH metric admits close to certain regions of spacetime, the Rindler metric as an approximation."
],
[
"The static and spherically symmetric black hole case",
"In the Schwarzschild coordinates $(t,r,\\theta ,\\phi )$ , we consider a static spherically symmetric four-dimensional spacetime with metric $ds^2=f(r)dt^2-\\frac{dr^2}{f(r)}-r^2d\\sigma ^{2}.$ As usual, $d\\sigma ^{2}=d\\theta ^2 + \\sin ^2 \\theta d\\varphi ^2$ is the line element on the unit 2-sphere $S^{2}$ , with $\\theta \\in [0,\\pi ]$ and $\\varphi \\in [0,2\\pi ]$ .",
"Moreover, let us consider the BH exterior with $r \\in ]r_h,+\\infty [$ , where $r=r_h$ is a simple root of $f(r)$ and defines the location of the BH event horizon.",
"Let us assume also that the background geometry is asymptotically flat and the tortoise coordinate $r_\\ast = r_\\ast (r)$ , defined as $dr_\\ast /dr = 1/f(r)$ , is a bijection from $]r_h,+\\infty [$ to $] -\\infty , +\\infty [$ .",
"Without loss of generality, we will consider motions on the equatorial plane $\\theta =\\pi /2$ .",
"A free-falling massless particle moves along null geodesics according to $-f(r)\\dot{t}^2+\\frac{1}{f(r)}\\dot{r}^2+r^2\\dot{\\varphi }^2=0,$ where, as done earlier in this paper, $\\dot{t}=dt/d\\lambda $ , $\\dot{r}=dr/d\\lambda $ , $\\dot{\\varphi }=d\\varphi /d\\lambda $ and $\\lambda $ is an affine parameter which describes null geodesics.",
"From symmetries of the BH spacetime, one can define integrals of motion, i.e.",
"energy $E$ and angular momentum $L$ of the massless particle, associated respectively with the Killing vectors $\\partial /\\partial t$ and $\\partial /\\partial \\varphi $ : $f(r)\\dot{t}=E\\quad ; \\quad r^2\\dot{\\varphi }=L.$ The equation of motion is easily deduced from (REF ) and reads $\\dot{r}^2+V_{\\textrm {eff}}(r)=E^2,$ where the effective potential $V_{\\textrm {eff}}$ is defined as $V_{\\textrm {eff}}(r)=\\frac{L^2}{r^2}f(r).$ A photon sphere (or “lightrings”, or “photon rings”) located at $r=r_c$ corresponds to a local maximum of $V_{\\textrm {eff}}(r)$ at $r=r_c$ such as $\\left.\\frac{d}{dr}V_{\\textrm {eff}}(r)\\right|_{r_c}=0 &\\Leftrightarrow & \\frac{2}{r_c}f_c=f^{\\prime }_c, \\\\\\left.\\frac{d^2}{dr^2}V_{\\textrm {eff}}(r)\\right|_{r_c}<0 &\\Leftrightarrow & f^{\\prime \\prime }_c - \\frac{2}{r_c^2}f_c<0.$ The subscript “$c$ ” means, above and in the following, that the quantity considered is evaluated at $r=r_c$ , and the superscripts “ $^{\\prime }$ ” and “ $^{\\prime \\prime }$ ” respectively mean the first and second derivatives with respect to $r$ .",
"A massless particle reaches the photon sphere when the turning point of its motion satisfies $E^2 = V_\\textrm {eff,c} $ , i.e.",
"$L/E = r_c/\\sqrt{f_c}$ , where $r_c/\\sqrt{f_c}=b_c$ is the critical impact parameter for massless particles to reach tangentially the photon sphere, before circling the BH at $r=r_c$ .",
"Moreover, at $r=r_c$ one also has $V^{\\prime \\prime }_{\\textrm {eff},c}=-2\\eta _c^2 \\frac{L^2}{r_c^4},$ where $\\eta _c = \\frac{1}{2}\\sqrt{4f_c-2r_c^2f^{\\prime \\prime }_c}.$ The study of instability associated with the circular orbits of massless particles on the photon sphere follows [3].",
"Indeed, writing down a second order Taylor series expansion of $V_{\\textrm {eff}}$ near the photon sphere at $r=r_c$ gives $V_{\\textrm {eff}}(r) \\simeq V_{\\textrm {eff},c} + \\frac{1}{2}V^{\\prime \\prime }_{\\textrm {eff},c} (r-r_c)^2,$ such as (REF ) becomes $\\dot{r}^2+V_{\\textrm {eff},c} +\\frac{1}{2}V^{\\prime \\prime }_{\\textrm {eff},c} (r-r_c)^2 = E^2.$ If the turning point is close enough to $r=r_c$ , one can consider that $E^2 \\approx V_{\\textrm {eff},c}$ .",
"The equation of motion then simplifies to $\\dot{r}^2 +\\frac{1}{2}V^{\\prime \\prime }_{\\textrm {eff},c} (r-r_c)^2 = 0,$ which can be written, in $(r,t)$ coordinates, as $\\left(\\frac{dr}{dt}\\right)^2 + \\frac{V^{\\prime \\prime }_{\\textrm {eff},c}}{2 \\dot{t}^2} (r-r_c)^2 = 0.$ Close to $r=r_c$ , one has $\\dot{t}=dt/d\\lambda \\approx E/f_c$ , and the equation of motion in the vicinity of the photon sphere simply reads $\\left(\\frac{dr}{dt}\\right)^2 - \\Lambda _c^2 (r-r_c)^2 = 0,$ where $|\\Lambda _c|=\\eta _c \\frac{\\sqrt{f_c}}{r_c}$ is the Lyapunov exponent associated with the unstable circular motion of a free-falling massless particle near the BH photon sphere."
],
[
"The near horizon region: the highly damped QNM",
"In this short section, we consider the well-known case of a purely radial motion in the line element $(\\ref {metric_BH})$ $ds^2=f(r)dt^2-\\frac{dr^2}{f(r)}.$ In the near-horizon region, one can approximate $f(r)$ by $f(r)=f^{\\prime }_h(r-r_h)$ at first order, with $f^{\\prime }_h=(df/dr)(r=r_h)$ .",
"In a coordinate systems for which $d\\rho =\\frac{dr}{\\sqrt{f^{\\prime }_h(r-r_h)}}$ and introducing the surface gravity $\\kappa =(1/2)f^{\\prime }_h$ , the Rindler approximation of the metric $(\\ref {metric_BH_radial})$ in the near horizon region follows easily $ds^2=\\kappa ^2\\rho ^2dt^2-d\\rho ^2$ The form $(\\ref {RindlerMetric})$ is obviously recovered from which an equation similar to $(\\ref {MasslessMotion2})$ is derived.",
"It should be noted that the equation of motion of a free-falling massless particle can also be obtained from $(\\ref {EqMotion})$ , with $L=0$ because $\\varphi $ is zero for purely radial motions towards the BH, which reads $\\dot{r}^2=E^2.$ With $(\\ref {EL})$ , one deduces $\\left(\\frac{dr}{dt}\\right)^2 - f(r)^2=0.$ In the near horizon region, $f(r)=f^{\\prime }_h(r-r_h)$ at first order in $r$ , and with the change of variable $(\\ref {rhoRindler})$ one obviously recovers $(\\ref {MasslessMotion2})$ in the form $\\left(\\frac{d\\rho }{dt}\\right)^2-\\kappa ^2\\rho ^2=0.$ All the results of section REF can now be applied to the near horizon case with the $SL(2,\\mathbb {R})$ algebra approach.",
"In particular, the highly damped QNF are given by a relation like $(\\ref {Rindler_QNF})$ $\\omega _{n}^{(\\pm )} \\approx \\mp i \\kappa n,$ for which the imaginary part is equally spaced as expected.",
"It should be noted that this approach does not give the real part of the QNF that appears in $(\\ref {HighlyDampedQNF})$ , which is related to the behavior of the BH metric near the singularity $r\\rightarrow 0$ .",
"Of course, although almost everything has already been said in the literature about the Hawking-Unruh effect in the near horizon region, this $SL(2,\\mathbb {R})$ algebraic approach is worth mentioning, because it gives easily the equally spaced overtone level of the imaginary part of the highly damped QNF.",
"Moreover, an interpretation that is less known is that the BH horizon appears also in this context as an unstable point, as for $(\\ref {MasslessMotion2})$ , here for purely radial null geodesics, in the phase space of an accelerated observer $(d\\rho /dt,\\rho )$ .",
"In this context, the surface gravity plays a role of a Lyapunov exponent, i.e.",
"an inverse characteristic time, that characterizes the (in)stability of the BH horizon for radial trajectories of massless particles seen from an accelerated observer $(t,\\rho )$ (see [12] and references therein for a very exhaustive study of this interpretation).",
"In this section, we reproduce the outline of the proof in [8] according to which, in the ultrarelativistic limit, the near photon sphere limit of the metric $(\\ref {metric_BH})$ is also a Rindler metric.",
"Then, from the above, all the results of section could also be easily applied.",
"We start by considering the motion of a free-falling test particle of mass $m$ following the geodesic line element $(\\ref {metric_BH})$ .",
"For a massive test particle to get very close to the photon sphere, we will need to consider the ultrarelativistic limit of its motion.",
"Without loss of generality, let us focus again on the equatorial plane $\\theta =\\pi /2$ of the metric $(\\ref {metric_BH})$ $ds^2=-f(r)dt^2+\\frac{dr^2}{f(r)}+r^2d\\varphi ^{2}.$ In the massive case, the integrals of motion read $f(r)\\left(\\frac{dt}{d\\tau }\\right)=\\frac{E}{m}\\quad ; \\quad r^2\\left(\\frac{d\\varphi }{d\\tau }\\right)=\\frac{L}{m}$ where $\\tau $ is the masssive particle proper time.",
"Using (REF ), the equation of motion for the test particle reads $m^2\\left(\\frac{dr}{d\\tau }\\right)^2 + U_\\textrm {eff}(r) = E^2,$ where $U_\\textrm {eff}(r)=f(r)\\left[\\frac{L^2}{r^2}+m^2\\right].$ The effective potential $U_\\textrm {eff}(r)$ has extrema located at $r=r_i$ such that $f^{\\prime }(r_i)-\\frac{2}{r_i^2}f(r_i)+\\frac{m^2}{L^2}f^{\\prime }(r_i)=0.$ We simplify the study by assuming that the particle angular momentum $L$ value is such that $U_\\textrm {eff}(r)$ admits a local maximum.",
"Let us call $r_0(L)$ the location of this local maximum.",
"In the limit $L \\gg 1$ , $(\\ref {r0})$ tends to $(\\ref {rc})$ , the local maximum coincides with the location of the photon sphere, i.e.",
"$r_0(L)$ tends to $r_c$ and $U_\\textrm {eff}(r_0(L))$ tends to $V_\\textrm {eff,c}$ .",
"In other words, a test particle (with energy $E$ and angular momentum $L$ ) coming from infinity, gets very close to the photon sphere if at least $&m^2& < E^2 \\approx U_\\textrm {eff}(r_0(L)),\\\\&L& \\gg 1 \\quad \\textrm {with} \\quad L/E \\quad \\textrm {finite} .$ The condition (REF ) implies that the turning point of the particle motion is located in the vicinity of the maximum of $U_\\textrm {eff}(r)$ .",
"The second condition () implies that the location of this maximum tends to the location of the photon sphere.",
"If the conditions (REF ) are both satisfied, then one has $E^2 \\approx U_\\textrm {eff}(r_0(L)) \\approx V_\\textrm {eff,c}$ , i.e.",
"$L/E \\approx r_c/\\sqrt{f_c}$ , which corresponds to the limit of an ultrarelativistic test particle.",
"Then, in the ultrarelativistic limit, the test particle has an impact parameter $b$ which tends, from above, to the critical impact parameter $b_c$ associated with massless particles motion, and the particle coming from infinity will get close to the photon sphere before moving away, back to infinity.",
"Let us now focus on the near-photon sphere limit of $(\\ref {geodesics_equatorial_plane})$ to describe the motion of the test particle in the ultrarelativistic limit.",
"We first use the constants of motions $(\\ref {EL_massive})$ for geodesic motions [15], to restrict ourselves to an equivalent effective geodesic line element in the $(t,r)$ -plane.",
"From $(\\ref {EL_massive})$ , one can write $r^2 d\\varphi ^2=f(r)^2\\frac{L^2}{E^2 r^2}dt^2.$ This allows to transform $(\\ref {geodesics_equatorial_plane})$ into a line element, that would give the same radial equation of motion $(\\ref {EqMotionMassive})$ , $ds^2=f(r)\\left(-1+\\frac{V_\\textrm {eff}(r)}{E^2}\\right)dt^2+\\frac{dr^2}{f(r)},$ where $V_\\textrm {eff}(r)$ is still, from a pure formal point of view, defined by expression $(\\ref {EffPotential})$ but with $L$ being now the angular momentum of the massive test particle.",
"It should be noted that the line elements (REF ) and (REF ) have the same magnitude for any geodesic motion, i.e.",
"for any given $E$ and $L$ .",
"Moreover, let us emphasize that (REF ) does not describe a purely radial motion in the BH background (in such case $\\varphi $ would have been constant, i.e.",
"$L=0$ , and the photon sphere would have had no effect on the test particle motion), but rather describes the effective non-radial geodesic motion of a test particle in the $(t,r)$ -plane of a static and spherically symmetric BH background, taking into account explicitly the effect of the centrifugal potential barrier in its time component.",
"Let us recall that we will not consider the case where the particle gets trapped into the BH, i.e.",
"here one has $b \\gtrsim b_c$ .",
"The near-photon sphere limit of $(\\ref {metric_BH2})$ requires the conditions $(\\ref {EL_conditions})$ to be satisfied, i.e.",
"$E^2 \\approx V_\\textrm {eff,c}$ , and is obtained from the lowest order Taylor series expansion around $r=r_c$ which does not cancel the time component of $(\\ref {metric_BH2})$ .",
"Using $(\\ref {Taylor_EffPotential})$ , the effective line element $(\\ref {metric_BH2})$ then becomes in this limit $ds^2 \\simeq -\\frac{V^{\\prime \\prime }_\\textrm {eff,c}}{2E^2}f_c(r-r_c)^2dt^2+\\frac{dr^2}{f_c}.$ Now, from $(\\ref {VeffSecond})$ and $(\\ref {Lyapunov})$ , and introducing the variable $\\rho = \\frac{r-r_c}{\\sqrt{f_c}} \\Leftrightarrow d\\rho = \\frac{dr}{\\sqrt{f_c}},$ one finally obtains a Rindler form of the effective line element near the photon sphere, which acts in this setting as an effective Rindler horizon $ds^2 \\simeq - \\Lambda _c^2 \\rho ^2 dt^2 + d\\rho ^2,$ where we have used $L^2/E^2 \\approx r_c^2/f_c$ because $E^2 \\approx V_\\textrm {eff,c}$ .",
"The Lyapunov exponent $|\\Lambda _c|$ associated with the massless particle motions around the photon sphere plays the role of a constant proper acceleration in the near-photon sphere limit of the line element (REF ), describing the test particle effective geodesic motion in the $(t,r)$ -plane, a role analogue to the role played by the surface gravity in the near-horizon limit."
],
[
"Massless scalar field in the BH metric",
"In this section, we look at the Klein-Gordon equation for a massless scalar field $\\Phi $ in a spacetime metric $(\\ref {metric_BH})$ .",
"After separation of variables, assuming a harmonic time dependence ($e^{-i\\omega t}$ ) for $\\Phi $ and the introduction of the radial partial wave functions $\\Phi _{\\ell \\omega }(r)$ with $\\ell =0,1,2,\\dots $ , the Klein-Gordon equation gives the well-known Regge-Wheeler equation $\\frac{d^2 \\Phi _{\\ell \\omega }}{d r_\\ast ^2} + \\left[\\omega ^2 - V_\\ell (r_\\ast )\\right] \\Phi _{\\ell \\omega } = 0.$ where $r_\\ast = r_\\ast (r)$ is the tortoise coordinate, introduced previously, and $V_{\\ell }(r)$ is the Regge-Wheeler potential defined as $V_{\\ell }(r)=f(r)\\left[\\frac{\\ell (\\ell +1)}{r^2}+\\frac{1}{r}f^{\\prime }(r)\\right].$ It should be noted that, for every $\\ell \\in \\mathbb {N}$ , $V_{\\ell }(r)$ admits a local maximum at $r=r_{0}(\\ell )$ which is close to the photon sphere located at $r=r_c$ .",
"Moreover, in the limit $\\ell \\gg 1$ , $r_0(\\ell ) \\approx r_c$ and $V_{\\ell }(r) \\approx V_{\\textrm {eff}}(r)$ .",
"Using the tortoise coordinate, we will denote $(r_\\ast )_{0,\\ell } = r_\\ast (r_0(\\ell ))$ the location of the maximum of $V_{\\ell }(r)$ .",
"Following [16], let us consider $V_\\ell (r_\\ast )$ around the location of its local extremum at $(r_\\ast )_{0,\\ell }$ , i.e.",
"around a location as close to the photon sphere as $\\ell \\gg 1$ .",
"A second order Taylor series expansion gives $V_\\ell (r_\\ast ) \\approx V_0(\\ell ) + \\frac{1}{2}V^{(2)}_0(\\ell ) \\left(r_\\ast - (r_\\ast )_{0,\\ell } \\right)^2,$ where we have used the notation $V_0^{(2)}(\\ell )=\\left(\\frac{d^2 V_{\\ell }(r_\\ast )}{dr_\\ast ^2}\\right)_{(r_\\ast )_{0,\\ell }}.$ Introducing $x$ , $h(\\ell ,\\omega )$ and $\\alpha (\\ell )$ such that $x &=& \\left(r_\\ast - (r_\\ast )_{0,\\ell } \\right),\\nonumber \\\\h(\\ell ,\\omega ) &=&\\omega ^2-V_0(\\ell ),\\nonumber \\\\\\alpha (\\ell ) &=& \\sqrt{-2V_0^{(2)}(\\ell )},$ equation (REF ) reads $H\\tilde{\\Phi }_{\\ell \\omega }(x) = h(\\ell ,\\omega )\\tilde{\\Phi }_{\\ell \\omega }(x)$ where $\\tilde{\\Phi }_{\\ell \\omega }(x)=\\Phi _{\\ell \\omega }(r_\\ast (x))$ and $H=-\\frac{d^2}{dx^2} - \\frac{\\alpha (\\ell )^2}{4}x^2$ is the time-independent “Hamiltonian” (here with dimension $L^{-2}$ ) governing the massless scalar field dynamics in the near-photon sphere limit.",
"In the coordinate $x$ , $H$ is in the form $(\\ref {Hamiltonian})$ , and then all the eigenvalue problem can be solved via algrebraic $SO(2,1)\\sim SL(2,\\mathbb {R})$ method.",
"For the weakly damped BH QNF (i.e.",
"$\\ell \\gg 1$ and $\\ell \\gg n$ ), it should be noted that one can make the substitution $\\ell (\\ell +1)=(\\ell +1/2)^2-1/4$ in $(\\ref {RWPotential})$ and look at $(\\ell +1/2)$ (instead of $\\ell $ ) as the quantity in which one will compute a Taylor series expansion of the Regge-Wheeler potential and its second derivative at $(r_{\\ast })_{0,\\ell }$ , in the limit $(\\ell +1/2)\\gg 1$ .",
"At lowest order in $\\ell +1/2$ , one has $\\alpha (\\ell ) &\\approx & 2\\eta _c \\frac{f_c}{r_c^2} \\left(\\ell +\\frac{1}{2}\\right),\\\\V_{0}(\\ell ) &\\approx & \\frac{f_c}{r_c^2}\\left(\\ell +\\frac{1}{2}\\right)^2.$ Equation $(\\ref {HEigenvalues})$ then reads $\\omega ^2=V_{0}(\\ell ) \\pm i \\alpha (\\ell )\\left(n+\\frac{1}{2}\\right),$ which gives, in the weakly damped regime ($(\\ell +1/2)\\gg 1$ and $\\ell \\gg n$ ), a equally spaced imaginary part of the BH QNF $\\textrm {Im}\\left(\\omega _{\\ell n}^{(\\pm )}\\right)\\approx -\\frac{i}{2}\\frac{\\alpha (\\ell )}{\\sqrt{V_0(\\ell )}}\\left(n+\\frac{1}{2}\\right)=-i|\\Lambda _c|\\left(n+\\frac{1}{2}\\right),$ with $|\\Lambda _c|=\\eta _c \\sqrt{f_c}/r_c=$ is the Lyapunov exponent that characterizes the instability of null motions on the photon sphere."
],
[
"Remarks on the amplitude of the overtone spacing",
"A relevant fact is that the amplitude of the overtone level spacing is related to a Lyapunov exponent that characterizes the (in)stability of the top of the underlying IHO potential barrier.",
"Rindler spacetime is an example of such behavior: the surface gravity in $(\\ref {RindlerSimple})$ is understood in this framework as a Lyapunov exponent, and reciprocally, the Lyapunov exponent in $(\\ref {Rindler_metric})$ can be interpreted as a locally constant acceleration for a certain family of observers.",
"It should be noted that in both cases, the Lyapunov exponent characterizes motions of massless particles that seem to play a fundamental role in this description.",
"Of course, as long as one has a Rindler approximation of a BH metric, the thermal aspects follow directly (see [12], [8] for more details).",
"As a Lyapunov exponent is interpreted as the inverse of a characteristic time, this allows to expect a fundamental link between “time-thermality-instability”.",
"Another remark is about the real part of the QNF in the highly damped regimes.",
"This approach does not allow to access such a real part, because it has been shown to be related to the behavior of the BH metric near its singularity $r\\rightarrow 0$ , which is not considered here."
],
[
"Conclusion",
"In this paper, we propose a simple way to understand the equally spaced overtone level of BH QNF both in the highly and weakly damped regimes.",
"To do so, we have shown that the resonant scattering problem in Rindler spacetime is deeply linked to the resonant scattering problem by an IHO potential, which in turn can be completely described through the $SL(2,\\mathbb {R})$ algebra.",
"The $SL(2,\\mathbb {R})$ algebra allows to show that the spectrum of the “Hamiltonian” is discrete, purely imaginary and indexed by a non negative integer.",
"This integer turns out to be simply the overtone of the QNF, and is shown to be equally spaced.",
"A general claim could follow: in any BH geometry, at last for static and spherically symmetric BH, if there exist some regions where the BH metric can be locally approximated as a Rindler spacetime, in some set of coordinates, then the underlying algebra should be $SL(2,\\mathbb {R})$ , and the imaginary part of the BH QNF, be it in the highly damped or weakly damped regimes, will be indexed by a non negative integer, i.e.",
"the overtone number, which will be equally spaced.",
"A very interesting fact is that we explicitly obtained this result for both highly and weakly damped QNM families, by looking at different regions of the BH spacetime.",
"This work can be easily extended to higher dimensional spacetime.",
"A future look at the Kerr BH is of interest.",
"The author would like to thank J.P. Provost and J.L.",
"Jaramillo for stimulating discussions.",
"The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002)."
]
] | 2212.05538 |
[
[
"Extending TrOCR for Text Localization-Free OCR of Full-Page Scanned\n Receipt Images"
],
[
"Abstract Digitization of scanned receipts aims to extract text from receipt images and save it into structured documents.",
"This is usually split into two sub-tasks: text localization and optical character recognition (OCR).",
"Most existing OCR models only focus on the cropped text instance images, which require the bounding box information provided by a text region detection model.",
"Introducing an additional detector to identify the text instance images in advance is inefficient, however instance-level OCR models have very low accuracy when processing the whole image for the document-level OCR, such as receipt images containing multiple text lines arranged in various layouts.",
"To this end, we propose a localization-free document-level OCR model for transcribing all the characters in a receipt image into an ordered sequence end-to-end.",
"Specifically, we finetune the pretrained Transformer-based instance-level model TrOCR with randomly cropped image chunks, and gradually increase the image chunk size to generalize the recognition ability from instance images to full-page images.",
"In our experiments on the SROIE receipt OCR dataset, the model finetuned with our strategy achieved 64.4 F1-score and a 22.8% character error rates (CER) on the word-level and character-level metrics, respectively, which outperforms the baseline results with 48.5 F1-score and 50.6% CER.",
"The best model, which splits the full image into 15 equally sized chunks, gives 87.8 F1-score and 4.98% CER with minimal additional pre or post-processing of the output.",
"Moreover, the characters in the generated document-level sequences are arranged in the reading order, which is practical for real-world applications."
],
[
"Introduction",
"Scanned receipt digitization aims at documenting text in receipts.",
"This process was formally defined as Scanned Receipts OCR and Information Extraction (SROIE) task in the ICDAR 2019 competition [1], which provides a benchmark dataset, called SROIE, and splits the task into localization and recognition sub-tasks.",
"Existing works focus on either the text detection [2], [3], [4] or character recognition [5], [6], [7], but there is no document-level OCR model for transcribing receipt images end-to-end without text localization.",
"Introducing an additional detector to identify the text instance images in advance is inefficient, and it requires post-processing to combine the instance-level sequences to obtain a document-level transcription.",
"Figure: Our proposed step-by-step finetuning strategy for adapting TrOCR to the document-level OCR.",
"“æ” indicates the separator token for dividing characters in each text-line.Recently, a pretrained Transformer-based OCR model TrOCR [8] was proposed which achieved state-of-the-art performance on the SROIE dataset.",
"However, TrOCR was trained using only text instance images, which makes its adaptation to document-level OCR challenging because of the great variation in input images, which include many more characters and more lines than the single line it was trained on.",
"To explore the potential of TrOCR for end-to-end document-level OCR without text localization, we propose a step-by-step finetuning strategy as shown in Fig.REF .",
"Specifically, we first randomly split the whole receipt image into image chunks whose size is closer to the original text instance images.",
"These are then used to finetune the TrOCR model for what we call “chunk-level” OCR, and we gradually increase the chunk size to introduce more difficult chunks containing more lines and characters.",
"Every time we use the model finetuned in the previous step to initialize the current model.",
"Finally, we train using the entire, un-chunked, receipt images to achieve document-level OCR.",
"The intuition behind this strategy is to progressively get the model to generalize its recognition ability to larger images.",
"We define the order of characters in the chunk-level label as top-left to bottom-right, and propose a method to construct the reference label for each chunk automatically.",
"We also include a text-line separator token in the constructed reference label, which aims to encode the line segmentation for the layout learning and also facilitate post-processing the model output into lines during inference.",
"We conduct experiments on the SROIE dataset for the document-level OCR.",
"We cannot directly compare our results with those in the literature, since they only focus on instance-level OCR.",
"Thus we construct two baseline finetuning methods for comparison.",
"The experimental results show that our method achieves better performance than the two baselines on both word-level and character-level metrics.",
"The main contributions of our work are summarized as follows: We propose a method to construct chunk-level reference labels automatically using only annotated instance-level labels.",
"This method can be easily applied to other OCR datasets.",
"The generated characters are arranged in the reading order with a unique text-line separator token for post-processing, which is practical for real-world applications.",
"We propose a step-by-step finetuning strategy to adapt TrOCR for document-level OCR, which can process the entire image and achieve competitive performance.",
"Figure: Finetuning the TrOCR model for chunk-level OCR."
],
[
"Related Works",
"CNN-Based OCR models Based on our target, we mainly focus on end-to-end OCR models which include two modules for detection and recognition, respectively.",
"Previous research treats these two problems independently [9], [10], [11] by combining a text detector with a recognition model.",
"Since the interaction between the two modules can complement each other to avoid the error propagation, recent research [12], [13], [14], [15] jointly optimizes the two modules by sharing the intermediate results.",
"However, all these models include a text detection module explicitly or implicitly whereas we aim to perform document-level OCR without extracting intermediate text regions.",
"Transformer-Based OCR models Recently, a Transformer-based model TrOCR [8] was proposed for the receipt OCR task.",
"TrOCR incorporates a vision transformer and a language model in its encoder-decoder architecture, which was trained on large-scale printed and handwritten OCR data for robust text recognition.",
"Several Transformer-based models were also proposed focusing on hand written text OCR [16], [17] or scene text OCR [18], [19], [20].",
"However, all these models are restricted to the transcription of the cropped text-line or instance images instead of the full images."
],
[
"TrOCR Model Architecture",
"We will first introduce TrOCR as the backbone model for our finetuning work.",
"TrOCR is a Transformer-based OCR model which consists of a pretrained vision Transformer encoder BEiT [21] and a pretrained language model decoder RoBERTa [22] as shown in Fig.REF .",
"To recognize characters in the cropped text instance images, the images are first resized into square boxes of size $384\\times 384$ pixels and then flattened into a sequence of 576 patches, which are then encoded by BEiT into high-level representations and decoded by RoBERTa into corresponding characters step-by-step.",
"TrOCR was pretrained with 684M textlines in English, which ensures the robust recognition ability for characters in various formats.",
"However, TrOCR can only handle cropped text instance images, which leads to the underperformance when finetuning the model directly using whole receipt images.",
"Therefore, a better finetuning strategy to adapt the model recognition from cropped images to full images is required."
],
[
"Chunk-Level OCR Finetuning",
"To leverage TrOCR for whole image text recognition, we propose to finetune the model with image chunks extracted from the whole images for chunk-level OCR.",
"As Fig.REF shows, our finetuning pipeline contains three modules: (i) randomly sample image chunks from the full receipt image; (ii) construct the label for the sampled chunks; and (iii) finetune the model with chunks and corresponding chunk labels.",
"We will introduce the random sampling and the finetuning process in this section.",
"Randomly sampling image chunks from whole images aims to obtain larger images for training the model.",
"The reason for introducing randomness is that we hope to extract different chunks from each image across different epochs, which can increase data diversity and improve model generalization.",
"Formally, to sample a chunk from a receipt image of width $W$ and height $H$ , we first set a hyper-parameter $L$ for the chunk numbers that we will split a receipt image into, then define the image chunk size whose width $w$ is always the same as the corresponding image width $W$ and height $h$ equal to $H/L$ .",
"With the determined chunk size, we randomly select an image chunk starting point $s$ on the y-axis whose value ranges from 0 to $H-H/L$ , and crop the chunk between $s$ and $s+h$ on the y-axis.",
"Last, we repeat the sampling $N$ times to extract multiple chunks from each receipt image.",
"With the randomly cropped chunk images, we can obtain corresponding chunk-level labels with the method that will be described in the next section, and use the chunks and labels for chunk-level OCR finetuning.",
"The chunk size determines the contents in the chunk which in turn affect the learning difficulty.",
"Therefore we feed the model with smaller chunks whose resolution is similar to the text instance images at the early stage, and increase the chunk size gradually to finetune the model with progressively more difficult image chunks.",
"Concretely, we start by setting $L$ with a large value which produces smaller chunks, then after the training is finished using the current $L$ , we start the next stage training with smaller $L$ , and every time we use the best model checkpoint from the previous training step to initialize the model in the current step.",
"Finally, we increase the chunk size to the full image size (L=1) for document-level OCR.",
"We use the notation Growing-Finetune for our proposed method.",
"[t] [1] Input: overlapping threshold $\\theta $ , merging threshold $\\delta $ , chunk numbers to split L, chunk numbers to sample N Output: chunk set $X$ and chunk label set $Y$ Data: input image $\\textit {V} = \\textit {W} \\times \\textit {H}$ Init $X\\leftarrow \\lbrace \\rbrace $ , $Y\\leftarrow \\lbrace \\rbrace $ , $n\\leftarrow 1$ , $h \\leftarrow H/L$ $n$ = 1 to $N$ Randomly select a starting point $s \\in (0,H-H/L)$ Chunk $x$ between $s$ and $s+h$ Gather boxes $B=\\lbrace b_{i}\\rbrace _{i=1}^{I}$ and labels $R=\\lbrace r_{i}\\rbrace _{i=1}^{I}$ Sort boxes and labels by the y-axis values of anchors $b_{i}$ in B filtering boxes $overlap(b_{i},x) \\le \\theta $ $B \\leftarrow B-b_{i}$ Init merged boxes $B^{^{\\prime }} \\leftarrow \\lbrace \\rbrace $ and labels $T^{^{\\prime }} \\leftarrow \\lbrace \\rbrace $ $b_{i}$ in $B$ merging text-line labels $b_{i} \\notin B{^{\\prime }}$ Init text-line label set $t_{i} \\leftarrow \\lbrace r_{i}\\rbrace $ $b_{j}$ in $B-b_{i}$ $height\\_overlap(b_{i},b_{j}) \\ge \\delta $ $t_{i} \\leftarrow $ $t_{i} + r_{j}$ Sort $t_{i}$ based on x-axis values of anchors Concat labels in $t_{i}$ with whitespace $T^{^{\\prime }} \\leftarrow T^{^{\\prime }} + t_{i}$ $B^{^{\\prime }} \\leftarrow B^{^{\\prime }} + b_{i}$ $y \\leftarrow $ Concat labels in $T^{^{\\prime }}$ with the separator æ $X \\leftarrow X + x$ $Y \\leftarrow Y + y$ return $X,Y$ Chunk-Level Label Construction"
],
[
"Chunk-Level Label Construction",
"Chunks usually contain multiple text-lines and some characters are split vertically in half as shown in Fig.REF .",
"Therefore, to construct labels for randomly cropped chunks, a definition for the character order in the label and which split characters should be included in the label is required.",
"In this paper, we define the top-left to bottom-right reading order for characters in the chunk as the correct order, and set a overlapping threshold $\\theta $ to include characters with an overlapping rate larger than $\\theta $ to make sure no unrecognizable split characters are mistakenly included in the label.",
"Based on these definitions, we construct the chunk-level label with the use of annotated texts and corresponding bounding boxes information as shown in Algorithm REF .",
"We first gather the $I$ bounding boxes in the randomly cropped image chunk and filter out boxes with overlapping rate less than $\\theta $ , and sort boxes as well as labels based on the y-axis values of the left-upper anchors of boxes.",
"To align boxes horizontally in the same line, we define a merging threshold $\\delta $ to merge boxes that overlap vertically over the threshold into text-line level labels, and sort the boxes in each group based on the x-axis values of the left-upper anchors from left to right.",
"Lastly, we concatenate labels of boxes in each group with white space as text-line labels, which are concatenated with text-line label separator token “æ\".",
"This character does not appear in any of the receipts and is used to encode the line segmentation for the receipt layout learning.",
"To clarify, we only use the boxes for the label construction and no box is used during inference."
],
[
"Settings",
"Dataset We use the SROIE (Scanned Receipts OCR and Information Extraction) dataset from the ICDAR 2019 competition for our experiments.",
"The SROIE OCR task focuses on text recognition of cropped receipt images.",
"There are 626 and 361 images in the training and testing set, respectively, which are annotated with ground truth bounding boxes and corresponding texts, and we keep the train and test data split the same as the TrOCR setting.",
"We randomly sample 60 images from the training set to construct the validation set.",
"For the chunk-level OCR setting, the training chunks are randomly sampled from each image in the training set, while the validation and testing chunks are sequentially cropped from each image to ensure the full image area is covered for the evaluation.",
"All the labels are obtained by the chunk-level label construction method in Section 3.3.",
"Hyper-parameters We use the trocr-base-printedhttps://huggingface.co/microsoft/trocr-base-printed model checkpoint finetuned with the original SROIE training data from Hugging Face [23] for our own finetuning experiments.",
"The boxes filtering threshold $\\theta $ , text-line merging threshold $\\delta $ and sampled chunk number $N$ for training data are 0.3, 0.5, and 20, respectively, which are determined by results on the validation set.",
"For the split chunk number $L$ , we first compute the distribution of text-line numbers of training images as shown in Fig.REF , then use the median number 30 as the initial value.",
"By decreasing the value to 15, 7, 4, 2 and 1, we train the model with increasingly larger chunks at subsequent stages.",
"We set the beam search size as 5 for the text generation.",
"Evaluation Metrics We use two evaluation metrics adopted in OCR tasks to evaluate the performance: Word-Level precision, recall, F1 (Word-Level PRF) and Character Error Rate (CER).",
"The Word-Level PRF focuses on correctly matched words in the hypothesis without considering the word order, while the CER focuses on the character-level substitutions, deletions, and insertions as: Figure: Distribution of the number of text lines in receipt images.$CER = (S + D + I) / (S+D+C)$ where $S$ , $D$ , $I$ , and $C$ are the number of substitutions, deletions, insertions, and correct characters, respectively."
],
[
"Baselines",
"Direct-Finetune This is a straight-forward method that uses the whole receipt image for the document-level OCR directly.",
"Concretely, we resize the whole image as input and split it into patches, and finetune the model with patches and document-level labels end-to-end.",
"Resizing the whole image to $384\\times 384$ pixels will cause a large variation in the average resolution of each character for the recognition.",
"Concatenate-Finetune This is a compromise strategy to keep the input resolution closer to the original setting, which splits the image equally into several chunks, and embeds each chunk into sequences that are concatenated in the temporal dimension to construct document-level inputs.",
"Since the linearly increased input length brings higher computational cost, we restrict the split number to 4 and interpolate the position embeddings to adjust for longer inputs.",
"Figure: Error analysis on the generated full-page receipt OCR texts.",
"Characters in the red, blue, and green colors represent the substitution, insertion, and deletion errors, respectively.",
"Black characters are the corresponding correct content.Figure: Generated Texts with Different Beam Search Size.",
"The underlined texts are deleted contents with the default beam size."
],
[
"Quantitative Analysis",
"Performance Comparison We first compare the performance of our method and two baselines.",
"As the results in Table REF show, our method outperforms the two baselines on all metrics, which indicates that the direct finetuning methods do not work well.",
"Moreover, the worst performance with Concatenate-Finetune also highlights the increased computational complexity on the longer inputs.",
"Compared with TrOCR, our results are worse since the evaluation for the longer and ordered document-level sequence is more strict, which also reveals the efficiency of our method to achieve good results without using the bounding box information for the localization.",
"Table: Model performance comparison with different finetuning strategies for whole document OCR.Table: Model performance with different chunk numbers per image.",
"Larger number L indicates smaller chunks on average.Chunk Size Analysis We then analyzed how the model performance changes according to the input chunk size.",
"Since the chunk size is determined by the chunk number $L$ , we test the performance under different $L$ and the variation is shown in Table REF .",
"We observed that performance improved with increased chunk size at first, since the larger chunk will introduce fewer split characters which reduces errors caused by the insertion and deletion, but the performance decreases when the size becomes larger, since images with more content and a longer target sequence significantly increases the learning difficulty, especially for the half-page and full-page settings which doubles the error rates compared with their previous settings.",
"The optimal trade-off is achieved at $L=15$ , where each chunk contains roughly 2 text-lines, on average.",
"It is worth noting that if the text detection in the baseline TrOCR misses 4% of ground truth regions, our best method results would be better than TrOCR.",
"Ablation Study Lastly, we analyze the influence of two main factors in our strategy with the ablation study as shown in Table REF .",
"By removing the separator token, we noticed the CER performance drop is minor.",
"While by sampling chunks sequentially without randomness, we found the performance dropped significantly on all metrics, which indicates the importance of randomness in bringing more diverse data and larger data size among all epochs for the convergence.",
"This may also be thought of as a form of data augmentation.",
"Table: Ablation study for the factor influence analysis."
],
[
"Qualitative Analysis",
"To understand the quality of generated document-level texts, we conduct the error analysis on generated full-page OCR texts with $L=1$ as shown in Fig.REF .",
"For the first example, our model generates a much longer sequence than instance-level TrOCR and achieved 0.13% error rate where only one character was mistakenly substituted.",
"This indicates the model can generate high-quality texts for full-image receipts.",
"On the other hand, for the second example, our model achieved a 14.82% error rate and made substitution errors on numbers and insertion or deletion errors on sequences.",
"We hypothesize that the insertions are caused by memorization during model training, and the deletions are caused by the small beam size.",
"We generate texts for the second example with the beam size 1 and 7, respectively, and the results in Fig.REF show that decoding with a larger beam size can successfully generate the missing contents which supports our hypothesis."
],
[
"Conclusions and Future Work",
"In this paper, we propose a step-by-step finetuning strategy and an automatic label construction method for adapting TrOCR to perform document-level receipt image OCR.",
"The finetuned model can handle the full image input and transcribe all characters into long ordered sequences.",
"Moreover, our method outperforms other straight-forward finetuning baselines, which indicates the efficiency of finetuning with image chunks of increasing size.",
"We observed the trade-off between performance and chunk size, and learned the importance of random sampling from the ablation study.",
"We expect this study can serve as a baseline for future studies that concentrate on document-level receipt OCR, using more recent Transformer models such as Donut [24]."
]
] | 2212.05525 |
[
[
"Crossover from exciton polarons to trions in doped two-dimensional\n semiconductors at finite temperature"
],
[
"Abstract We study systematically the role of temperature in the optical response of doped two-dimensional semiconductors.",
"By making use of a finite-temperature Fermi-polaron theory, we reveal a crossover from a quantum-degenerate regime with well-defined polaron quasiparticles to an incoherent regime at high temperature or low doping where the lowest energy \"attractive\" polaron quasiparticle is destroyed, becoming subsumed into a broad trion-hole continuum.",
"We demonstrate that the crossover is accompanied by significant qualitative changes in both absorption and photoluminescence.",
"In particular, with increasing temperature (or decreasing doping), the emission profile of the attractive branch evolves from a symmetric Lorentzian to an asymmetric peak with an exponential tail involving trions and recoil electrons at finite momentum.",
"We discuss the effect of temperature on the coupling to light for structures embedded into a microcavity, and we show that there can exist well-defined polariton quasiparticles even when the exciton-polaron quasiparticle has been destroyed, where the transition from weak to strong light-matter coupling can be explained in terms of the polaron linewidths and spectral weights."
],
[
"Introduction",
"The notion of a quantum impurity, where a particle is dressed by excitations of a quantum gas, was first introduced by Landau in 1933 [1] to describe the behavior of conduction electrons in a dielectric medium.",
"The properties of this dressed impurity (or polaron), such as mobility or effective mass, are changed with respect to those of an isolated impurity, leading to strong modifications of, e.g., electrical and thermal transport.",
"Despite nearly a century of work, the polaron problem continues to attract significant interest, with noticeable realizations in ultracold atomic gases [2], [3], [4], $^3$ He impurities in $^4$ He [5], and the Kondo effect generated by localized magnetic impurities in a metal [6], just to cite a few examples.",
"Recently, the absorption and emission spectra of doped two-dimensional (2D) transition metal dicalchogenide (TMD) monolayers [7], [8], [9], [10], [11], [12] have been interpreted in terms of a Fermi-polaron model [13], [14], [15], [16], [17], [18].",
"Here, the optically generated exciton is dressed by excitations of a 2D Fermi gas of charge carriers (electrons or holes) induced by either gating or natural doping of the TMD monolayer.",
"This leads to the formation of “attractive” and “repulsive” Fermi polarons, where the exciton attracts or repels the surrounding charge carriers, respectively.",
"Furthermore, when such a TMD monolayer is embedded in a microcavity [7], [19], [20], [9], [21], the strong coupling between light and matter can lead to the formation of Fermi polaron polaritons.",
"The intense recent activity on this topic is motivated by several features of exciton-polaron quasiparticles.",
"The dressing of polaritons by a Fermi gas can boost their pairwise interactions or non-linearities [22], [20], which can potentially be exploited to reach the polariton blockade regime [23].",
"Another interesting aspect of this configuration is the possibility of manipulating dressed excitons or polaritons by external electric and magnetic fields [24], [25], [26], [27].",
"Finally, exciton polarons can be used as a probe of strongly correlated states of electrons, such as Wigner crystals in a TMD monolayer [28], [29], quantum Hall physics in TMD monolayers [26], fractional quantum Hall states in proximal graphene layers [30], and correlated Mott states of electrons in a Moiré superlattice [31].",
"The theoretical analysis of the Fermi polaron in 2D semiconductors has so far mainly focused on the zero-temperature limit While the results in Refs.",
"[74], [15] have been derived at finite temperature, the former only considered infinite exciton mass, while the latter did not analyze the consequences of temperature on the polaron description..",
"However, a natural question to ask is how the system changes with temperature, since this has been shown to strongly modify the nature of the Fermi polaron in cold-atom experiments [33].",
"In particular, there is a competing picture to that of the Fermi polaron based on few-body complexes such as excitons and trions, which provides a description for the photoluminescence of the attractive branch in doped semiconductors at finite temperature [34], [35], [36].",
"In this paper, we use a finite-temperature variational approach developed in the context of cold atoms [37] to reveal the important role that temperature plays in the exciton-polaron problem.",
"In particular, we demonstrate that, when the temperature becomes large compared to the Fermi energy of charge carriers, the attractive Fermi polaron merges with a continuum of states comprised of a bound trion and unbound Fermi-sea-hole—the so-called trion-hole continuum.",
"Here, the attractive branch ceases to be a well-defined quasiparticle and it crosses over into an incoherent regime dominated by the trion-hole continuum.",
"We discuss the implications of this crossover for the existence of polariton quasiparticles when the semiconductor is strongly coupled to light in a microcavity.",
"By contrast, we find that temperature does not qualitatively change the nature of the repulsive branch at typical doping levels, only its polaron properties.",
"We find that the destruction of the attractive polaron quasiparticle has little effect on the energy and spectral weight, but it strongly modifies the linewidth and the overall shape of the attractive peak in both absorption and photoluminescence.",
"In particular, we observe that the attractive branch evolves from a symmetric Lorentzian shape to a strongly asymmetric profile with an exponential tail below the trion energy.",
"In the latter trion-dominated regime, our theory becomes perturbatively exact since it corresponds to the lowest order term in a quantum virial expansion [38].",
"Note that, even though our model is formulated to describe doped 2D semiconductors, our results can be easily generalized to 2D atomic Fermi gases [39], [40], [41], and they can straightforwardly be extended to the three-dimensional case.",
"The paper is organized as follows: In Sec.",
"we introduce the model describing excitons and electrons in a doped TMD monolayer.",
"In Sec.",
"we describe the variational approach for impurities at finite temperature, we relate it to the $T$ -matrix approach, and we define absorption and photoluminescence.",
"In Sec.",
"we illustrate the results for optical absorption and photoluminescence, comparing the case of finite temperature against the zero temperature limit.",
"We also carry out a perturbatively exact quantum virial expansion to relate the results of this work with those of the accompanying paper [38].",
"In Sec.",
"we extend the formalism and results to the case where the doped TMD monolayer is embedded into a microcavity to allow for strong light-matter coupling.",
"Finally, in Sec.",
"we conclude and present future perspectives of this work."
],
[
"Model",
"We consider the following model Hamiltonian describing a doped 2D semiconductor such as a TMD monolayer: H = H0 + H0X +Hint , H0= k (k -)c†kck , H0X= k Xkx†kxk , Hint= -vAkk'qx†k+qc†k'-q ck'xk , where $\\mathcal {A}$ is the system area (throughout this paper we set $\\hbar = k_B= 1$ ).",
"The Hamiltonian $\\hat{H}_0$ describes the fermionic medium-only part, i.e., a Fermi gas of either electrons or holes, in terms of the fermionic operator $\\hat{c}^{\\dag }_{{\\bf k}}$ that creates a charge carrier with mass $m$ and dispersion $\\epsilon _{{\\bf k}}= |{\\bf k}|^2/2m \\equiv k^2/2m$ .",
"For concreteness, we consider the specific case where the charge carriers are electrons, but our results can be trivially extended to the case of hole doping.",
"For simplicity, we treat electrons as non-interacting, an assumption which becomes exact in the high-temperature, low-doping limit [38].",
"We furthermore ignore the spin/valley degree of freedom such that all the electrons are spin polarized but distinguishable from the electron within the exciton (see Ref.",
"[42] for a theoretical analysis of the case where the charge carriers are indistinguishable).",
"This minimal model is reasonable for describing the polaron and trion physics in TMD monolayers, such as MoSe$_2$ monolayers [7], [14], [18].",
"The medium is described within the grand canonical ensemble, where the chemical potential $\\mu $ is related to the excess electron density $n=N/\\mathcal {A}$ by = T (eEF -1 ), EF =2 m n , where $E_F$ is the Fermi energy and $\\beta = T^{-1}$ the inverse temperature.",
"While we use a grand canonical formalism for the medium, it is convenient to consider only a single excitonic impurity in the canonical formalism [37], as in $\\hat{H}_{0X}$ .",
"Formally, this corresponds to considering uncorrelated excitons, and this description is thus appropriate for a low density of excitons, i.e., the quantum impurity limit.",
"Following previous works [7], [14], [43], we treat the exciton as a structureless boson, with corresponding creation operator $\\hat{x}^{\\dag }_{{\\bf k}}$ , mass $m_X$ and free-particle dispersion $\\epsilon _{X{\\bf k}}=k^2/2m_X$ .",
"This is because in TMD monolayers the optically generated exciton is tightly bound and its binding energy is larger than all other relevant energy scales of the problem.",
"Note that, throughout this paper, energies are measured with respect to that of the exciton at rest.",
"The $H_{int}$ term () describes the short-range attractive interaction between electrons and excitons, which can be approximated as a contact interaction [44], [18] with coupling strength $v>0$ up to a high-momentum cutoff $\\Lambda $ .",
"The renormalization of such an interaction can be carried out by relating the bare parameters to the trion binding energy $\\varepsilon _T$ : $\\frac{1}{v} = \\frac{1}{\\mathcal {A}}\\sum _{{\\bf k}}^{\\Lambda } \\frac{1}{\\varepsilon _T+\\epsilon _{X{\\bf k}} + \\epsilon _{{\\bf k}}} \\; .$ Note that all our results are independent of cutoff since we formally take $\\Lambda \\rightarrow \\infty $ .",
"Our formalism is applicable to a general quantum impurity in a 2D Fermi gas, including both ultracold atomic gases and doped semiconductors.",
"To be concrete, in the following we will consider the experimentally relevant case of electron-doped MoSe$_2$ monolayers, where $\\varepsilon _T\\simeq 25$ meV [7], [12] and $m_X/m=2.05$ [45].",
"In this system one can readily achieve doping densities $E_F$ in the range of 0–40 meV [7]."
],
[
"Finite-temperature ansatz",
"At finite temperature, one can employ the variational approach for impurity dynamics developed in Ref.",
"[37] in the context of the Fermi polaron problem in ultracold atomic gases.",
"This variational approach has been successfully used to model dynamical probes such as Ramsey spectroscopy [46], [37] and Rabi oscillations [47], [48], as well as static thermodynamic properties such as the impurity contact [33], [49].",
"For completeness, we review the approach here, formulating it for the 2D semiconductor problem.",
"We consider the case of zero impurity momentum, relevant for evaluating absorption and emission.",
"The generalization to finite impurity momentum is discussed in Appendix .",
"The starting point of the variational approach [37] is the time-dependent impurity operator that approximates the exact operator in the Heisenberg picture, $\\hat{x}_{{\\bf 0}}^{}(t) = e^{i\\hat{H}t} \\hat{x}_{{\\bf 0}}^{} e^{-i\\hat{H}t}$ .",
"We choose the form $\\hat{x}_{{\\bf 0}}^{}(t) \\simeq \\varphi _{0}^{}(t) \\hat{x}_{{\\bf 0}}^{} + \\frac{1}{\\mathcal {A}} \\sum _{{\\bf k},{\\bf q}} \\varphi _{{\\bf k}{\\bf q}}^{}(t) \\hat{c}_{{\\bf q}}^{\\dag } \\hat{c}_{{\\bf k}}^{} \\hat{x}_{{\\bf q}-{\\bf k}}^{}\\; ,$ which is written in terms of the time-dependent variational coefficients $\\varphi _{0}^{}(t)$ and $\\varphi _{{\\bf k}{\\bf q}}^{}(t)$ .",
"The truncated form of this operator is similar to that of the Chevy ansatz [50] employed for the zero-temperature state, and describes an impurity dressed by a single excitation of the fermionic medium.",
"The time-dependent exciton operator (REF ) does not coincide with the exact solution of the Heisenberg equation of motion, and thus we determine the variational coefficients by minimizing the error function $\\Delta (t) = \\langle \\hat{e}(t) \\hat{e}^{\\dag }(t) \\rangle _{\\beta } \\equiv [\\hat{\\rho }_0 \\hat{e}(t) \\hat{e}^{\\dag }(t)]\\; ,$ with respect to $\\varphi _{0}^{*}(t)$ and $\\varphi _{{\\bf k}{\\bf q}}^{*}(t)$ .",
"Here, the trace is over medium-only states, $\\hat{e}(t) = i\\partial _t \\hat{x}_{{\\bf 0}}^{}(t) - [\\hat{x}_{{\\bf 0}}^{}(t),\\hat{H}]$ is an error operator, and $\\hat{\\rho }_0= e^{-\\beta \\hat{H}_{0}}/Z_0$ is the medium-only density matrix with $Z_0=\\text{Tr}[e^{-\\beta \\hat{H}_{0}}]$ the medium partition function in the grand canonical ensemble.",
"By considering the stationary solutions, $\\varphi _{0}^{}(t)= \\varphi _{0}^{} e^{-i E t}$ and $\\varphi _{{\\bf k}{\\bf q}}^{}(t) = \\varphi _{{\\bf k}{\\bf q}}^{} e^{-i E t}$ , we obtain the following eigenvalue problem: E0 = -vA2k,q fq(1-fk) kq Ekq = EXkq kq - v0 -vAk'(1-fk') k'q .",
"Here, $E_{X{\\bf k}{\\bf q}} = \\epsilon _{X{\\bf q}-{\\bf k}} + \\epsilon _{\\bf k}- \\epsilon _{\\bf q}$ and we have used the Fermi-Dirac distribution for the electron occupation, $\\langle \\hat{c}^{\\dag }_{{\\bf k}} \\hat{c}_{{\\bf k}}^{} \\rangle _{\\beta }=f_{\\bf k}= [e^{\\beta (\\epsilon _{{\\bf k}}-\\mu )}+1]^{-1}$ , and for the hole occupation, $\\langle \\hat{c}_{{\\bf k}}^{} \\hat{c}^{\\dag }_{{\\bf k}}\\rangle _{\\beta }=1-f_{\\bf k}$ .",
"Note that we have dropped terms that vanish when $\\Lambda \\rightarrow \\infty $ ; for instance, since $v\\sim 1/\\ln \\Lambda \\rightarrow 0$ , terms like $\\frac{v}{\\mathcal {A}}\\sum _{{\\bf q}^{\\prime }} f_{{\\bf q}^{\\prime }} \\varphi _{{\\bf k}{\\bf q}^{\\prime }}^{}$ also go to zero as $\\Lambda \\rightarrow \\infty $ .",
"The set of equations in () constitutes an eigenvalue problem which can be solved to give a set of eigenvalues $E^{(n)}$ and associated eigenvectors $\\varphi _{0}^{(n)}$ and $\\varphi _{{\\bf k}{\\bf q}}^{(n)}$ , with $n$ a discrete index.",
"We require that the corresponding stationary operators $\\hat{x}_{{\\bf 0}}^{(n)} = \\varphi _{0}^{(n)} \\hat{x}_{{\\bf 0}}^{} + \\frac{1}{\\mathcal {A}} \\sum _{{\\bf k},{\\bf q}} \\varphi _{{\\bf k}{\\bf q}}^{(n)} \\hat{c}_{{\\bf q}}^{\\dag } \\hat{c}_{{\\bf k}}^{} \\hat{x}_{{\\bf q}-{\\bf k}}^{}\\; ,$ are orthonormal under a thermal average, $\\langle \\hat{x}_{{\\bf 0}}^{(n)} \\hat{x}_{{\\bf 0}}^{(m)\\dag } \\rangle _{\\beta } = \\delta _{n,m}$ , implying that $\\varphi _{0}^{(n)} \\varphi _{0}^{(m)*} + \\frac{1}{\\mathcal {A}^2}\\sum _{\\textbf {k},\\textbf {q}} f_{\\bf q}(1-f_{\\bf k}) \\varphi _{{\\bf k}{\\bf q}}^{(n)} \\varphi _{{\\bf k}{\\bf q}}^{(m)*} = \\delta _{n,m} \\; .$ The stationary operators thus form a complete basis within which we can expand the approximate impurity operator (REF ), giving $\\hat{x}_{{\\bf 0}}^{}(t) = \\sum _n \\varphi _{0}^{(n)*} \\hat{x}_{{\\bf 0}}^{(n)} e^{-i E^{(n)} t}\\; ,$ where $\\varphi _{0}^{(n)*} = \\langle \\hat{x}_{{\\bf 0}}^{} \\hat{x}_{{\\bf 0}}^{(n)\\dag }\\rangle _{\\beta }$ and where we have used the boundary condition $\\hat{x}_{{\\bf 0}}^{}(0) = \\hat{x}_{{\\bf 0}}^{}$ .",
"The exciton retarded Green's function in the time domain, $\\mathcal {G}_{X} (t) = -i \\theta (t) \\langle [\\hat{x}_{{\\bf 0}}^{}(t), \\hat{x}_{{\\bf 0}}^{\\dag }] \\rangle _{\\beta }=-i \\theta (t) \\langle \\hat{x}_{{\\bf 0}}^{}(t) \\hat{x}_{{\\bf 0}}^{\\dag } \\rangle _{\\beta },$ can be evaluated approximately within the variational ansatz (REF ) by using Eq.",
"(REF ).",
"By taking the Fourier transform into the frequency domain we obtain: $G_{X} (\\omega +i0) = \\sum _n\\frac{|\\varphi _0^{(n)}|^2}{\\omega -E^{(n)}+i0} \\; ,$ where the small imaginary part originates from the Heaviside function $ \\theta (t)$ in the retarded Green's function $\\mathcal {G}_{X} (t)$ ."
],
[
"Exciton self-energy and $T$ matrix",
"As discussed above, solving the eigenvalue problem () allows us to evaluate the exciton Green's function.",
"It turns out that it is numerically convenient to instead consider the exciton self-energy $\\Sigma (\\omega )$ , which is related to the Green's function via $G_{X}(\\omega ) =\\frac{1}{\\omega - \\Sigma (\\omega )}\\;.$ The expression for the exciton self-energy can be derived by manipulating the eigenvalue problem () [37], following the same procedure valid at zero temperature [51] — for completeness we describe this in Appendix .",
"In this way, the exciton self-energy reads $\\Sigma (\\omega )=\\frac{1}{\\mathcal {A}}\\sum _{{\\bf q}} f_{\\bf q}\\mathcal {T}({\\bf q}, \\omega +\\epsilon _{{\\bf q}})\\; ,$ where the inverse of the $T$ matrix is defined as $\\mathcal {T}^{-1}_{}({\\bf q}, \\omega ) =-\\frac{1}{v} - \\frac{1}{\\mathcal {A}} \\sum _{{\\bf k}} \\frac{1-f_{\\bf k}}{\\omega - \\epsilon _{{\\bf k}} - \\epsilon _{X{\\bf k}-{\\bf q}} + i0}\\; .$ The same expression (REF ) can also be derived by using a diagrammatic expansion within the ladder approximation [51], [37].",
"Thus, our variational approach provides an additional theoretical foundation for the ladder diagrams.",
"It is profitable to separate the vacuum contribution to the $T$ matrix, describing the electron-exciton scattering in the absence of a surrounding Fermi gas, from the many-body contribution.",
"To this end, we note that the logarithmic divergence of the second term in (REF ) cancels with that of the inverse contact interaction constant $v^{-1}$ (REF ), allowing the vacuum contribution $\\mathcal {T}_0$ to be calculated analytically [3]: T-1(q, ) = T-10(q, )-mb(q, ) T-10(q, ) = mr2 (-T- q22mT+i0) mb(q, ) = -1A k fk- k - Xk-q + i0 , where $m_T=m+m_X$ is the trion mass and $m_r=mm_X/m_T$ is the exciton-electron reduced mass.",
"Note that the vacuum $T$ matrix (REF ) has a pole at the bound state, i.e., the trion energy, $\\omega =-\\varepsilon _T+\\epsilon _{T{\\bf q}}$ , where $\\epsilon _{T{\\bf q}} = \\frac{q^2}{2m_T}$ is the trion kinetic energy (for a summary of the trion properties at finite momentum and finite doping, see Appendix ).",
"The many-body contribution $\\Pi _{mb}({\\bf q}, \\omega )$ to the $T$ matrix has to be evaluated numerically—for details on the numerical procedure see Appendix ."
],
[
"Absorption and photoluminescence",
"The optical absorption coincides with the exciton spectral function (up to a frequency-independent prefactor): $A(\\omega )=-\\frac{1}{\\pi }\\Im G_X(\\omega +i0) \\; .$ Indeed, in the linear-response regime, the spectral function is equivalent to the transfer rate from an initial state $|n\\rangle $ containing no excitons (the impurity vacuum) to a final state $|\\nu \\rangle $ containing a single exciton.",
"Here, the impurity vacuum and single-impurity states are eigenstates of the Hamiltonian, i.e., $\\hat{H} |n\\rangle =\\hat{H}_{0} |n\\rangle = E_n |n\\rangle $ and $\\hat{H} |\\nu \\rangle = E_\\nu |\\nu \\rangle $ .",
"Using Fermi's golden rule, we have $A(\\omega ) = \\sum _{n,\\nu } \\langle n| \\hat{\\rho }_{0} |n \\rangle |\\langle \\nu | \\hat{x}_{{\\bf 0}}^\\dag |n \\rangle |^2 \\delta (E_{\\nu } - E_n - \\omega ) \\; ,$ where $\\hat{\\rho }_0$ is the medium-only partition function introduced above.",
"By using the completeness of the $\\lbrace |\\nu \\rangle \\rbrace $ basis, one can easily show that Eq.",
"(REF ) coincides with (REF ) — see Ref. [52].",
"Using the definition (REF ), it is now straightforward to show that the exciton optical absorption satisfies the sum-rule: $\\int _{-\\infty }^{\\infty }d\\omega \\, A(\\omega )=1\\;.$ In order to evaluate the photoluminescence, we instead consider the opposite situation, i.e., an initial state $|\\nu \\rangle $ containing the medium and the exciton, and a final state $|n\\rangle $ after the exciton has recombined to emit a photon.",
"Here, we have assumed that the exciton density is low enough such that each exciton can be treated individually.",
"The transfer rate at thermal equilibrium is then given by $P(\\omega ) = \\sum _{n,\\nu } \\langle \\nu | \\hat{\\rho } |\\nu \\rangle |\\langle n| \\hat{x}_{{\\bf 0}}^{} |\\nu \\rangle |^2 \\delta (E_{\\nu } - E_n - \\omega ) \\; ,$ where $\\hat{\\rho } = e^{-\\beta \\hat{H}}/Z_{int}$ is the density matrix associated with the interacting exciton and medium system () and $Z_{int} = \\sum _\\nu *{e^{-\\beta \\hat{H}}}{\\nu }$ the associated partition function.",
"It is straightforward to show that the photoluminescence satisfies the following sum rule $\\int _{-\\infty }^{\\infty }d\\omega \\,P(\\omega ) = \\text{Tr} [\\hat{\\rho } \\hat{x}_{{\\bf 0}}^{\\dag } \\hat{x}_{{\\bf 0}}^{}] \\; .$ Using the properties of the delta function, the absorption $A(\\omega )$ and photoluminscence $P(\\omega )$ can be related by a detailed balanced condition [52]: $P(\\omega ) = \\displaystyle \\frac{Z_0}{Z_{int}} e^{-\\beta \\omega } A(\\omega ) \\; .$ The thermodynamic, Boltzmann-type scaling between absorption and emission profiles (REF ) is also known as the Kubo-Martin-Schwinger relation [53], [54], the Kennard-Stepanov relation [55], [56], [57] or the van Roosbroeck-Shockley relation [58], depending on the context within which it has been studied, and it applies to a broad range of systems, including semiconductors [59], [60], [61].",
"It relies on the assumption that the population of excited states, here excitons, has thermalized at a temperature $T$ before the emission and that they are otherwise uncorrelated.",
"Note also that the thermalization temperature $T$ can be different from the system lattice (cryostat) temperature."
],
[
"Numerical implementation",
"Even though one only has to evaluate two momentum integrals to obtain the exciton Green's function in Eq.",
"(REF ), namely the integrals in Eqs.",
"(REF ) and (REF ), some comments about the numerical procedure are necessary.",
"For the optical absorption (REF ), the numerical convergence of the integrals is much improved by shifting the frequency to the complex plane, $\\omega \\mapsto \\omega + i\\eta $ .",
"Apart from helping with convergence, this shift provides a simplified description of the exciton's intrinsic broadening due to effects beyond those included in the Hamiltonian such as recombination and disorder.",
"In the following, we have used the typical value $\\eta =0.04\\varepsilon _T \\simeq 1$ meV.",
"Including $\\eta $ implies that the exciton spectral function decays as a Lorentzian at low and high energies.",
"However, one cannot evaluate the photoluminescence using this procedure.",
"Indeed, by using the detailed balance condition (REF ), the photoluminescence diverges at infinitely low frequencies if one uses a finite value of $\\eta $ to evaluate the absorption, since the Boltzmann occupation increases more rapidly than the Lorentzian decay of absorption.",
"This means that photoluminescence needs to be evaluated by first calculating absorption at $\\eta =0$ and then multiplying by the Boltzmann occupation.",
"The effects of the exciton intrinsic decay time can be re-introduced at the end of the calculation by convolving the photoluminescence with a Lorentzian profile with broadening $2\\eta $ .",
"This procedure is described in detail in Appendix .",
"Figure: (a) Spectral function A(ω)A(\\omega ) at zero temperature as a function of doping and energy.",
"Blue and purple dashed lines are respectively the attractive (AA) and repulsive (RR) polaron energies, while the dashed red lines are the boundaries of the trion-hole continuum E ± E_{\\pm } (see text).",
"(b) Spectral function A(ω)A(\\omega ) at temperature T=50T=50 K≃0.17ε T \\simeq 0.17 \\varepsilon _T.",
"Dashed lines are the zero-temperature energies as in panel (a) — for the trion-hole continuum we plot only the upper boundary E + E_+.",
"Solid lines are the attractive (blue) and repulsive (purple) branch energies at finite temperature.",
"(c,d) Doping dependence of the spectral weights ZZ and half linewidths Γ\\Gamma extracted from the spectral function (see Appendix for details of how these are determined) at T=0T=0 (dashed) and T=50T=50 K≃0.17ε T \\simeq 0.17 \\varepsilon _T (solid).",
"In panel (d), note that the constant value of Γ A \\Gamma _A at T=0T=0 coincides with the intrinsic broadening."
],
[
"Results",
"We now illustrate our results for both optical absorption and photoluminescence.",
"In order to stress the differences between the zero and finite temperature cases, let us first briefly summarize what is known about polarons at zero temperature."
],
[
"Zero temperature",
"Numerous theoretical works have so far analyzed the zero-temperature properties of an impurity interacting with a free Fermi gas, in the context of both semiconductor heterostructures [13], [7], [14], [15], [16], [17], [18] and ultracold atomic gases [2], [4].",
"The main polaron properties at $T=0$ are summarized in Fig.",
"REF .",
"In panel (a) we plot the doping and energy dependence of the spectral function.",
"One observes that the optical absorption is dominated by two polaron quasiparticle resonances: the attractive polaron branch at lower energy $E_{A}$ and the repulsive polaron branch at higher energy $E_R$ .",
"The energies $E_{A,R}$ are evaluated from the positions of the absorption peaks in Fig.",
"REF .",
"Here, both polaron branches are quasiparticles in the sense that $E_{A,R}$ satisfy the condition [62] $E_{A,R} = \\Re \\Sigma (E_{A,R}) \\; ,$ which coincides with a pole of the exciton Green's function when the corresponding imaginary part of the self-energy $\\Im \\Sigma (E_{A,R})$ becomes small.",
"Note that the repulsive branch eventually stops satisfying this condition when $E_F\\gtrsim 3\\varepsilon _T$ [63], and thus ceases to be a polaron quasiparticle.",
"This large doping regime is not analyzed in this work.",
"In the limit of vanishing doping, the attractive branch energy recovers the trion energy $E_A \\rightarrow -\\varepsilon _T$ , while the repulsive branch energy reduces to the exciton energy, $E_R \\rightarrow 0$ .",
"(We remind the reader that we measure energies with respect to that of the exciton at rest).",
"However, differently from the trion, whose energy blueshifts with doping due to its interactions with the surrounding electrons [see Ref.",
"[64] and Eq.",
"()], the attractive polaron branch redshifts with doping.",
"At the same time, the repulsive polaron branch blueshifts with doping.",
"Thus, at zero temperature, as soon as doping is finite, the attractive and repulsive polaron quasiparticles are no longer the trion and the exciton, respectively, even if they recover the corresponding energies at low doping.",
"Furthermore, within the single particle-hole-excitation ansatz (REF ), the effective mass of the attractive polaron [65], [66] does not evolve into the trion mass at low doping.",
"As far as the coupling to light of the polaron branches is concerned, at very low doping the repulsive branch retains all the spectral weight and the attractive branch is dark, as is also the case for the trion.",
"However, when doping increases, there is a transfer of oscillator strength from the repulsive to the attractive branch.",
"This is shown in Fig.",
"REF (c), where we plot the doping dependence of the spectral weights $Z_{A,R}$ for each branch.",
"At low doping $E_F \\ll \\varepsilon _T$ , $Z_A$ grows linearly with $E_F$ , which agrees with the doping dependence of the trion oscillator strength [43].",
"There are also important differences between the attractive and repulsive polaron branches at zero temperature, even though they both satisfy Eq.",
"(REF ) for values of doping up to $E_F \\sim \\varepsilon _T$ .",
"The attractive polaron is always a sharp resonance, with a Lorentzian broadening coinciding with the intrinsic broadening $2\\Gamma _A=2\\eta $ — see Fig.",
"REF (d).",
"By solving the eigenvalue problem () at zero temperature, one finds that the attractive branch energy coincides with the lowest eigenvalue energy, $E_A=E^{(n=1)}$ , which is separated from the energy of the excited states $n>1$ by a gap.",
"By contrast, the repulsive branch does not correspond to a specific eigenstate; rather it is composed of a continuum of eigenstates with closely spaced eigenvalues, resulting in a polaron quasiparticle with finite lifetime.",
"As a consequence, its broadening $2\\Gamma _R$ grows monotonically with $E_F$ and only coincides with $2\\eta $ at small doping — see Fig.",
"REF (d).",
"In between the attractive and repulsive branches, one can observe in Fig.",
"REF (a) a continuum of trion and Fermi-sea-hole states with a well-defined relative momentum ${\\bf q}$ in the variational function $\\varphi _{{\\bf k}{\\bf q}}^{}$ .",
"The boundaries of this so-called trion-hole continuum can be evaluated from the energy of a trion (see Appendix ) plus a hole separately.",
"If the hole has zero momentum ${\\bf q}={\\bf 0}$ , then, because of momentum conservation, the trion is also at zero momentum and the upper boundary of the trion-hole continuum is $E_{+} = E_T^{({\\bf 0},E_F)}=-\\varepsilon _T + \\frac{m_T}{m_X} E_F\\; .$ Conversely, the energy of the trion-hole lower bound is $E_{-} = E_T^{({\\bf k}_F,E_F)} -E_F\\; ,$ where both the hole and trion are at ${\\bf q}={\\bf k}_F=k_F \\hat{\\textbf {n}}$ , with $\\hat{\\textbf {n}}$ an arbitrary direction, and $E_T^{({\\bf k}_F,E_F)}$ is the trion energy [which can be evaluated from Eq.",
"(REF )].",
"Thus, at zero temperature and finite doping, the attractive branch is separated from the trion-hole continuum by an energy gap.",
"Note that, while this is a consequence of considering a single excitation of the medium, our results are consistent with those of diagrammatic quantum Monte Carlo [67] which demonstrated that there is a region of anomalously low spectral weight (a “dark continuum”) above the narrow attractive polaron branch.",
"Furthermore, calculations that incorporate Coulomb interactions between charges also obtain a suppression of spectral weight of this continuum [15].",
"Figure: (a) Spectral function A(ω)A(\\omega ) at finite doping E F =0.4ε T E_F=0.4\\varepsilon _T as a function of temperature and energy.",
"The solid lines are the energies of the attractive (AA, blue) and repulsive (RR, purple) branches, while the dashed (red) line is the T=0T=0 upper boundary of the trion-hole continuum E + E_{+} in Eq. ().",
"Dot-dashed lines are the AA and RR energies at E F =0.04ε T E_F=0.04\\varepsilon _T.",
"(b,c) Temperature dependence of spectral weight ZZ and half linewidth Γ\\Gamma evaluated from the exciton spectral function (see Appendix ) at E F =0.04ε T E_F=0.04\\varepsilon _T (dot-dashed) and E F =0.4ε T E_F=0.4\\varepsilon _T (solid).",
"Note that, in panel (c), the constant value of Γ R \\Gamma _R at E F =0.04ε T E_F=0.04\\varepsilon _T is approximately given by the intrinsic broadening η\\eta ."
],
[
"Finite temperature",
"We now discuss the effect of temperature on the optical response.",
"Figure REF (b-d) allows a comparison between the doping-dependent properties of optical absorption at zero and finite temperature.",
"Both the energies and spectral weights of attractive and repulsive branches have a very weak dependence on temperature in the regime $T\\lesssim \\varepsilon _T$ .",
"The branch energies are slightly redshifted, while $Z_A$ ($Z_R$ ) is slightly smaller (larger) compared to the zero-temperature case.",
"This small variation with temperature is also observed in Fig.",
"REF (a,b) for a fixed doping.",
"The most important difference at finite temperature is the behavior of the trion-hole continuum, which subsumes the attractive branch when $T\\gtrsim E_F$ , i.e., at sufficiently low doping or sufficiently high temperature.",
"This can be clearly seen from Fig.",
"REF (b), where we no longer observe a sharp lower bound of the trion-hole continuum at low doping because, at finite temperature, the unbound hole belonging to the trion-hole continuum can thermally occupy any momentum state.",
"However, the upper bound of the trion-hole continuum is still clearly visible and approximately follows the zero-temperature expression $E_{+}$ (REF ).",
"Similarly, for a fixed doping in Fig.",
"REF (a), we observe that the trion-hole continuum is only well-separated from the attractive branch at low temperatures.",
"Since the spectral weight of the trion-hole continuum is small, its merging with the attractive branch only slightly affects the attractive peak energy.",
"However, the disappearance of the attractive polaron quasiparticle strongly modifies the attractive-branch linewidth $2\\Gamma _A$ .",
"In particular, we observe in Fig.",
"REF (d) that $\\Gamma _A$ has a striking non-monotonic dependence at low doping, while it decreases towards its zero-temperature value (corresponding to the intrinsic broadening $\\eta $ ) when $E_F$ increases.",
"Likewise, increasing the temperature at fixed doping can substantially increase $\\Gamma _A$ from $\\eta $ , as shown in Fig.",
"REF (c).",
"As we will discuss in Sec.",
"REF , this behavior signals a crossover from a coherent Fermi polaron regime to an incoherent trion-dominated regime, where there no longer exists a well-defined attractive quasiparticle that is separated from the trion-hole continuum.",
"The repulsive branch, on the other hand, remains a polaron quasiparticle with a finite lifetime (broadening) for the dopings considered in this work ($E_F \\lesssim \\varepsilon _T$ ).",
"In particular, we see that temperature does not change the nature of the repulsive branch in Figs.",
"REF and REF , but it can lead to a faster increase of the half linewidth $\\Gamma _R$ with increasing doping [Fig.",
"REF (d)].",
"Figure: (a) Spectral function A(ω)A(\\omega ) and (b) Lorentzian convolved photoluminescence P ¯(ω)\\bar{P}(\\omega )for different dopings E F E_F and at a fixed temperature of T=50T=50 K≃0.17ε T \\simeq 0.17 \\varepsilon _T.",
"The attractive branch photoluminescence peaks are rescaled to unity.Figure: (a) Spectral function A(ω)A(\\omega ) and (b) Lorentzian convolved photoluminescence P ¯(ω)\\bar{P}(\\omega )for different values of the temperature TT and at a fixed doping E F =0.1ε T E_F=0.1\\varepsilon _T.",
"The attractive branch photoluminescence peaks are rescaled to unity.Figure: (a) Unconvolved photoluminescence P(ω)P(\\omega ) and (b) Lorentzian convolved photoluminescence P ¯(ω)\\bar{P}(\\omega ) for different dopings E F E_F, at a fixed temperature of T=50T=50 K≃0.17ε T \\simeq 0.17 \\varepsilon _T, and for a frequency range around the attractive branch only.",
"The photoluminescence peaks are rescaled to unity.",
"In panel (a) the vertical dashed lines are the upper boundary of the trion-hole continuum E + E_{+} at zero temperature (see text).We now discuss the shape of the optical response profiles and how they evolve with either doping or temperature.",
"Figures REF and REF display both the absorption and the Lorentzian convolved photoluminescence (see Appendix ) at fixed temperature and fixed doping, respectively.",
"The repulsive polaron quasiparticle shows up approximately as a Lorentzian symmetric profile in both absorption and photoluminescence spectra, with a full width at half maximum (FWHM) that increases with increasing $E_F$ .",
"Note that the FWHM of the repulsive branch only has a weak dependence on temperature in Fig.",
"REF since $E_F$ has been fixed to a low value such that the intrinsic width $2\\eta $ dominates [see Fig.",
"REF (c)].",
"By contrast, the shape of the attractive branch is strongly modified by temperature: in the low-temperature (high-doping) regime $E_F > T$ , it is described by a Lorentzian with FWHM $2\\eta $ , while for $E_F < T$ , it develops a strongly asymmetric shape with an exponential tail below the trion energy.",
"This evolution in the asymmetry of the attractive branch once again indicates a crossover from a Fermi-polaron quasiparticle to a continuum of trion states.",
"In Fig.",
"REF we further analyze the shape of the attractive peak at low doping $E_F \\lesssim T$ , comparing the Lorentzian convolved photoluminescence $\\bar{P} (\\omega )$ with the “bare” photoluminescence $P(\\omega )$ , where we have removed the effects of any intrinsic exciton broadening.",
"In panel (a), we observe a sharp onset of the photoluminescence which approximately coincides with the upper boundary of the trion-hole continuum at zero temperature, $E_+$ .",
"Thus, according to Eq.",
"(REF ), it blueshifts with increasing doping.",
"As shown in Fig.",
"REF (b), any intrinsic broadening $\\eta $ only smooths out the sharp onset, while it has little effect on the position of the peak.",
"When $E_F$ increases, the sharp onset tends to disappear as the attractive peak redshifts and detaches from the trion-hole continuum.",
"Our calculated profiles for the attractive branch are in excellent quantitative agreement with recent experiments in the high-temperature (low-doping) regime [12], as discussed in the accompanying paper [38].",
"The exponential tail of the asymmetric attractive peak has previously been modelled within a trion picture for the case of photoluminescence [68], [69], [70], [71].",
"There, the tail is ascribed to the kinetic energy of remaining electrons after the exciton within each trion has decayed into a photon.",
"This description in terms of electron recoil can be formally derived from our theory in the limit of weak interactions, as we will discuss in Sec.",
"REF ."
],
[
"Loss of the attractive polaron quasiparticle: polaron to trion-hole continuum crossover",
"In this section, we use the pole condition in Eq.",
"(REF ) to characterize the crossover from a well-defined polaron quasiparticle to a trion-hole continuum with increasing temperature (decreasing doping).",
"In order to find the values of temperature and doping at which this crossover occurs, we plot in Fig.",
"REF the local maximum value of the function $\\omega - \\Re \\Sigma (\\omega )$ for $\\omega < 0$ and identify the curve of doping versus fugacity $z=e^{\\beta \\mu } = e^{\\beta E_F} -1$ at which this maximum value is zero.",
"For $E_F \\lesssim \\varepsilon _T$ , we find that this occurs roughly when $z \\sim 1$ and thus $E_F \\sim 0.7 T$ .",
"On the left of this curve, we lose the attractive polaron quasiparticle, as the condition (REF ) cannot be satisfied, i.e., $E_A - \\Re \\Sigma (E_A)\\ne 0$ .",
"On the right of this curve, instead, the system is in the polaron regime where (REF ) is satisfied.",
"In order to further illustrate this crossover, we compare the results for the polaron energies, spectral weights and linewidths extracted from the spectral function with those obtained by treating the polaron as a well-defined quasiparticle [62].",
"In the latter case, the polaron properties can be obtained directly from the expression of the impurity self-energy.",
"Close to a quasiparticle resonance, the exciton Green's function can be approximated as $G_X(\\omega ) \\operatornamewithlimits{\\simeq }_{\\omega \\simeq E_{j}} \\frac{Z_{j}}{\\omega -E_{j} +i\\Gamma _{j}} \\; ,$ where $j=A,R$ is the two-branch index, and the quasiparticle energy $E_{j}$ is a solution of Eq.",
"(REF ).",
"The pole weight or residue $Z_{j}$ is $Z_{j} = \\left(1 - \\left.\\displaystyle \\frac{\\partial \\Re \\Sigma (\\omega )}{\\partial \\omega }\\right|_{E_{j}}\\right)^{-1} \\; ,$ and the polaron damping rate is $\\Gamma _{j} = - Z_{j}\\Im \\Sigma (E_{j})\\; .$ We compare the results for $E_j$ , $Z_j$ , and $\\Gamma _j$ obtained with both methods in Fig.",
"REF .",
"We observe that the positions of the poles coincide to high accuracy with those of the spectral function maxima — see panel (a).",
"For the repulsive branch, both the spectral weight and the broadening from the quasiparticle theory are in good agreement with those evaluated from the spectral function, even when the linewidth is non-negligible.",
"By contrast, for the attractive branch, the results depart from one another when we approach the (gray) region $E_F \\lesssim 0.7 T$ where there is no attractive quasiparticle, and the quasiparticle description breaks down since, according to Eq.",
"(REF ), $1/Z_A\\rightarrow 0$ when $\\rm {max}[\\omega -\\Re \\Sigma (\\omega )]=0$ .",
"In the following section, we analyze the system properties well inside the trion-hole continuum (gray) regime, where the attractive branch is no longer a polaron quasiparticle.",
"Here, for temperatures $T \\gtrsim E_F$ , we can apply a systematic quantum virial expansion.",
"This connects the results of this work with those obtained in the accompanying paper [38].",
"Figure: Maximum value of ω-ℜΣ(ω)\\omega - \\Re \\Sigma (\\omega ) for ω<0\\omega < 0 as a function of the Fermi gas fugacity z=e βμ z=e^{\\beta \\mu } and doping.",
"The black dashed line describes the values of zz and E F E_F at which this maximum is zero.",
"On the left of this curve (blue area) the attractive branch is not a polaron quasiparticle (no QP), while on the right (red area) the attractive branch is a well defined polaron quasiparticle (QP).Figure: (a) Polaron energies E A,R E_{A,R}, (b) spectral weights Z A,R Z_{A,R}, and (c) half linewidths Γ A,R \\Gamma _{A,R}, evaluated from the exciton spectral function (solid lines) (as described in Appendix ) and from quasiparticle expressions (symbols).",
"The attractive branch stops to be a quasiparticle resonance in the gray region at low doping when E F ≲0.7TE_F \\lesssim 0.7 T.In panel (c) we plot Γ A,R -η\\Gamma _{A,R}-\\eta is order to compare the numerical results with the analytical estimate of the repulsive branch broadening evaluated at η=0\\eta =0 at small doping (dashed line), derived within the virial expansion in Sec.",
".Temperature is fixed at T=50T=50 K≃0.17ε T \\simeq 0.17 \\varepsilon _T."
],
[
"Virial expansion and connection to the trion wave function at high temperature or low doping",
"As discussed, at high temperature or low doping such that the fugacity $z= e^{\\beta \\mu }\\lesssim 1$ , the attractive polaron quasiparticle disappears and the attractive branch only consists of a broad continuum.",
"In this limit, one can formally apply a perturbatively exact quantum virial expansion in the fugacity [38].",
"We will now briefly discuss how this expansion is related to the polaron theory, and how this allows us to demonstrate that the trion picture results of Refs.",
"[68], [12] are contained within the polaron formalism.",
"When $T \\gg E_F$ and $z \\simeq \\beta E_F\\ll 1$ , we can formally expand the Fermi occupation function which then coincides with the Boltzmann distribution $f_{\\bf k}\\simeq z e^{-\\beta \\epsilon _{{\\bf k}}}$ .",
"Likewise, to leading order in $z$ we have ${\\mathcal {T}}\\simeq {\\mathcal {T}}_0$ — see Eq.",
"(REF ).",
"Within this expansion, the exciton self-energy in Eq.",
"(REF ) becomes ()zAqe-q T0(q,+q).",
"In other words, to leading order in the fugacity, the self-energy is determined by two-body interactions weighted by a Boltzmann distribution.",
"Focusing first on the attractive branch, the dominant contribution to the self-energy arises from the pole of the $T$ matrix at the trion energy.",
"We therefore expand the $T$ matrix for $\\omega \\simeq -\\varepsilon _T + \\epsilon _{T{\\bf q}}$ , with the result T0(q,+q) ZT+q-Tq+T+i0, where $Z_T\\equiv 2\\pi \\varepsilon _T/m_r$ .",
"The $T$ matrix can be expressed in terms of the vacuum ($E_F=0$ ) trion wave function $\\tilde{\\eta }_{{\\bf q}_r}^{({\\bf 0})}$ for a contact electron-exciton interaction (see Appendix ).",
"In our case, the relative momentum is ${\\bf q}_r={\\bf q}m_X/m_T$ , and therefore the kinetic energy of the relative motion is $\\epsilon _{r{\\bf q}_r}=\\epsilon _{\\bf q}-\\epsilon _{T{\\bf q}}$ .",
"Using the expression for the vacuum trion wave function in the center of mass frame, $\\tilde{\\eta }_{{\\bf q}_r}^{({\\bf 0})}= \\displaystyle \\frac{\\sqrt{Z_T}}{\\varepsilon _T + \\epsilon _{r{\\bf q}_r}}$ (see Appendix ), we have T0(q,+q) |(T+rqr)qr(0)|2+rqr+T+i0.",
"Here, the numerator is derived by manipulating the relation between the trion wave function and $Z_T$ , and is momentum independent.",
"However, this is a special property of the contact electron-exciton interaction, and Eq.",
"(REF ) in fact yields the correct generalization for an arbitrary electron-exciton interaction that leads to the formation of a trion.",
"In other words, it can be applied for any realistic trion wave function.",
"For details of the generalization to arbitrary interactions, see Appendix .",
"Using the approximation for the vacuum $T$ matrix, Eq.",
"(REF ), the self-energy in Eq.",
"(REF ) becomes $\\Sigma _{A}(\\omega <-\\varepsilon _T) \\simeq \\frac{z}{\\mathcal {A}}\\sum _{\\bf q}e^{-\\beta \\epsilon _{\\bf q}} |\\tilde{\\eta }_{{\\bf q}_r}^{({\\bf 0})}|^2 \\\\\\times \\left[{\\mathcal {P}}\\frac{|\\epsilon _{r{\\bf q}_r}+\\varepsilon _T|^2}{\\omega +\\epsilon _{r{\\bf q}_r}+\\varepsilon _T}-i\\pi \\omega ^2 \\delta (\\omega +\\epsilon _{r{\\bf q}_r}+\\varepsilon _T)\\right]\\;,$ where ${\\mathcal {P}}$ denotes the principal value.",
"This explicitly relates the self-energy calculated within the polaron theory to the trion wave function in the regime where the fugacity is small, and importantly it applies for any realistic trion wave function.",
"A similar approach involving trion wave functions has been used to calculate absorption [34].",
"Note that the real part diverges logarithmically when $\\omega +\\varepsilon _T\\rightarrow 0^-$ , and therefore it cannot in general be neglected close to the onset of the attractive branch.",
"For the repulsive branch, we can again apply Eq.",
"(REF ) to find the leading contribution at small fugacity.",
"In the regime where $E_F\\ll \\varepsilon _T$ , the width of the repulsive branch is much smaller than the trion binding energy, and to lowest order in the fugacity, we can simply evaluate the repulsive polaron self-energy within the repulsive branch by taking $\\omega =0$ .",
"We then use the fact that, to logarithmic accuracy, the logarithmic behavior at small momentum is generic for the vacuum $T$ matrix for any short-range interaction [72], [73] and thus we have R(0) zA2mrqe-q(T/rqr)+i z(m/mr) T2+2(eET)[(eET)-i] , where $\\gamma _{\\rm E}\\simeq 0.5772$ is the Euler-Mascheroni constant.",
"Here, we have evaluated the integral by noting that the integral is dominated by $\\epsilon _{\\bf q}\\sim 1/\\beta $ for small $\\beta \\varepsilon _T$ , and the inclusion of $\\gamma _{\\rm E}$ originates from expanding the integral to the first subleading order in $\\beta \\varepsilon _T$ .",
"The self-energies in Eqs.",
"(REF ) and (REF ) can now be directly inserted into the Dyson equation (REF ) to yield the absorption in Eq.",
"(REF ) or the photoluminescence in Eq.",
"(REF ).",
"To be explicit, within the virial expansion we have the exciton spectral function A() = -1Im [(--T)-A()+1-R(0)].",
"This yields a broad continuum for the attractive branch, where the spectral weight vanishes at $\\omega +\\varepsilon _T\\rightarrow 0^-$ , and when we are far below this onset, we have $\\Sigma _{A}(\\omega )/\\omega \\rightarrow 0$ .",
"Therefore we have an exponential tail modulated by the trion wave function: A()*- z e(+T)mTmX|2mr|+T|(0)|2.",
"The peak of the absorption is between the onset and the tail, and therefore it will not correspond to the vacuum trion energy.",
"This can lead to an overestimate of the trion binding energy in experiments [12], as shown in the accompanying paper [38].",
"From Eq.",
"(REF ), we find that the repulsive branch is a Lorentzian of width $\\Gamma _{R}=\\pi (m/m_r) E_F/[\\pi ^2+\\ln ^2(e^{\\gamma _{\\rm E}}\\beta \\varepsilon _T)]$ .",
"We have compared in Fig.",
"REF (c) this expression against the numerical evaluation of the repulsive branch half linewidth $\\Gamma _R - \\eta $ (where we have removed the effect of the intrinsic exciton lifetime), finding excellent agreement at low doping, inside the region of validity of the virial expansion.",
"For the photoluminescence, we find P() -Z0Zint1Im [(--T)e--A()+1-R(0)], where we have used the fact that the width of the repulsive branch is much smaller than the temperature, and thus the repulsive branch is very weakly modified by the Boltzmann factor."
],
[
"Connection to the trion theory of electron recoil",
"Finally, we discuss how our theory relates to the calculation of electron recoil in previous trion-picture calculations such as by Esser et al., Refs.",
"[68], [69].",
"As we shall demonstrate, that theory corresponds to a weak-interaction limit of the low doping/high temperature version of our polaron theory.",
"To see this, we take the limit of a small self-energy in the Dyson equation in Eq.",
"(REF ) as follows: GX()1+12() .",
"Since the $1/\\omega $ terms only have a pole at $\\omega =0$ (i.e., at the bare exciton energy), the attractive branch is obtained from the imaginary part of the self-energy.",
"The detailed balance equation (REF ) then implies that the corresponding PL from the attractive branch is PA()-1Z0Zint e-2ImA() Z0Zint zAqe-(+q)|qr(0)|2(+rqr+T) Z0Zint z e(m+mTT)mX|2mr|+T|(0)|2(--T) , where we used Eq.",
"(REF ).",
"Up to frequency-independent prefactors, this precisely matches the result of Esser et al.",
"Thus, the trion-picture PL is already contained within the polaron picture.",
"However, as we have already argued, the trion picture calculation assumes that we are in a weakly interacting limit where the self-energy is much smaller than the frequency.",
"This assumption manifestly breaks down close to the onset of the attractive branch, where the real part of the self-energy diverges, and hence the previous trion picture calculation fails to correctly describe the onset and shape of the attractive branch.",
"On the other hand, it correctly identifies the exponential tail of the PL, which is dominated by the imaginary part of the self-energy.",
"For the repulsive branch, the trion picture again uses the weak-interaction limit of the Dyson equation (REF ), this time neglecting even the second term on the right hand side.",
"This results in PR()Z0Zint(), with a small correction that counteracts the oscillator strength transferred to the attractive branch [69].",
"We see that the trion picture fails to describe the Lorentzian shape of the repulsive polaron, which arises from the exciton-electron scattering states as well as the full Dyson series.",
"We now extend the formalism of Sec.",
"to describe recent experiments where a doped TMD monolayer was embedded into a microcavity [7], [19], [20], [9], [21].",
"In this case, the strong coupling between light and matter can lead to the formation of Fermi polaron-polaritons [74], [7].",
"A particularly important question is whether there still exists strong coupling to light either because of temperature induced broadening effects or because the attractive branch does not correspond to a well-defined quasiparticle.",
"To describe the light-matter coupled system, we add two terms describing cavity photons and the photon-exciton interaction to the Hamiltonian (): H = H0 + H0X+ H0C+ Hint+HXC H0C= k Cka†kak HXC=2k( x†kak + H.c. ) .",
"Photons are described by the bosonic creation operator $\\hat{a}^{\\dag }_{{\\bf k}}$ and, in a cavity, they acquire a quadratic dispersion $\\epsilon _{C{\\bf k}} = \\delta + {\\bf k}^2/2m_C$ [75], where $\\delta $ is the photon-exciton energy detuning.",
"Typically the photon mass in a planar microcavity is $m_C = 10^{-5} m_X$ [75].",
"The term $\\hat{H}_{XC}$ () describes an exciton recombining to emit a photon and vice versa, with a coupling strength given by the Rabi splitting $\\Omega $ [75].",
"In monolayer TMDs, this is typically in the range of 1-2 $\\varepsilon _T$ [76], [77].",
"In order to derive the photon Green's function and the optical absorption in the strong coupling regime, one can follow the same procedure employed in Sec. .",
"The difference is that now we formulate a variational ansatz for the time-dependent photon operator, with an analogous form to (REF ): $\\hat{a}_{{\\bf 0}}^{}(t) \\simeq \\alpha _{0}^{}(t) \\hat{a}_{{\\bf 0}}^{} + \\frac{1}{\\mathcal {A}} \\sum _{{\\bf k},{\\bf q}} \\varphi _{{\\bf k}{\\bf q}}^{}(t) \\hat{c}_{{\\bf q}}^{\\dag } \\hat{c}_{{\\bf k}}^{} \\hat{x}_{{\\bf q}-{\\bf k}}^{}+ \\varphi _{0}^{}(t) \\hat{x}_{{\\bf 0}}^{} \\; .$ We neglect the dressing of the photon operator by a particle-hole excitation $\\frac{1}{\\mathcal {A}} \\sum _{{\\bf k},{\\bf q}} \\alpha _{{\\bf k}{\\bf q}}^{}(t) \\hat{c}_{{\\bf q}}^{\\dag } \\hat{c}_{{\\bf k}}^{} \\hat{a}_{{\\bf q}-{\\bf k}}^{}$ .",
"This term involves photon recoil, and therefore implies energies far off resonance from the exciton and trion energies because of the extremely small mass of the photon.",
"To obtain the eigenvalue problem for the light-matter coupled system, we introduce the error operator corresponding to the photon $\\hat{e}(t) = i\\partial _t \\hat{a}_{{\\bf 0}}^{}(t) - [\\hat{a}_{{\\bf 0}}^{}(t),\\hat{H}]$ and minimize the error function, Eq.",
"(REF ), with respect to the variational coefficients $\\alpha _{0}^{*}(t)$ , $\\varphi _{{\\bf k}{\\bf q}}^{*}(t)$ , and $\\varphi _{0}^{*}(t)$ .",
"Considering the stationary solutions, we find E0 = 2 0 -vA2k,q fq(1-fk) kq Ekq = EXkq kq - v0 -vAk'(1-fk') k'q E0 = 0 + 2 0 , where we have again neglected terms that vanish in the limit $\\Lambda \\rightarrow \\infty .$ These equations reduce to those for the exciton polaron, Eq.",
"(), when we take $\\Omega =0$ and $\\alpha _0^{}=0$ .",
"By following an analogous derivation to that in Sec.",
", one can easily demonstrate that the retarded photon Green's function in the frequency domain is given by: $G_{C} (\\omega ) = \\sum _n\\frac{|\\alpha _0^{(n)}|^2}{\\omega -E^{(n)}+i0} \\; .$ For the same reason that we can neglect the particle-hole dressing of the photon operator in the ansatz (REF ), the expressions for the exciton self-energy $\\Sigma (\\omega )$ in Eqs.",
"(REF ) and (REF ) are not affected by the coupling to light.",
"Therefore, we can derive the coupled exciton and photon Green's functions in the strong coupling regime by inverting the matrix [78] $\\mathbb {G}(\\omega ) = \\begin{pmatrix}\\omega -\\Sigma (\\omega ) & -\\Omega /2 \\\\-\\Omega /2 & \\omega -\\delta \\end{pmatrix}^{-1} \\; ,$ and evaluating the diagonal elements, giving: GX() =G11()=1- () - (/2)2- GC() =G22()=1--(/2)2-() .",
"The expression () for the exciton Green's function in terms of the self-energy can also be obtained by following the same procedure as in Appendix , where we have first eliminated the photon amplitude by solving Eq.",
"(), i.e., $\\alpha _0 = \\frac{\\Omega }{2} \\varphi _0^{} (E-\\delta )^{-1}$ .",
"Finally, in the strong coupling regime, optical absorption coincides (up to a frequency-independent prefactor) with the photon spectral function [79]: $A_{C}(\\omega )=-\\frac{1}{\\pi }\\Im G_C(\\omega ) \\; ,$ which we will focus on in the following.",
"We plot in Fig.",
"REF the finite-temperature photon spectral function at normal incidence as a function of energy and detuning $\\delta $ .",
"When $T \\ll E_F \\lesssim \\Omega $ , the strong coupling to light leads to three polariton branches, the lower (LP), middle (MP), and upper polariton (UP), as can be seen in Fig.",
"REF (a,c).",
"The existence of these polariton modes is connected to the attractive and repulsive exciton branches in the presence of doping.",
"We can capture the behavior of the Fermi polaron-polaritons by employing a three coupled oscillator model, which yields the following simplified expression for the photon Green's function GC() = .",
"(- H)-1 |11 H = -i C A/2 R/2 A/2 EA - iA 0 R/2 0 ER - iR .",
"Here, we have explicitly included a cavity photon lifetime $1/\\eta _C$ and we have used the extracted parameters for the exciton-polaron branches in the absence of coupling to light, namely the energies $E_{A,R}$ , spectral weights $Z_{A,R} = (\\Omega _{A,R}/\\Omega )^2$ , and half linewidths $\\Gamma _{A,R}$ .",
"Note that we cannot evaluate the polariton branch energies as the (complex) eigenvalues of the matrix $\\tilde{H}$ [80].",
"Rather, we have to evaluate first the photon spectral function (REF ) and then determine the polariton energies from the photon spectral function peak positions.",
"The comparison in Fig.",
"REF between the LP, MP and UP energies obtained in this way and the full calculation demonstrates essentially perfect agreement.",
"Remarkably, we find that a strong enough light-matter coupling can produce well-defined polariton quasiparticles (where $\\Re \\tilde{G}_{C}^{-1} (\\omega )=0$ ) for the lower and middle branches even when there is no attractive polaron quasiparticle and Eq.",
"(REF ) is not satisfied.",
"However, once the linewidths $2\\Gamma _{A,R}$ approach $\\Omega _{A,R}$ , the energy splitting between branches closes, indicating a loss of strong coupling.",
"We observe this in the low-doping regime for LP and MP [Fig.",
"REF (b)], and in the high-doping regime for MP and UP [Fig.",
"REF (c,d)].",
"Within the three coupled oscillator model, this transition from weak to strong light-matter coupling approximately occurs when $2\\Gamma _{A,R} \\gtrsim \\Omega _{A,R}$ , as expected, though there are small deviations when the attractive branch is no longer a well-defined polaron quasiparticle.",
"To conclude, the effect of temperature on this transition can be easily accounted for by considering its effect on the quasiparticle linewidths and spectral weights (see Sec.",
")."
],
[
"Conclusions and perspectives",
"We have studied the optical properties of a doped two-dimensional semiconductor at a finite temperature using a Fermi-polaron approach involving a single excitation of the fermionic medium.",
"Our results reveal that the attractive branch can experience a smooth transition from a regime where it is a well-defined quasiparticle to a regime where is subsumed into a broad continuum of trion-hole scattering states.",
"This crossover results in a strong change in the spectral lineshape and can be driven by either decreasing doping or increasing temperature, but it cannot occur at zero temperature.",
"While the Fermi polaron theory is able to capture both limits, theories based on the trion wavefunction necessarily only apply in the limit where there is no a well-defined quasiparticle.",
"In particular, we formally show that the trion theory corresponds to a weak-interaction limit of our finite-temperature Fermi polaron theory.",
"In the regime of strong light-matter coupling, we have shown how the temperature can modify the properties of the Fermi polaron-polaritons.",
"We demonstrate that the strong-to-weak coupling crossover observed at finite temperature for the attractive branch at low enough doping and the repulsive branch in the high doping regime can be explained in terms of the linewidths and spectral weights of the two branches.",
"In future studies, it would be interesting to investigate how the quasiparticle transition of the attractive branch, driven by either temperature or doping, would affect the interactions between exciton impurities.",
"In particular, common descriptions of polaron-polaron interactions use Landau's theory of dilute solutions [81], which assumes well-defined quasiparticles.",
"Such interactions could, for instance, be measured using coherent multidimensional spectroscopy on gated 2D materials, similarly to recent experiments on intrinsically doped samples [82].",
"We gratefully acknowledge useful discussions with Dmitry Efimkin.",
"AT and FMM acknowledge financial support from the Ministerio de Ciencia e Innovación (MICINN), project No.",
"AEI/10.13039/501100011033 (2DEnLight).",
"FMM acknowledges financial support from the Proyecto Sinérgico CAM 2020 Y2020/TCS-6545 (NanoQuCo-CM).",
"JL and MMP are supported through Australian Research Council Future Fellowships FT160100244 and FT200100619, respectively.",
"JL, BM and MMP also acknowledge support from the Australian Research Council Centre of Excellence in Future Low-Energy Electronics Technologies (CE170100039)."
],
[
"Finite impurity momentum",
"For completeness, we generalize the finite-temperature polaron formalism illustrated in Sec.",
"to absorption and photoluminescence at finite momentum.",
"This can in principle be measured in doped semiconductors using angle-resolved photoemission spectroscopy [83], [84].",
"Similarly to Eq.",
"(REF ), we approximate the exciton operator in the Heisenberg picture at finite momentum $\\hat{x}_{{\\bf Q}}^{}(t) = e^{i\\hat{H}t} \\hat{x}_{{\\bf Q}}^{} e^{-i\\hat{H}t}$ as $\\hat{x}_{{\\bf Q}}^{}(t) \\simeq \\varphi _{0}^{({\\bf Q})}(t) \\hat{x}_{{\\bf Q}}^{} + \\frac{1}{\\mathcal {A}} \\sum _{{\\bf k},{\\bf q}} \\varphi _{{\\bf k}{\\bf q}}^{({\\bf Q})}(t) \\hat{c}_{{\\bf q}}^{\\dag } \\hat{c}_{{\\bf k}}^{} \\hat{x}_{{\\bf Q}+{\\bf q}-{\\bf k}}^{}\\; .$ The derivation then follows similarly to the zero momentum case.",
"We minimize the error function $\\Delta _{{\\bf Q}}(t) = \\langle \\hat{e}_{{\\bf Q}}^{}(t) \\hat{e}_{{\\bf Q}}^{\\dag }(t) \\rangle _{\\beta }$ , where $\\hat{e}_{{\\bf Q}}^{}(t) = i\\partial _t \\hat{x}_{{\\bf Q}}^{}(t) - [\\hat{x}_{{\\bf Q}}^{}(t),\\hat{H}]$ , obtaining the following eigenvalue problem: E(Q)0(Q) = XQ 0(Q) -vA2k,q fq(1-fk) kq(Q) E(Q)kq(Q) = EXQkq kq(Q) - v0(Q) -vAk'(1-fk') k'q(Q) , where $E_{X{\\bf Q}{\\bf k}{\\bf q}} = \\epsilon _{X{\\bf Q}+{\\bf q}-{\\bf k}} + \\epsilon _{\\bf k}- \\epsilon _{\\bf q}$ .",
"The exciton Green's function can thus be written in terms of the eigenvalues $E^{({\\bf Q},n)}$ and eigenvectors $\\varphi _{0}^{({\\bf Q},n)}$ as $G_{X} ({\\bf Q}, \\omega ) = \\sum _n\\frac{|\\varphi _0^{({\\bf Q},n)}|^2}{\\omega -E^{({\\bf Q},n)}+i0} \\; .$ Equivalently, the exciton Green's function can be written in terms of the exciton self-energy at finite momentum GX(Q,) =1- XQ - (Q,) (Q,) =1Aq fqT(q+Q, +q) , where the inverse of the $T$ matrix is defined in Eq.",
"(REF ).",
"This follows the same procedure as in the case of ${\\bf Q}={\\bf 0}$ , which is described in Appendix .",
"The absorption of a photon with momentum ${\\bf Q}$ is given by $A({\\bf Q},\\omega ) =-\\frac{1}{\\pi }\\Im G_X({\\bf Q},\\omega ) \\; .$ Absorption and photoluminescence can be connected using detailed balance conditions as derived in the main text, starting from the Fermi's golden rule definitions: A(Q,) = n, n| 0 |n || xQ†|n |2 (En - ) P(Q,) = n, | ||n| xQ ||2 (En - ) .",
"Thus, the detailed balance condition is identical to that at zero momentum: $P({\\bf Q},\\omega ) = \\displaystyle \\frac{Z_0}{Z_{int}} e^{-\\beta \\omega } A({\\bf Q},\\omega ) \\; .$"
],
[
"Exciton self-energy",
"In this appendix, we demonstrate that the exciton self-energy can be derived starting from the eigenvalue equations () and that it coincides with the expression (REF ) that can be derived within a $T$ matrix formalism.",
"Let us start from the eigenvalue problem () which can be rewritten in terms of the auxiliary function $\\chi _{{\\bf q}}^{}$ : q =vAk (1-fk) kq E 0 =-1Aq fqq E kq = EXkq kq-v0- q .",
"Introducing Eq.",
"() into Eq.",
"() and solving for $\\chi _{\\bf q}$ we obtain $\\chi _{{\\bf q}}^{} = \\varphi _{0}^{} \\left[\\frac{1}{v}+\\frac{1}{\\mathcal {A}}\\sum _{{\\bf k}} \\frac{1-f_{\\bf k}}{E-E_{X{\\bf k}{\\bf q}}} \\right]^{-1} \\; ,$ where we have used that $(v/\\mathcal {A}) \\sum _{\\bf k}(1-f_{{\\bf k}})/(E-E_{X{\\bf k}{\\bf q}}) \\rightarrow -1$ in the limit $\\Lambda \\rightarrow \\infty $ .",
"Substituting (REF ) into Eq.",
"(), one thus finally obtains an implicit equation for the energy $E$ : $E = - \\frac{1}{\\mathcal {A}}\\sum _{{\\bf q}} f_{\\bf q}\\left[\\frac{1}{v} +\\frac{1}{\\mathcal {A}}\\sum _{{\\bf k}}\\frac{1-f_{\\bf k}}{E-E_{X{\\bf k}{\\bf q}}} \\right]^{-1} \\; .$ This expression coincides with the pole of the exciton Green's function (REF ) and therefore we can identify the self-energy term correcting the exciton energy because of the exciton-electron interaction as the term: $\\Sigma (E) \\equiv - \\frac{1}{\\mathcal {A}}\\sum _{{\\bf q}} f_{\\bf q}\\left[\\frac{1}{v} +\\frac{1}{\\mathcal {A}}\\sum _{{\\bf k}}\\frac{1-f_{\\bf k}}{E -E_{X{\\bf k}{\\bf q}}} \\right]^{-1} \\; .$ Recalling that $E_{X{\\bf k}{\\bf q}} = \\epsilon _{X{\\bf q}-{\\bf k}} + \\epsilon _{\\bf k}-\\epsilon _{\\bf q}$ , we obtain exactly the expression (REF ) in the main text.",
"By a completely equivalent procedure, we can arrive at the self-energy in the case of finite exciton momentum, Eq.",
"()."
],
[
"Trion",
"In this appendix, we summarize for completeness the known trion properties within our model Hamiltonian ().",
"At zero temperature, a trion with momentum ${\\bf Q}$ in the presence of a Fermi sea $|FS\\rangle = \\prod _{|{\\bf q}|<k_F} c_{{\\bf q}}^\\dag |0\\rangle $ is described as: $|T_2^{({\\bf Q})}\\rangle = \\displaystyle \\frac{1}{\\sqrt{\\mathcal {A}}}\\sum _{|{\\bf k}|>k_F} \\eta _{{\\bf k}}^{({\\bf Q})} \\hat{x}^{\\dag }_{{\\bf Q}-{\\bf k}} \\hat{c}_{{\\bf k}}^\\dag | FS\\rangle \\; .$ The trion wave function $\\eta _{{\\bf k}}^{({\\bf Q})}$ satisfies the Schrödinger equation [64] $E_T^{({\\bf Q},E_F)} \\eta _{{\\bf k}}^{({\\bf Q})} = \\left(\\epsilon _{X{\\bf Q}-{\\bf k}} + \\epsilon _{{\\bf k}}\\right) \\eta _{{\\bf k}}^{({\\bf Q})} - \\frac{v}{\\mathcal {A}}\\sum _{|{\\bf k}^{\\prime }|>k_F}^{\\Lambda } \\eta _{{\\bf k}^{\\prime }}^{({\\bf Q})} \\; ,$ and the trion energy can be evaluated by solving the implicit equation: $\\displaystyle \\frac{1}{v} = \\displaystyle \\frac{1}{\\mathcal {A}}\\sum _{|{\\bf k}|>k_F}^{\\Lambda } \\displaystyle \\frac{1}{-E_T^{({\\bf Q},E_F)} + \\epsilon _{X{\\bf Q}-{\\bf k}} + \\epsilon _{\\bf k}} \\; .$ Note that some care has to be taken when comparing the trion energy with that of a bare exciton, since the Fermi sea in Eq.",
"(REF ) involves one less electron.",
"This difference has been taken into account in Eqs.",
"(REF ) and (REF ) of the main text.",
"We now discuss the solution of Eq.",
"(REF ) in various limits.",
"When $E_F=0$ , it is profitable to introduce the relative momentum in the center of mass frame ${\\bf q}_r$ , so that the electron momentum becomes ${\\bf k}= {\\bf q}_r + {\\bf Q}_c$ , with ${\\bf Q}_c=\\frac{m}{m_T} {\\bf Q}$ , where $m_T = m_X + m$ is the trion mass, and the exciton momentum becomes ${\\bf Q}-{\\bf k}= {\\bf Q}_X - {\\bf q}_r$ , with ${\\bf Q}_X= \\frac{m_X}{m_T} {\\bf Q}$ .",
"Now one has that relative and center of mass kinetic energies factorize, $\\epsilon _{X{\\bf Q}-{\\bf k}} + \\epsilon _{{\\bf k}} = \\epsilon _{X{\\bf Q}_X-{\\bf q}_r} + \\epsilon _{{\\bf q}_r+{\\bf Q}_c} = \\epsilon _{r{\\bf q}_r} + \\epsilon _{T{\\bf Q}}$ , where $\\epsilon _{r{\\bf q}_r} = \\frac{q_r^2}{2m_r}$ , $m_r = m m_X/m_T$ is the trion reduced mass, and $\\epsilon _{T{\\bf Q}} = \\frac{Q^2}{2m_T}$ .",
"Thus, the trion energy and wave function are given by ET(Q,EF=0) = -T + TQ qr(Q) = ZTT + rqr = qr(0) , where $Z_T = 2\\pi \\varepsilon _T/m_r$ and where $\\tilde{\\eta }_{{\\bf q}_r}^{({\\bf Q})} = \\eta _{{\\bf q}_r + {\\bf Q}_c}^{({\\bf Q})}$ .",
"At finite doping and zero momentum ${\\bf Q}={\\bf 0}$ , Eqs.",
"(REF ) and (REF ) can also be solved exactly to give ET(0,EF) = - T+mTmXEF k(0) = ZTT - mTmX EF + rk .",
"At finite doping and finite momentum, Eq.",
"(REF ) can be solved analytically [66] to find an implicit equation for $E_T^{({\\bf Q},E_F)}$ .",
"Here, one finds that the trion acquires a finite momentum when $E_F > \\frac{m_X m_r}{m^2} \\varepsilon _T$ .",
"We can now analyze the trion coupling strength to light.",
"At zero doping, $E_F=0$ , trions do not couple to light, and thus cannot be probed optically.",
"This is because the matrix element between a trion state and a cavity photon plus a majority particle at zero momentum $|C+1\\rangle = \\hat{a}_{\\bf 0}^\\dag \\hat{c}_{\\bf 0}^\\dag |0 \\rangle $ of the light-matter interaction term $\\sum _{{\\bf k}} \\hat{x}^{\\dag }_{{\\bf k}}\\hat{a}^{}_{{\\bf k}}$ is given by $\\langle T_2^{({\\bf 0})} | \\sum _{{\\bf k}} \\hat{x}^{\\dag }_{{\\bf k}}\\hat{a}^{}_{{\\bf k}} |C+1\\rangle = \\displaystyle \\frac{1}{\\sqrt{\\mathcal {A}}} \\eta _{{\\bf 0}}^{({\\bf 0})}\\; .$ Taking the squared amplitude of this matrix element, we see that the coupling to light of a single trion scales as $1/\\mathcal {A}$ , which is vanishingly small.",
"On the other hand, if we have $N$ electrons within $\\mathcal {A}$ , then the coupling to light scales instead as $N/\\mathcal {A}=n\\sim E_F$ .",
"Thus, even though the coupling per electron vanishes, the net effect of having an electronic medium is to create a continuum of states, with a total spectral weight that scales as $E_F$ , in agreement with Ref.",
"[43]."
],
[
"Numerical evaluation of the many-body $T$ matrix and the exciton self-energy",
"In contrast to the vacuum $T$ matrix (REF ), the many-body correction in Eq.",
"(REF ) cannot be evaluated analytically.",
"The approach we illustrate in the following allows us to treat it in a simpler way and to evaluate it numerically, even when the intrinsic exciton linewidth $2\\eta $ is set to zero.",
"Let us start by re-writing (REF ) in an equivalent form: $\\Pi _{mb}({\\bf q},\\omega ) = - \\frac{1}{\\mathcal {A}} \\sum _{{\\bf k}} \\displaystyle \\frac{f_{{\\bf k}+ \\frac{m}{m_T}{\\bf q}}}{\\omega - \\epsilon _{{\\bf k}+\\frac{m}{m_T}{\\bf q}} - \\epsilon _{X{\\bf k}-\\frac{m_X}{m_T}{\\bf q}} + i0^+} \\\\= - \\int \\frac{dkk}{2\\pi } \\frac{\\int \\frac{d\\theta }{2\\pi } f_{{\\bf k}+ \\frac{m}{m_T}{\\bf q}}}{\\omega - \\epsilon _{\\bf q}\\frac{m}{m_T} - \\epsilon _{\\bf k}\\frac{m_T}{m_X} + i0^+}\\;.$ The $k$ -integral can then be evaluated numerically by applying the Sokhotski–Plemelj theorem dxF(x)x + i0+=- iF(0)+Pdx F(x)x , where $\\mathcal {P}[\\dots ]$ is the integral principal part.",
"As far as the ${\\bf q}$ -integral for evaluating the exciton self-energy (REF ) is concerned, it can be re-written in the following equivalent form by defining $y=q^2$ : $\\Sigma (\\omega ) =\\int \\frac{dy}{4\\pi } \\frac{f_{\\sqrt{y}}}{ \\mathcal {T}^{-1} (\\sqrt{y} , \\omega + \\epsilon _{\\sqrt{y}})} \\; .$ This integral has a pole when $\\Re \\mathcal {T}^{-1} = 0 = \\Im \\mathcal {T}^{-1}$ .",
"Using a model involving contact interactions in 2D means that such a pole always exists, and furthermore it is a simple pole [85].",
"Thus, we define $y^*= y^*(\\omega )$ to be the pole of the $T$ matrix.",
"In this case, we can apply the Cauchy residue theorem to write: $\\Sigma (\\omega ) = - i\\pi \\; \\text{sign}\\left[\\Im \\mathcal {T}^{-1} (\\sqrt{y^*}, \\omega + \\epsilon _{\\sqrt{y^*}})\\right] \\frac{\\Theta (y^*)}{4\\pi } \\mathcal {R}es \\left[\\frac{f_{\\sqrt{y}}}{\\Re \\left[\\mathcal {T}^{-1}(\\sqrt{y},\\omega + \\epsilon _{\\sqrt{y}})\\right]}\\right]_{y^*}\\\\+ \\mathcal {P} \\int \\frac{dy}{4\\pi } \\frac{f_{\\sqrt{y}}}{\\Re \\mathcal {T}^{-1}(\\sqrt{y},\\omega +\\epsilon _{\\sqrt{y}}) + i\\Im \\mathcal {T}^{-1} (\\sqrt{y},\\omega + \\epsilon _{\\sqrt{y}})} \\; ,$ where the principal part prescription is used in the vicinity of the pole at $y^*$ , and the residue can be evaluated as $\\mathcal {R}es \\left[\\frac{f_{\\sqrt{y}}}{\\Re \\left[\\mathcal {T}^{-1}(\\sqrt{y},\\omega + \\epsilon _{\\sqrt{y}})\\right]}\\right]_{y^*}\\\\= \\displaystyle \\frac{f_{\\sqrt{y^*}}}{\\displaystyle \\left| \\frac{\\partial \\Re \\mathcal {T}^{-1}(\\sqrt{y},\\omega +\\epsilon _{\\sqrt{y}}) }{\\partial y}\\right|_{y^*}}\\; .$ From the expression of the exciton self-energy, we can get the exciton Green's function (REF ), absorption (REF ), and photoluminesence (REF ).",
"These results allow us to evaluate the absorption and photoluminescence without any intrinsic homogeneous broadening for the exciton.",
"For absorption alone, the integration method illustrated here is unnecessary, as one can conveniently shift the frequency to the complex plane, $\\omega \\mapsto \\omega + i\\eta $ , where $\\eta $ is related to the exciton decay rate, giving well converged results.",
"However, as explained in the main text, for the luminescence one has to adopt the numerical method illustrated here.",
"Thus, in order to compare with experiments, we introduce the effect of lifetime broadening at the end by convolving the photoluminescence with a Lorentzian profile with broadening $\\eta $ : $\\bar{P}(\\omega ^{\\prime }) = \\int _{-\\infty }^{\\infty } d\\omega ^{\\prime } P(\\omega ^{\\prime }) \\displaystyle \\frac{1}{\\pi } \\displaystyle \\frac{\\eta }{(\\omega - \\omega ^{\\prime })^2 + \\eta ^2}\\; .$ Figure: Spectral function A(ω)A(\\omega ) at temperature T=50T=50 K≃0.17ε T \\simeq 0.17\\varepsilon _T for two different dopings: E F =0.04ε T E_F= 0.04\\varepsilon _T (a) and E F =0.8ε T E_F=0.8 \\varepsilon _T (b).",
"From the spectral function profile we extract the attractive and repulsive polaron energy E A,R E_{A,R} as the peak position, thelinewidth 2Γ A,R 2\\Gamma _{A,R} as the peak FWHM, and the polaron spectral weight Z A,R,continuum Z_{A,R,continuum} as the area under the peak.While at low doping in panel (a) the trion-hole continuum is merged with the attractive branch (which, here, ceases to be a quasiparticle resonance) and Z continuum =0Z_{continuum} = 0, at larger doping in panel (b) the polaron resonances are separated by the trion-hole continuum with a finite spectral weight Z continuum ≠0Z_{continuum} \\ne 0."
],
[
"Extracting polaron properties at finite temperature",
"In this appendix we illustrate how to extract the polaron energies, spectral weights, and half linewidths of Figs.",
"REF and REF from the spectral function profile $A(\\omega )$ .",
"This is illustrated in Fig.",
"REF .",
"The polaron energy $E_{A,R}$ is evaluated as the spectral function peak position, the linewidth $2\\Gamma _{A,R}$ as the full width at half maximum (FWHM) and the spectral weight as the area under the peak.",
"We have used the location of the spectral function minima as limits for the integrals evaluating the spectral weights: If there is only one minimum, this is the upper (lower) bound for evaluating $Z_A$ ($Z_R$ ), while if there are two minima, these are the limits for evaluating the trion-hole continuum spectral weight $Z_{continuum}$ .",
"This criterion is the origin of the discontinuity of the residues in Figs.",
"REF , REF , and REF .",
"Note that, in the cases shown in Fig.",
"REF , at low doping of panel (a) the trion-hole continuum does not appear because is merged with the attractive branch, which in this case, ceases to be a polaron quasiparticle resonance in the sense explained in Sec.",
"REF .",
"This can also be appreciated by the non-Lorentzian and asymmetric form of the spectral function around $E_A$ .",
"By contrast, at larger doping in panel (b), the attractive branch is separated from the continuum, is a polaron quasiparticle in this case and recovers the Lorentzian symmetric shape."
],
[
"Relationship between electron-exciton $T$ matrix and the trion wave function",
"While Eq.",
"(REF ) was derived for the special case of contact exciton-electron interactions, it is in fact general and can be applied for any realistic trion wave function.",
"To see this, we note that the generalization of Eq.",
"(REF ) to arbitrary two-body transition operator $\\hat{T}_0$ is ()zAqe-q T0(+q)Q,qr.",
"We have defined the two-body state ${{\\bf Q},{\\bf q}_r}\\equiv \\hat{c}_{\\bf q}^\\dag x_{\\bf 0}^\\dag {vac}$ in terms of the relative momentum ${\\bf q}_r$ and the total momentum ${\\bf Q}$ which are related to the electron momentum via ${\\bf q}_r+{\\bf Q}m/m_T={\\bf q}$ and the exciton momentum via $-{\\bf q}_r+{\\bf Q}m_X/m_T={\\bf 0}$ .",
"To evaluate the matrix element of the transition operator for an electron-exciton interaction that yields a trion bound state, we use the relationship between the Green's operator and the transition operator at energy $E$ : G(E) = G0(E)+G0(E)T(E)G0(E).",
"Here, $\\hat{G}(E)=\\frac{1}{E-\\hat{H}+i0}$ and $\\hat{G}_0(E)=\\frac{1}{E-\\hat{H}_0-\\hat{H}_{0X}+i0}$ (which, for the two-body problem, should be evaluated in the canonical ensemble, effectively taking $\\mu =0$ ).",
"Close to the trion resonance, we can neglect the first term, and we therefore have the spectral representation T0(E)Q,qr G0-1(E)G(E)G0-1(E)Q,qr =(E-q)2G(E)Q,qr (E-q)2|qr(0)|2E+T-TQ+i0 (-T+TQ-q)2|qr(0)|2E+T-TQ+i0, where in the third step we approximated the expectation value by inserting the trion state, and in the last step we used the pole condition.",
"Taking $E=\\omega +\\epsilon _{\\bf q}$ and using ${\\bf Q}={\\bf q}$ we see that Eq.",
"() reduces to Eq.",
"(REF ) which demonstrates that it holds for arbitrary electron-exciton interactions that lead to trion formation.",
"In the special case of contact interactions, the numerator of Eq.",
"() reduces to $Z_T$ due to Eq.",
"(), and therefore it reproduces the pole expansion in Eq.",
"(REF ) as it should.",
"Note that the fact that the $T$ matrix in this case is independent of the relative momentum is a special feature of contact interactions."
]
] | 2212.05635 |
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