Keldysh approach for nonequilibrium phase transitions in quantum optics: Beyond the Dicke model in optical cavities

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Citation Torre, Emanuele, Sebastian Diehl, Mikhail Lukin, Subir Sachdev, and Philipp Strack. 2013. “Keldysh approach for nonequilibrium phase transitions in quantum optics: Beyond the Dicke model in optical cavities.” Physical Review A 87 (2) (February 21). doi:10.1103/ PhysRevA.87.023831. http://dx.doi.org/10.1103/PhysRevA.87.023831.

Published Version doi:10.1103/PhysRevA.87.023831

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:11688797

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Keldysh approach for non-equilibrium phase transitions in quantum optics: beyond the Dicke model in optical cavities

1, 2, 3 1 1 1 Emanuele G. Dalla Torre, ⇤ Sebastian Diehl, Mikhail D. Lukin, Subir Sachdev, and Philipp Strack 1Department of Physics, Harvard University, Cambridge MA 02138 2Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria 3Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria (Dated: January 16, 2013) We investigate non-equilibrium phase transitions for driven atomic ensembles, interacting with a cavity mode, coupled to a Markovian dissipative bath. In the thermodynamic limit and at low-frequencies, we show that the distribution function of the photonic mode is thermal, with an e↵ective temperature set by the atom-photon interaction strength. This behavior characterizes the static and dynamic critical exponents of the associated su- perradiance transition. Motivated by these considerations, we develop a general Keldysh path integral approach, that allows us to study physically relevant nonlinearities beyond the idealized Dicke model. Using standard di- agrammatic techniques, we take into account the leading-order corrections due to the finite number of atoms N. For finite N, the photon mode behaves as a damped, classical non-linear oscillator at finite temperature. For the atoms, we propose a Dicke action that can be solved for any N and correctly captures the atoms’ depolarization due to dissipative dephasing.

I. INTRODUCTION portional to the atom-photon interaction strength. The static and dynamic critical exponents at the high-temperature phase Much interest has recently been directed towards under- transitions are analogous to those in a conventional laser (or, standing many-body dynamics in open systems away from more precisely, optical parametric oscillator) threshold. thermal equilibrium. This subject is not new, as the analogies To treat the interplay between external driving, dissipation between threshold phenomena in dynamical systems, such as and many-body interactions in a more general setting, we next the laser, and the conventional phase transitions have been rec- develop a unified approach to describe phase transitions in ognized over 40 years ago. However, recent experiments with open quantum systems based on Keldysh path integrals [15– ultracold atoms in optical cavities o↵er intriguing possibili- 21]. This approach is used to analyze the driven Dicke model ties to explore the physics of strongly interacting atom-photon in the presence of finite-size e↵ects and atomic dissipative systems far away from thermal equilibrium from a new van- processes. Both perturbations are non-linear and cannot be tage point. Many fundamental concepts of condensed matter treated by the usual quantum-optical methods. Instead, we ap- physics, ranging from classification of phase transitions to the ply non-perturbative techniques, specific to the path-integral universal behavior of correlation functions in the vicinity of approach. We find that the low-frequency dynamics is ther- quantum critical points in the presence of driving and dissipa- mal even in this case, allowing for an e↵ective equilibrium tion, need to be revisited in light of these developments. description. In this paper, we investigate non-equilibrium phase tran- We expect the Keldysh approach to be directly applicable sitions for driven atomic ensembles interacting with a cavity to other dissipative models such as the recently discussed cen- mode that is subject to dissipation, focusing specifically on tral spin model [22] or fermionic lattice models [23–26]. We the dynamical superradiance transitions and associated self- believe the Keldysh calculations are not more involved, and organization of the atoms observed in Refs. [1–3]. Due to the sometimes simpler, than those of the usual quantum optics interplay of external driving, Hamiltonian dynamics and dissi- frameworks [27, 28]. At the same time, they facilitate an pative processes, the observed Dicke superradiance transitions easy comparison to other phase transitions of condensed mat- [1, 2] exhibit several properties [4–7] which are not present in ter physics. One of the objectives of the present paper is to the closed Dicke model [8–11]. Recently, it was shown that make the Keldysh approach more accessible to the broader other more interesting quantum many-body phases, such as quantum optics community. At the same time, we hope that quantum spin and charge glasses with long-range, random in- the Keldysh perspective will be helpful for condensed mat- teractions [12–14] mediated by multiple photon modes, could ter physicists to understand driven dissipative atom-photon potentially be simulated with many-body cavity QED. systems–especially in view of the qualitatively di↵erent en- arXiv:1210.3623v2 [cond-mat.quant-gas] 15 Jan 2013 In what follows we first review the physics of the non- ergy scales and bath properties in quantum optics. equilibrium Dicke transition in optical cavities with con- The paper is organized as follows. In Sec. II we intro- ventional techniques of quantum optics. Using linearized duce the Dicke model and perform a brief analysis with lin- Heisenberg-Langevin equations, we demonstrate that at low earized Heisenberg-Langevin equations pointing out that the frequencies, close to the phase transition, the system evolves relevant low-frequency correlations are thermal. In Sec. III, into a thermal state with a high e↵ective temperature, pro- we map the operators of the master equation and the associ- ated Liouvillians for dissipative processes to the field content of an equivalent real-time, dissipative Keldysh action S [a⇤, a] with a⇤, a the photon field variables. In Sec. IV we introduce ⇤ [email protected] the atomic degrees of freedom and study the thermodynamic 2 limit N of the open Dicke model. We define a non- 2. Dissipation: In addition to the coherent dynamics gener- equilibrium!1 distribution function F(!) and compare it to the ated by the Hamiltonian Eq. (1), there is a dissipative contri- equilibrium case, finding that both diverge at low-frequencies bution consisting of cavity loss and dissipative processes for as 1/!. In Sec. V we study the e↵ects of a finite size N. Com- the atoms. In the rotating frame, this vacuum is e↵ectively bining analytic and numerical methods, we derive the critical out-of-equilbrium, and leads to the non-equilibrium Marko- scaling of the photon number as function of N, and find it to be vian master equation (cf. Appendix A), equivalent to an equilibrium system at finite temperature and distinct from the zero temperature case. In Sec. VI we propose @t⇢ = i[H,⇢] + ⇢, (2) L an e↵ective method to describe the e↵ects of single atom de- Here ⇢ is the density matrix and the Liouville operator in cay across the phase transition of the Dicke model. We again L find that the distribution function is thermal, but with renor- Lindblad form malized couplings and, in particular, a di↵erent critical cou- ⇢ = ↵ 2L↵⇢L↵† L↵† L↵,⇢ , (3) pling gc. Sec. VII concludes the paper with a summary of our L ↵ { } main results and some final remarks. X ⇣ ⌘ where the curly brackets , denotes the anti-commutator and { } L↵ is a set of Lindblad or quantum jump operators. In the II. THERMAL NATURE OF THE OPEN DICKE present work we consider two types of disspative processes: TRANSITION cavity photon loss and single atom dissipative dephasing. The former is modeled by the Liouvillian: The Dicke Hamiltonian [8, 9] describes N two-level sys- x z tems or “qubits” represented by Pauli matrix operators , cav ⇢ = (2ˆa⇢aˆ† aˆ†aˆ,⇢ ) , (4) i i L { } coupled to a quantized photon mode represented by bosonic creation and annihilation operatorsa ˆ†,ˆa: where  is an e↵ective decay rate (inverse lifetime) of a cavity photon of the order a few MHz [2]. Modelling the dissipative N N dynamics of the atoms depends on the specific implementa- !z z g x H = !0aˆ†aˆ + + aˆ† + aˆ . (1) tion of the driven Dicke model usually involving local pro- 2 i i i=1 pN i=1 X X ⇣ ⌘ cesses of each two-level atom separately. In Sect. VI, we account for dissipative dephasing of the atoms in an approxi- Here ! is the photon frequency, ! the level-splitting of the 0 z mate way by resorting to a simplified e↵ective low frequency qubits, and g the qubit-photon coupling, assumed to couple model. all qubits uniformly to the photon. Eq. (1) is invariant under x x an Ising-type Z2-transformation,a ˆ aˆ and i i . In the thermodynamic limit N , and! for suciently! strong qubit-photon coupling g, the!1 ground state of Eq. (1) sponta- A. Heisenberg-Langevin analysis neously breaks this Ising symmetry and exhibits a phase tran- sition to a “superradiant” phase with a photon condensate aˆ . We now study the above non-equilibrium Dicke model h i In the context of ultracold dilute gases in optical cavities, using conventional quantum optical techniques, namely, the Dimer et al. [4] proposed to implement the qubits using two Heisenberg-Langevin equations of motion. We will later re- hyperfine states of the atoms and showed that, close to the peat and extend these calculations using the Keldysh path in- transition, the relevant Hilbert space can be exactly mapped tegral approach, in the following Sections. The master equa- to Eq. (1). Inspired by the work of Dimer et al. [4], the tion (2–3) with Hamiltonian (1) and cavity dissipation (4) is qubit states of the Dicke model were realized using two collec- equivalent to the equations of motion: tive motional degrees of the Bose-Einstein condensate of the ˙ ig x atoms in the cavity [2, 29], see [30] for a review. In that case aˆ = i!0aˆ aˆ i + , pN F !z becomes a collective recoil frequency and the two Dicke states are components of a dynamically forming charge den- + + ig z ˙ i = i!zi (ˆa + aˆ†), sity wave. pN i This open realization of the Dicke model in optical cavities z ig + with pumped atoms is di↵erent from the closed system Dicke ˙ i = 2 (i i)(ˆa + aˆ†). (5) pN model Eq. (1), due to the interplay of coherent drive and dis- sipation: Here the force = (t) is a stochastic Markovian operator F F 1. Coherent drive: The photon-atom coupling g de- satisfying (t) †(t0) = 2(t t0) and (t)† (t) = 0. hF F i hF F i scribes the scattering of pump-photons and rotates, as func- This term is needed in order to preserve the commutation re- tion of time, at the pump frequency !p. To obtain the time- lation [ˆa(t), aˆ†(t)] = 1, which would otherwise exponentially independent Dicke model (1), one has to move to a rotat- decrease. See for example Ref. [32] for a detailed study of the ing frame, where the explicit time-dependence of the origi- single-atom case, N = 1. nal Hamiltonian is “gauged” away[31]. In this frame, the pa- To analyze the dynamics below the superradiance threshold rameter !0 appearing in Eq. (1) is the cavity-pump detuning in the limit of N , we assume that the atoms are fully ! = ! ! , where ! the bare cavity frequency. polarized S z = 1!1z N/2, and neglect non-linear 0 c p c 2 i i ⇡ P 3 terms in the equations of motion. The resulting operator of the field to an equilibrium situation in the lab frame. equations can be solved exactly in the Fourier domain as has been done in detail in the work of Dimer et al. [4] and we will not repeat their calculations here. We just add B. Photon flux exponent one simple point to their comprehensive analysis, namely that the relevant low-frequency dynamics of the photons The Langevin equation (6) becomes dynamically unstable occurs in the presence of a finite e↵ective temperature. at the Dicke transition, correspondent to the point where ↵ This is most easily seen from the equation of motion for vanishes, or equivalently to the critical coupling the real-valued phase-space coordinate x(!) = (ˆa(!) + aˆ ( !))/ p2! . Defining stochastic force operators f (!) = † 0 !2 + 2 1 (!) ( i(! + !)) + ( !) ( + i(! !)) , we g = 0 ! . p2! 0 † 0 c z (11) 0 F F s 4!0 can writeh the equation of motion as i 2 Upon approaching the Dicke transition, the number of pho- 4g !z!0 ⌫x (  + i!)2 + !2 x(!) = f (!) . (6) tons diverges as g gc , where ⌫x is called the “photon 0 !2 !2 | | z ! flux exponent” [5]. For the present non-equilbrium Dicke transition, it was found [5, 6] that ⌫x = 1, in contrast to The force operator satisfies the equilibrium case of a quantum phase transition at zero ⌫ = / 2 + !2 + !2 temperature[11], where x 1 2. We now explain that this 1 0 discrepancy is due to the finite e↵ective temperature of the 2 f (!) f (!0) + f (!0) f (!) =  (! + !0) . (7) h i !0 low-frequency fluctuations of the system. The photon number is related to the fluctuations of x by At low frequencies,we can neglect high-order terms in !. Eq. (6) becomes identical to the Langevin equation of a classi- 2 2 n + 1 = 2! x2 + p2 = 2! 1 + x2 , (12) cal particle in a harmonic potential with oscillation frequency h i 0 h i h i 0 !2 h i ↵, defined by 0 0 1 ⇣ ⌘ B C @B AC 2 Here we defined p = i(ˆa aˆ†)/ p2!0 and, by repeating 2 2 2 4g !0 ↵ =  + !0 , (8) the above derivation of the Langevin equation, observed that !z 2 2 2 2 2 p = ( /!0) x . We can compute x using an equi- libriumh i partitionh functioni equivalent [33,h 34]i to the (low- and friction constant 2. In the same low-frequency approxi- frequency limit) of the Langevin equation (6): mation, the correlation function of the stochastic force opera- tors on the right-hand-side of Eq. (7) becomes identical to the F 1 “noise correlations” provided by an equilibrium classical bath Z = exp , with F = ↵2 x2 . (13) T e↵ 2 at a non-zero temperature x ! !2 + 2 2 Performing the Gaussian integral we obtain e↵ 0 gc Tx = = . (9) 4!0 !z 2 T e↵ 1 + = + ! x , 2 n 1 2 1 2 0 2 (14) In contrast to the temperature in an equilibrium problem, h i ! ↵ ⇠ g gc 0 0 1 | | the low-frequency e↵ective temperature here is not a global B C B C property of the system, but is in general observable depen- leading to the correct photon@ fluxA exponent ⌫x = 1. dent. In Sec. IV D, we present a systematic, generalizable The above results indicate that, from the point of view of way to extract low-frequency e↵ective temperatures based on phase transitions, it is incorrect to call the driven Dicke tran- observable-dependent fluctuation-dissipation relations. We sition a quantum phase transition even though it is “made already here quote the e↵ective temperature for the atoms (see of quantum ingredients” (two collective motional states of a Sec. VI for the details of the computation): BEC [2]). Instead, it should be regarded as a classical phase transition belonging to the dynamical universality class of 2 2 e↵ + !z the classical Ising model with no conserved quantities and T = , (10) infinite-range interactions, a mean-field version of the “Model 4!z A” of Hohenberg and Halperin [35]. The e↵ect that dissi- where !z is the recoil energy and an e↵ective single atom de- pation induces a finite e↵ective temperature is not new. In e↵ e↵ cay rate. Because T , Tx , the di↵erent parts of the driven several other condensed matter systems [36–38], the coupling system thus do not equilibrate to each other and, although the to a non-equilibrium bath typically admixes the pure many- dominant low-frequency correlations are thermal, the system body states of the closed system, transforming pure quan- is not in a global thermal state. We remark that our defini- tum phase transitions into thermal phase transitions (see also tion of e↵ective temperature does not coincide with the one [16, 17, 22, 39, 40]). What is perhaps more surprising is that commonly used in laser theory, as discussed in Sec. IV D: the neither the low-frequency e↵ective temperature for the pho- former relates the fluctuations of the field to its response in tons, nor for the atoms, is set by the cavity loss rate , but the rotating frame, while the latter compares the fluctuations instead set by the atom-photon interaction g and the e↵ective 4 single atom parameters, respectively. III. KELDYSH APPROACH FOR CAVITY VACUUM As a side remark, we note that, being complex-valued ob- jects, the photons have two normal modes. One quadrature In this section, we introduce the real-time Keldysh formal- is thermally amplified and diverges at the Dicke transition. ism and fix our notation by considering the case of a single Its orthogonal quadrature is quantum squeezed [4], remains electromagnetic mode in an open cavity (without atoms). We gapped at the transition and therefore does not influence the also indicate how to relate the operators of the master equa- thermal nature of the phase transition. These attenuated and tion (2) to the field content and choice of time contour in a amplified quadratures arise naturally as the eigenmodes of the Keldysh action (see also Refs. [15, 18, 42]). photon correlation function (see Sec. IV B). The decay of a single boson (the cavity photon) into a con- tinuum of modes (the external vacuum) is described by the master equation

@ ⇢ = i[! aˆ†aˆ,⇢] + (2ˆa⇢aˆ† aˆ†aˆ,⇢ ). (20) t 0 { } C. Dynamic critical exponent This equation results from the (fully unitary) Heisenberg equation for the coupled system-bath setting, where the sys- In addition to the photon flux exponent, we identify a sec- tem is described by the degrees of freedoma ˆ, aˆ† and the bath ond indicator of criticality, the dynamical exponent. This ex- by a continuum of harmonic modes. Eq. (20) is obtained by ponent governs the decay of the two-time correlations close eliminating (“integrating-out”) the bath in the Born-Markov to criticality. Going back to the Langevin equation (6) and and rotating wave approximation (cf. e.g. [27]). The integra- keeping the !2 terms we obtain: tion of the bath variables gives rise to an e↵ective evolution including dissipative terms. Performing the same program in V!2 + 2i! + ↵2 x(!) = f (!), (15) path integral formulation, we obtain the Markovian dissipa- 2 2 2 tive action ⇣ !0⌘ +  + ! f (!) f (!0) =  (! + !0) , (16) h i ! 1 0 S a = dt a+⇤ (i@t !0)a+ a⇤ (i@t !0)a (21) where we defined a dimensionless parameter V = 1 + Z1 ⇣ 2 3 i[2a+a⇤ (a+⇤ a+ + a⇤ a )] . 4!0g /! . For simplicity we further approximate V 1 and z ⇡ obtain the correlation function [41] In the path integral formalism, the quantum mechanical⌘ opera- tors are replaced by fluctuating, time-dependent and complex- xˆ(t), xˆ(0) = iGK (t) (17) h{ }i xx valued fields (we omit the time argument for notational sim- d! (!2 + 2 + !2) plicity). The fact that the density matrix can be acted on from = i!t 0 i e 2 2 2 2 2⇡ 2!0[(2!) + (↵ ! ) ] both sides, as reflected in the Heisenberg commutator struc- Z t ture of the original evolution equation, finds its counterpart in e = m2 !2 + 2 + ↵2 cos pm2t the presence of a forward (+) and backward (-) component of ! m2 ↵2 0 8 0 the fields. The former is associated to an action on the den- h ⇣ ⌘ ⇣ ⌘ +  pm2 !2 + 2 ↵2 sin pm2t , sity matrix from the left, and the latter to the right. Indeed, in 0 the first of line of Eq. (21) there is a relative minus sign be- ⇣ ⌘ ⇣ ⌘i where we defined m2 = ↵2 2. In the vicinity of the transition, tween the terms involving the two components, reflecting the for 0 <↵<, the frequency m becomes purely imaginary and Heisenberg commutator structure of Eq. (2). The terms in the the oscillatory behavior in the above expression disappears. second line instead display the characteristic Lindblad form; As already mentioned, this is a generic feature of dissipative the “jump” or “recycling” term is represented by an explicit phase transitions (see App. C). For suciently large times and coupling of the two contours. approaching the transition ↵ 0 (where only the closest pole ! It is convenient [19–21] to introduce “center-of-mass” and to zero contributes), we then obtain “relative” field coordinates, acl = (a+ + a )/ p2, aq = (a+ !2+2 a )/ p2. These new coordinates are often referred to as “clas- 0 t/⇠t xˆ(t), xˆ(0) = 2 e . (18) h{ }i 8!0↵ sical” and “quantum” fields, because the first can acquire an The correlation time: expectation value while the second one cannot in the absence of sources. In this basis, and going to frequency space, we 2 1 write ⇠t = (19) ↵2 ⇠ g g ⌫t c A 1 | | 0[G ] (!) acl S = a⇤ , a⇤ , is governed by a dynamical exponent ⌫ = 1. a ( cl q) R 1 K (22) t ! [G ] (!) D (!) aq Z 0 1 ! The remainder of the paper is dedicated to the development B C @B 1 d! AC of a unified, and generalizable, Keldysh approach. The Dicke where we used the notation ! = 2⇡ , and acl,q(t) = i!t ! 1 model will be used as a prototypical test object and we com- ! e acl,q( ). This classical-quantumR R basis is often referred pare our results to those of other approaches, where available. to as RAK basis: the entries are the inverse Retarded (lower R 5 left) and Advanced (upper right) Green’s functions, and the the bosonic degrees of freedom inverse Keldysh component. The RAK action Eq. (22) can be easily inverted to deliver the photonic Green’s functions d! (!) = [ˆa, aˆ†] = 1 . (29) 2⇡ Aaa† h i K R A 1 1 Z G (!) G (!) 0[G ] (!) A = 1 , This “sum rule” is an exact property of the theory valid in– G (!)0 [GR] (!) DK(!) ! ! and out–of– equilibrium. In our example of one cavity mode, (23) 2 where the Keldysh Green’s functions is a matrix product (!) = . (30) Aaa† (! ! )2 + 2 0 GK(!) = GR(!)DK(!)GA(!) . (24) For the open cavity of Eq. (21) the RAK inverse Green’s func- B. Cavity correlation function tions are

1 R/A R/A K K The correlation function encodes the system’s internal cor- [G ] = ! !0 +⌃ , D =⌃ , (25) a a relations, for example the frequency-resolved photon spec- with the “self-energies” trum of the intracavity photon fields. In the steady state, the photon correlation function is related to the Keldysh Green’s ⌃A = i, ⌃R =+i, ⌃K = 2i. (26) function by a a a K It is a key property of a Markovian system that the Keldysh aa† (t) = aˆ(t), aˆ†(0) = aˆ(t)ˆa†(0) + aˆ†(0)ˆa(t) = iG (t). K C h{ }i h i component ⌃a, in Eq. (26) is frequency independent. As can (31) be seen from App. A, this is due to a separation of scales be- Here the last identity is valid only in the specific case of a tween (i) the large pump (!p) and cavity frequency (!c), both optical frequencies in the Tera Hertz range ( 1014 Hz cor- scalar Keldysh Green’s function. At equal times this relation responding to temperatures T 104K), and⇠ (ii) the charac- results in ⇠ teristic frequencies of the electromagnetic vacuum outside the K d! K 12 (0) = 2 aˆ†aˆ + 1 = iG (t = 0) = i G (!) .(32) cavity (. 10 Hz corresponding to temperatures T . 300K). Caa† h i 2⇡ Z In the literature, it is often argued that a frequency inde- For the single decaying cavity mode, characterized by pendent, nonzero inverse Keldysh component indicates an ef- Eqs. (22-26), it is easy to show that (!) = (!) and fective finite temperature state. The Markovian lossy cavity aa† aa† the frequency integral over the KeldyshC Green’sA function is is a simple counterexample: Even though the inverse Keldysh unity yielding aˆ aˆ = 0. As expected, the steady state corre- component is constant 2i, the state is pure and the e↵ective † sponds to the cavityh i vacuum. temperature zero, as we⇠ will argue below. We next introduce the key propagators that encode the sys- tems’ response and correlation functions. In equilibrium, the C. Comparison with a closed system at equilibrium two are rigidly related by the Bose (or Fermi) distribution function; out-of-equilibrium, no such a priori knowledge is available, and it is important to distinguish them. In the absence of dissipation, the cavity becomes an iso- lated harmonic oscillator. Its inverse Green’s functions are still given by (22) with the self-energies serving only as regu- larization parameters. At equilibrium,

⌃A = i✏, ⌃R =+i✏, A. Cavity spectral response function a,EQ a,EQ ! ⌃K (!) = 2i✏ coth . (33) a,EQ 2T The spectral response function encodes the system’s  response to active, external perturbations such as time- Here ✏ 0 at the end of the calculation and T is the actual modulated external fields coupling to spin or charge operators, temperature.! Note that the Keldysh component is odd with for example. The spectral response function is the di↵erence K K respect to the frequency ⌃a,EQ( !) = ⌃a,EQ(!), while in the between the retarded and advanced Green’s function: Markovian system (26) it is even. Using Eq. (33), we obtain (!) = i(GR(!) GA(!)). (27) A EQ 2✏ In the scalar case considered here, we have (!) = lim = 2⇡(! !0), (34) Aaa† ✏ 0 (! ! )2 + ✏2 ! 0 R ✏ aa† (!) = 2ImG (!), (28) EQ 2 ! ! A (!) = lim coth T = 2⇡coth T (! !0), Caa† ✏ 0 (! ! )2 + ✏2 2 2 The frequency-integrated spectral response function is nor- ! 0 malized to unity, because of the exact commutator relation of In this noninteracting case, the spectral response function is 6 fully centered at the isolated mode with frequency !0.We In case of a decaying cavity, the Green’s functions are observe that formally, the thermodynamic equilibrium limit scalars and we can easily invert (35) to obtain can be seen as a situation with an infinitesimal loss, replacing K  ✏, and the replacement in the inverse Keldysh component G (!) aa† (!) ! ! F(!) = = C = 1. (37) 2i 2i✏ coth . GR(!) GA(!) (!) ! 2T Aaa† confirming that the cavity vacuum has a zero e↵ective tem- perature. Moreover, as for a pure quantum state in the equilib- ↵ D. Cavity distribution function and low-frequency e ective rium case at T = 0, the distribution function here also squares temperature to a unit matrix, F2(!) = 1.

The response and correlations allow us to define a An important di↵erence between the zero temperature equi- fluctuation-dissipation relation, by introducing the distribu- librium case and Markovian case appears in the sign of the tion function F(!): distribution function. In the former case (as for any equilib- rium distribution), F(!) is anti-symmetric with respect to the K R A G (!) = G (!)F(!) F(!)G (!) (35) frequency !. On the contrary, for a Markovian bath the distri- 1 1 DK(!) = [GR(!)] F(!) F(!)[GA(!)] , bution function is symmetric with respect to !; one signature , of a strongly-out-of-equilibrium system. where the equivalence holds due to Eq. (24). At thermal equi- librium the distribution F is universal and equals to the unit matrix times

EQ ! ! F (!) = coth 2T = 2nB( T ) + 1, EQ FT=0(!) = sign(!), IV. KELDYSH APPROACH FOR PHOTON OBSERVABLES EQ 2T F! T (!) + ... (36) ⌧ ⇡ ! 1 with the Bose distribution nB(x) = (exp x 1) . The unit ma- We now analyze the Dicke model Eq. (1) with the path trix in field space signals detailed balance between all subparts integral approach explained in the previous section. We in- of the system. clude cavity photon loss but defer the inclusion of dissipa- In the present case, the system is out-of-equilibrium due to tive processes for the atoms to Sec. VI. Assuming homoge- its driven and dissipative nature. A notion of a temperature neous qubit-cavity coupling, one can use the large-N strategy is not a priori meaningful: Neither must the driven system of Emary and Brandes [11]. We introduce collective large-N = 1 N z x = 1 N + + equilibrate to an external heat bath with temperature T (in our spin operators S z 2 i=1 i and S 2 i=1( i i) , to write the Dicke model (1) in terms of one large spin coupled case, due to the separation of scales underlying the Markov P P approximation, this temperature would be e↵ectively T = 0 to the cavity photon mode, compared to the system scales), nor do the di↵erent subparts 2g x of the system have to equilibrate with respect to each other. In H = !0aˆ†aˆ + !zS z + S aˆ† + aˆ . (38) this work, we argue that a notion of a temperature nevertheless pN ⇣ ⌘ emerges as a universal feature of the low frequency domain We then express the spin in terms of a Holstein-Primako↵ of Markovian systems. It is introduced by computing the F [11, 43] boson operator bˆ, defined by S = N/2 + bˆ†bˆ, matrix through Eq. (35) and comparing the low-frequency be- z S + = pN nˆbˆ pN(1 + nˆ/(2N))bˆ , and S x = (S + + S )/2. havior of its eigenvalues with the equilibrium result of Eq. † † Neglecting unimportant⇡ constants we obtain the normal or- (36). In particular, if F has a thermal infrared enhancement dered Hamiltonian: 1/! for small frequencies, its dimensionful coecient is ⇠ ↵ identified as an e ective temperature. This notion of a “low H = !0aˆ†aˆ + !zbˆ†bˆ (39) frequency e↵ective temperature” (LET) becomes particularly 1 +g aˆ + aˆ† bˆ + bˆ† bˆ†(bˆ† + bˆ)bˆ . relevant in the vicinity of a phase transition, where the spec- 2N R, A tral weight encoded in G G is concentrated near zero fre- At N , the⇣ last, non-quadratic⌘⇣ term vanishes⌘ and the quency. Below, we will use this concept to establish a con- problem!1 reduces to a linear system of two coupled bosonic de- nection of Markovian quantum systems to the classical the- grees of freedom one of which (the cavity modea ˆ) decays into ory of dynamical universality classes according to Hohenberg a Markovian bath. As outlined in section III, we can trans- and Halperin [35]. Moreover, we find that, while all governed form the Liouvillian Eq. (4), with Hamiltonian Eq. (39), into /! by the 1 divergence in the distribution function, di↵erent an equivalent Keldysh action with subparts of the system exhibit di↵erent LETs. In contrast to the global temperature present in thermodynamic equilibrium, S = S a + S b + S ab (40) the LET is not an external parameter but rather a system im- manent quantity, determined by the interplay of unitary and 0 ! !z bcl S b = (bcl⇤ , bq⇤) ! !z 0 bq dissipative dynamics. Z! ! ! 7 where S a is given by Eq. (22) and the interaction in terms of with the action the “classical” and “quantum” fields reads 1 A 1 0[G4 4] (!) S N = V†(!) 1 ⇥ V8(!) . 8 R K 2 ! [G4 4] (!) D4 4 S ab = g [(aq + aq⇤)(bcl + bcl⇤ ) + (acl + acl⇤ )(bq + bq⇤)] Z 0 ⇥ ⇥ 1 ! B C (47) Z @B AC 1 (a + a⇤ )(b + b⇤) + (a + a⇤)(b + b⇤ ) b⇤ b + b⇤b The subscript N stands for normal phase and the dagger de- cl cl q q q q cl cl cl cl q q 4N notes transposition and complex conjugation. The block† en- ( h ih i tries are 4 4 Green’s functions given by + (acl + acl⇤ )(bcl + bcl⇤ ) + (aq + aq⇤)(bq + bq⇤) bcl⇤ bq + bq⇤bcl . ⇥ ) ! ! + i 0 g g h ih (41)i 0 1 0 ! ! i g g R 0 [G4 4] (!) = 0 1 We now demonstrate that the static saddle point solutions of ⇥ g g ! !z 0 B C this action reproduce the results of other approaches [4]. Vary- B g g 0 ! !z C B C ing S with respect to the quantum components of the fields K B C D4 4 = 2iB diag(,, 0, 0). (48) C ⇥ @ A and substituting acl(t) = p2a0, bcl(t) = p2b0, aq = 0, bq = 0, we obtain the coupled equations To obtain the photon-only action, we now integrate out the Holstein-Primako↵ field b and get a Keldysh func- @S 1 K = ( ! + i)a g 1 b2 2b = 0 , (42) tional integral that goes only over the photon fields: Z = 0 0 0 0 iS [a ,a] @a⇤ 2N D a , a e photon ⇤ . with the photon-only action q ! { ⇤ } @S 3 2 1 = ! b g 1 b (a + a⇤) = 0 . (43) R A z 0 0 0 0 0[G2 2] (!) @bq⇤ 2N S photon[a⇤, a] = A†(!) 1 ⇥ A4(!) . ! 4 R K ! [G2 2] (!) D2 2(!) Z 0 ⇥ ⇥ 1 where we chose b = b⇤. These saddle-point equations ad- B C 0 0 B C (49) mit solutions with non-zero “ferromagnetic” moment b0, and @ A super-radiant photon condensate a0, The photon four-vector collects the classical and quantum field components 2 2 2 2 N g gc g gc b = a = p2N (44) acl(!) 0 ± 2 g2 0 ± ! i r s p 0 a⇤ ( !) A (!) = cl . (50) 4 0 a (!) 1 for atom-photon couplings larger than a critical value B q C B a⇤( !) C B q C B C 2 2 B C !0 +  and the block entries are 2 2@ photon Green’sA functions which g > gc = !z , (45) ⇥ s 4!0 we now analyze one-by-one. in agreement with the results known from the literature[5, 6]. We are now going to integrate-out the atomic field b and ob- tain an e↵ective action describing the photons in the normal A. Photon spectral response function phase (g < gc), where a0 = b0 = 0. In App. B, we give the corresponding expressions in the superradiant phase. The inverse retarded Green’s function of the photons is

1 R [G2 2] (!) = ⇥ ! ! + i +⌃R(!) ⌃R(!) 0 , ⌃R( !) ⇤ ! ! i + ⌃R( !) ⇤ 0 ! In the limit of N , we can safely neglect the terms h i h i (51) proportional to 1/N in!1 (40). We can re-write Eq. (40) as 8 8 ⇥ matrix multiplying the 8-component fields: where the interaction induced photon-self energy reads

2 acl(!) 2g ! ⌃R(!) = z . (52) a⇤ ( !) 2 2 cl ! !z 0 bcl(!) 1 B C B bcl⇤ ( !) C The characteristic frequencies of the system are defined by V8(!) = B C (46) 1 B aq(!) C R B C the zeros of the determinant [G2 2] (!), corresponding to the B ! C ⇥ B aq⇤( ) C poles of the response function GR (!). Due to the symmetry B C 2 2 B bq(!) C ⇥ B C B bq⇤( !) C R ⇤ R B C x G2 2( !) x = G2 2(!) , (53) B C ⇥ ⇥ @B AC h i 8

15 � � = � � � ≪ �� Im[�] Im[�] 10 5 −� −� � � −� −� � � " Ω !

Re[�] Re[�] ! 0 aa A !5 −� −� !10 !15 � � ≾ �� � � = �� !2 !1 0 1 2 Im[�] Im[�] Ω

−� −� � � −� −� � � Re[�] Re[�] 15

−� −� 10 " Ω ! ! aa

C 5 FIG. 1. (Color online) Schematic plot of the position of the poles of the retarded Green’s function Eq. (54). (a) At zero coupling g = 0, two poles can be associated with the photonic mode ! = ! + i 0 ± 0 and two with the atomic mode ! = ! . (b) In the presence of ± z !2 !1 0 1 2 a finite coupling g, the modes hybridize and the corresponding fre- quencies are shifted in opposite directions. (c) When approaching the Ω transition, two solutions become purely imaginary and correspond to damped modes. (d) At the transition point g = gc one of the poles A ! FIG. 2. (Color online) Photon spectral response function aa† ( ) approaches zero, making the system dynamically unstable. C ! ! and correlation function aa† ( ) as a function of real frequencies . Numerical parameters: ! = ! = 1,= 0.2 (leading to g 0.51), 0 z c ⇡ and g=0 (dotted), 0.25 (dashed), 0.5 (solid). the poles come in pairs, such that = , meaning that { } { ⇤} they either are pure imaginary or come in pairs with opposite B. Photon correlation function real part. The explicit solution of

1 2g2! 2 2g2! 2 The inverse Keldysh component of the action Eq. (49) is 0 = = ! + z (!+i)2 z R 0 !2 !2 !2 !2 det[G2 2(!)] z z ⇥ ! ! 2i 0 (54) DK = (55) 2 2 02i yields four poles, schematically plotted in Fig. 1. Note that, in ⇥ ! the vicinity of the phase transition, two poles become purely K ! = imaginary. This phenomenon seems to apply to generic dissi- and the Keldysh Green’s function G2 2( ) R K A ⇥ pative transitions as we further describe in App. C. Indeed it G2 2(!)D2 2G2 2(!) is a 2 2 matrix. ⇥ ⇥ ⇥ ⇥ has been previously observed for a dissipative critical central K = K T The first diagonal element of G2 2, aa† i(1 0)G2 2(1 0) spin model [22]. Overdamping of collective modes, due to a corresponds to the photon correlation⇥ C function defined⇥ in similar mechanism, has also been found in dissipative multi- Eq. (31) and is plotted in Fig. 2. As noted above in Eq. (32), mode systems in symmetry broken phases [16, 17]. its frequency integral gives the steady-state photonic occupa- tion and it can be shown to diverge at the Dicke transition The imaginary part of the first diagonal element of GR (!) 2 2 according to Eq. (12). This result will be explicitly derived in corresponds to the photon spectral response function ⇥ , aa† Sec. V C using a low-frequency e↵ective description of GK. defined in (27), and is plotted in Fig. 2 for di↵erent valuesA The photon number diverges in steady state despite the fact of the coupling g. In the absence of atom-photon coupling that the system undergoes photon loss, and no explicit photon g = 0 (dotted curve) there is a single resonance peak at fre- pumping occurs within the model. The reason is that the cou- quency ! = ! , broadened by the cavity decay rate . A finite 0 pling constant g is an e↵ective parameter, which in any con- coupling g (dashed curve) “collectively Rabi-splits” the reso- crete physical realization microscopically involves a coherent nance [44] in two distinct peaks, corresponding to two distinct laser drive process compensating for the loss. poles of the system. Upon approaching the Dicke transition (solid curve), the spectral weight is shifted towards the low- The matrix structure of the Keldysh Green’s function K frequency pole; a precursor to the superradiant cavity mode. G2 2(!) can be conveniently exploited to compute the quadra- ⇥ 9 ture fluctuations for a general phase angle ✓ ! of the order of the atomic detuning !z and vanishes for large frequencies. i✓ i i✓ i✓ K e x (!)x ( !) = e e G (!) , (56) Nevertheless, also in the driven Dicke model, the experi- ✓ ✓ 2 2 i✓ out h i 4!0 ⇥ e mentally relevant homodyne spectrum G (!) of the cavity ! 2 2 ⇣ ⌘ output field shows noise reduction below the⇥ vacuum level in where x✓ is defined by the ✓⇤ quadrature [4]. Following the standard “input-output theory” [45, 47], it is possible to show that the homodyne 1 i✓ i✓ x✓ = e a + e a† . (57) spectrum can be linked to the Keldysh response function via p2!0 ⇣ ⌘ The corresponding equal time, frequency-integrated fluctua- 2 2i 0 R tions x✓(t)x✓(t) = x✓(!)x✓( !) are plotted in Fig. 3 and out ! i G2 2(!) = G2 2(!) 12 2 12 2, (59) h i h i 0 2i ⇥ ⇥ ⇥ diverge at the Dicke transition for all angles ✓ except for ✓⇤ ⇥ ! R defined by 2 where M M† M. By applying the transformation (56) 1 out|| || ⌘ ✓⇤ = ⇡ tan (!0/) . (58) to G2 2, it is then possible to compute the fluctuations of the output⇥ quadrature x (!)x ( !) . This quantity has h out,✓ out,✓ i The angle ✓⇤ can also be obtained as the phase angle for the been studied in detail in Ref. [4]: at the Dicke transition, non-diverging eigenmode of the zero-frequency limit of the the zero-frequency component of ✓ = ✓⇤ tends to the maxi- Keldysh correlation function, thereby naturally yielding the mally attenuated value of 1/(4!0). It should be noted, how- attenuated and amplified quadratures alluded to in Sec. II. ever, that the equal-time, frequency integrated fluctuations

equal-time xout,✓⇤ (t)xout,✓⇤ (t) are zero and therefore not attenuated below Here in the case of the driven Dicke model, the h i photon fluctuations of x✓ inside the cavity are independent of vacuum level. the atom-photon coupling⇤ and in particular are not attenuated below the vacuum noise level (the g = 0 limit, without atoms in the cavity). This is di↵erent from the case of the optical C. Comparison with a closed system at equilibrium parametric oscillator [45, 46] where, at threshold, the equal- time fluctuations of the non-diverging intra-cavity quadrature are reduced to 50% of the vacuum level. This di↵erence can In the case of a closed system at equilibrium, the retarded be traced back the di↵erent frequency dependencies of the ef- Green’s function is the same as Eq. (51), up to the replace- fective driving term in both situations. In the parametric oscil- ment  ✏, and letting ✏ 0. The corresponding spectral response! function (!) is! similar to the one shown for the lator, the driving of the cavity occurs via a classically-treated Aaa† photon pump laser, and the driving amplitude is typically set Markovian case, with the narrow peaks substituted by delta- to a constant coherent field amplitude. In the present case of functions at the resonant frequencies. The Keldysh compo- the driven Dicke model, the e↵ective driving of the cavity is nent of the inverse Green’s function reads, at zero tempera- mediated by the atoms via virtual absorption and emission of ture: photons from the pump laser into the cavity. The correspond- 2 2 2 K 2i✏sign(!)0 ing driving term g !z/(!z ! ) is maximal for frequencies D2 2, EQ = . (60) ⇠ ⇥ 0 2i✏sign(!) ! Again note the di↵erent symmetry under frequency reflection 1.0 with respect to (55). 0.8

0.6 # D. Photon distribution function and low-frequency e↵ective 2 Θ

x temperature

" 0.4

0.2 We now show that the above mentioned di↵erence between the Markovian case and the equilibrium case leads to the gen- 0.0 eration of a “low frequency e↵ective temperature” (LET) for 0.0 0.5 1.0 1.5 2.0 2.5 3.0 the former. For this purpose, we calculate the distribution ma- Θ trix F, defined in Eq. (35). To this end, recall that at thermal equilibrium, F = coth(!/2T)1: it exponentially approaches FIG. 3. Frequency integrated equal-time correlations (x (t))2 as unity at high frequencies ( ! T), and diverges as 2T/! at h ✓ i | | function of the angle ✓, for di↵erent values of the coupling strength low frequencies ( ! T). For our Markovian problem, using | |⌧ approaching the transition at gc 0.51: g=0 (dotted), 0.25 (dashed), Eq. (49) we find: ⇡ 1 0.5 (solid). The fluctuations in the quadrature ✓⇤ = ⇡ tan (! /) 0 ⇡ 1.768 are independent of the coupling strength. Numerical parame- 2 g2! F = + z , (61) ters: !0 = !z = 1.0,  = 0.2 z ! !2 !2 x z 10

4 spectral properties in this regime (cf. Fig. 2). At higher fre- quencies, it finally tends to one, following the non-equilibrium 3 curve f Te↵/(! !z). The approach to the quantum regime f 1 is⇡ polynomial, unlike the exponential approach in the " equilibrium⇡ case. In App. D, we show that the thermal 1/! Ω

! 2 #

f divergence, leading to a finite LET, is generic for Markovian systems. 1 We note that our definition of an e↵ective temperature is not the one commonly used in the context of laser theory [28]. 0 In this context, the e↵ective temperature is used only to de- !2 !1 0 1 2 scribe the fluctuations of the photonic field, as compared to the Ω equilibrium fluctuations in the lab frame. As a consequence, the divergence of the photon number at the phase transition FIG. 4. (Color online) Positive eigenvalue of the distribution func- is always associated to a diverging e↵ective temperature. In tion F(!) for the same parameters as in Fig. 2. At low frequencies contrast, our low-frequency e↵ective temperature (LET) de- the distribution diverges as 2Te↵ /!, where the finite e↵ective tem- scribes the ratio between the fluctuations and the response of 2 perature is proportional to the photon-atom interaction, Te↵ g . the system in the rotating frame. It is finite at the transition ⇠ and, as we will see, allows us to map the Dicke transition to an existing dynamical universality class of equilibrium systems. where z and x are Pauli matrices. The F matrix is hermitian and traceless, so its two eigenvalues are real and opposite: V. FINITE-N CORRECTIONS FROM A KELDYSH AND 2g2/! 1 2 LANGEVIN PERSPECTIVE ! = + z . f ( ) 1 2 (62) ± ± s ! 1 (!/!z) ! We now move beyond the quadratic theory, by consider- Fig. 4 shows the behavior of the positive eigenvalue in two ing the e↵ects of finite N. Based on the formalism developed points of the phase diagram, nearby and far away from the in the previous section, we approach this problem by scal- transition. In both cases, at low frequencies the eigenvalue ing analysis, diagrammatic technique, mapping into a low- diverges as 1/!. Exploiting the analogy to the equilibrium frequency e↵ective Langevin equation. As will be seen be- case, we obtain the LET low, these methods show quantitative agreement with a Monte

2 Carlo solution of the original Master equation, highlighting g the utility of the present formalism. Te↵ = . (63) !z

We find that Te↵ is not proportional to the decay rate . The temperature is rather proportional to the e↵ective interaction A. Scaling analysis between the spin and the photon. g is the scale that leads to a competition of unitary and dissipative dynamics, and the Up to this point we have considered only the thermody- LET is a measure of this: For g = 0, the steady state of the namic limit N . In this limit, the resulting theory is dissipative part of the dynamics (empty cavity) is an eigenstate quadratic and can!1 be studied by Keldysh means as well as by of the Hamiltonian, while this is no longer the case for any the Heisenberg-Langevin method. Corrections due to a finite finite g. In the limiting case of g 0, the LET goes to zero. ! N introduce non-quadratic terms into the problem and require A closer inspection of (62) reveals an important di↵erence a more careful study. The present path-integral approach al- between the Markovian bath and thermal equilibrium, related lows us to develop a diagrammatic approach and to resum all to the presence of a second energy scale, !z. If !z Te↵, this leading-order corrections in an organized fashion. A similar energy scale does not a↵ect the crossover between the quan- approach has been used to study the instability of an optical tum and classical regimes, which then proceeds monotonously parametric oscillator in Ref.s [46, 48, 49], where however the similar to an equilibrium problem. If, on the other hand, emerging low-frequency thermal nature of the problem has !z Te↵, the quantum-classical crossover of the Marko- not been discussed. vian⌧ bath occurs in an unusual way, highlighting the non- Leading 1/N corrections to the Hamiltonian of the Dicke equilibrium nature of the problem. Starting from a divergence model are easily obtained by retaining the first-order terms in at zero frequency, the distribution function (62) decreases as the Holstein-Primako↵ approximation (see Eq. (39)). These Te↵/!, in analogy to an equilibrium system at finite temper- terms are expressed using the Keldysh formalism in Eq. (41) ature. Then, instead of monotonously decreasing towards the and contain products of four fields. In a diagrammatic descrip- quantum regime where f 1, it exhibits a second divergence tion (see Fig. 5), they correspond to four-point vertices. These ⇡ at ! = !z. Since the spectral weight vanishes suciently fast vertices can be “classical” if they contain only one quantum in this regime, the correlation functions still remain finite; the field (either bq or aq), or “quantum” if they contain three of pole in F accounts for a di↵erent scaling of correlations and them. The former type can be casted into a semi-classical 11

that the appropriate transformation is: (a) (c) quantum (d) Σ (b) classical × N N N0 = (68) � 2 � � !

� � � With this modification, the theory including leading 1/N cor- rections becomes scale invariant at the critical point. Consider now for example the photonic fluctuations x2 . This object h cli FIG. 5. (Color online) Non-equilibrium diagrammatic expansion of can be made scale invariant if multiplied by 1/N1/2, indicating the Dicke model to leading order in 1/N. The dashed lines indi- that cate “quantum” fields and the dotted lines “classical” fields. (a) Bare Green’s function of the photons (red) and of the atoms (blue). (b-c) 1 n x2 N1/2 . (69) Leading order 1/N corrections: classical vertices contain only one h i⇡2h cli⇠ quantum field, while quantum vertices contain more than one (three in this case). (d) One loop correction to the retarded Green’s func- Remarkably, the same scaling relation holds for the optical tion. parametric oscillator [49], but is here obtained in the frame- work of the Dicke model. In fact, recent numerical calcula- tions on this model [50] suggested a di↵erent scaling relation ↵ description of the problem, while the latter describe genuine n N , with ↵ = 0.41, in contrast to the present analysis. h i⇠ quantum corrections. To further supplement our analytical result, we now con- Before going into the calculations, let us first study the rel- sider the e↵ects of next-to-leading-order corrections, stem- evance (in the sense of the renormalization group (RG)) of ming from higher-order terms of the Holstein-Primako↵ ex- k 2+2k the classical and quantum vertices with respect to the critical pansion. Their general form is g/N dt . For any k, the / k 1+2k point in the thermodynamic limit (N and g = gc). From most relevant term is the classical vertexR g N dt q cl . the low-frequency expansion of Eq. (76)!1 we obtain that, at this Under the scaling transformation, this term is multiplied by 1 k R point, the photonic Keldysh action corresponds to: . This shows that all terms with k > 1 are irrelevant at a tree level and cannot modify the above scaling relations. 0 2i@t xcl(t) Before proceeding, we briefly compare the present analy- S xx = dt xcl(t) xq(t) , (64) 2i@t 8iTx xq(t) sis with the zero temperature equilibrium case. There, the Z ⇣ ⌘ ! ! Keldysh component of the action would correspond to 4i ! , | | where Tx is defined in Eq. (9). This action is invariant under leading to the scaling transformation xcl pxcl and xq the scaling transformation ! ! pxq. As a consequence, both classical and quantum vertices t t, scale in the same manner, and the latter cannot be disregarded. ! To compute the finite size scaling of expectation values we ob- x (t) p x (t), cl ! cl serve that 1 xq(t) xq(t). (65) g g ! p dt 3 dt . (70) N ! N0 Repeating the same analysis for the atomic field b we again Z Z find that, under the scaling transformation, the classical com- In order to preserve the scale invariance, we therefore need to ponent bcl is increased by a factor p and the quantum com- renormalize N by ponent b decreased by the same factor. Using the scaling q 3 N 1/3 relation (65), we find that N N0 = n N . (71) ! )hi⇠ g g dt 2 dt 3 , (66) N cl cl cl q ! N cl q This relation is known in the literature and has been shown to Z Z be valid for the zero temperature case, both analytically [51] g g 3 dt clqqq dt cl. (67) and numerically [52]. As we explained, it does not hold for N ! N q Z Z the (non-equilibrium) thermal case presented here. Here are one of the a, a⇤, b, b⇤ fields. Eq. (66) indicates that the classical vertex is relevant in the RG sense, while the quantum vertex is at most marginal. In the limit of N 1 its B. Diagrammatic calculations contribution is very small at low frequencies and can be ne- glected. In contrast, the e↵ects of the classical vertex grow as We now use the Keldysh approach to explicitly compute we approach the transition and need to be taken into account. the photon occupation across the transition in the presence of The above scaling transformation can be used to derive 1/N corrections. As discussed in the previous section, the the finite-size scaling of expectation values. For this task, it (bare) Keldysh and retarded propagators of the system are 4 is convenient to combine the scaling transformation with a 4 matrices. In this language, quartic corrections correspond⇥ renormalization of the system size N, such that overall the to forth-order tensors of total size 44 = 256, which we will relevant vertex remains unchanged. Using Eq. (66) we find denote as M¯ . In our case the relevant corrections are (see Eq. 12

(41)) 4.5 g (a + a⇤)(b + b⇤ ) + (b + b⇤)(a + a⇤ ) b b⇤ 4N ! q q cl cl q q cl cl cl cl 4 1.6 R h i 3.5 1.4 + bqbcl⇤ + bq⇤bcl (acl + acl⇤ )(bcl + bcl⇤ ) , (72) +1 )

3 c 1.2 +1 〉 〉 n

⇣ ⌘ n 〈 and the tensor M¯ contains 16 identical entries, M¯ i, j,k,l = 〈 2 2.5 1

g/(4N). ln( 2 2 0.8 The leading-order 1/N corrections can be computed us- ing standard diagrammatic techniques. When constructing 1.5 0.6 one-loop diagrams one needs to remember that a vertex con- 1 0.4 0.6 0.8 1 2 3 4 nects fields at equal time. Because any field satisfies g/g ln(N) R c q(t)q(t) = 0 and G (0) = q(t)cl(t) = 0, to obtain a non-vanishingh i loop one needs toh connect twoi classical edges of the vertex. Thus, one loop corrections renormalize only the FIG. 6. (Color online) LEFT: Photon occupation in the vicinity of retarded and advance Green’s function, as shown in Fig. 5 (d). the Dicke transition. The circles (o) correspond to the one-loop re- The analytic expression of the self energy is: summation, obtained using the Keldysh diagrammatic technique for N = 10 (blue), 40 (red). The dashed lines correspond to the e↵ective R K equilibrium theory, Eq. (84), derived from the Langevin equation. ⌃ (!) = iM¯ d!0 (!0) , (73) · G The crosses (+) correspond to the Monte Carlo solution of the orig- Z inal master equation. The solid curve corresponds to the mean field where the operator “ ” indicates the tensorial product includ- solution, Eq. (14), valid in thermodynamic limit N . RIGHT: !1 ing all allowed permutations· of the indices. The dressed Photon occupation at the critical point as function of the system size. Green’s functions should be computed in a self-consistent The circles (o) and crosses (+) represent respectively diagrammatic manner. The resummation over all one-loop irreducible di- and Monte Carlo results. The dashed line corresponds to the e↵ective agrams leads to the Dyson equation: equilibrium theory (85). Numerical parameters: !z = 2.0, !0 = 1.0,  = 1.0, giving gc = 1.0. K(!) = R(!)DK(!) R(!) † (74) G G G 1 1 [ R(!)] = GR(!) 1 +⌃R(h!)GR(!i) (75) where G R K h i !2 + 2 + !2 Here G and D are explicitly given in Eq. (48). The resulting K 0 R 1 1 D (!) = 2i , [G (!)] = , predictions for the photon occupation are shown by circles in xx ! xx R 0 det[G2 2(!)] Fig. 6. ⇥ (77) R and the determinant of G2 2(!) is given by Eq. (54). As well known, any quadratic⇥ Keldysh action is equiva- lent to a linear Langevin equation. Starting from a generic ↵ C. E ective low-frequency Langevin approach and mapping quadratic action (76) one introduces a Hubbard-Stratonovich to a thermal ensemble “noise” field f (!) to obtain: f ( !) f (!) In this subsection we derive a simple description of the = R ! 1 ! ! ! . S xf 2 [Gxx( )] xcl( ) f ( ) xq( ) K photon-only action (49), focusing on the x = (a + a⇤)/ p2! D (!) 0 Z! xx quadrature, by mapping its Keldysh action to a stochastic n o (78) equation. Using the basic theorems of thermodynamics, we will then convert it into an e↵ective equilibrium free energy, (This action is equivalent to Eq. (76), as can be explicitly and obtain an analytic expression for the number of photons shown by performing the Gaussian integral over f (!), and us- at the critical point. ing GR(!) = GA( !) – see Ref. [20, 21] for more details.) In- K iS xf We first consider the N limit where (as we already side the Keldysh partition function Z = D xq; xcl; f e , !1 the integration over x then takes the form of{ a delta function} saw in Sec. II A using the Heisenberg-Langevin approach), q R the Langevin equation coincides with the equation of mo- with argument tion of a classical particle in an harmonic confinement cou- R ! 1 ! = ! . pled to an equilibrium bath at finite temperature. Using the [Gxx( )] xcl( ) f ( ) (79) Keldysh formalism and starting from Eq. (49), we replace The remaining last part of the action Eq. (78) involves only a = p! /2(x + ip) and a = p! /2(x ip) and integrate 0 ⇤ 0 f (!) and can be thought of as the statistical weight of a Gaus- out the p-component, to obtain: sian random variable with correlations A 1 0[G (!)] xcl(!) K S = x ( !) x ( !) xx , f (!) f (!0) = iDxx(!)(! + !0) . (80) xx cl q R 1 K h i ! [G (!)] D (!) xq(!) Z 0 xx xx 1 ! ⇣ ⌘ B C (76) Eq. (79) then becomes a stochastic equation of motion for @B AC 13 the bosonic field x(!) = xcl(!)/ p2, identical to the Langevin cavity loss (4). Specifically, we apply the Monte Carlo Wave- equation (6) obtained in Sec. II A. Function (MCWF) method [53], as implemented in the open- We now include non-linearities for the photon dynamics source C++QED library [54]. (Specific parameters: num- arising from a finite number of atoms N. Our starting point ber of trajectories Ntraj = 10, time step dt = 1, number is the frequency limit of Eq. (79), (2i! + ↵2)x(!) = f (!), of time steps T = 400). The resulting curves are shown where ↵ is defined in Eq. (8). To this equation we add the in Fig 5. We emphasize that no fitting parameters were most relevant non-linear term in the from of a frequency inde- used when comparing the di↵erent methods. As expected, pendent cubic term: the numerically solution is closer to the predictions of the Keldysh non-equilibrium diagrammatic technique than to the 2 3 3 2@t + ↵ x(t) + x (t) = f (t). (81) low-frequency thermal e↵ective theory. Remarkably, the dif- ference between these two approaches is minimal at the tran- ⇣ ⌘ This equation defines the dynamical critical theory of an Ising sition, in agreement with our identification of the transition as transition with no conserved quantities, the so-called “Model driven by equilibrium thermal fluctuations. A”, for n = 1 degrees of freedom and d = 0 dimensions [35]. The frequency can be determined from the microscopic the- ory, by demanding the e↵ective Langevin description to re- produce the same saddle-point as the original action Eq. (40). Using x = (a + a⇤)/ p2!0, Eq. (8), and Eq. (44) we obtain: VI. KELDYSH APPROACH FOR ATOM OBSERVABLES 2 2 2 2 2 2 ↵ 2!0(g g ) 2( + ! ) = 2 pN c 3 = 0 . 3 2 + !2 ) N! s p 0 z (82) We now analyze the single-atom observables of the open As expected, the parameter vanishes in the thermodynamic Dicke model using a method which is valid for arbitrary val- limit N , where the mean-field linearized description ues of the number of atoms, N. To this end, we represent !1 (79) becomes exact. each of the N atoms by a real field variable `, with the in- The stationary state dictated by the Langevin equation (81) dex ` ranging over all the atoms ` = 1, .., N. Our method re- is equivalent to an equilibrium system with free energy lies on generalizing each ` to have M components, a` with a = 1 ...M, and then taking the large M limit; even though we 1 1 are interested in the M = 1 case, the large M limit is expected F(x) = ↵2 x2 + 3 x4 . (83) 2 4 to properly describe the physics of models with long-range in- teractions [12, 55]. Although we will also consider the large ↵ Here we recall that is defined in Eq. (8) and vanishes at N limit in the present section in the interest of comparing with the transition, while is defined in Eq. (82) and captures the previous results, it is important to note that the present method 1/N corrections. Steady-state expectation values are com- 2 F(x)/T does not require the large N limit, and is valid for general val- 2 dx x e e↵ puted through the thermal average x = F(x)/T . ues of N. Also, in the interests of simplicity, we will not write h i R dx e e↵ out the a index, and directly present the large M approxima- In particular, the photon number is: R tion in the context of the physical M = 1 case. 2 2 n + 1 = 2! 1 + x2 This single-atom representation of the Ising spins allows h i 0 !2 h i 0 0 1 to treat the qualitative e↵ects of atom dissipative dephasing B 2 C 2 F(x)/Te↵ B  C dx x e within a simplified “friction model” for ` which we explain = ! @ + A , 2 0 1 2 (84) below. This process couples directly to the local Ising degrees ! dx e F(x)/Te↵ 0 0 1 R of freedom of the single atoms. The same is true for disorder B C where in the first identity@B we usedAC R the mean field relation due to spatial variations of the qubit-photon couplings [12]. 2 2 2 2 In such cases, one cannot employ the single large-N Holstein- p = x  /!0. At the critical point ↵ = 0 and the inte- gralh i is easilyh i evaluated: Primako↵ representation of the Dicke model.

2 We proceed by introducing into the path integral N La-  Te↵ (3/4) 2 n c + 1 = 2!0 1 + grange multipliers `, corresponding to a suitable Fourier h i !2 3 (1/4) 2 0 0 1 s representation of the delta function [12, 56], (` 1) = B C i(2 1) B 2 C 2 d exp ` . On the closed time contour, this amounts to @B ( + !AC )!z (3/4) = pN 0 . (85) adding the following expression to the action: 3 / s !0 (1 4) R N Here (3/4)/(1/4) 0.338 is the ratio of two Gamma func- 1 2 2 ⇡ S , = `,+(t) `,+(t) 1 `, (t) `, (t) 1 . tions. ± 2!z t Z X`=1 " # To evaluate the precision of the Keldysh and Langevin ⇣ ⌘ ⇣ (86)⌘ methods, we compare their predictions with the solution of the Master equation associated with the Dicke model (1) with Moving to the “classical/quantum” notation and adding the 14 bare action of the atoms we obtain: quency independent term is ruled out for ⌃A,R. The above re- N sults for Markovian baths should be compared with the results 1 1 cl,`(!) for atoms in equilibrium, S , = cl,`( !) q,`( !) G , ! q,`(!) z ! `=1 Z X ⇣ ⌘ ! ⌃A (!) = i✏!, ⌃R (!) =+i✏!, N ,EQ ,EQ 1 ! + q,`(t), (87) K ! ⌃,EQ(!) = 2i✏!coth , (92) z Zt `=1 2T X  !2 +⌃A ! 1 q,` cl,` ,`( ) where one also lets ✏ 0 at the end of the calculation. We G , = 2 R K . ! ,` +⌃ (!) ,` +⌃ (!) ! cl ,` q ,` ! here analyze the Dicke model in terms of the atomic degrees of freedom alone. One can exactly integrate out the pho- Note that cl,` can be associated with the excitation energy tons from the action (89). This is conveniently done by go- of the Ising spins and is to be determined self-consistently. ing to a coordinate representation of the photons: aq/cl(t) = p R/A/K The atom self-energies ⌃ (!) will be explained below. ,` !0 x (t) + ip (t) , and first performing the integration Finally, we have for the atom-cavity interaction: 2 q/cl q/cl qover ⇣pq/cl and subsequently⌘ over xq/cl. We obtain, with S , N given by Eq. (87), the atom-only action, g S a = +,`(t) a+(t) + a+⇤ (t) ,`(t) a (t) + a⇤ (t) . t 2 S [,] = S , + S , 2 , (93) Z X`=1 g h (88) i 1 N g2 S ,g2 = (94) The Keldysh action for the full Dicke model (1) then becomes 2 ! N ⇥ Z `X,m=1 0 A(!) (!) S [a,,] = S + S , + S , (89) cl,m a a cl,`( !) q,`( !) R K , (!) (!) q,m(!) ! ! with the various terms given by Eqs. (22,87, 88). ⇣ ⌘ We model local, single-atom damping in a simple e↵ec- where the matrix entries are tive way which is consistent with symmetry properties of our 2! R(!) = A(!) ⇤ = 0 , real-valued Ising oscillators . The atoms are subject to decay 2 2 (! + i) !0 into photon modes outside the cavity and possible other damp- h i ing mechanisms like s-wave scattering with other momentum- 2i !2 + 2 + !2 K ! = 0 . modes, trap loss or finite-size dephasing [57]. As a result ( ) 2 (95) (!⇣ i)2 !2 ⌘ some fraction of the atoms leave the two-density mode Hilbert 0 space which maps to the Dicke model; others may be sponta- neously scattered back in. Representing the atoms by a com- Our analysis of the above theory will rely on the approxi- plex field ⇤, , we subsume the above processes into Marko- mation of substituting the N Lagrange multipliers by a single vian decay of the atoms with the self-energies e↵ective field q/cl,` q/cl. This “spherical” approximation for the Lagrange multiplier! becomes exact in the limit of a ⌃A = i, ⌃R =+i, ⌃K = 2i, (90) large number of internal spin components M . It can be ⇤ ⇤ ⇤ shown that the critical behavior is not qualitatively!1 modified with an e↵ective single-atom decay rate. Our e↵ective for any finite value of M including the Ising case M = 1 of the real-valued Ising field in Eq. (87) may be viewed as the present paper [55]. real component of the originally complex boson q/cl(t) = The above method is valid for arbitrary values of N, and we 1 ˜ ˜ will describe the general N solution below in Section VI E. 2 q/cl(t) + iq/cl(t) . Integrating out the -component, However, first we present a method which eciently treats q ⇣ ⌘ 2 2 2 the N limit. We decouple Eq. (94) with a Hubbard- A R K ! + + !z !1 ⌃ (!) = i!, ⌃ (!) =+i!, ⌃ (!) = i . Stratonovich field `(!) `(!) and integrate out the - ! 2 z field. We assume to be time-independent$ and spatially uni- (91) form (!) /(2⇡) , and the resulting Keldysh partition ` ! !,0 Note that this simple model for dissipative dephasing couples function to the x projection of the atomic states and does not specify K i N [ ,] Z = D De 2⇡ S , (96) the states of the z projection of the spins. We emphasize, however, that the form of the dissipative self-energies is dic- Z tated by the combination of low-frequency expansion and the obtains a prefactor of the number of atoms N in the exponent real-valued nature of the Ising field . In particular, a fre- multiplying the action 15

q 1 i [ ,] = + g2 R(0) + K(0) 2 + ln ⌃K(!) + !2 +⌃A(!) !2 +⌃R(!) (97) S ! cl q 2 q 2 q q cl cl z ! Z! " #! 4 ⇣ ⌘ ⇣ ⌘⇣ ⌘ g !z K R 2 2 R R K R K 2 ⌃ (0) q (0) + 2cl (0) q (0) cl + (0) q q (0) cl + (0) q . K 2 q 4 q ⌃ (0) q + cl " ⇣ ⌘⇣ ⌘ ⇣ ⌘ ⇣ ⌘ # ⇣ ⌘

Taking N , we now extract the phase diagram, response photon-atom coupling and shifts the Dicke transititon to large and correlation!1 functions, and the value of the order parame- values of the coupling. This e↵ect can be understood in terms ter using a saddle-point approximation. This can be obtained of the atom’s depolarization, leading to a reduction of the ef- by requiring the derivatives with respect to q and q to be fective number of atoms contributing to the super-radiant tran- zero, and then substituting q = 0, cl = , q = 0, cl = . sition. The derivative with respect to q constrains –by construction Eqs. (98) and (100) determine the value of ferromagnetic Eq. (86)– the frequency integral of the Keldysh Green’s func- order parameter as well. When approaching the phase tran- tion to be equal to unity: sition from above (g g ), vanishes as c

@ 2 d! K S = 0 = iG(!) = 1 (98) 3(g + gc) @q )hi 2⇡ = pg gc . (104) Z s 4g2 with Compared with a closed system at equilibrium, Eq. (108), the ! ⌃K(!) order parameter for this open Markovian system vanishes with K z G(!) = enhanced amplitude but with the same mean-field like square- 2 !2 +⌃A(!) !2 +⌃R(!) root exponent. 2 2 ⇣ g ! ⌘⇣ 2 ⌘ 2⇡i(!) z R(0) 2 . (99) 42 ⇣ ⌘ The saddle-point condition for the order parameter yields A. Atom spectral response function

@ g2! R(0) S = 0 1 z = 0 . (100) The single-atom retarded and advanced Green’s function @ ) 2 are determined by the derivative of (97) with respect to q " # cl and reads To determine the position of the Dicke transition, we need to ! R A ⇤ z compute the saddle point value of cl = in the normal and G(!) = G(!) = , (105) in the superradiant phases and equate the two values. In the 2 !2 +⌃R(!) h i normal (N) phase is determined by (98) and (99) with = 0, ⇣ ⌘ from which follows the spectral response function 2 + !2 z R N = . (101) (!) = 2ImG (!) (106) 3 A Note that naively taking 0 does not reproduce the equilib- as the expected frequency-resolved signal from the atoms after ! rium value for N , cf. also Subsec. VI C. In the ferromagnetic local, time-modulated density perturbations. Fig. 7 displays phase (FM) the order parameter acquires a finite expectation the characteristic Lorentzian shape of the spectral response value , 0 and, to fulfill Eq. (100), we need to require the function peaked at p, broadened by single atom decay ⇠ argument of the square bracket to be zero: . The single atom response is smooth across the transition. This is not to be confused with the roton-type mode softening 2 g !0!z observed by Mottl et al. [57] that pertains to the collective = , (102) FM 2 2 atomic density excitations for a finite number of atoms. !0 +  where we have used Eq. (95) for R(0). At the phase bound- aries both (101) and (102) must hold, leading to: B. Atom correlation function

2 + !2 2 + !2 z 0 The atom correlation function is given by the Keldysh gc = . (103) tv⇣ 3!⌘⇣0!z ⌘ Greens function Eq. (99)

Therefore, the spontaneous emission weakens the e↵ective (!) = iGK (!), (107) C 16

20 4

10 2 " " Ω !

0 Ω 0 ! ΦΦ f A !10 !2

!20 !4 !2 !1 0 1 2 !2 !1 0 1 2 Ω Ω

10 FIG. 8. (Color online) Distribution function F(!) for the same numerical parameters as Fig. 7. The expression for the distribution 8 function Eq. (110) is independent of the atom-photon coupling g.

" 6 Ω ! ΦΦ

C 4 obtains the equilibrium analogs of Eqs. (103,104):

2 EQ 1 gc = p!z!0, 0 2 EQ 1 !2 !1 0 1 2 = pg gc . (108) pg Ω We note that, both dissipative channels, cavity photon loss FIG. 7. (Color online) Atomic spectral response function (!) and atomic dissipative dephasing, shift the critical value of the A and atomic correlation function (!) in the normal phase (g < g , C c coupling. The amplitude with which the ferromagnetic order green, dashed) and in the superradiant phase (g = 1.2gc, black, solid). parameter vanishes is also di↵erent. Other parameters used: !0 = !z = 1, =  = 0.2 (leading to Note that in a model for “one-way” spontaneous emission gc 0.6). The arrow illustrates the delta-function contributions from + ⇡ coupling to and starting from a fully polarized atomic Eq. (99) in the superradiant phase. state as assumed in Sec. II A, one would recover the equilib- rium limit for 0. As explained above, our dissipative dephasing model for! spontaneous emission couples to x and thereby assumes a mixed state of the atoms (similar to a many- K G is defined in (99), and is exhibited in Fig. 7. Even before body paramagnet). the onset of the superradiance peaks (black and blue-dashed arrows) for g gc, the correlation function has finite weight at ! = 0. This is the non-equilibrium signature of the dissipative D. Atom distribution function and low-frequency e↵ective dephasing of the pumped atoms due to coupling to the vacuum temperature outside the cavity (a continuum of modes with characteristic frequencies orders of magnitudes lower than the optical pho- tons the pumped atoms emit when they spontaneously decay We now execute the procedure of subsection III D to cal- and absorb). culate the e↵ective temperature of the atoms. With the atom Green’s functions and the simple model for atom decay pre- sented above, one finds

K 2 2 2 (!) G(!) ! + + ! 1 ! = C = = z , F( ) R A (!) G (!) G (!) 2!z ! A (109) C. Comparison with a closed system at equilibrium leading to the e↵ective temperature

Note that taking  0 and 0 in Eq. (103) does not 2 + !2 ! ! e↵ z reproduce the equilibrium value for a closed system. This T = , (110) 4!z is due to non-commuting limits of making the Markov ap- proximation and performing the integral to fulfill the sum rule which is independent of the coupling strength to the photons Eq. (98). To obtain gEQ one needs to use the equilibrium bath (cf. also Fig. 8). For ! , the e↵ective temperature is c ⌧ z self-energies Eqs. (33,92) from the start of the calculation and set by the recoil energy of the atoms ER = !z/2. Within our 17

e↵ model, T also does not depend on the cavity loss rate , con- We now compute finite size N corrections to the single atom trary to what obtains for the model of Ref. [13]. In our case the spectral response function, thereby underlining the strength reason for this is the careful treatment of the thermodynamic of Keldysh path integrals to perform systematic approxima- limit N limit leading to Eqs. (93,97). This limit ensures tion schemes. It should be noted that, in contrast to the that the only!1 photon-induced self-energies for the atoms occur photons’ correlations computed in the previous section, the for zero-frequency quantities (the weight of (!) in Eq. (99)). single-atom correlation functions do not diverge at the transi- tion and do not need to be regularized by the number of atoms N. Thus, for typical cavity QED experiments where the num- E. General N solution for the spectral response function ber of atoms is of the order of 105 106, deviations from the N limit will not be observed in the single-atom observ- !1 The results presented above refer to the N limit. How- ables. Nevertheless, the few-body regime might become in- ever, as we noted earlier, this limit is not really!1 necessary, and teresting in future applications. the methods of this section can produce general N results re- To study the finite-size e↵ects, we write Eq. (93) as a RAK lying only on the M limit. matrix of N N-matrices: !1 ⇥

!2 +⌃A ! N ( ) 1 2 A 0 ` g (!) !z m 2 cl,m(!) cl,`( !) q,`( !) !2 +⌃R(!) , (111) ! 0 1 2 R 1 K 1 2 K 1 q,m(!) `,m=1 ! `m 2 g (!) ! ⌃ (!)`m 2 g (!) ! Z X ⇣ ⌘ B z z C B C @B AC where we have not rescaled the coupling g by N and used the saddle-point values for the other variables. The bottom-left el- 30 R ement of the N N matrix inverse gives G(!), from which 20 follows the spectral⇥ response function (see Eq. (28). To invert this matrix we note that all its diagonal and o↵-diagonal ele- " 10 Ω ments are separately equal to each other. Using this property, ! 0 ΦΦ we obtain that the local response function is: A !10 1 ! !20 GR (!) = 1 z + N !2 + i! !30 ! 1 !z . (112) !1.0 !0.5 0.0 0.5 1.0 N !2 + i! 1 N! g2R(!) 2 z Ω We observe from this expression that as N , only the first term survives, shown in Fig. 7. At finite !1N an additional mode appears in the single-atom spectrum, which vanishes as FIG. 9. (Color online) Finite atom number signatures in the single atom, local spectral response function. Numerical parameters used: N becomes large as shown in Fig. 9. The presence of two g = 0.4gc,  = = 0.2. Line coding: N = 2, black curve; N = 3, modes, the atomic and photonic branch, also emerges from an blue-dashed ; N = 4 purple-dashed-dotted; N = 5, red-dashed; N = analysis in terms of collective, polaritonic variables [11]. 6, orange-dashed. The second peak at ! 0.4 for the black-solid ⇡ curve is pushed to higher energies until for N & 6 only the dominant peak at ! 0.25 remains. ⇡ VII. CONCLUSION

In this paper, we presented a path integral approach for the systems with spatially fluctuating degrees of freedom such as non-equilibrium steady-states of driven quantum systems cou- disorder [12, 14]. In these correlated quantum many-body pled to Markovian baths, such as ultracold atoms in optical situations, Master equation approaches are typically limited cavities. In the past, these systems have more often been de- to relatively small number of atoms and a recipe to compute scribed using a Master equation formalism. We believe that disorder-averaged quantities does not seem to exist. our Keldysh approach allows an easier comparison with other We first applied our formalism to the cavity vacuum equilibrium and non-equilibrium (classical and quantum) sys- (Sec.III) and subsequently added atomic qubits, interacting tems. While some of the results presented here are actually with the cavity through a Dicke interaction, and computed the new, and not just known results re-phrased in a new approach, key observables for both the photons (Sec. IV and V) and the full utility of our approach will become clear then com- atoms (Sec.VI). The key novelties of our analysis are: puting thermodynamics and critical properties of large, open (i) The fluctuation-dissipation relation of a single cavity cou- 18 pled to a Markovian bath in the rotating frame, Eq. (37), dif- 1176/1-1, by the Austrian Science Fund (FWF) through SFB fers from the thermal-equilibrium case. In the former case the FOQUS F4016-N16 and the START grant Y 581-N16, by the bath contains both positive and negative frequency, while in European Commission (AQUTE, NAMEQUAM), and by the the latter it can contain only positive frequency, leading to a Institut fur¨ Quanteninformation GmbH. di↵erent symmetry with respect to ! !. (ii) Nevertheless, in the presence of a drive,! the low-frequency distribution functions of the photons and atoms is thermal- like and diverges as 1/!, allowing the definition of a low- Appendix A: Self-energy of an open cavity in the rotating frame frequency e↵ective temperature⇠ (LET). The LET of the pho- ↵ tons, Eq. (63), and of the atoms, Eq. (110), are however di er- In this appendix, we discuss how the coherent drive with a ent, highlighting the non-equilibrium nature of the problem. frequency scale !p, which exceeds all other frequency scales, (iii) At higher frequencies, the distribution functions display justifies the form of the Markovian dissipative action Eq. (21), non-equilibrium and quantum behaviors. For example, the which in particular displays frequency independent terms only photon distribution contains a gapped mode, Eq. (58), whose which are -correlated in time and neglect memory e↵ects. quantum fluctuations remain identical to the zero temperature case throughout the transition. Our starting point is the Hamiltonian of a single boson a,

(iv) The thermal-like divergence of the distribution func- coupled to a continuum of vacuum fields k, via tions determines the critical properties of the “superradiant” = ! + ! † + + † . phase transition. In particular, the photon number diverges as H0 ca†a k k k gk(a† k a k) (A1) 1/2 k 1/ g gc for N and scales as N for g = gc. Both X results| coincide| with!1 the equilibrium behavior of a Landau- Here gk is the coupling constant between the cavity boson and Ginzburg model at finite temperature, Eq. (83), and di↵er the external vacuum, and we neglected counter-propagating from the well studied zero-temperature case (where one ob- terms of the form a† †. Eq. (A1) is quadratic in , allowing tains 1/ g g 1/2 and N1/3). k k | c| us to analytically integrate-out the vacuum fields and obtain (v) Dissipative dephasing processes involving single atoms a cavity-only action of the form (22). If we assume that the can also be studied using non-perturbative techniques. As vacuum fields are kept at an equilibrium temperature Text = long as the symmetries of the original model are preserved, 300K, the corresponding entries are a Dicke transition is still expected, but its position may be R 1 strongly renormalized even for small decay rates, Eq. (103), [G (!)] = ! !c ! + iK(!), due to the depolarization of the atomic ensemble. K ! (vi) Within the nonlinear sigma model approach (Section VI), D = 2iK(!) coth . (A2) 2T we can obtain the spectral properties of the single atoms for ext ! general finite values of N across the phase transition in the Here ! corresponds to the Lamb shift and can be absorbed dissipative Dicke model. in a finite renormalization of !c. The function K(!) = In the future, it will be interesting to apply our approach g2 (! ! ) is the spectral density of the vacuum. k | k| k to dissipative quantum glasses coupled to Markovian (and The inverse Green’s functions (A2) describe the cavity P other) baths such as potentially achievable in multi-mode op- mode in the lab frame. In practice, it is often more convenient tical cavities [12–14] or circuit QED [58]. It would also be to move to a frame rotating with a constant frequency, in our desirable to obtain a more general classification of conditions case corresponding to the pump frequency !p. In this frame, under which quantum phase transitions of closed systems are the photons are described by Eq. (A2) with ! !p + !. turned into thermal phase transitions by dissipation–and per- Next, we apply the equivalent of the Wigner-Weisskopf! ap- haps to find counterexamples by engineered dissipation along proximation, by Taylor expanding the inverse Green’s func- the lines of Refs. [39, 40]. tion in small ! to zero order. This approximation is justified by the energy scale separation discussed in the text. We obtain

R 1 [G (!)] = ! ! ! ! + iK(! ), ACKNOWLEDGMENTS c p p !p DK = 2iK(! ) coth . (A3) p 2T We thank M. A. Baranov, F. di Piazza, S. Gopalakrishnan, ext ! B. Halperin, E. Kessler, D. Marcos, J. Otterbach, P. Rabl, ! The factor coth p plays the role of the 2n + 1 factor ap- H. Ritsch, L. Sieberer, H. Tureci, and P. Zoller for useful 2Text discussions and M. Buchhold, H. Ritsch, D. Marcos and S. pearing in the finite⇣ temperature⌘ extension of the master equa- Gopalakrishnan for critical comments on the manuscript. tion (2). To be precise, the two expressions coincide only for This research was supported by the U.S. National Science !c = !p. For most experiments the pump frequency is any- Foundation under grant DMR-1103860, by the U.S. Army way much higher than the external temperature and we can ! Research Oce Grant W911NF-12-1-0227, by the Center approximate coth p = 1. Under this approximation, Eq. 2Text for Ultracold Atoms (CUA), by the Multidisciplinary Univer- (A3) becomes equivalent⇣ ⌘ to Eqs. (25-26) with sity Research Initiative (MURI), by the Packard Foundation, ! = ! + ! ! ,= K(! ). (A4) by the DARPA OLE program, by the DFG under grant Str 0 c p p 19

Appendix B: Photon-only action in the superradiant phase with complex coecients and a low frequency spectrum

iIm[z ⌫ y µ] ( z 2 y 2)↵2 (Im[z ⌫ y µ])2 In the superradiant (SR) phase, the field a (b) is given ! = ⇤ ⇤ ± | | | | ⇤ ⇤ . ± ( z 2 y 2) by the sum of a time-independent component a0 (b0) and a p | | | | fluctuating term. The action governing the fluctuating terms (C3) can be obtained from Eq. (40) by substituting a a + a 0 Without dynamic renormalization e↵ects, z = 1 and (b b + b) (we choose a , b real without loss! of general- 0 0 0 all other frequency coecients are zero, so they will ity).! At N , we end up with the quadratic action !1 be generically much smaller than one (more precisely, A 1 Im[z] , Re[y] , Im[y] 1), and in particular z 2 y 2. (In 1 0[G4 4] (!) S SR = V†(!) 1 ⇥ V8(!) | | | | | |⌧ | | || 8 R K the large N open Dicke model, they are exactly zero.) A mass 2 ! [G4 4] (!) D4 4 Z ⇥ ⇥ ! gap, i.e. the scale that characterizes the action at zero fre- (B1) quency, provides a measure of the distance from the phase 1 1 transition and reads R A † with the Green’s functions [G4 4] (!) = [G4 4] (!) = ⇥ ⇥ ↵2 R 1 ! = = ⌫ 2 µ 2 ⇣ ⌘ det G2 2 ( 0) 0 (C4) ! !0 + i 0 g¯ g¯ ⌘ ⇥ | | | | 0 ! ! i g¯ g¯ (the last inequality must hold for a stable physical system). 0 , 0 g¯ g¯ ! !z 2!z !z 1 Approaching the phase transition, this gap shrinks to zero, B C B g¯ g¯ !z ! !z 2!z C such that the frequencies must become purely imaginary as B C B C a generic feature of a phase transition in the presence of dis- @B AC sipation. Indeed, in a situation where the phase transition is K = ,, , . D 2i diag( 0 0) (B2) driven by a competition within the Hamiltonian sector of the problem by a quantity g, the dominant g dependence is con- Here the 8-vector V8(!) is defined in the analogous way to tained in ↵2(g) (more precisely, Re[⌫], Re[µ]), while the dissi- V8(!) of Eq. (46) and pative scales (Im[⌫], Im[µ]) do not strongly depend on g and 3g 3g2 + g2 remain essentially at their bare, finite values even at the tran- g¯ = g b2 c , (B3) N 0 ⇡ 4g sition point. (In the open Dicke model, only the real parts are 2 2 modified, while the imaginary parts exactly remain at their 4g g gc ¯ bare values.) In such a situation, as ↵(g gc) 0, we may !z = b0(a0 + a0⇤) 4!z . (B4) N ⇡ g2 expand the square root in Eq. (C3) in ↵2!, identifying! that pa- rameter as the distance from the phase transition. We note that the principal change to the spectral response and correlation function in the superradiant phase (!) = (!) , (B5) Appendix D: 1/! divergence in Markov distribution functions Aaa† Aa,a† 2 (!) = (!) + a !, (B6) Caa† Ca,a† 0 0 We here show that the 1/! pole in the photon distribution is the -function peak at ! = 0 in the correlation function due function at low frequency, and the associated low-frequency to coherent photons (“photon condensate”). e↵ective temperature (LET), is indeed a generic feature of Markovian non-equilibrium systems. To this end, we consider the low-frequency regime, where Appendix C: Damped dynamics near the phase transition in the spirit of a systematic derivative expansion the inverse retarded Green’s function of the photon takes the form

R 1 We argue based on a systematic low-frequency expansion G = (! + i⌫2)z H, H = ⌫11 + µ1x + µ2y,(D1) of the inverse retarded Green’s function that the overdamped dynamics observed in the vicinity of the phase transition is where ⌫1,µ1(⌫2,µ2) denote the real (imaginary) part of ⌫, µ. generic for systems where a phase transition is driven by a We set z = 1, y = 0 here, which in principle contribute at (!), and anticipate that this omission will not alter the qual- competition within the Hamiltonian sector, while dissipative O dynamics acts as a “spectator”. To see this, we (i) write the itative results. The hermitean part H represents Hamiltonian most general form of the inverse retarded Green’s function dynamics, the antihermitean iz decay. In a derivative ex- pansion, the most general form⇠ of the Keldysh component is 1 p(!) o(!) R K [G2 2] (!) = , (C1) D = 2i(11 + 2x). (D2) o⇤( !) p⇤( !) ⇥ ! Solving the fluctuation-dissipation relation, we obtain and (ii) use that the phase transition is governed by low fre- quency behavior and an expansion in powers of the frequency 1 1 1 2µ1 F(!) = z µ1x + µ2y + 1 + 2y .(D3) is appropriate, ⌫2 ! ⌫2 ⌫2 h ⇣ ⌘ i p(!) = ⌫ + z!, o(!) = µ + y! (C2) Crucially, this confirms the 1/! divergence behavior of the 20 distribution function. Allowing for the most general form of frequency dependent terms to DK only leads to subleading R 1 G of the frequency expansion in terms of a finite imaginary corrections in F. Therefore, the 1/! pole at low frequency, part of z and finite y only results in subleading corrections: and the associated scale generated in this regime, the LET, is also in this case, lim! 0[! F(!)] const. Clearly, adding a generic feature of Markovian non-equilibrium systems. ! · !

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