PHYSICAL REVIEW A 91, 033836 (2015)

Real photons from vacuum fluctuations in optomechanics: The role of polariton interactions

Marc-Antoine Lemonde and Aashish A. Clerk Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8 (Received 14 January 2015; published 30 March 2015)

We study nonlinear interactions in a strongly driven optomechanical cavity, in regimes where the interactions give rise to resonant scattering between optomechanical polaritons and are thus strongly enhanced. We use a Keldysh formulation and self-consistent perturbation theory, allowing us to include self-energy diagrams at all orders in the interaction. Our main focus is understanding how nonequilibrium effects are modified by the polariton interactions, in particular the generation of nonzero effective polariton temperatures from vacuum fluctuations (both in the incident cavity drive and in the mechanical dissipation). We discuss how these effects could be observed in the output spectrum of the cavity. Our work also provides a technical toolkit that will be useful for studies of more complex optomechanical systems.

DOI: 10.1103/PhysRevA.91.033836 PACS number(s): 42.50.Wk, 42.50.Ex, 07.10.Cm

I. INTRODUCTION our previous study, we go beyond a simple lowest-order treatment and introduce a self-consistent perturbation theory. The rapidly growing field of cavity optomechanics seeks This corresponds to an infinite partial resummation of self- to understand the interaction between photons and mechanical energy diagrams, and extends the range of couplings and motion in driven electromagnetic cavities, hopefully in truly temperatures that we can address. quantum regime [1,2]. The past few years have seen many Our work also address a new set of physical phenomena. breakthroughs, including laser cooling of mechanical motion Reference [17] focused on understanding the cavity density to the ground state [3,4], and the optomechanical generation of states (DOS), the quantity probed in optomechanically of squeezed light leaving the cavity [5–7]. Although the induced transparency (OMIT) experiments [19,29–31]. Here, basic radiation pressure interaction between a mechanical we instead focus on the nonequilibrium state of this system: resonator and cavity photons is intrinsically nonlinear, almost What are the effective temperatures of the optomechanical all the remarkable achievements in the field to date rely on polaritons? Of particular interest is how vacuum fluctuations working in strongly driven regimes where the dynamics is can result in nonzero effective temperatures. Such effects essentially linear. To see truly nonlinear effects in the simplest are often termed “quantum heating” [32–36]; the simplest setting, one needs to achieve a single photon, single-phonon example is the amplification of zero-point fluctuations by a optomechanical coupling g which exceeds both the mechanical parametric amplifier. Although often described using disparate frequency ω and the cavity damping rate κ [8–11]. With M terms, this physics has been studied in a wide variety of the exception of experiments using cold atoms [12,13], this systems ranging from driven nonlinear oscillators [32–35], parameter regime remains challenging for experiments. superconducting circuits [37,38], and circuit quantum elec- Nonlinear effects which only require g ∼ κ can be achieved trodynamics [39] to phase transitions in driven-dissipative in slightly more complex settings where the nonlinear interac- many-body systems [27]. It also sets a limit to the minimum tion becomes resonant. This can occur in an optomechanical mechanical temperature achievable using cavity cooling [40]. setup with two optical modes [14,15]. Alternatively, this can Here, we show that quantum heating effects in an optome- occur in a standard single-mode optomechanical cavity which chanical system lead to observable signatures in the output is driven, such that the interaction becomes resonant in a basis spectrum of the cavity, i.e., the spectrum of output light that of dressed states (so-called optomechanical polaritons). This would be measured using a photomultiplier. Unlike an OMIT occurs both in regimes of weak driving [16,17] and strong experiment, such an experiment probes both the DOS of driving [17]; similar physics can also occur in membrane-in- the cavity (as modified by the optomechanical interactions), the-middle–style optomechanical systems [18]. Note that such as well as the effective temperature of the optomechanical strong driving regimes, where the drive-enhanced optome- polaritons. This “quantum heating” is already present at the chanical coupling exceeds dissipative rates, has been achieved level of the linearized theory of the optomechanical cavity; in several experiments [19–21]. here, we investigate how it is modified by the nonlinear In this paper, we both expand upon and extend the results interactions. Our work is timely, given that the generation previously reported in Ref. [17]. We consider a standard of real photons from mechanical vacuum fluctuations was single-cavity optomechanical system that is driven (possibly recently probed experimentally by Lecocq et al. [41]. strongly) in a regime where the nonlinear interaction is resonantly enhanced. We again address this system using a Keldysh formulation [22,23] and perturbation theory; we begin this paper by providing a more complete discussion of A. Main findings this approach to optomechanics, emphasizing the subtleties Our work has many technical aspects to it that will involved in treating dissipation. Our work thus contributes hopefully be an aid in further studies of nonlinear quantum to a growing body of work using the Keldysh technique optomechanics (e.g., optomechanical lattices [42–46]). It also to address various quantum optics contexts [24–28]. Unlike predicts several additional physical phenomena, in particular:

1050-2947/2015/91(3)/033836(19) 033836-1 ©2015 American Physical Society MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

1. Polariton thermalization In the standard linearized theory of a driven optomechanical cavity, quantum heating leads to very different effective tem- peratures for the two polariton modes (i.e., normal modes of the linearized theory) [cf. Eqs. (24)]. The nonlinear interaction tends to dilute this effect, as it allows energy exchange between the polaritons and favors their thermalization. We study this competition in detail for two representative cases: a laser drive at the red-mechanical sideband (see Fig. 8), and a laser drive detuned further to the red where nonlinear effects are more important (see Fig. 10). The effects of the nonlinear interaction on this quantum heating physics can be seen experimentally in FIG. 1. (Color online) (a) A generic optomechanical cavity: a the cavity output spectrum by tuning the nonlinear interactions cavity mode of frequency ωC and damping rate κ is driven by a laser into and out of resonance (see Fig. 9). at frequency ωL, and is coupled via radiation pressure to a damped mechanical mode (resonance frequency ωM , damping rate γ ). The temperature of the mechanical bath is TM . (b) The single-photon optomechanical interaction g can be made resonant in the basis of 2. New instabilities optomechanical polaritons; the resulting enhanced interaction shown schematically and described in Eq. (8). For a red-detuned laser, we find that a leading-order treatment of the nonlinear interaction suggests a different kind of parametric instability not present in the linearized theory, II. SYSTEM, TREATMENT OF DISSIPATION, where one polariton mode acts as an incoherent pump mode AND LINEARIZED THEORY for the other (see Sec. VD). Including higher-order terms A. Hamiltonian in the polariton basis stabilizes the system as expected (see Fig. 11). We consider a standard optomechanical system where the frequency of a driven cavity mode is modulated linearly by the position of a mechanical resonator (Fig. 1). The Hamiltonian governing its dynamics is given by [1,2] 3. Two-phonon cavity heating ˆ = † + ˆ† ˆ + † ˆ + ˆ† We show that nonlinear heating of the cavity (as manifest H ωC aˆ aˆ ωM b b gaˆ aˆ(b b ) −iωLt † in the cavity spectrum) is greatly enhanced for red-detuned + i(a¯ine aˆ − H.c.) + Hˆdiss. (1) laser drives near the second mechanical sideband (a detuning ˆ  ≈−2ωM ); this is a simple consequence of the lower op- Here, aˆ is the cavity mode, with frequency ωC and b the tomechanical polariton being mostly phononic in this regime, mechanical mode with frequency ωM . The parameter g is the causing effects to be enhanced by the typically large phonon single-photon optomechanical coupling and a¯in is proportional lifetime (see Fig. 12). to the amplitude of a classical drive at frequency ωL. Finally, Hˆdiss describes dissipation due to the coupling to bosonic environments (both for the mechanics and the cavity). Going into a rotating frame at the drive frequency and displacing the cavity field by its classical value, induced by the coherent drive B. Organization of the paper − {i.e., aˆ(t) → e iωLt [a¯ + dˆ(t)]}, the Hamiltonian of Eq. (1) can The remainder of this paper is organized as follow. We be expanded into a quadratic part, known as the linearized begin in Sec. II by introducing the basic driven optomechanical optomechanical Hamiltonian, and a nonlinear interaction term system studied in this work, and reviewing the linearized such that theory. We also pay careful attention to the coupling to ˆ ˆ ˆ ˆ dissipative baths, and to how quantum heating effects can arise H = HL + HNL + Hdiss, (2a) even without nonlinearity. In Sec. III, we introduce the basic † † † † HˆL =−dˆ dˆ + ωM bˆ bˆ + G(dˆ + dˆ )(bˆ + bˆ ), (2b) aspects of the Keldysh technique as applied to optomechanics, ˆ = ˆ† ˆ ˆ† + ˆ focusing first on the linearized system. In Sec. IV,we HNL gd d(b b). (2c) develop a Keldysh perturbation theory to treat the nonlinear In the above, we have defined the laser detuning  = ω − optomechanical interaction, and introduce our self-consistent L ωC and the many-photon coupling constant G = ag¯ ;wetake approach. In Sec. V, we discuss how the nonlinear interaction a,g¯ > 0 without loss of generality.1 modifies the nonequilibrium physics of the system. We present a physical picture where interaction effects can be mapped onto a coupling to additional “self-generated” dissipative baths. We also investigate in this section the new kind of parametric 1The mean value of aˆ (a¯) is determined by solving the classical heating which arises with a red-detuned laser. In Sec. VI, equations of motion following from Eq. (1). We also shift the we discuss the observable consequences of our predictions mechanical lowering operator bˆ [i.e., bˆ=0inEqs.(2)] to account on the cavity output spectrum. We present our conclusions in for the static radiation pressure force. We also redefine the detuning Sec. VII. accordingly.

033836-2 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015)

Our general approach is to diagonalize the quadratic part B. Resonant polariton interactions ˆ of the Hamiltonian HL (thus treating it exactly), and then As discussed in previous works [16,17] (and later in [48]), ˆ treat the terms due to nonlinear interaction (HNL)asa we can enhance the effects of even a weak single-photon perturbation. Unlike treatments based on polaron-transformed coupling g by tuning G and  such that the nonlinear processes Hamiltonians [8,9], we do not require the coherent driving in Eq. (6) that scatter a + polariton into two − polari- of the cavity to be so small that it too can be treated B tons (∝ g−−+) become resonant. This requires E+[,G] = perturbatively. We focus exclusively on a red-detuned laser 2E−[,G]; for a given laser detuning in the range  ∈ (<0), and only require the cavity drive to be weak enough − − 2 2 [ 2ωM , ωM /2], this can always be achieved by tuning so that G < −ωM /4 ≡ G . For drives stronger than this crit G = Gres[], where critical value (or for a blue-detuned laser), the linearized coherent Hamiltonian Hˆ is unstable, and corresponds to a 2 2 − 4 + 4 L 17 ωM 4  ωM detuned parametric amplifier driven beyond threshold. This G [] ≡ √ . (7) res − critical value also coincides with the onset of the well-known (10 ωM ) static optomechanical instability [1,47]. Operating near this Once G is tuned to achieve this resonance condition, instability has been investigated as an alternative promising one can show using standard perturbation theory that all way to enhance the nonlinear interaction [18]. Note that this nonresonant nonlinear processes are suppressed by a factor instability is also closely analogous to the superradiant phase of κ/(E+ − E−) ∝ κ/ωM compared to the resonant process, transition in the driven Dicke model studied in [27]. where κ is the cavity damping rate (see [17] for more details). 2 Focusing on <0 and G < −ωM /4, Eq. (2b) can be In addition, in this regime, the relative modification of the diagonalized via a Bogoliubov transformation to yield polariton energies and wave functions due to the linear terms 2 in Eq. (6) are strongly suppressed by a factor of (g/ωM ) . Hˆ = E cˆ† cˆ . (3) Thus, if we focus on parameters near this resonant regime, L σ σ σ σ =± in the resolved sideband regime (κ/ωM 1), and for weak nonlinear interaction (g/ωM 1), we can both ignore the renormalization of polariton energies and wave functions, and Here, cˆ± destroys an excitation in the eigenmode of Hˆ with L make a rotating-wave approximation on Eq. (6), keeping only energy E± > 0, given by the resonant interaction. The coherent system Hamiltonian in this regime thus reduces to 1 2 2 2 2 1/2 E± = √ ω +  ± ω − 2 − 16G2ω . † M M M ˆ = † + + 2 Heff Eσ cˆσ cˆσ g˜(cˆ+cˆ−cˆ− H.c.), (8) (4) σ =± Note that E− tends to zero as G approaches the critical value at where g˜ = gB is the effective nonlinear coupling [see the instability G .AsHˆ does not conserve the total number −−+ crit L Eq. (A8) in Appendix A]. Shown in Fig. 2(a) is the dependence of photons and phonons, the operators cˆ−,cˆ+ mix photon and of G as a function of laser detuning , as well as the behavior phonon annihilation and creation operators: res of g˜ on , when G is tuned to be Gres. For the rest of this paper, † † bˆ = α −cˆ− + α +cˆ+ + α¯ −cˆ− + α¯ +cˆ+, (5a) we focus on the dynamics governed by the effective coherent b, b, b, b, ˆ † † Hamiltonian Heff. dˆ = αd,−cˆ− + αd,+cˆ+ + α¯ d,−cˆ− + α¯ d,+cˆ+. (5b) C. Coupling to dissipative reservoirs The coefficients α ± andα ¯ ± are functions of /ω b/d, b/d, M We now turn to a more careful consideration of the effects and G/ωM ; their explicit form in the case  =−ωM is given in Eqs. (A1)–(A4) of Appendix A. As the excitations described of dissipation on our system. As is standard in optomechanics, both the mechanics and the cavity are taken to be linearly by cˆ−,cˆ+ have both phononic and photonic components, we refer to them as polaritons in what follows. coupled to independent, Markovian bosonic baths (i.e., baths Having diagonalized the linearized optomechanical Hamil- with constant DOS over the frequency range of interest). This is analogous to the approach taken in input-output treatment of tonian, we now express the nonlinear Hamiltonian HˆNL in the polariton basis. We obtain interactions that do not conserve the dissipation (see, e.g. [49,50]). As we are interested in possibly total number of polaritons large many-photon optomechanical couplings G, care must still be taken, as the eigenstates of our coherent Hamiltonian † † † † † are polaritons, not individual photons or phonons; similar ˆ = A + B + HNL gσσ σ cˆσ cˆσ cˆσ gσσ σ cˆσ cˆσ cˆσ H.c. issues have recently been addressed in strongly-coupled circuit σ,σ ,σ =± QED systems [51]. Moreover, the strong driving of the cavity + (A−cˆ− + A+cˆ+ + H.c.). (6) can also lead directly to “quantum heating” effects, which manifest themselves directly in the treatment of the cavity A/B dissipation. We describe these effects more in what follows. Here, the constants gσσ σ and Aσ are all proportional to g [see Eqs. (A7)–(A9) of Appendix A]. Note that the linear terms in 1. Cavity dissipation in the presence of optomechanical HˆNL (∝ Aσ ) arise from normal ordering HˆNL in the polariton basis; physically, while the zero-polariton state is the vacuum coupling and driving of HˆL, this is no longer true when we include the nonlinear Consider first the interaction between the cavity and interaction. its dissipative reservoir. In the original laboratory frame,

033836-3 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

Note that we transform both cavity and bath operators. The result is that in our interaction picture, terms in Eq. (9) which conserve excitation number are time independent, whereas the remaining excitation nonconserving terms are rapidly oscillation at a frequency ±2ωL ≈±2ωC (ωC ||). Since ωC is much larger than all other energy scales in the rotating frame (i.e., ωC Eσ ,||,ωM ,g,κ), and since the bath oscil- lators necessarily have positive energies ωj > 0, these rapidly oscillating terms can never become resonant. We can thus safely make a rotating-wave approximation, and drop them. With this rotating-wave approximation, the cavity-bath Hamiltonian then becomes time independent. Re-writing the cavity annihilation operator as per Eq. (2b), one obtains a standard rotating-wave system-bath interaction ˆ = − ˆ† ˆ = ˆ† ˆ Hκ (ωj ωL)fj fj ω˜ j fj fj , (11) j j κ † Hˆ int = i (fˆ dˆ − fˆ dˆ†). (12) κ 2πρ j j C j Note crucially that in this final rotating frame, the trans- formed bath frequenciesω ˜ j can be negative (they extend down to −ωL). While this may seem innocuous, things become more interesting when we rewrite this interaction in terms of polariton operators [cf. Eqs. (5)]: κ † Hˆ int = i fˆ (α cˆ + α¯ cˆ† ) + H.c. (13) κ 2πρ j d,σ σ d,σ σ C j,σ

FIG. 2. (Color online) (a) Gres, the value of the many-photon The anomalous terms which create or destroy two excitations coupling which leads a resonant nonlinear interaction, as a function of here should not be dropped; as the polariton energies Eσ laser detuning  [cf. Eq. (7)]; the corresponding effective nonlinear ωL ∼ ωC , these terms can be resonant in our interaction g˜ coupling [cf. Eq. (8)] is also plotted. (b), (c) Coefficients relating the picture, as bath modes havingω ˜ j < 0 can be involved. optomechanical polaritons (eigenstates of the linearized Hamiltonian) Physically, such processes involve the creation of both a to original photon and phonon operators [cf. Eqs. (5)] as a function polariton and a bath excitation, while at the same time a = of laser detuning  with G Gres[]. In (b), we plot the “normal” (classical) drive photon is absorbed. Such heating processes coefficients which relate polariton destruction operators to photon = | ˆ†|  thus involve the interplay of the system drive and the bath and phonon destruction operators, i.e., αb,σ 0,0 cˆσ b 0,0 , with zero-point fluctuations, and are at the heart of quantum |0,0 being the polariton vacuum. In (c), we plot the “anomalous” activation [32–35]. coefficients, i.e.,α ¯ b,σ =0,0|cˆσ bˆ|0,0; these coefficients are directly related to the existence of “quantum heating” effects in the linearized theory [cf. Eq. (15)]. 2. Quantum heating and effective temperature The anomalous, excitation nonconserving terms in Eq. (13) the cavity-bath interaction will have the following generic can lead to polariton heating even if the cavity dissipation form [49]: is at zero temperature. We can naturally associate an effective temperature to this heating by computing golden rule transition † κ † − ˆ int Hˆ = ω fˆ fˆ ,Hˆ int =i (fˆ − fˆ )(aˆ + aˆ †), rates [50]. Consider first the lower-energy polaritons. Hκ κ j j j κ 2πρ j j will cause transitions uphill in energy between an initial state j C j having N − 1 polaritons and a final state having N polaritons 2 (9) at a rate N,N−1 ∝ N|α¯ d,−| . Similarly, it will cause transitions ˆ downhill in energy from the N to N − 1 polariton state at a where fj is the anihiliation operator for cavity-bath mode j 2 rate N−1,N ∝ N|αd,−| . If these transitions were due to a (frequency ωj ), κ the damping rate of the photons inside the ρ bath in true thermal equilibrium at temperature T , detailed cavity, and C is the bath DOS. As we consider a Markovian = − bath, we take κ and ρ to be frequency independent. balance dictates that N,N−1/ N−1,N exp( E−/kB T ). In C our case, we can use the ratio of these rates to define the We next transform to an interaction picture at the drive cav frequency via the unitary effective temperature T− of the cavity dissipation as seen by ⎡ ⎛ ⎞⎤ the − polaritons: † † 2 ˆ = ⎣− ⎝ + ˆ ˆ ⎠⎦ − cav N,N−1 α¯ d,− U exp iωLt aˆ aˆ fj fj . (10) E−/kB T− ≡ = e 2 . (14) j N−1,N αd,−

033836-4 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015)

If these transitions were the only dynamics of the − polaritons, As the mechanics is undriven, the bath Hamiltonian and they would indeed cause them to reach a thermal state at bath-system interaction are unchanged under the transforma- cav temperature T− , with a thermal occupancy tion of Eq. (10) to the rotating frame at the laser frequency. This 2 time, we rewrite the interaction in terms of polariton operators 1 α¯ − cav = = d, using Eqs. (5) before making further approximations. This nB [E−,T− ] cav . (15) E−/kB T− − 2 − 2 e 1 αd,− α¯ d,− difference from the treatment of cavity dissipation stems from the fact that unlike ω , ω is comparable to E .Itisthus We thus see the two crucial ingredients needed to obtain a C M σ crucial to go to the eigenstates basis of polaritons before nonzero effective temperature in a bosonic system where the assessing which terms may be safely dropped. In the polariton physical bath temperature is zero. We needed both coherent basis, we have drive (yielding effective negative energy bath modes in our interaction picture) and coherent parametric-amplifier type γ † Hˆ int = i (gˆ − gˆ )(α + α¯ )(cˆ + cˆ† ). (19) interactions (i.e., coherent interactions which do not conserve γ 2πρ j j b,σ b,σ σ σ M j,σ=± particle number, and hence yieldα ¯ d,− = 0). In a completely analogous fashion, one can also associate an effective temper- We can now consider the role of terms in Eq. (19) that do not ature T cav describing the quantum heating of the + polaritons. + conserve excitation number. In contrast to our treatment of the We stress that these effective temperatures have nothing to do cavity dissipation, here such anomalous terms can be dropped with nonlinear interactions. in a rotating-wave approximation. As there is no mechanical For a concrete example of this effective temperature drive, there are no effective negative-energy mechanical bath physics, consider the special case of a cavity drive at the red modes, and hence these terms can never be made resonant. mechanical sideband  =−ω . One finds [cf. Eqs. (A1)– M Thus, we finally obtain a simple rotating-wave interaction (A4)] √ between the mechanical bath and the polaritons: − ± 2 cav (1 1 2G/ωM ) ± | =− = √ nB [E ,T± ]  ωM . (16) int γ † 4 1 ± 2G/ωM Hˆ = i [(α + α¯ )gˆ cˆ + H.c.]. (20) γ 2πρ σ σ j σ cav M j,σ=± Note that nB [E−,T− ] diverges when G approaches the → onset of parametric instability G ωM /2. In this limit, As there are no excitation nonconserving terms in Eq. (20), → cav E− 0, and the divergence of nB [E−,T− ] is equivalent to it is easy to confirm that the effective temperature of the cav  =| | a fixed effective temperature T− ωM /4  /4 near the mechanical dissipation seen by the polaritons is simply equal instability. Such behavior is generic for quantum heating near to the physical temperature of the mechanical dissipation. parametric instabilities. As already emphasized, quantum heating requires both the The type of quantum heating phenomena described here is presence of coherent parametric-amplifier (paramp) type in- generic, and is found in a variety of related systems, though teractions and the presence of a coherent drive. Here, while the the generic nature of the mechanism is often not appreciated. coherent paramp interactions are present, there is no coherent Analogous effective temperatures also arise in mean-field driving of the mechanics; as such, there is no quantum heating treatments of driven dissipative phase transitions. For example, effects involving the mechanical bath. We see that even if the study of the driven-dissipative Dicke model in Ref. [27] both the cavity and mechanical baths have identical physical finds an effective temperature near the transition identical to temperatures, the polaritons see them as having different that quoted after Eq. (16). The physical origin is analogous effective temperatures. The driven nature of the system thus to that in our system: it arises from the interplay of coherent gives us interesting nonequilibrium physics even at the level parametric-amplifier interactions combined with a coherent of the linearized (i.e., quadratic Hamiltonian) theory. linear driving. We end this section with a caveat on the validity of treating dissipation via Markovian baths. For the cavity dissipation, 3. Mechanical dissipation this is an excellent approximation, as we are always probing We now turn to the interaction between the mechanical the bath in a narrow interval of width ∼Eσ around ωC ,an resonator and its dissipative bath. The starting interaction is interval over which the bath DOS can be treated as constant analogous to Eq. (9) for the cavity dissipation (recall that Eσ ωC ). In contrast, a similar statement does not hold for the mechanical dissipation: we will be probing the † γ † ˆ = ˆ int = − ˆ + ˆ† Hγ ωj gˆj gˆj , Hγ i (gˆj gˆj )(b b ), mechanical bath at frequencies Eσ which could be significantly 2πρM j j different from the mechanical frequency ωM . As such, it is (17) not aprioriobvious that the bath spectral density can be treated as flat. For simplicity, we will nonetheless use the where gˆj is a bath annihilation operator, γ the mechanical Markov bath approximation for the mechanics in what follows damping rate, and ρM is the constant DOS of the (Markovian) (consistent with the majority of works in optomechanics). bath. In what follows, we consider the mechanical bath to be For the weak dissipation limit of interest, the main effects at temperature TM and define the mean number of excitations of a nonflat bath spectral density could be easily incorporated inside the bath at ωM as into our calculations. One would simply make the mechanical contribution κ to the intrinsic polariton decay rates [cf. n¯M ≡ n [ω ,T ], (18) M th B M M Eq. (23)] proportional to the mechanical bath density of states with nB [ω,T ] being the Bose-Einstein distribution. at the relevant polariton energy.

033836-5 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

D. Lindblad master equation and effective polariton dissipation For further insight, it is useful to use the form of the system- bath couplings in Eqs. (12) and (20) to derive an approximate Linblad master equation for the dynamics of polaritons in our system. While we do not use such a master equation in our analysis, it provides a useful comparison point. Using the standard derivation (see, e.g., [49]) valid for weakly coupled Markovian baths, we obtain

ρˆ˙(t) =−i[Hˆeff,ρˆ(t)] κ− 0 κ− 0 † + (n¯− + 1)D[cˆ−]ˆρ + n¯−D[cˆ−]ˆρ, 2 2 κ+ 0 κ+ 0 † + (n¯+ + 1)D[cˆ+]ˆρ + n¯+D[cˆ+]ˆρ, (21) 2 2 with the Lindblad superoperator D[cˆσ ]ˆρ defined as ≡ † − † − † D[cˆσ ]ˆρ 2cˆσ ρˆ(t)cˆσ cˆσ cˆσ ρˆ(t) ρˆ(t)cˆσ cˆσ . (22) The effective polariton damping rates appearing in Eq. (21) are = + 2 + 2 − 2 ≡ mech + cav κσ γ (αb,σ α¯ b,σ ) κ αd,σ α¯ d,σ κσ κσ . (23)

mech cav Here, we have introduced κσ (κσ ) as the contribution to the damping rate of the σ polariton coming from the interaction with the mechanical resonator (cavity) dissipative bath. The corresponding effective bath thermal occupancies are 1 0 = + 2 + 2 n¯σ γ (αb,σ α¯ b,σ ) nB [Eσ ,TM ] κα¯ d,σ (24a) =± κσ FIG. 3. (Color online) (a) Damping rates κσ of the σ polari- = mech cav cav tons [cf. Eq. (23)] in the linearized theory (i.e., g˜ 0), as a function κ nB [Eσ ,TM ] + κ nB Eσ ,T = σ σ σ . (24b) of the detuning . For each , we tune the cavity control drive mech + cav κσ κσ to maintain G = Gres(). Near  =−2ωM ,the− (+) polariton damping tends to γ (κ) as it is mostly phononlike (photonlike); Here, TM is the (physical) temperature of the mechanical bath =− cav the converse is true near  ωM /2. (b) Thermal occupation [cf. Eq. (18)], Tσ is the temperature of the cavity bath as seen 0 numbers n¯σ of the effective bath coupled to the σ polaritons [cf. by the σ polariton [cf. Eq. (15)], and the different coefficients = = M Eq. (24)] also when g˜ 0andG Gres. Here, n¯th characterizes the α are given in Eq. (5). As expected, Eq. (24b) represents a (physical) temperature of the mechanical bath [cf. Eq. (18)]. (c) The bosonic mode coupled independently to two dissipative baths. 0 = temperatures corresponding to the thermal occupation n¯σ (kB 1) in An analogous expression holds for the effective mechanical the same regime as panels (a) and (b). For each curve, γ/κ = 10−4 occupancy used to describe cavity-cooling experiments [40]. and ωM /κ = 50. Note that we have made a standard secular approximation, allowing us to drop dissipative terms that do not conserve the number of each polariton independently in Eq. (21); this is (see Fig. 2) and becomes very sensitive to thermal fluctuations of the mechanical bath. valid for the regime of interest Eσ ,|E+ − E−| κ,γ . The thermal occupation number of the effective baths given Returning to zero physical bath temperature, another in Eqs. (24) and their corresponding temperatures, defined striking feature in Fig. 3(c) is the sudden drop in effective 0 ≡ 0 temperature for the more phononlike polariton branch at the as nB [Eσ ,Tσ ] n¯σ , are plotted in Fig. 3 as a function of the detuning . Anticipating our interest in nonlinear edges of the detuning interval. As it will be of interest in what follows, we discuss this behavior for detunings near interactions, for each  we adjust the control laser amplitude =− G = G  2ωM in more detail; here, one sees a sudden drop in so that res [i.e., the value that will make the nonlinear 0 0 →− → − interaction resonant, cf. Eq. (7)]; this can be done for any n¯− and T−.As 2ωM , Gres 0, and the polariton 0  ∈ [−2ω , − ω /2]. One sees that even when the physical becomes simply a phonon. Expanding n¯− to lowest order in M M = bath temperature is zero, quantum heating effects can yield Gres/ωM for TM 0 and γ κ, one gets effective polariton temperatures as large as ∼0.1 quanta (solid G2 1 res κ κ 9 ω2 curves). At nonzero physical temperature (dashed curves), n¯0 = α¯ 2 ≈ M . (25) − d,− G2 these quantum heating effects persist, but are swamped by the κ− γ + 8 res κ 9 ω2 contribution of mechanical noise at the edges of the detuning M range considered. This is simply because near the limits of the We see there are two competing effects associated with detuning range, one polariton species is almost all phononic nonzero Gres. The numerator reflects the parametric heating

033836-6 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015) associated with the linearized optomechanical interaction. the unperturbed DOS of each polariton, given by The denominator in contrast reflects that the phononlike 1 0 =− R polariton has its lifetime decrease as Gres increases and it ρσ [ω] Im Gσ [ω] . (29) becomes more photonlike; this is just standard optomechanical π optical damping. The result is that the net quantum heating is Finally, GK (t) is known as the (unperturbed) Keldysh Green’s 2 ∼ σ maximized for [Gres/ωM ] γ/κ, corresponding to a laser function. It encodes knowledge of the energy distribution + ∼ 0 detuning  2ωM γ/κ; the maximum occupancy n¯σ that function of each polariton (as we will see more clearly in 1 can be obtained is 8 [as seen in Fig. 3(b)]. the following). With these definitions in hand, one could follow the III. KELDYSH DESCRIPTION OF THE LINEARIZED standard approach used in input-output theory and derive the SYSTEM Heisenberg-Langevin equations from the coherent Hamilto- nian of Eq. (8) and the particular form of the system baths As shown in the previous section, the driven nature of coupling given by Eqs. (12) and (20). From there, one can the system leads to nonequilibrium physics even at the level directly get the bare Green’s functions by calculating Eqs. (28). of the linearized theory (i.e., the two polariton species see This approach has been used in our previous work [17]. Here, different effective temperatures). Consequently, we need to we instead follow a different but equivalent route to obtain the use the Keldysh formalism [23] in order to describe the bare Green’s functions and Keldysh action. dynamics and to properly construct a perturbation theory that The goal here is to write the different Green’s functions treats the nonlinear interaction present in Hˆ [Eq. (8)]. In eff such that our description of the linearized theory in the Keldysh this section, we quickly introduce this approach, considering formalism is completely equivalent to the master equation of first the linearized Hamiltonian of Eq. (2b) and the couplings Eq. (21). As discussed before, Eq. (21) describes two indepen- with the environment given by Eqs. (12) and (20). We will dent bosonic modes with simple Markovian damping rates κ also map the resulting free Keldysh theory onto the simple σ [cf. Eq. (23)] and energies E [cf. Eq. (4)]. Consequently, master equation previously derived [Eq. (21)], again working σ the retarded (advanced) Green’s functions, which give the in the regime where polaritons have a well-defined energy (i.e., response functions of the system, have to adopt the simple κ E ). We stress, however, that even without interactions, σ σ following form: the Keldysh approach is more general than a Lindblad-style ∗ 1 master equation, as it is not restricted to Markovian baths. R = A = Gσ [ω] Gσ [ω] , (30) In the Keldysh formalism, we represent our linearized op- ω − Eσ + iκσ /2 tomechanical system by a field theory which is general enough to allow the system to be in an arbitrary, nonequilibrium state. i.e., a polariton has an energy Eσ and a lifetime 1/κσ . In this field theory, there are two time-dependent fields (classi- We now construct the Keldysh Green’s functions using the cal and quantum) corresponding to each annihilation operator same approach. As is standard [23], we define a distribution in the original theory. Consequently, the quadratic action that function f [ω] that relates the Keldysh and the retarded Green’s functions, such that conserves the number of particles of our two independent bosonic modes (polaritons) will have the following general GK [ω] ≡−2i(2f [ω] + 1)Im GR[ω] . (31) form: σ σ ∞ ∞ This function f [ω] parametrizes the occupation of different = ∗ ∗ SL dt dt (cσ,cl(t),cσ,q(t)) polariton energy eigenstates. As an example, for a system −∞ −∞ σ =± in thermal equilibrium at a certain temperature, f [ω] would be the corresponding Bose-Einstein distribution and Eq. (31) − cσ,cl(t ) × G 1(t − t ) . (26) would be an exact statement of the fluctuation-dissipation σ c (t ) σ,q theorem [23]. In the particular case studied here, the free Here, the cσ,q/cl(t) are complex functions of time and polaritons have sharply peaked single-particle DOS ρσ [ω][cf. −1 − Gσ (t t ) is the (operator) inverse of the unperturbed (i.e., Eqs. (29) and (30) with κσ Eσ ], such that for the σ polariton, g˜ = 0) Green’s function. The latter is given by a 2 × 2matrix the function f [ω]ofEq.(31) can be approximated as f [ω]  K R f [Eσ ]. Finally, if we insist that the average occupancy of the Gσ (t) Gσ (t) Gσ (t) = . (27) polariton matches that in the master-equation description, then A = 0 Gσ (t)0 we must have f [Eσ ] n¯σ . We thus have In terms of Heisenberg picture operators, each element is K =− 0 + R Gσ [ω] 2i 2n¯σ 1 Im Gσ [ω] . (32) defined as Describing the linearized theory in the Keldysh formalism R = A ∗ ≡−  †  Gσ (t) Gσ (t) iθ(t) [cˆσ (t),cˆσ (0)] , (28a) using the bare Green’s functions (30) and (32) is thus † completely equivalent to the Lindblad master equation (21). GK (t) ≡−i{cˆ (t),cˆ (0)}, (28b) σ σ σ We recall that the two assumptions underlying these two where the expectations are taken with respect to the initial equivalent descriptions are that the polaritons have sharply density matrix without nonlinear interaction (g˜ = 0). Here, peaked DOS (i.e., dissipation is weak) and that coupling R A + − Gσ (t) and Gσ (t) are the standard unperturbed retarded and between the and polaritons due to dissipation is negligible advanced Green’s functions, which govern the linear-response [i.e., secular approximation made to derive Eq. (21), see properties of the unperturbed system. They are also related to Sec. II D].

033836-7 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

Finally, we note that it is possible to derive exact Langevin equations from the linear Keldysh action given in Eq. (26). Briefly, one first decouples the quadratic quantum-field terms via an exact Hubbard-Stratonovich transformation; this intro- env duces new fields ξσ (t) which have a Gaussian action. One can then exactly do the integrals over quantum and classical fields. The resulting functional delta function corresponds to the Langevin equations κ ∂ c (t) =− iE + σ c (t) − ξ env(t), (33) t σ σ 2 σ σ

env where the noise ξσ (t) is Gaussian with zero mean. The only nonzero noise correlation functions are given by env env ∗ 0 ξ t ξ t = κ n¯ + / δ t − t δ . σ ( ) σ ( ) σ σ 1 2 ( ) σ,σ (34) FIG. 4. (Color online) Diagrams representing the self-energies As expected, Eqs. (33) and (34) represent two uncoupled up to second order in g˜ in the Keldysh formalism. The classical damped harmonic oscillators each in contact with their vertices are composed of only one quantum component, while the quantum vertex is composed of three. respective finite-temperature Markovian baths.

versions of the Langevin equations in Eq. (33): IV. KELDYSH PERTURBATIVE TREATMENT OF POLARITON INTERACTIONS κ− ∗ env ∂ c− =− iE− + c− − 2igc˜ +c− − ξ− , (37a) t 2 A. Self-energies and dressed Green’s functions Having established the Keldysh formulation of the lin- κ+ env ∂ c+ =− iE+ + c+ − igc˜ −c− − ξ+ , (37b) earized optomechanical theory, we can now address the effects t 2 of the nonlinear interaction as a perturbation. We assume throughout this section that the drive laser has been tuned where the autocorrelation functions of the ξσ (t) noise are unchanged by the nonlinear interactions. We have suppressed to make the nonlinear interaction resonant, i.e., G = Gres, and thus consider only the resonant interaction process given in the explicit time dependence of fields here for clarity. Eq. (8). We will not use these approximate quantum Langevin First, the action generated by the nonlinear interaction in equations further, but proceed in a way that does not neglect terms that are cubic in the quantum fields. As we are interested Hˆeff [Eq. (8)] in the cl-q basis is in weak nonlinear couplings g˜, we will compute the self-energy ∞ = √g˜ ∗ + ∗ ([ω]) of our Keldysh Green’s functions perturbatively to SNL dt(c+,q c−,clc−,cl 2c+,clc−,q c−,cl 2 2 −∞ order g˜ . At this order, all relevant scattering processes (see ∗ Fig. 4) conserve the number of polaritons independently + c c− c− + c.c.), (35) +,q ,q ,q (i.e., the self-energies are diagonal in the +/− index). where the time dependence of the fields is implicit for Consequently, the Dyson equation that gives the Green’s clarity. Diagrammatically, each term in the nonlinear action functions in presence of interactions can be separately written corresponds to a vertex shown in Fig. 4(a). The vertices with for each polariton: a single quantum field correct the classical saddle point − GK GR 1 K A σ [ω] σ [ω] − σ [ω] σ [ω] ∂(S + S ) ∞ = G [ω] 1 − . L NL = R − −1 GA σ R ∗ dt [G+(t t )] c+,cl(t ) σ [ω]0 σ [ω]0 ∂c+ (t) = −∞ ,q c±,q 0 (38) GA,R,K g˜ Here, we used σ [ω] to distinguish the full Green’s + √ c−,cl(t)c−,cl(t), (36a) 2 functions (i.e., including the effects of g˜) from the unperturbed ones GA,R,K[ω]. + ∞ σ ∂(SL SNL) R −1 ∗ = dt [G−(t − t )] c−,cl(t ) ∂c− (t) −∞ ,q c±,q =0 1. Retarded self-energies and interaction-induced √ + ∗ polariton damping 2gc˜ +,cl(t)c−,cl(t) (36b) The diagrams related to the second-order retarded self- and thus correspond to a classical nonlinear potential. The energies are shown in Fig. 4(b). From these, one straight- term with three quantum fields is more of a purely quantum forwardly calculates effect. It could be interpreted as an effective nonlinearity of κ− κ− + κ+ the quantum noise. R = Ceff −[ω] − κ−+κ+ , (39a) 4 − + − − + One can still derive Langevin equations from the resulting ω (E E ) i 2 nonlinear action if one ignores the terms which are cubic in R eff κ+ κ− +[ω] = C+ , (39b) quantum fields. In this approximation, one obtains modified 2 ω − 2E− + iκ−

033836-8 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015)

equilibrium, the fact that E+ >E− guarantees this factor is int positive, yielding − > 0. Our system, however, is not in true thermal equilibrium: as discussed in the previous section, it is 0 0 possible to have (n¯− − n¯+) < 0 by having suitably different effective temperatures for the two polariton species. One thus finds that the interactions can lead to negative damping: int − < 0. The physics of this regime and the possibility of true instability are discussed further in Sec. VD. The retarded polariton self-energies presented here directly lead to an interaction-induced modification of the polariton DOS, ρσ [ω], given by 1 ≡− GR ρσ [ω] Im σ [ω] π 1 1 FIG. 5. (Color online) Solid lines: leading-order-in-g effective =− Im . (42) Ceff − + − R cooperativities σ associated with the nonlinear interaction [defined π ω Eσ iκσ /2 σ [ω] in Eq. (41)], as a function of the detuning in the resonant regime (G = Ceff =− Signatures of g in the DOS (and corresponding changes Gres[]). One sees that the σ are enhanced near  2ωM .The eff inset zooms on the region where C− goes below zero, which signals to OMIT-style experiments) were the focus of our previous the possibility of a new kind of instability, as discussed explicitly in work [17]. For strong enough nonlinear coupling g,the Sec. VD. Circles: results of the self-consistent perturbation theory self-energies turn from simply describing an extra broadening described in Sec. IV B, which includes diagrams at all orders in g. of the polaritons to describing the coherent hybridization of the Ceff |+ |− − For all values of σ shown here, the approach converges and is in resonant and , polariton states. The key consequence good agreement with numerical simulations of the Lindblad master of this is that the single peak in the + polariton DOS splits. equation [cf. Eq. (21)]. The parameters used are γ/κ = 10−4, g = κ, Further details about this splitting (and how it can be measured = M = ωM 50κ,andn¯th 0. via an OMIT-type experiment) can be found in Ref. [17].

A ={ R }∗ 2. Keldysh self-energies with σ [ω] σ [ω] . As discussed in detail in [17], R the self-energies σ [ω] describe the hybridization between We now turn to the Keldysh self-energies, which are also the near-resonant |+ and |−,− polariton states. The res- directly calculated from the diagrams of Fig. 4(c). Here, we onant nonlinear processes underlying this hybridization [see parametrize each Keldysh self-energy via a thermal occupancy int Fig. 4(b)] are responsible for the sharply peaked self-energies factor n¯σ associated with the interaction, defined such that of Eqs. (39)(κ E ). We have introduced effective co- σ σ K ≡− int + R Ceff σ [ω] 2i 2n¯σ 1 Im σ [ω] . (43) operativities σ to parametrize how strong the decay rates resulting from these processes are (i.e., imaginary part of the int This parametrization is always possible if we let the n¯σ to be self-energy) on resonance, compared to the intrinsic polariton frequency dependent. However, we find them to be frequency linewidth. Defining independent and given by int =− R σ [ω] 2Im σ [ω] , (40) n¯0 (n¯0 + 1) (n¯0 )2 int = + − int = − n¯− 0 0 , n¯+ 0 . (44) we have n¯− − n¯+ 2n¯− + 1 int − 2 0 − 0 eff − [E+ E−] 16g˜ (n¯− n¯+) We stress that these results (as well as the self-energy results C− ≡ = , (41a) κ− κ−(κ− + κ+) above) are based on only keeping the nonlinear polariton interaction in Eq. (8), and thus assume that E+  2E−. int 2 0 + eff + [2E−] 4g˜ (2n¯− 1) C+ ≡ = . (41b) κ+ κ−κ+ B. Self-consistent calculation eff eff The definitions of C− ,C+ are analogous to the definition The self-energy results discussed so far (and in Ref. [17]) of the standard optomechanical cooperativity C = 4G2/κγ as only retain diagrams to leading order in g. To capture higher- the cavity-induced “optical damping” of the mechanics to the order effects and effectively resum diagrams at all orders in intrinsic mechanical damping. The effective cooperativities perturbation theory, one can make the diagrams in Fig. 4 are plotted in Fig. 5 as a function of the detuning (keeping self-consistent. One simply replaces all internal propagators G = Gres for all detunings). in the diagrams by full dressed propagators. The self-energy A crucial feature of the interaction-induced polariton thus becomes a functional of the full-dressed Green’s function, damping described by Eqs. (41) is their explicit temperature and the Dyson equation becomes a self-consistent equation dependence. This is a direct consequence of the multiparticle for the full Green’s function. Solving this self-consistent nature of the relevant decay process. For the − polariton, we Dyson equation allows us to capture a particular ensemble of int 0 0 have − ∝ (n¯− − n¯+), as expected for a bosonic polarization processes at all orders in g. Note that a related self-consistent bubble; a similar damping rate is found for an oscillator Keldysh approach was previously used to study a nonlinear coupled quadratically to an oscillator bath [52]. In true thermal parametric amplifier near threshold [24–26].

033836-9 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

R − In practice, one solves such self-consistent equations Here, σ (t t ) is the Fourier transform of the retarded env iteratively: in each step, one calculates the self-energy using self-energies given in Eqs. (39). The noise functions ξσ (t) int the current versions of the full Green’s functions, and then uses and ξσ (t) describe complex independent Gaussian noise env these to update the Green’s functions which will be used for processes with zero mean. ξσ (t) describes the intrinsic the self-energy calculation in the next iteration. Applying this polariton dissipation as discussed in Sec. III; its correlators iterative procedure until convergence solves the self-consistent are given in Eq. (34). The only nonzero correlator of the new Dyson equation. To improve the accuracy of our approach, we noise ξ int(t)is σ have applied this iterative strategy for the results shown in ∗ int int = int − int + Fig. 5 to Fig. 12. For each calculation, we have performed ξσ (t) ξσ (t ) σ (t t ) n¯σ 1/2 δσ,σ , (47) 20 iterations, which turn out to be more than enough to get int − where σ (t t ) is the Fourier transform of the frequency- excellence convergence of the Green’s functions. Note that by dependent interaction-induced polariton damping given in int using this self-consistent approach, n¯σ , as defined in Eq. (43), Eq. (40). becomes frequency dependent. These Langevin equations reproduce the intuitive picture As expected, the self-consistent approach does not converge sketched above: each polariton species is now effectively if the nonlinear interactions become too strong. Convergence coupled to two independent dissipative environments, with is not solely controlled by the magnitude of g˜, but is primarily int corresponding damping rates κσ and σ [ω], and correspond- determined by the effective cooperativities Ceff defined in 0 int σ ing thermal occupancies n¯σ and n¯σ . The first bath corresponds Eqs. (41). These cooperativities involve both the magnitude of to intrinsic dissipation (i.e., the intrinsic mechanical and cavity g˜ and the polariton occupancies, reflecting the fact that large dissipation), whereas the second is due to polariton-polariton temperature can also enhance the importance of nonlinearity. interactions. As shown in the figures, we find that when the self-consistent int It is worth stressing that the induced damping rates σ [ω] approach converges, it also is in excellent agreement with are in general sharply peaked functions of frequency, due to full numerical simulations of the Linblad master equation the resonant nature of the relevant scattering process. For some describing the system, Eq. (21). int parameters, the width of σ [ω] can even be much smaller than the width of the density of state ρσ [ω]; e.g., the + polaritons for  near −2ωM . In contrast, there are other cases where the V. INFLUENCE OF POLARITON INTERACTIONS ON density of state is much sharper than the interaction-induced NONEQUILIBRIUM EFFECTS dissipation rate, as is the case for the + polariton for  near A. Interaction-induced effective environment −ωM /2. As a result, the “interaction-induced” baths cannot always be considered as Markovian. From our previous discussion, we see that the nonlinear int interaction gives rise to self-energies which modify the single- The thermal occupancies n¯σ associated with the particle properties of polaritons. The imaginary part of the interaction-induced environments are plotted in Fig. 6 as a retarded self-energies describe interaction-induced polariton function of the detuning , in the interesting case where all int intrinsic dissipation (i.e., mechanical bath, cavity bath) are at damping rates σ [ω] [cf. Eq. (40)], whereas the Keldysh self-energies describe interaction-induced heating effects, with zero temperature. int associated thermal occupancy factors n¯σ [cf. Eq. (44)]. This suggests that in terms of single-particle properties, the effects B. Interaction-induced quantum heating of interactions are equivalent to having coupled the linearized We now discuss in more detail the behavior of Eq. (44) optomechanical system to new dissipative baths. In what int which gives the thermal occupancies n¯σ of the effective follows, we make this picture of an “interaction-induced interaction-induced dissipative baths introduced in the pre- effective environment” explicit. vious subsection. For simplicity, we focus on the case of exact First, note that all single-particle polariton properties of our resonance, where G = Gres and hence E+ = 2E−. Consider system are described by the effective quadratic action first the case where the linear-theory polariton dissipation is in thermal equilibrium at temperature T , i.e., n¯0 = n [E ,T ]. ∞ eq σ B σ eq c (t ) int = eff = ∗ ∗ G−1 − σ,cl In this case, it is easy to confirm that for each polariton, n¯σ Sσ dt dt (cσ,cl(t),cσ,q(t)) σ (t t ) , 0 −∞ cσ,q(t ) n¯σ , i.e., the interaction-induced dissipation also corresponds to the same temperature T . Thus, if without interactions the (45) eq polaritons start in equilibrium at the same temperature, then the same is true with interactions. where G−1(t − t ) is the Fourier transform of the inverse of the σ The actual situation is, however, more complicated: due 2 × 2 matrix of the full Green’s functions given in Eq. (38). to quantum heating effects, the effective temperatures of the Further, as discussed in Sec. III, this quadratic action is two polariton species are different even without interactions. completely equivalent to a set of linear Langevin equations. n¯0 and n¯0 are thus not related as they would be in thermal Using the standard derivation [23], the effective action in − + equilibrium; this can be parametrized as Eq. (45) is equivalent to the Langevin equations 0 2 0 (n¯−) 0 ∞ n¯ ≡ + δn¯ . (48) κ + 0 + =− + σ − R − 2n¯− + 1 ∂t cσ (t) iEσ cσ (t) dt σ (t t )cσ (t ) 2 −∞ Thermal equilibrium and the condition E+ = 2E− would im- − env − int 0 = 0 = ξσ (t) ξσ (t). (46) ply δn¯+ 0; δn¯+ 0 means that even in the linearized theory,

033836-10 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015)

environments describe quantum heating physics. Together, they will determine the total number of polaritons produced by quantum heating and, more specifically, the energy distribution function of the polaritons. We define this distribution function in the standard manner, as an energy-dependent distribution eff function n¯σ [ω]. This quantity is defined via the full Keldysh and retarded polariton Green’s functions GK ≡− eff + GR σ [ω] 2i 2n¯σ [ω] 1 Im σ [ω] . (50) If our polaritons were in thermal equilibrium at temperature eff Teq, then the distribution function n¯σ [ω] would simply be the Bose-Einstein distribution nB [ω,Teq]. In contrast, in our system this function will be determined by the thermal FIG. 6. (Color online) Effective thermal occupancies for the var- occupancies of the two effective baths, and the strength of the ious baths coupled to the two polariton species, as a function of couplings (i.e., damping rates) to each. Using the expression  with G = Gres[]. Solid lines: occupancies associated with the of the dressed Green’s functions G[ω] coming from the “interaction-induced” baths to leading order in g [cf. Eqs. (44)]. Dyson equation [cf. Eq. (38)] and the relation between the Circles: same, but calculated using the all-orders self-consistent self-energies given in Eq. (43), one finds approach of Sec. IV B. Dashed lines: occupancies of the intrinsic polariton baths [cf. Eqs. (24)]. Note the leading-order occupancy int[ω]n¯int + κ n¯0 n¯eff[ω] = σ σ σ σ . (51) for the − polariton interaction bath (n¯int) diverges when n¯0 = n¯0 σ int − − + [ω] + κσ 0 0 σ (/ωM ≈−0.8) and becomes negative when n¯+ > n¯− (/ωM  −0.8), as shown in the inset. This divergence persists in the self- This is exactly the simple expression that would be expected consistent theory, but occurs at smaller-magnitude detunings. This for a free bosonic mode coupled independently to two baths; negative occupancy signals the possibility of a different kind of the same form holds for the linear theory [cf. Eqs. (24)]. instability, as discussed explicitly in Sec. VD. All curves are plotted In order to focus our attention on the contribution of the for n¯M = 0[cf.Eq.(18)], γ/κ = 10−4, ω /κ = 50, and g = κ. eff th M nonlinear interaction to n¯σ [ω], we rewrite Eq. (51)usingthe int int expressions for σ [ω] [cf. Eqs. (39) and (40)] and n¯σ [cf. the two polaritons experience different effective temperatures Eq. (44)]. Doing so, one gets (see Fig. 3). γ 2 Using this definition, the thermal occupancy of the − n¯eff[ω] = n¯0 + I σ , (52) σ σ σ − 2 + 2 polariton interaction-induced bath becomes (ω ωσ ) γσ + 0 2 with int 0 0 (1 2n¯−) n¯− = n¯− + δn¯+ . (49) n¯0 (1 + n¯0 ) − δn¯0 (1 + 2n¯0 ) Ceff − − + − I = n¯int − n¯0 σ , (53a) σ σ σ 1 + Ceff Thus, a deviation from true thermal equilibrium in the linear σ κ− + κ+ theory (i.e., without polariton interactions) causes the occu- = + Ceff = + Ceff int γ− 1 − ,γ+ κ− 1 + , (53b) pancy of the interaction-induced bath n¯σ and the intrinsic bath 2 0 n¯σ (linear-theory dissipation) to deviate from one another. This ω− = E+ − E−,ω+ = 2E−. (53c) is not surprising: in this case, the nonlinear interaction between the two polariton species tends to favor their thermalization, The contribution from the interaction-induced environment and hence transfers energy from the high-temperature species appears as a sharp Lorentzian in the polariton distribution to the low-temperature species. functions. This is a direct consequence of the resonant nature Finally, we also stress that even in the case where the of the relevant nonlinear scattering process. = eff intrinsic mechanical and cavity dissipation is at zero tem- For exact resonance (E+ 2E−), both n¯σ [ω] and the perature (i.e., the system only experiences vacuum noise), single-particle DOS ρσ [ω] [cf. Eq. (42)] are peaked at Eσ , int the interaction-bath thermal occupancies n¯σ will be nonzero, so that the nonlinear interaction heating effects are maximal. 0 and are in general different from n¯σ . This is shown explicitly Even in this case, though, the frequency dependence of the eff in Fig. 6. We thus see that interactions change the effective interaction contribution to n¯σ [ω] can be very different than temperature associated with quantum heating effects. that of the polariton DOS. In this fully resonant case, the polaritons distribution functions evaluated at ω = Eσ adopt C. Polariton energy distribution functions the following simple form: The picture established so far is that our optomechanical Ceffn¯int + n¯0 n¯eff[E ] = σ σ σ . (54) polaritons are each effectively coupled to two independent σ σ Ceff + 1 effective environments, one of which is self-generated and σ due to the nonlinear optomechanical interaction. In the limit Equation (54) is plotted in Fig. 7 as a function of the laser eff where both the intrinsic cavity and mechanical dissipative detuning  and is compared to n¯σ [Eσ ] obtained using the M = baths are at zero temperature (n¯th 0), both these effective self-consistent approach (cf. Sec. IV B).

033836-11 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

limit where γ κ, one can simplify the instability condition, eff i.e., C− < −1, to + 2 0 0 κ−(κ− κ+) κ− n¯+ > n¯− + ≈ , (56) 16g˜2 16g˜2 and the final mean number of − polaritons becomes 2 0 eff 16g˜ n¯+ n¯− [E−] ≈ . (57) 2 2 0 κ− − 16g˜ n¯+ The surprising negative damping occurring here can be understood as the result of a parametric instability arising directly from the polariton interaction. In fact, Eqs. (56) and (57) have exactly the same form as the instability condition and the mean number of excitations that one would find for a degenerate parametric amplifier (DPA) pumped near degeneracy [49,50]. In a DPA, a pump-mode photon scatters into signal mode photons, and the pump mode is coherently driven. In our system, the − polariton plays the role of the signal mode in a DPA, while the + polariton plays the role of a pump mode that has been incoherently driven by noise. Despite this incoherent driving, the form of the above equations is the same as a coherently driven DPA (see Appendix B). FIG. 7. (Color online) (a) Solid lines: net polariton occupancies Note that nondegenerate parametric amplifier instability can eff in the presence of nonlinear interaction n¯σ [Eσ ], calculated using be realized in a linearized optomechanical system driven = leading-order self-energies, as a function of  with G Gres[]. with a blue-detuned laser (see, e.g., [53,54]). In contrast, the Circles: same, but calculated using the self-consistent approach (cf. instability described here occurs for a red-detuned drive. = Sec. IV B). Dashed curves: occupancies for g 0, i.e., calculated Note that our discussion here is based solely on using in the linearized theory. In each case, the mechanical temperature is the leading-order results for the polariton self-energies. In- zero (n¯M = 0) and g = κ. (b) Same as (a), but now with n¯M = 100 th th cluding higher-order effects via our self-consistent approach and g = 0.1κ; main plot and inset show different ranges of . Near can dramatically change the onset and magnitude of the −2ωM , one sees an important contribution to the mean number of + polaritons due to nonlinear interaction; this contribution is greatly interaction-induced negative damping. We discuss this more enhanced by temperatures, as discussed in Sec. VI D. All the curves in Sec. VI C. −4 are plotted for γ/κ = 10 and ωM /κ = 50. VI. OBSERVABLE SIGNATURES OF QUANTUM HEATING EFFECTS D. Nonlinear parametric heating Among the more striking nonequilibrium behaviors possi- In the previous sections, we have demonstrated how 0 0 quantum heating effects can lead to a finite density of op- ble in the linear theory is the possibility of having n¯+ > n¯−, i.e., the thermal occupancy of the higher-energy + polariton tomechanical polaritons at zero temperature; we also discussed exceeds that of the − polariton. We discuss this regime in how these quantum heating effects can be modified by the more detail here, focusing on the exactly resonant case where nonlinear interaction. In this section, we discuss how these E+ = 2E−. effects lead to observable signatures in the light leaving We start by recalling that the total damping of the − the optomechanical cavity. We first relate the cavity output polariton is spectrum to the polariton distribution functions, and then discuss specific parameter regimes where the heating effects tot int eff κ− [E−] = κ− + − [E−] = κ−(1 + C− ). (55) are most prevalent. In addition, we propose a way to effectively control the strength of the nonlinear interaction in experiments From Eqs. (41), we see that if we have the occupancy inversion by tuning in and out the resonance condition (i.e., by varying n¯0 > n¯0 , then Ceff < 0, and hence the contribution of the + − − G at fixed detuning ). Doing so, one can explicitly isolate nonlinear interaction to the damping rate of the − polaritons and observe the nonlinear interaction signatures in the cavity becomes negative. This is at first glance surprising: we have output spectrum. opened a scattering process for the − polariton via the nonlinear interaction, and yet we get an increase in its lifetime. eff A. Polariton energy distribution functions We also have the possibility of an instability if C− < −1. eff From Fig. 5, one sees that a negative C− occurs for To measure polariton occupancies, we consider a measure- detunings near  =−0.5ωM . In that regime, the − polariton ment of the flux of photons leaving our cavity (assuming a is mostly photonic and the + polariton is mostly phononic. single-sided cavity, and that the reflected classical drive tone is Consequently, by having a high intrinsic mechanical bath filtered away). The spectrum of this flux is given in the standard M temperature n¯th 1, one naturally can achieve the inverted manner [49] by the normal-ordered cavity spectrum (also 0 0 occupancy regime where n¯+ n¯−. In that case and in the known as the “lesser” Green’s function within the Keldysh

033836-12 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015) technique), In what follows, we consider signatures of quantum heating ∞ in the spectrum for particularly interesting choices of the drive iωt † Sd [ω] ≡ dt e dˆ (0)dˆ(t). (58) laser detuning . −∞ Rewriting Eq. (58) in terms of polariton Green’s functions B. Limit of zero-temperature dissipation: Effects of nonlinear yields interaction on quantum heating = 2 eff The first studied limit is for a mechanical bath at zero tem- Sd [ω] 2π αd,σn¯σ [ω]ρσ [ω] M = perature (n¯th 0), where the finite number of polaritons inside σ =± the optomechanical cavity exclusively comes from quantum + 2 eff − + − heating. We focus on two regimes: the red sideband drive, 2π α¯ d,σ(n¯σ [ ω] 1)ρσ [ ω]. (59) σ =± i.e.,  =−ωM , where both polaritons are equal mixture of photons and phonons, and the asymmetric polaritons regime, Here, the coefficients αd,σ andα ¯ d,σ are the change-of-basis where the polaritons are not equal combinations of photon coefficients introduced in Eq. (5a) and plotted in Fig. 2, and and phonon. For the latter regime, we chose  =−1.8ωM the polariton DOS, ρσ [ω], is defined in Eq. (42). We are still as a representative laser detuning. The results predicted for working in a rotating frame with respect to the laser drive the cavity driven on the red sideband have the advantage = frequency ωL, hence, ω 0 implies output photons leaving to be robust to temperatures since both polaritons have an at the laser frequency. The negative frequency term means M important photon part; this implies that even for finite n¯th , that removing a photon at frequency ω can involve creating quantum heating is still the prevalent source of polaritons. a polariton at frequency −ω, as expected from the presence In contrast, a laser detuned at  =−1.8ωM leads to more of “anomalous” terms in Eq. (5b). Finally, Eq. (59) reflects striking modifications of the polaritons energy distribution the fact that the only nonzero polariton Green’s functions are Ceff =− Ceff =− since σ [ 1.8ωM ] > σ [ ωM ](cf.Fig.5). In those that conserve the number of polaritons independently, as both cases, we show that the nonlinear interaction modifies the discussed in Secs. III and IV. energy distribution of the polaritons as it tends to thermalize Using the same assumptions, we also derive the cavity DOS the two species. and its energy distribution function: = 2 − 2 − 1. Results for symmetric polaritons (red sideband drive) ρd [ω] αd,σρσ [ω] α¯ d,σρσ [ ω] , (60) σ =± In Fig. 8, we plot the cavity DOS [Eq. (60)], the cavity spectrum [Eq. (59)], its energy distribution function [Eq. (61)], 1 S [ω] eff = d = n¯d [ω] . (61) and the corresponding effective temperature [Eq. (62)] for  2π ρd [ω] −ωM . The linearized theory (g = 0) is compared to the case From the cavity energy distribution, we can use the Bose- where g = κ. For the latter interacting case, we present results Einstein distribution to define an effective cavity temperature obtained from three different methods: the leading order in g˜ (kB = 1) self-energies [Eqs. (42) and (52)], the self-consistent approach described in Sec. IV B, and, finally, a numerical simulation of ω T eff[ω] ≡ , (62) the Lindblad master equation given in Eq. (21). By comparing d + 1 ln 1 eff the three different approaches, one sees that for g = κ, higher- n¯d [ω] order corrections captured by the self-consistent approach play which is always possible if we let the effective temperature to an important role for the effective distribution functions and be frequency dependent. the effective temperatures [Figs. 8(e)–8(h)]. In contrast, the As discussed extensively in [17], the polariton DOS DOS is already well described at the leading order in g˜, which ρσ [ω] can be directly measured in an OMIT-style experiment is in agreement with Ref. [17]. [19,29–31], where one measures the reflection of a weak The splitting of the + polariton resonance in ρd [ω] and additional probe tone incident on the cavity. The cavity Sd [ω] near ω = E+ arises from the hybridization between the spectrum in contrast also yields information on polariton states |+ and |−,−; the resulting hybridized states become occupancies. As the polariton energies Eσ are well separated, spectrally resolved for g  κ (see [17] for more details). The the output spectrum will have a series of peaks corresponding same hybridization phenomena give rise to a resonance in to the emission or absorption of a given polariton species. eff eff − = n¯d [ω] and Td [ω]atthe polariton frequency ω E− [see The magnitude of these peaks is directly proportional to the Figs. 8(e) and 8(g)]. occupancy of the given polariton. In Fig. 9, we plot the number of photons leaving the cavity It is also useful to look at the total number of photons near each polariton resonance, as defined in Eq. (63), and due to a given polariton resonance, which we can obtain show that one can effectively isolate the effects of nonlinear by integrating the output spectrum around the corresponding interaction. To do so, one varies G around Gres such that the resonance. We thus introduce nonlinear interaction gets amplified by a factor of ωM /κ when ω +δω = 0 dω the nonlinear process becomes resonant, i.e., for G Gres, tot ≡ n¯d [ω0,δω] Sd [ω], (63) compared to the off-resonant case, i.e., G − G  κ.Away − 2π res ω0 δω from resonance, the number of photons leaving the cavity is where ω0 will be taken to be E±, and δω will be taken to be in good approximation given by the linearized theory (dashed M = larger than the spectral width of the given polariton resonance. lines in Fig. 9). For the parameters here (n¯th 0 and γ/κ 1),

033836-13 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

FIG. 9. (Color online) (a) Output photon flux in a bandwidth of 5κ near the − polariton resonance E− [cf. Eq. (63)] as a function of the FIG. 8. (Color online) (a), (b) Cavity DOS near the − polariton many-photon coupling G, and for a control laser detuning  =−ωM . resonance and the + polariton resonance, respectively, for a cavity (b) Same as (a) but near the + polariton resonance E+. Note that G driven on the red sideband and when the nonlinear interaction is can be varied by simply tuning the amplitude of the control laser. resonant (i.e., G = Gres). We work in the frame rotating at the drive On resonance, i.e., G = Gres, the nonlinear effects are enhanced by frequency so that ω = 0 refers to the drive frequency while E+ = a factor of ωM /κ = 50 compared to the off-resonance case, i.e., 2E− = 63.24κ in this frame. The light dashed curves represent the (G − Gres)  κ. The light dashed curves represent the linearized linearized theory, the green dotted ones represent the results to leading theory (g = 0), the solid blue curves are for the self-consistent orders in g˜ [Eq. (42) for the DOS], the full blue curves are the results approach as described in Sec. IV B, and the black dashed curves are for of the self-consistent approach as described in Sec. IV B, and the the numerical simulation of the Lindblad master equation shown (21), black curves are for the numerical simulation of the Lindblad master but this time, using the full nonlinear part of the Hamiltonian in equation shown in Eq. (21). (c), (d) Cavity spectrum in the same Eq. (6) that includes all the nonresonant nonlinear processes. For all = −4 M = conditions; the results to leading orders in g˜ are given in Eqs. (52) the curves, we used γ/κ 10 and n¯th 0. and (59). (e), (f) Cavity energy distribution function [Eq. (61)] and (g), (h) the corresponding (frequency-dependent) effective temperatures − the − polariton is more phononlike while the + is more [Eq. (62)]. The parameters used for all the curves are γ/κ = 10 4 and = photonlike. This particular asymmetry leads to larger values ωM /κ 50, which leads to the leading-order effective cooperativities eff eff eff of C than in the red sideband regime, mainly because of the [i.e., Eq. (41)] C− = 0.18 and C+ = 2.46. σ long lifetime of the phononlike polariton (cf. Fig. 5). Due to Ceff these larger σ , the results obtained to the leading order in the linearized theory leads to [see Eqs. (16) and (A4)] g˜ are not sufficient to recover the numerical simulation of the 2 Lindblad master equation [cf. Eq. (21)] even for the DOS. It tot 2 0 1 (G/ωM ) n¯ E±, κ ≈ α n¯ ≈ . is then crucial to use the self-consistent approach to properly d [ 5 ] d,± ± ± (64) 8 1 2G/ωM describe the effects of nonlinear interaction. Moreover, from =− Figure 9 clearly shows the thermalization between po- Figs. 10(g) and 10(h), one sees that unlike the case  ωM , laritons brought about the nonlinear interaction. Without the nonlinear interaction cools down the − polaritons and interactions, at  =−ω the − polaritons have a lower heats up the + polaritons. This is also consistent with a partial M =− effective temperature than the + polaritons [cf. Fig. 3(c)]. thermalization, as for  1.8ωM , without interactions the − When G is near Gres, the nonlinear interaction “turns on” effective temperature of the polaritons is greater than that and allows the two polariton species to exchange energy and of the + polaritons [cf. Fig. 3(c)]. partially thermalize (i.e., interactions heat up the − polaritons while cooling down the + polaritons.) C. Parametric amplification of the − polaritons While our emphasis in this work has been on quantum 2. Results for asymmetric polaritons heating effects involving zero-temperature dissipation, our In Fig. 10, we plot the same functions as in Fig. 8,butin approach can also conveniently describe thermal nonlinear the case where the laser detuning is  =−1.8ωM (and again, phenomena. Perhaps the most striking example of this occurs G = Gres). For this more negative detuning, the polaritons for detuning  ≈−ωM /2, where leading-order perturbation are no longer an equal mixture of photons and phonons: theory predicts the presence of a parametric instability at finite

033836-14 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015)

= FIG. 10. (Color online) Same as Fig. 8 but for laser detuning FIG. 11. (Color online) Same as Fig. 8 but for laser detuning  M −0.65ωM and finite temperature n¯ = 650. In these circumstances,  =−1.8ωM . In that case, E+ = 2E− = 92.08κ in the rotating th frame and the corresponding leading order [i.e., Eq. (41)] effective leading-order perturbation theory predicts that the system is close eff eff to a parametric instability (see Sec. VD). Note the mechanical cooperativities C− = 1.92 and C+ = 5.40. damping rate here is larger than in previous plots (γ = 10−3κ), as this enhances the pumping effects of mechanical temperature. For ωM = 50κ and g = κ, the leading-order effective cooperativities [cf. temperatures (see Sec. VD). In that regime, the nonlinear eff eff Eq. (41)] are C− =−0.97 and C+ = 0.75. In the frame that rotates interaction acts as an incoherently pumped degenerate para- at the drive frequency, E+ = 2E− = 53.34κ. One sees from these metric amplifier with the + polariton (mainly phonon) being plots the crucial importance of higher-order-in-g terms. the (incoherent) pump and the − polariton (mainly photon) being the signal mode. predicts a photon occupancy at the − polariton resonance In Fig. 11, we show the cavity DOS [cf. Eq. (60)], the cavity eff of n¯ [E−] ≈ 50, in the self-consistent approach, one only spectrum Sd [ω] [cf. Eq. (59)], and the corresponding energy d eff obtains a value 1.3. This is of course still much larger distribution function n¯d [ω] [cf. Eq. (61)] for a laser detuning M than what would be obtained without interaction; in that case,  =−0.65ωM and n¯ = 650. Combined with a damping th − eff  rate of the mechanical resonator γ = 10 3κ, its resonant n¯d [E−] 0. frequency ωM = 50κ and g = κ, one obtains, to leading order eff in g, an effective cooperativity C− =−0.97 [cf. Eq. (41)]. D. Effective two-phonon absorption eff As discussed in Sec. VD,forC− =−1, the leading-order Another striking example of temperature-enhanced nonlin- perturbation theory predicts a parametric instability caused by ear effects occurs for a laser detuning  ≈−2ωM .Inthis eff the nonlinear interaction. For C−  −1, one thus expects an regime, the − polaritons are mostly phonons, which leads R important narrowing of the cavity DOS as well as an important to a thermal “stimulated emission” enhancement of −[ω], heating of the cavity near the − polariton resonance. These leading to a strong modification of the cavity DOS. This predictions from the leading-order self-energy are shown in physics follows from Eqs. (39), and was discussed extensively Fig. 11. in Ref. [17]. However, as shown in Fig. 7, one also obtains Not surprisingly, higher-order corrections (as captured by significant nonlinearity-induced heating of the cavity in this the self-consistent self-energy) are especially important in this regime, as we now describe. regime and strongly contribute to prevent the system from Recall that if  =−2ωM , then Gres = 0 and the + going unstable. More precisely, it is the hybridization between (−) polaritons are exactly photons (phonons). A necessary the states |+ and |−− that competes with the parametric consequence is that the amplitude for the resonant nonlinear amplification of the − polaritons; the high number of − interaction vanishes, g˜ = 0. One thus ideally wants a detuning polaritons leads to an important modification of the energy close to, but not exactly equal to, −2ωM such that 0

033836-15 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015)

sideband [19,29–31], except that here, it is the nonlinear interaction that is involved. In Fig. 12(b), we show that nonlinear interaction greatly modifies the cavity spectrum; no output light is predicted by the linearized theory while a sharp signal is produced when nonlinear interaction is considered. The effective two-phonon absorption becomes a very important process in the high-temperature limit. In that case, the nonlinear interaction becomes easier to observe, but the phenomenon tends to become purely classical. In order to support this statement, we present a classical treatment of this phenomenon in Appendix D and show that it succeeds to recover the right dependence in temperatures and single- photon coupling constant g of the two-phonons absorption signature in the cavity spectrum at the lowest order in g.

VII. CONCLUSIONS In this work, we have described the effects of nonlinear interaction on the nonequilibrium state of the optomechanical cavity. We have shown the tendency of the nonlinear inter- action to thermalize the polaritons, that it can lead to a new parametric instability for a red-detuned laser drive as well FIG. 12. (Color online) (a) Signatures of the effective two- as a temperature-enhanced effective two-phonon absorption. phonon absorption in the cavity DOS near the + resonance; the inset In addition to these results, we have presented in details shows a larger range of frequency than the main plot. The light-blue many technical aspects with the aim to provide the proper dashed curve represents the linearized theory (g = 0), the green tools to investigate nonlinear effects in more complicated dotted one represents the results to leading order in g [Eqs. (42), (52), optomechanical systems. This work also opens the path to and (59)], and the full dark-blue and black curves are the results of a more detailed characterization of this parametric instability M = the self-consistent approach (S.C.) of Sec. IV B for n¯th 100 and 0, and further investigations of its consequences. respectively. (b) Signatures of the effective two-phonon absorption in the cavity spectrum in the same circumstances than (a). It shows the striking temperature-enhanced effect of nonlinear interaction in ACKNOWLEDGMENTS the + polariton population. The parameters used are γ/κ = 10−4 and We thank N. Didier and F. Beaudoin for helpful discussions. ωM /κ = 50. This work was supported by NSERC and FQRNT. spectrum simplifies to 2 2 M 2 2 APPENDIX A: CHANGE OF BASIS FOR A DETUNING κ G g 16 n¯ κ− =−ω S [ω] ≈ th . ON THE RED SIDEBAND ( M ) d 3 − 2 + 2 γ ωM κ + 8 G2 κ (ω E+) κ− 1 2 In this appendix, we show particular examples of the 9 ωM γ transformation used to go from the photon and phonon basis (65) to the polariton basis for  =−ωM . As discussed near Eq. (25), the competition between the We start with the change-of-basis coefficients αb/d,± and parametric heating associated with the linearized optome- α¯ b/d,± introduced in Eqs. (5): chanical interaction and the standard optomechanical optical 1 ω + E± damping leads to an optimal value of G/ωM where the =±√ M√ 2 = 9 γ αb,± , (A1) heating is maximal. In the optimal case, (G/ωM ) 16 κ and 8ωM E± g 2 M 2 S [E+] ∼ ( ) (n¯ ) . Details of the calculations that lead to d κ th 1 ωM − E± Eq. (65) are presented in Appendix C. α¯ b,± =±√ √ , (A2) 8ω E± In Fig. 12, the cavity DOS and the cavity spectrum near M + + ± the polariton resonance is plotted for a laser detuning near = √ 1 ωM√ E − M αd,± , (A3) 2ωM and for finite mechanical bath temperatures n¯th .In 8ωM E± Fig. 12(a), we show the sharp dip in the density of state due 1 ωM − E± to nonlinear interaction, also described in Ref. [17]. Note, α¯ d,± = √ √ , (A4) however, that effects of higher order in g, captured in the 8ωM E± self-consistent approach, considerably modify the predictions with made in Ref. [17], where only effects to leading order in g were considered. This sharp feature is completely analogous E± = ω ± G/ω , to the optomechanical induced transparency (OMIT) observed M 1 2 M (A5) in the optomechanical cavity weekly driven on the red Gres/ωM = 3/10. (A6)

033836-16 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015)

From these coefficients, we can express the different effective given by ∂−cˆ− = 0. In that case, we have A/B nonlinear coupling constants g and A± introduced in σσ σ 0 Eq. (6). Here, we only show few examples: 4ig˜ n¯+ † 2 cˆ− =− cˆ− − √ ξˆ−, (B4) A A A κ− κ− g++− + g+−+ + g−++

= + + − + − 2 2 −1 † g[αd, α¯ d, (αb, α¯ b, ) ⇒ cˆ− = √ (1 −|Q| ) (Qξˆ− − ξˆ−), (B5) κ− + (¯αd,−αd,+ + α¯ d,+αd,−)(αb,+ + α¯ b,+)] G g ω 3/4 E− − E+ + (E− + 2E+) with = √ M ωM , 4 2 E− E+ 0 4ig˜ n¯+ (A7) Q ≡ . (B6) κ− ≡ B † g˜ g−−+ Imposing the right commutation relation [cˆ−,cˆ−] = 1, one finds = g[αd,−α¯ d,−(αb,+ + α¯ b,+) 2 + (¯αd,−α¯ d,+ + αd,−αd,+)(αb,− + α¯ b,−)] † κ−(1 −|Q| ) [ξˆ−,ξˆ−] = , (B7) G G g ω 3/4 1 + 2 E− + 1 − E+ 4 =− √ M ωM ωM , † 4 2 E+ E− which has the right units since ξˆ−ξˆ− represents a rate at (A8) which excitations are coming in. In the case of Gaussian noise † with zero mean (i.e., ξˆ−ξˆ−=0), as studied all along this 2 2 work, we find that the mean number of − polaritons inside the A− = g 2αd,−α¯ d,− + α¯ − + α¯ + (αb,− + α¯ b,−) d, d, optomechanical cavity is + (¯αd,−αd,+ + α¯ d,+αd,−)(αb,+ + α¯ b,+) 2 2 0 † |Q| 16g˜ n¯+ E− G  = = 3/4 − + + + − − cˆ−cˆ− . (B8) g ωM 2 ω 1 E ω (E E ) −| |2 2 − 2 0 = √ M M . 1 Q κ− 16g˜ n¯+ 4 2 E− E+ | |2 = (A9) From this result, one immediately sees that for Q 1, the mean number of polaritons diverges. This condition sets the threshold above which the parametric amplifier goes unstable. More precisely, the system becomes unstable when APPENDIX B: PARAMETRIC AMPLIFIER DESCRIPTION 2 In this appendix, we derive the condition to get an 0 κ− n¯+ > . (B9) instability from the parametric amplifier Hamiltonian and get 16g˜2 the corresponding mean number of signal excitations. These calculations are based on the formalism introduced in [50]. These results have exactly the same form as Eqs. (56) and (57), Starting with the nonlinear interaction in the effective but here, we have explicitly used the fact that the + polaritons 0 Hamiltonian of Eq. (8) and considering the hypothetical case are coherently pumped such that cˆ+=n¯+. 0 where the + mode is coherently pumped, such that cˆ+=n¯+, we get, in the mean-field approximation, APPENDIX C: EXPANSION OF THE PHOTON SPECTRUM FUNCTION FOR ≈−2ωM ˆ ≈ 0 + Heff g˜ n¯+(cˆ−cˆ− H.c.). (B1) In this appendix, we show details of the calculation that leads to Eq. (65) starting from Eqs. (52) and (59). Here, the ≈− ∼ M Using standard input-output theory, one derives the following limit of interest is for  2ωM , G/ωM γ/κ 1, n¯th Ceff + equation of motion: 1, and + 1. In these circumstances, photons are mainly polaritons, such that Eq. (65) for frequencies near E+ reduces to κ− √ ∂t cˆ− = i[Hˆeff,cˆ−] − cˆ− − κ−ξˆ− (B2) 2 | ≈ eff Sd [ω] ω≈E+ 2πn¯+ [ω]ρ+[ω]. (C1) √ 0 † κ− =−2g˜ n¯+cˆ− − cˆ− − κ−ξˆ−. (B3) ≈− ∼ 2 Also, for  2ωM and G/ωM γ/κ 1, the mean num- ber of photons inside the cavity without nonlinear interaction 0 ˆ n¯+ is negligible compared to the nonlinear contribution. Thus, Here, ξ− represents an incoming field, which includes noise eff − from Eq. (52), we see that n¯+ [ω] reduces to a sharply coming from the bath coupled to the polaritons [as in ≈ ≈ + ∼ Eq. (33)]. Since we will focus only on quantities evaluated at peaked Lorenztian with width γ+ κ− γ (1 opt) γ ≈ 8 G2 ω = E− and that Hˆ of Eq. (B1) is written in the interaction with the optical damping rate opt 2 κ. Consequently, eff 9 ωM picture where E− = 0, we can seek for the particular solution ρ+[ω][Eq.(42)], which has a characteristic width of κ+ ≈ κ,

033836-17 MARC-ANTOINE LEMONDE AND AASHISH A. CLERK PHYSICAL REVIEW A 91, 033836 (2015) can be evaluated at E+. Doing so, one gets We solve these nonlinear coupled equations of motion via a perturbation approach. Starting with a sinusoidal displace- eff S [ω]| ≈ ≈ 2πn¯ [ω]ρ+[E+], (C2) d ω E+ + ment x(t) = x0sin(ωM t), we get the consequent evolution of the optical mode and, then, the perturbed displacement 2 1 2 of the mechanical resonator. This is a good approximation only + + = ≈ ρ [E ] eff . (C3) π κ+(1 + C+ ) πκ in the weak-coupling regime (G κ) where the eigenstates have a well-defined number of photons and phonons. This eff 0 Now, we can write n¯+ [ω] as (we have dropped n¯+ and method is thus restricted to the extreme detuning case ( ≈ + = − explicitly used E 2E ) −2ωM ) studied in Sec. VI D. 2 Following this method, one finds that the steady-state cavity κ− n¯eff[ω] ≈ n¯intCeff , (C4) + + + 2 2 field is given by (ω − E+) + κ− +∞ with i(n−m)J ()J () n m i[(n+m)ωM −ωL]t 2 0 2 a(t) = iain e . (D3) 4g˜ (n¯−) κ + − int eff =−∞ i(nωM ) n¯+ C+ = . (C5) n,m 2 κ−κ+ Thus, Here, Jn(x) are the Bessel functions of the first kind and where, in the small displacement limit,  ≡ x0 ωC plays the role of the 2 0 2 2 L0 ωM 16 g˜ (n¯−) κ− perturbation parameter. S [ω]| ≈ ≈ . (C6) d ω E+ 2 2 2 κ κ− (ω − E+) + κ− Note that by computing the radiation pressure force acting on the mechanical resonator in Eq. (D2) and by expanding γ n¯M Again, for this particular limit, we can approximate n¯0 ≈ th  − κ− to the lowest order in , one recovers the standard optical and g˜2 ≈ ( G )2g2, such that we get Eq. (65), i.e., damping and optical spring constant coming from the linear ωM interaction [cf. Eq. (2b)] in the weak-coupling regime [40]. | κSd [ω] ω≈E+ We focus here on the peak in the cavity spectrum that 2 2 M 2 2 comes from the nonlinear interaction. According to Eq. (58), κ G g 16 n¯ κ− ≈ th . (C7) the proper definition of the classical counterpart of the photon opt 2 2 γ ωM κ + (ω − E+) + κ− 1 γ spectrum to use is 1 ∞ APPENDIX D: CLASSICAL TREATMENT OF THE ≡ iωt ∗ +  Scl[ω] dt e δa (t0)δa(t0 t) t0 . (D4) EFFECTIVE TWO-PHONON ABSORPTION ωC −∞ As shown in the Sec. VI D, the nonlinear process cor- Here, the fluctuations of the cavity field δa(t)aregivenby responding to the absorption of two phonons by a photon keeping only the n =−m terms in Eq. (D3). The mean from the classical drive is greatly enhanced by temperature.   value ... t0 means taking the average over all initial time In particular, even if the nonlinear interaction directly comes t0 and the coefficient 1/ωC is there to ensure that units match from single-photon dynamics, one can show that in the high- the definition of the photon spectrum used in the quantum temperature limit, this phenomenon becomes purely classical. treatment [cf. Eq. (58)]. In what follows, we present an accurate classical description Using Eq. (D3) and focusing only on the frequency range of this limit. near 2ωM with  ≈−2ωM , one finds that in the good cavity The classical Langevin equations for the cavity field limit (κ ωM ), the classical cavity spectrum reads as coupled to a mechanical resonator via radiation pressure force in the limit of weak displacements are G 2 g 2 ≈ M 2 − Scl[ω] 16 n¯th πδ(ω 2ωM ), (D5) κ x(t) ωM κ −iωLt a˙(t) = − − iωC 1 − a(t) + ia¯ine , (D1) 2 L0 where we have used the definition of the single-photon xZPFωC 2 coupling constant g ≡ , with xZPF being the zero-point γ |a(t)| L0 + + 2 = x¨(t) x˙ ωM x . (D2) motion of the mechanical resonator and the semiclassical 2 mL0 1 2 2 = M M relation 2 mωM x0 ωM n¯th valid for n¯th 1. Here, m is the mass of the mechanical oscillator, L0 is the If one takes opt = 0 and γ → 0inEq.(65), Sd [ω ≈ E+] = length of the cavity when x(t) 0, and the damping rate and Scl[ω ≈ 2ωM ] agree. The classical approach fails in getting κ has been introduced following the standard input-output the width of the peak since the mechanical damping rate is formalism [50]. The cavity field is normalized such that the not taken into account while calculating the effects of the 2 steady-state mean energy inside the cavity is U¯cav =|a¯| = mechanical motion on the cavity field. Moreover, the optical | |2 κ 2 + 2 κ a¯in /[( 2 )  ], where a¯ is defined as the steady-state damping does not come out in this result in the level of − mean value of the cavity field a(t) = [a¯ + δa(t)]e iωLt . approximation we have used.

[1] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. [2] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity Phys. 86, 1391 (2014). Optomechanics (Springer, Berlin, 2014).

033836-18 REAL PHOTONS FROM VACUUM FLUCTUATIONS IN . . . PHYSICAL REVIEW A 91, 033836 (2015)

[3] J. Teufel, T. Donner, D. Li, J. Harlow, M. Allman, K. Cicak, [29] G. S. Agarwal and S. Huang, Phys.Rev.A81, 041803 A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, Nature (2010). (London) 475, 359 (2011). [30] S. Weis, R. Riviere,` S. Deleglise,´ E. Gavartin, O. Arcizet, A. [4] J. Chan, T. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010). S. Groblacher,¨ M. Aspelmeyer, and O. Painter, Nature (London) [31] A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. 478, 89 (2011). Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, Nature [5] D. W. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, (London) 472, 69 (2011). andD.M.Stamper-Kurn,Nature (London) 488, 476 (2012). [32] M. Marthaler and M. I. Dykman, Phys.Rev.A73, 042108 [6] A. H. Safavi-Naeini, S. Groblacher,¨ J. T. Hill, J. Chan, M. (2006). Aspelmeyer, and O. Painter, Nature (London) 500, 185 (2013). [33] M. I. Dykman, Phys. Rev. E 75, 011101 (2007). [7] T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. [34] I. Serban and F. K. Wilhelm, Phys. Rev. Lett. 99, 137001 (2007). Regal, Phys. Rev. X 3, 031012 (2013). [35] M. I. Dykman, M. Marthaler, and V. Peano, Phys.Rev.A83, [8] A. Nunnenkamp, K. Børkje, and S. M. Girvin, Phys. Rev. Lett. 052115 (2011). 107, 063602 (2011). [36] F. R. Ong, M. Boissonneault, F. Mallet, A. C. Doherty, A. Blais, [9] P. Rabl, Phys. Rev. Lett. 107, 063601 (2011). D. Vion, D. Esteve, and P. Bertet, Phys. Rev. Lett. 110, 047001 [10] A. Kronwald, M. Ludwig, and F. Marquardt, Phys. Rev. A 87, (2013). 013847 (2013). [37] J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, Phys. [11] A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, 133601 Rev. Lett. 103, 147003 (2009). (2013). [38] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. [12] K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, Johansson, T. Duty, F. Nori, and P. Delsing, Nature (London) Nature (London) 4, 561 (2008). 479, 376 (2011). [13] F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science [39] V. Peano and M. Thorwart, Phys. Rev. B 82, 155129 (2010). 322, 235 (2008). [40] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. [14] M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Rev. Lett. 99, 093902 (2007). Phys.Rev.Lett.109, 063601 (2012). [41] F. Lecocq, J. D. Teufel, J. Aumentado, and R. W. Simmonds, [15] P. Komar, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. arXiv:1409.0872. Rabl, P. Zoller, and M. D. Lukin, Phys.Rev.A87, 013839 [42] D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, (2013). New J. Phys. 13, 023003 (2011). [16] K. Borkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, [43] M. Schmidt, V. Peano, and F. Marquardt, arXiv:1311.7095. Phys.Rev.Lett.111, 053603 (2013). [44] W. Chen and A. A. Clerk, Phys.Rev.A.89, 033854 (2014). [17] M.-A. Lemonde, N. Didier, and A. A. Clerk, Phys.Rev.Lett. [45] A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. 111, 053602 (2013). Groblacher,¨ and O. Painter, Phys.Rev.Lett.112, 153603 (2014). [18] X. Xu, M. Gullans, and J. M. Taylor, Phys. Rev. A 91, 013818 [46] V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, (2015). arXiv:1409.5375. [19] J. Teufel, D. Li, M. Allman, K. Cicak, A. Sirois, J. Whittaker, [47] A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. and R. Simmonds, Nature (London) 471, 204 (2011). Walther, Phys.Rev.Lett.51, 1550 (1983). [20] S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, [48] Y.-C. Liu, Y.-F. Xiao, Y.-L. Chen, X.-C. Yu, and Q. Gong, Phys. Nature (London) 460, 724 (2009). Rev. Lett. 111, 083601 (2013). [21] E. Verhagen, S. Deleglise, S. Weis, A. Schliesser, and T. [49] C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Kippenderg, Nature (London) 482, 63 (2010). Markovian and Non-Markovian Quantum Stochastic Methods [22] A. Kamenev and A. Levchenko, Adv. Phys. 58, 197 (2009). with Applications to Quantum Optics, Springer Series in [23] A. Kamenev, Field Theory of Non-Equilibrium Systems Synergetics (Springer, Berlin, 2004). (Cambridge University Press, Cambridge, UK, 2011). [50] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and [24] C. J. Mertens, T. A. B. Kennedy, and S. Swain, Phys.Rev.A48, R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010). 2374 (1993). [51] F. Beaudoin, J. M. Gambetta, and A. Blais, Phys.Rev.A84, [25] C. J. Mertens, J. M. Hasty, H. H. Roark, D. Nowakowski, and 043832 (2011). T. A. B. Kennedy, Phys.Rev.A52, 742 (1995). [52] M. I. Dykman and M. A. Krivoglaz, Phys. Status Solidi 68, 111 [26] O. Veits and M. Fleischhauer, Phys. Rev. A 55, 3059 (1997). (1975). [27] E. G. Dalla Torre, S. Diehl, M. D. Lukin, S. Sachdev, and P. [53] F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys.Rev.Lett. Strack, Phys. Rev. A 87, 023831 (2013). 96, 103901 (2006). [28] L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Phys. Rev. [54] M. Ludwig, B. Kubala, and F. Marquardt, New J. Phys. 10, B 89, 134310 (2014). 095013 (2008).

033836-19