Nonlinear interaction effects in a strongly driven optomechanical cavity
NNonlinear interaction effects in a strongly driven optomechanical cavity
Marc-Antoine Lemonde, Nicolas Didier,
1, 2 and Aashish A. Clerk Department of Physics, McGill University, Montr´eal, QC Canada H3A 2T8 D´epartement de Physique, Universit´e de Sherbrooke, Sherbrooke, QC Canada J1K 2R1 (Dated: Apr. 15, 2013)We consider how nonlinear interaction effects can manifest themselves and even be enhanced ina strongly driven optomechanical system. Using a Keldysh Green’s function approach, we calculatemodifications to the cavity density of states due to both linear and nonlinear optomechanical in-teractions, showing that strong modifications can arise even for a weak nonlinear interaction. Weshow how this quantity can be directly probed in an optomechanically-induced transparency typeexperiment. We also show how the enhanced interaction can lead to nonclassical behaviour, asevidenced by the behaviour of g correlation functions. PACS numbers:
Introduction–
The field of cavity optomechanics in-volves understanding and exploiting the quantum in-teraction between a mechanical resonator and photonsin a driven electromagnetic cavity. It holds immensepromise for both fundamental studies of large-scale quan-tum phenomena as well as applications to quantum infor-mation processing and ultra-sensitive detection, and hasseen remarkable progress in the past five years. High-lights include the use of radiation pressure forces to coola mechanical resonator to close to its motional groundstate [1, 2] and experiments where the mechanical motioncauses squeezing of the light leaving the cavity [3, 4].As remarkable as this progress has been, it has re-lied on strongly driving the optomechanical cavity toenhance the basic dispersive coupling between photonsand mechanical position. While the resulting interac-tion can be made larger than even the dissipative ratesin the system [5–7], it is purely bilinear in photon andphonon operators. As a result, it cannot convert Gaus-sian state inputs into non-classical states or give riseto true photon-photon interactions. Recent theoreticalwork has addressed physics of the nonlinear interactionin weakly driven systems [8, 9]. Unfortunately, one findsthat effects are suppressed by the small parameter g/ω M .In this paper, we now consider nonlinear interac-tion effects in an optomechanical system that (unlikeRefs. [8, 9]) is also subject to a strong laser drive; we con-sider effects of this driving beyond simple linear-response.We find somewhat surprisingly that one can use thestrong drive to enhance the underlying single-photon in-teraction. Using non-equilibrium many-body perturba-tion theory (based on the Keldysh technique (see, e.g.,[4])), we calculate how these effects modify the cavitydensity of states, and hence the cavity’s response to anadditional weak probe laser. This response is exactlythe quantity measured in so-called optomechanically-induced transparency (OMIT) experiments [6, 11–13].We find striking modifications of the OMIT spectrum,effects which can be attributed to the nonlinear interac-tion causing a hybridization between one and two polari- ton states (with the polaritons being joint mechanical-photonic excitations). We also find the possibility ofenhanced polariton-polariton interactions, which lead inturn to non-classical correlations (as measured by a g correlation function). System–
The standard Hamiltonian of a driven op-tomechanical cavity is ˆ H = ˆ H + ˆ H diss , with ( (cid:126) = 1)ˆ H = ω C ˆ a † ˆ a + ω M ˆ b † ˆ b + g (cid:16) ˆ b † + ˆ b (cid:17) ˆ a † ˆ a +( √ κ ¯ a in ( t )ˆ a † + h.c. ) . (1)Here ˆ a is the cavity mode (frequency ω C , damping rate κ ), ˆ b is the mechanical mode (frequency ω M , dampingrate γ ), and g is the optomechanical coupling. ˆ H diss de-scribes dissipation of photons and phonons by indepen-dent baths; ¯ a in ( t ) is the amplitude of the drive laser.We consider the standard case of a continuous-wavedrive (i.e. ¯ a in ( t ) ∝ e − iω L t ), and work in a rotating frameat the laser frequency ω L . We further make a displace-ment transformation, writing ˆ a = e − iω L t (cid:16) ¯ a + ˆ d (cid:17) , where¯ a is the classical cavity amplitude induced by the laserdrive. Letting ∆ = ω L − ω C , the coherent Hamiltoniannow takes the form ˆ H = ˆ H + ˆ H withˆ H = − ∆ ˆ d † ˆ d + ω M ˆ b † ˆ b + G ( ˆ d + ˆ d † )(ˆ b + ˆ b † ) , (2)ˆ H = g ˆ d † ˆ d (ˆ b + ˆ b † ) . (3) G = g ¯ a is the drive-enhanced many-photon optomechan-ical coupling; we set g, ¯ a > a (cid:29) g (cid:28) κ, ω M . It is then standard to neglectthe effects of ˆ H . In the absence of any driving, a sim-ple perturbative estimate suggests that the effects of ˆ H enter as g /ω M , where the factor of ω M corresponds to avirtual state with one extra (or one less) phonon. Thisconclusion can be made more precise by exactly solvingthe coherent, undriven system using a polaron transfor-mation [8, 9]. Thus, in this standard regime, one can ig-nore ˆ H , leaving only ˆ H , which is easily diagonalized asˆ H = (cid:80) σ = ± E σ ˆ c † σ ˆ c σ . Here ˆ c + , − describe the two normalmodes of the system. As these modes have both photon a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r and phonon components, we refer to them as polaritonsin what follows. Their energies are: E ± = 1 √ (cid:18) ∆ + ω ± (cid:113) (∆ − ω ) − G ∆ ω M (cid:19) / . (4)For ∆ (cid:39) − ω M and G ≥ κ, γ , the polariton energy split-ting can be resolved experimentally [5–7]. Polariton interactions–
Unlike previous work, we wishto retain the effects of the nonlinear interaction ˆ H , butalso consider the effects of a large drive (and hence a largemany-photon coupling G ). To proceed, we will treat theeffects of ˆ H in perturbation theory. We use a KeldyshGreen’s function (GF) approach which is able to describethe non-equilibrium nature of the system. The linearHamiltonian ˆ H along with the dissipative terms in ˆ H diss define the free GFs of the system, which describe thepropagation of polaritons in the presence of dissipation.Written in the polariton basis, the nonlinear interactionˆ H gives rise to number-non-conserving interactions,ˆ H = (cid:88) σ,σ (cid:48) ,σ (cid:48)(cid:48) (cid:16) g Aσσ (cid:48) σ (cid:48)(cid:48) ˆ c † σ ˆ c † σ (cid:48) ˆ c † σ (cid:48)(cid:48) + g Bσσ (cid:48) σ (cid:48)(cid:48) ˆ c † σ ˆ c † σ (cid:48) ˆ c σ (cid:48)(cid:48) + h.c. (cid:17) , (5)where the coefficients g A/Bσσ (cid:48) σ (cid:48)(cid:48) ∝ g [14]. Note normalordering ˆ H in terms of polariton operators introducessmall quadratic and linear terms which modify the diag-onalized form of ˆ H (see EPAPS for details [14]).We start by considering how single-particle propertiesare modified by the nonlinear interactions; such proper-ties can be directly probed by weakly driving the cavitywith a second probe laser (i.e. an OMIT experiment [11–13]) or by measuring the mechanical force susceptibility.Understanding these properties amounts to calculatingthe self-energy Σ[ ω ] of both the polaritons due to ˆ H .We have calculated all self-energy processes to second or-der in g . Our approach captures both the modification ofspectral properties due to the interaction (i.e. the modifi-cation of the cavity and mechanical density of states), aswell as modifications of the occupancies of the mechanicsand cavity. While our approach is general, we will focuson the most interesting case of a high mechanical qualityfactor γ (cid:28) ω M , a cavity in the resolved sideband regime ω M > κ , and a strong cavity drive, G (cid:38) κ .Our full second-order self-energy calculation finds thatfor most choices of parameters, the polariton self-energiesscale as g /ω M and thus have a negligible effect forthe typical case where g (cid:28) ω M . However, effects aremuch more pronounced if one adjusts parameters so that E + = 2 E − . This condition makes the term in ˆ H whichscatters a + polariton into two − polaritons (and vice-versa) resonant. It can be achieved for any laser detuning∆ in the range ( − ω M , − ω M /
2) by tuning the amplitude¯ a in of the driving laser so that the many-photon optome- -0.40 -0.35 0.25 0.300.00.10.20.30.40.50.6 FIG. 1: Main: + polariton resonance in the cavity density ofstates, for various values of the nonlinear interaction strength g (as indicated), as obtained from Eq. (12) (with the inclusionof energy shifts from ˆ H NR [14]). For all plots, the laser drive isat the red sideband ∆ = − ω M , and G = 0 . ω M to ensure theresonance condition E + = 2 E − ; we also take ω M /κ = 50, T =0 and γ = 10 − κ . The peak splitting signals the hybridizationof a + polariton with two − polaritons. The dashed curve isthe result of a master-equation simulation for g = κ [14].Inset: Full density of states, same parameters, showing theasymmetry between + and − polariton resonances. chanical coupling G = G res , where G res [∆] ≡ (cid:113) − ω + 4 ω / (cid:16) (cid:112) ∆ ω M (cid:17) . (6)In this regime, the dominant physics is well describedby the approximation ˆ H (cid:39) ˆ H eff withˆ H eff = (cid:88) σ = ± E σ ˆ c † σ ˆ c σ + ˜ g (cid:16) ˆ c † + ˆ c − ˆ c − + h.c. (cid:17) + ˆ H NR , (7)ˆ H NR = (cid:88) σ = ± (cid:32) δ σ ˆ c † σ ˆ c σ + (cid:88) σ (cid:48) = ± U σσ (cid:48) ˆ c † σ ˆ c † σ (cid:48) ˆ c σ (cid:48) ˆ c σ (cid:33) . (8)The second term in ˆ H eff corresponds to making arotating-wave approximation on the nonlinear interac-tion ˆ H in Eq. (5), retaining only the resonant process;˜ g = g B −− + ∝ g is the corresponding interaction strength(see inset of Fig. 2 to see how ˜ g varies with ∆). The termsin ˆ H NR describe the small (i.e. ∝ g /ω M ) residual effectsof the non-resonant interaction terms in Eq. (5); we treatthem via straightforward second-order perturbation the-ory (i.e. a Schrieffer-Wolff transformation). They play norole in the extreme good-cavity limit ω M (cid:29) κ [14]. Green functions for resonant nonlinear interactions–
Focusing on the resonant interaction regime defined byEq. (6), and using the simplified Hamiltonian in Eq. (7),we obtain simple expressions for the retarded GFs of thesystem. The retarded photon GF in the displaced, rotat-ing frame is defined as G Rdd [ ω ] = − i (cid:90) ∞−∞ dtθ ( t ) (cid:68) [ ˆ d ( t ) , ˆ d † (0)] (cid:69) e iωt , (9)with similar definitions for the polariton retarded GF G Rσσ [ ω ] ( σ = ± ). As usual, ρ d [ ω ] = − Im G Rdd [ ω ] /π de-scribes the cavity density of states; G Rdd [ ω ] also deter-mines the reflection coefficient in an OMIT experiment(see Fig. 2). A standard linear response calculation [14]shows that the elastic OMIT reflection coefficient is givenby r [ ω pr ] = 1 − iκ cp G Rdd [ ω pr ], where ω pr is the frequencyof the weak probe beam, and κ cp is the contribution tothe total cavity κ from the coupling to the drive port.In the limit of interest where κ (cid:28) E σ , thereare no off-diagonal polariton GFs or self-energies [14].As a result, G Rdd [ ω ] will be given as G Rdd [ ω ] = (cid:80) σ (cid:16) C σ G Rσσ [ ω ] + D σ (cid:2) G Rσσ [ − ω ] (cid:3) ∗ (cid:17) , where the change-of-basis coefficients C σ , D σ are given in [14]. The Dysonequations for the polariton retarded GFs are G Rσσ [ ω ] = (cid:0) ω − E σ + iκ σ / − Σ Rσσ [ ω ] (cid:1) − , (10)where κ σ is the effective damping rate of the σ polariton[14]. Using the effective Hamiltonian in Eq. (7), a stan-dard Keldysh calculation yields that to second order in g , the polariton self-energies take the simple forms:Σ ++ [ ω ] = 2˜ g (1 + 2¯ n − ) ω − E − + iκ + , (11a)Σ −− [ ω ] = 4˜ g (¯ n − − ¯ n + ) ω − ( E + − E − ) + i ( κ + + κ − ) / . (11b)Here, ¯ n σ is the effective thermal occupancy of the σ po-lariton [14]; for ˜ g = 0, we have (cid:104) ˆ c † σ ˆ c σ (cid:105) = ¯ n σ . We havetaken the limit g/ω M →
0, and hence neglected the ef-fects of the non-resonant terms ˆ H NR in Eqs. (11); theexplicit corrections due to these terms are given in thesupplemental information [14].Eqs. (11) are central results of this work. Eq. (11a) de-scribes the fact that a single + polariton can resonantlyturn into two − polaritons, and describes the hybridiza-tion between these states that occurs for large enough g . To see this explicitly, we consider the case of exactresonance (i.e. E + = 2 E − ) and write G R + [ ω ] = 12 (cid:88) η = ± − iη κ − − ˜ κ + δ + ω − E + + i κ − +˜ κ + + ηδ + , (12) δ + = (cid:113) g (1 + 2¯ n − ) − (2˜ κ − − ˜ κ + ) / . (13)For ˜ g (cid:38) κ , we see that the + polariton GF has twopoles, corresponding to the new hybridized eigenstates.We stress that these eigenstates do not correspond to afixed excitation number. Note that unlike the undrivensystem [8, 9], the effects of the nonlinear interaction canbe significant even if g (cid:28) ω M . Also note that the reso-nant coupling between | + (cid:105) and | − −(cid:105) states is enhancedat finite temperature by a standard stimulated emissionfactor (1 + 2¯ n − ). The form of this GF and self-energyare reminiscent to the photon GF for ordinary OMIT, where a photon can resonantly turn into a phonon [11];however, that effect does not involve any temperature-dependent enhancement. Σ −− in Eq. (11b) describes aprocess where the propagating − polariton of interest in-teracts with an already-present − polariton to turn intoa +. As this process requires an existing density of po-laritons, it is strongly suppressed at low temperatures.We note that it is possible to use resonance to en-hance the nonlinear optomechanical interaction withoutstrong driving, if one instead considers a system wheretwo cavity modes interact with a single mechanical res-onator [15–17]. Our approach has the benefit of only re-quiring a single cavity mode; further, for drive detuningsnear ∆ = − ω M , it also has a natural resistance againstmechanical heating, as mechanical contribution to thepolariton temperature scales as γ ¯ n th /κ , where ¯ n th is themechanical thermal occupancy. While a low temperatureis not essential for the density-of-states effects describedabove, it is essential for the correlation effects discussedbelow. Finally, for superconducting microwave cavities,the cavity linewidth κ has a strong contribution fromtwo-level fluctuators, and thus improves if one stronglydrives the cavity (as in our scheme). Red-sideband drive–
For a detuning ∆ = − ω M , thepolariton resonance occurs when G = 0 . ω M . For thisdetuning, both polaritons are almost equal mixtures ofphoton and phonon operators. One finds κ σ = ( κ + γ ) / g (cid:39) − . g .Because ˆ H does not conserve the number of photons andphonons, the polaritons are not eigenstates of ˆ d † ˆ d +ˆ b † ˆ b ; asa result, even at zero temperature, the effective thermaloccupancies scale as ¯ n σ ∝ ( G/ω M ) (cid:28) g is increased from zero.For g = 0, one sees two symmetric peaks correspondingto the two polaritons, i.e. the well known normal-modesplitting [18, 19]. As g increases, these peaks develop amarked asymmetry. For g ∼ κ , a clear splitting of the +peak occurs, corresponding to the resonant hybridizationof one and two polariton states. Fig. 1 also shows resultsof a numerical (but non-perturbative) master-equationsimulation [14], showing our analytic approach is reliableeven for moderately strong g . Large-detuned drives–
The resonant-polariton inter-action is also interesting for drives far from the red-sideband, where the value of G res (cid:28) ω M . For a laserdetuning near the minimum possible value ∆ = − ω M atwhich resonance is possible (and setting G = G res ), thepolaritons are each either almost entirely phonon or pho-ton, implying a very small value of ˜ g ∝ gG/ω M . However,as the − polariton is now almost purely phononic, itssmall damping rate and potentially large thermal occu-pancy enhances the self-energy in Eq. (11a) (i.e. κ − (cid:39) γ ,and ¯ n − corresponds to the mechanical thermal occu-pancy). We can quantify these effects by considering thevalue of ρ d [ ω = E + ], which will be suppressed by the FIG. 2: Reflection coefficient for a weak probe beam incidentat a frequency ω pr (defined in the lab frame), as measuredin an OMIT experiment (upper inset). We take a one sidedcavity with κ cp /κ = 0 . g = 0 . κ . For each curve, ∆ islabelled, and G = G res [∆]. Remaining parameters are thesame as Fig. 1. Lower inset: behaviour of g , G res and C eff asa function of detuning ∆ of the main laser drive. hybdriziation physics described here. One finds: ρ d [ E + ] = 2 /πκ +
11 + C eff , C eff = 4˜ g (1 + 2¯ n − ) κ + κ − . (14)For a large detuning, the effective cooperativity scales as C eff ∝ C ( g/ω M ) , where C = 4 G /κγ is the standardmany-photon coupling cooperativity. Thus, in the large-detuned regime, resonant polaritons interactions allowone to amplify the effects of the nonlinear interaction bya factor C . Fig. 2 shows the evolution of the OMIT re-flection coefficient (which reflects the structure in ρ d [ ω ])as the detuning ∆ is varied, while keeping G tuned tothe resonant value G res (∆). Induced Kerr interaction–
The nonlinear interactionin the resonant regime defined by Eqs. (6)-(7) leads toa strongly enhanced two-particle interaction between − polaritons, mediated by the exchange of a + polari-ton (Fig. 3). In a weakly-driven optomechanical sys-tem, Eq. (3) implies that phonons can mediate an effec-tive photon-photon interaction; however, as the virtualphonon is off-resonance, this interaction U ∝ g /ω M . Incontrast, the resonance condition E + = 2 E − yields aninduced interaction U res ∝ ˜ g /κ , an enhancement by alarge factor ∝ ω M /κ .To assess the effects of the polariton-polariton in-teractions, we weakly drive our system with a secondprobe tone, and consider the g correlation functions g u = (cid:104) ˆ u † ˆ u † ˆ u ˆ u (cid:105) / (cid:104) ˆ u † ˆ u (cid:105) , where u = b, d, c + , c − . g u isa measure of interaction induced correlations; g ≤ − polaritons when the resonance con-dition E + = 2 E − is achieved, we expect that if the cavityis driven at the E − resonance, g − will drop below 1.This is indeed the result of a numerical, master-equation FIG. 3: Inset: Resonant interaction between − polaritons.Main: Numerically-calculated g correlation function for − polaritons ( g − ) and phonons ( g b ), in the presence of anadditional weak probe laser (frequency ω pr ). Here, g = κ ,∆ = − ω M , G = 0 . ω M = G res . We have taken ω M /κ → ∞ to suppress non-resonant interaction effects. The probe am-plitude is (cid:15) = 0 . κ ( g − ), (cid:15) = 0 . κ ( g b ). Both phonon andpolariton g functions drop below 1 due to the interactions, in-dicating non-classical correlations despite the fact g/ω M (cid:39) based calculation (see Fig. 3 and [14]). An analytic cal-culation based on a reduced state-space (similar to thatin Ref. [17]) reproduces these results. For a weak probedrive at the E − frequency, it yields [14]: g − = 11 + 4˜ g /κ − . (15)One also finds non-classical correlations for photons andphonons. Shown in Fig. 3 is the phonon g function g b (for same parameters); it clearly drops below 1. Thedouble-peak structure of this curve is the result of thedrive inducing correlations between − and + polaritons;it also occurs in the behaviour of (cid:104) ˆ b † ˆ b (cid:105) (see EPAPS formore details [14]). Conclusions–
We have presented a systematic ap-proach for describing nonlinear interaction effects ina driven optomechanical system, identifying a regimewhere a resonance enhances interactions between polari-tons. We have discussed how this would manifest itselfin a OMIT-style experiment, as well as in g correlationfunctions. The polariton interactions we describe couldbe extremely interesting when now considered in latticesystems, or when considering the propagation of pulses.We thank W. Chen and A. Nunnenkamp for useful dis-cussions. This work was supported by CIFAR, NSERCand the DARPA ORCHID program under a grant fromAFOSR. Note added–
During the preparation of this pa-per, we became aware of a related work by Børkje, Nun-nenkamp, Teufel and Girvin. [1] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. All-man, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehn-ert, and R. W. Simmonds, Nature , 359 (2011).[2] J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill,A. Krause, S. Gr¨oblacher, M. Aspelmeyer, and O. Painter,Nature , 89 (2011).[3] D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy,N. Brahms, and D. M. Stamper-Kurn, Nature , 476(2012).[4] A. H. Safavi-Naeini, S. Groeblacher, J. T. Hill, J. Chan,M. Aspelmeyer, and O. Painter, arXiv:1302.6179 (2013).[5] S. Groeblacher, K. Hammerer, M. R. Vanner, and M. As-pelmeyer, Nature , 724 (2009).[6] J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois,J. D. Whittaker, and R. W. Simmonds, Nature , 204(2011).[7] E. Verhagen, S. Deleglise, S. Weis, A. Schliesser, andT. Kippenberg, Nature , 63 (2012).[8] P. Rabl, Phys. Rev. Lett. , 063601 (2011).[9] A. Nunnenkamp, K. Børkje, and S. M. Girvin, Phys. Rev.Lett. , 063602 (2011).[4] A. Kamenev and A. Levchenko, Advances in Physics ,197 (2009). [11] G. S. Agarwal and S. Huang, Phys. Rev. A , 041803(R)(2010).[12] S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Ar-cizet, A. Schliesser, and T. Kippenberg, Science , 1520(2010).[13] A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichen-field, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, andO. Painter, Nature , 69 (2011).[14] See EPAPS for additional information on the normal-mode transformation, the self-energy calculation, the nu-merical master-equation simulation and the calculation of g correlation functions. (2013).[15] M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Mar-quardt, Phys. Rev. Lett. , 063601 (2012).[16] K. Stannigel, P. Komar, S. Habraken, S. Bennett, M. D.Lukin, P. Zoller, and P. Rabl, Phys. Rev. Lett. ,013603 (2012).[17] P. Komar, S. D. Bennett, K. Stannigel, S. J. M.Habraken, P. Rabl, P. Zoller, and M. D. Lukin,arXiv:1210.4039v1 (2012).[18] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin,Phys. Rev. Let. , 093902 (2007).[19] J. Dobrindt, I. Wilson-Rae, and T. Kippenberg, PhysRev Lett , 263602 (2008). SUPPLEMENTAL INFORMATIONLANGEVIN EQUATIONS FOR LINEARIZED OPTOMECHANICAL SYSTEM IN THE POLARITONBASIS
We start by diagonalizing the linearized optomechanical Hamiltonian given in Eq. (2) of the main text, working asalways in a displaced interaction picture set by the laser drive on the cavity. Introducing (cid:126)X = (cid:2) ˆ b ˆ d ˆ b † ˆ d † (cid:3) T , (cid:126)Y = (cid:2) ˆ c − ˆ c + ˆ c †− ˆ c † + (cid:3) T , (S1)the diagonalization can be expressed in terms of a 4 × U : (cid:126)Y = U · (cid:126)X. (S2) U can be found by standard means, though its general form is both cumbersome and unenlightening; we define V = U − . It is slightly less unwieldy in the special (but relevant) case of a drive at the red-detuned mechanicalsideband ∆ = − ω M . In this case, we have simply:ˆ c ± = 1 (cid:112) ω M E ± (cid:104) ( E ± − ω M )( ˆ d † ± ˆ b † ) + ( E ± + ω M )( ˆ d ± ˆ b ) (cid:105) , (S3)while the inverse transformation is defined by: V j = 1 √ ω M (cid:104) − ω M + E − √ E − , ω M + E + √ E + , − ω M − E − √ E − , ω M − E + √ E + (cid:105) (S4) V j = 1 √ ω M (cid:104) ω M + E − √ E − , ω M + E + √ E + , ω M − E − √ E − , ω M − E + √ E + (cid:105) (S5)Next, we include the coupling of our system to the cavity and mechanical dissipative baths in the standard way,treating these baths as Markovian over frequencies of interest. Consider first the coupling to the cavity bath. Priorto making displacement and interaction-picture transformations, the system-bath coupling has the form (see, e.g.,Ref. 1): ˆ H κ = (cid:88) j ω j ˆ f † j ˆ f j , ˆ H κ,int = − i (cid:114) κ πρ c (cid:88) j (cid:16) ˆ f j − ˆ f † j (cid:17) (cid:0) ˆ a + ˆ a † (cid:1) , (S6)where ˆ f j is the lowering operator for a bath mode, and ρ c is the density of states of bath modes (which we treatto be constant over frequencies of interest). In now moving to an interaction picture at the laser frequency, wealso transform the bath modes, i.e. ˆ f j → ˆ f j e − iω L t . Formally, the interaction picture transformation involves aunitary ˆ U ( t ) = exp (cid:104) − iω L t (cid:16) ˆ a † ˆ a + (cid:80) j ˆ f † j ˆ f j (cid:17)(cid:105) and transforms the bath Hamiltonian to ˆ H κ = (cid:80) j ( ω j − ω L ) ˆ f † j ˆ f j .In this interaction picture, the counter-rotating terms in ˆ H κ,int will explicitly oscillate at ± ω L = ± ω c + ∆).Even if we now write our photon operators ˆ a in terms of polariton operators, there is no possibility of having theseterms becoming resonant, as the cavity frequency is much larger than any other frequency scale in the problem (i.e. ω c (cid:29) | ∆ | , ω M , E + , E − ). As such, one can safely make a rotating-wave approximation in the photon basis, resultingin a standard system-bath interaction which is stationary in the interaction picture:ˆ H κ,int = i (cid:114) κ πρ c (cid:88) j (cid:16) ˆ f † j ˆ d − ˆ d † ˆ f j (cid:17) = − i (cid:114) κ πρ c (cid:88) j (cid:104) ˆ f † j (cid:16) V ˆ c − + V ˆ c + + V ˆ c †− + V ˆ c † + (cid:17) − h.c. (cid:105) (S7)We have also made the displacement transformation ˆ a = ˆ d + ¯ a as discussed in the main text. Note that in writing ˆ d interms of polariton operators via Eq. (S2), we obtain terms of the form ˆ f † j ˆ c † σ , which can cause polariton heating evenif the cavity bath is at zero temperature. Such terms are physical and must be retained. Formally, In the interactionpicture the bath now has negative frequency modes which can make such processes resonant. In more physical terms,the combination of zero-point bath fluctuations with the cavity driving can excite the polaritons. This mechanismhas been discussed in other contexts under the name “quantum activation” by various authors [2, 3].We turn now to the mechanical bath, where the basic interaction Hamiltonian can be written in an analogous way:ˆ H γ = (cid:88) j ω j ˆ g † j ˆ g j , ˆ H γ,int = − i (cid:114) γ πρ m (cid:88) j (cid:16) ˆ g j − ˆ g † j (cid:17) (cid:16) ˆ b + ˆ b † (cid:17) . (S8)Here, ˆ g j is a lowering operator for a mechanical bath oscillator, and ρ m is the density of states of mechanical bathmodes (which we also treat to be constant over frequencies of interest). As there is no direct driving of the mechanicalresonator, the analysis here is somewhat simpler. We first re-write the phonon operator ˆ b in the polariton basis, and then make a rotating-wave approximation. The justification is that counter-rotating terms such as ˆ g † j ˆ c † σ can never bemade resonant; as the mechanical resonator is not driven, there is no quantum activation mechanism involving themechanical bath. We thus obtain ˆ H γ,int (cid:39) (cid:88) σ (cid:114) κ mσ πρ m (cid:88) j (cid:16) ˆ g † j ˆ c σ + h.c. (cid:17) , (S9)where ( V = U − ) κ M − = γ ( V + V ) , κ M + = γ ( V + V ) . (S10)Having now established the correct system-bath coupling Hamiltonians in the polariton basis we wish to use, wecan derive the Heisenberg-Langevin equations for our system in the standard manner. As each bath couples to both+ and − polaritons, one obtains off-diagonal damping terms, (e.g. the mechanical bath produces a force on the +polariton that is proportional to the amplitude of the − polariton). Such terms will dynamically couple + and − polaritons, and can be included in our theory in a straightforward manner (i.e. by including Green functions that areoff-diagonal in the polariton index). However, in the weak-dissipation limit of interest ( κ, γ (cid:28) E + , E − , | E + − E − | ),the mixing effects induced by such terms is strongly suppressed. As such, we will drop off-diagonal damping terms,resulting in the form: ∂ t ˆ c σ ( t ) = − (cid:16) iE σ + κ σ (cid:17) ˆ c σ ( t ) − √ κ σ ˆ ξ σ ( t ) , (S11)where the polariton damping rates κ σ are given by: κ − = κ M − + κ (cid:104) ( V ) − ( V ) (cid:105) , κ + = κ M + + κ (cid:104) ( V ) − ( V ) (cid:105) . (S12)The noise operators ˆ ξ σ ( t ) are each linear combinations of the input noise emanating from the mechanical and cavitybaths. In the interaction picture we use, we will be sensitive to noise in the cavity bath at frequencies near ω c ,and noise in the mechanical bath at frequencies near ω M . In the limit of interest where the physical temperature T (cid:28) (cid:126) ω c /k B , there will be no thermal noise incident from the cavity bath at the frequencies of interest. Also, aswe focus on regimes where the polariton damping rates are much smaller than their energies, we can treat the noiseoperators as being white noise (as is standard in input-output theory treatments). We thus have (cid:68) ˆ ξ † σ ( t ) ˆ ξ σ ( t (cid:48) ) (cid:69) = n σ δ ( t − t (cid:48) ) , (cid:68) ˆ ξ σ ( t ) ˆ ξ † σ ( t (cid:48) ) (cid:69) = (1 + n σ ) δ ( t − t (cid:48) ) , (S13)where the effective temperatures of the two noises are given by: n − = 1 κ − (cid:16) κ M − n B [ E − ] + ( V ) κ (cid:17) , n + = 1 κ + (cid:16) κ M + n B [ E + ] + ( V ) κ (cid:17) . (S14) n B [ E σ ] denotes the Bose-Einstein distribution function evaluated at energy E σ and the mechanical bath temperature.Finally, one also finds that ˆ ξ + and ˆ ξ − are correlated with one another. Similar to the situation of off-diagonaldamping terms, such noise correlations could easily be included in our theory; however, as they play no role in theregime of interest where E σ , | E + − E − | (cid:29) κ, γ , we drop them in what follows. One also finds that anomalous noisecorrelators can be non-zero (e.g. (cid:104) ˆ ξ σ ( t ) ˆ ξ σ (cid:48) (0) (cid:105) ). Again, while such terms can be retained in our theory, they play norole in the weak-damping regime of interest, and we hence drop them in what follows. UNPERTURBED POLARITONS GFS
The standard definitions of the three relevant GFs needed in the Keldysh technique are ( σ stands for ± , but similardefinitions hold for ˆ c σ = ˆ b or ˆ d ): G Rσσ (cid:48) [ ω ] ≡ − i (cid:90) ∞−∞ dtθ ( t ) (cid:68)(cid:104) ˆ c σ ( t ) , ˆ c † σ (cid:48) (0) (cid:105)(cid:69) e iωt , (S15) G Aσσ (cid:48) [ ω ] ≡ i (cid:90) ∞−∞ dtθ ( − t ) (cid:68)(cid:104) ˆ c σ ( t ) , ˆ c † σ (cid:48) (0) (cid:105)(cid:69) e iωt , (S16) G Kσσ (cid:48) [ ω ] ≡ − i (cid:90) ∞−∞ dt (cid:68)(cid:110) ˆ c σ ( t ) , ˆ c † σ (cid:48) (0) (cid:111)(cid:69) e iωt . (S17)The retarded and advanced GFs keep track of spectral information, whereas the Keldysh Green function G K alsokeeps track of the occupancy of states. For the linearized (non-interacting) theory, the GFs are easily obtained fromthe Langevin equations in Eq. (S11). These free GFs (which we denote by G ) are diagonal (only non-zero for σ = σ (cid:48) )and given by: G Rσσ [ ω ] = 1 ω − E σ + i κ σ , (S18) G Kσσ [ ω ] = (2 n σ + 1) (cid:0) G Rσσ [ ω ] − G Aσσ [ ω ] (cid:1) , (S19)and G Aσσ [ ω ] = (cid:2) G Rσσ [ ω ] (cid:3) ∗ .Finally, as there are no off-diagonal polariton Green functions, we can use Eq. (S2) to write the photon retardedGreen function G Rdd [ ω ] as: G Rdd [ ω ] = (cid:88) σ (cid:16) C σ G Rσσ [ ω ] + D σ (cid:2) G Rσσ [ − ω ] (cid:3) ∗ (cid:17) , (S20)where C − = V , C + = V , D − = V , D + = V . (S21)As the Green functions remain diagonal in the polariton index even with the nonlinear interaction (in the regime ofinterest, see below), the above relation also holds for the full Green functions (i.e. including the self-energy associatedwith g ). ˆ H IN THE POLARITON BASIS
Using the change of basis matrix V = U − (c.f. Eq.(S2)), we can re-write the non-linear interaction ˆ H in Eq. (3)of the main text in the polariton basis via:ˆ H = g (cid:104) V (cid:126)Y (cid:105) (cid:104) V (cid:126)Y (cid:105) (cid:16)(cid:104) V (cid:126)Y (cid:105) + (cid:104) V (cid:126)Y (cid:105) (cid:17) . (S22)Expanding this equation allows one to obtain the interaction coefficients g A,Bσσ (cid:48) σ (cid:48)(cid:48) in terms of g and matrix elements of V . In particular, the coefficient ˜ g of the resonant interaction process ˆ c †− ˆ c †− ˆ c + will be given by:˜ g = g ( V V ( V + V ) + V V ( V + V ) + V V ( V + V )) . (S23)As the normal-mode transformation described by U mixes raising and lowering operators, ˆ H will not be normal-ordered when written in terms of polariton operators (even though it is normal ordered when written in terms ofphoton and phonon operators). Normal-ordering ˆ H in the polariton basis yields the form:ˆ H =: ˆ H : + ( A − ˆ c − + A + ˆ c + + h.c. ) (S24)where the colons indicate normal-ordering in the polariton basis, and the constants A σ ∝ g . The first term is thenormal-ordered polariton interaction written in Eq. (5) in the main text. We next make a unitary displacement FIG. 4: (a) Feynman diagrams describing the dominant + and − polariton self-energies (to second order in g ) in the casewhen the resonance condition E + = 2 E − is met. We have shown diagrams for the retarded self-energies; similar diagramsalso determine the advanced and Keldysh self-energies. The structure of propagators in Keldysh space is indicated by writingexplicit classical ( cl ) and quantum ( q ) indices at the ends of each Green function. (b) The non-resonant interaction terms induceeffective two-particle interactions between polaritons, as described by H NR in Eq. (8) of the main text. These two-particleinteractions modify the resonant self-energies depicted in Fig. (4); shown above are the diagrams describing the modificationof the + polariton self-energy by the effective interaction U −− . The summation of these ladder diagrams results in the simpleenergy shift indicated in Eqs. (S27),(S28). transformation of the form ˆ c σ → ¯ c σ + ˆ c σ , where the constants ¯ c σ are chosen to eliminate all linear-in-ˆ c σ terms in theHamiltonian; to leading order in g , ¯ c σ = − A σ /E σ . The resulting coherent Hamiltonian has the form:ˆ H = : ˆ H : + (cid:2) ˆ c †− ˆ c † + (cid:3) Z (cid:20) ˆ c − ˆ c + (cid:21) + (cid:18)(cid:2) ˆ c − ˆ c + (cid:3) Z (cid:20) ˆ c − ˆ c + (cid:21) + h.c. (cid:19) . Here Z , Z are 4 × g /E σ ∼ g /ω M . The quadratic terms on the RHS describecorrects to the linear Hamiltonian ˆ H = (cid:80) σ E σ ˆ c † σ ˆ c σ arising from the displacement transformation. In principle, onecould combine these with the terms in ˆ H , re-diagonalize the resulting quadratic Hamiltonian, obtaining a new basisof non-interacting polaritons. However, to leading order in g /ω M , all that is needed is to retain the diagonal elementsof Z , which simply shift the polariton energies but do not change their wavefunctions. These energy shifts (denotes (cid:15) + , (cid:15) − ) are just absorbed into the definition of our free Green functions. We are then left with a normal-orderedinteraction Hamiltonian ˆ H in the polariton basis which can be addressed perturbatively. PERTURBATIVE TREATMENT
We have calculated the full Keldysh self-energy corresponding to ˆ H to order g , without any further approximation.We have also done this calculation in the case where G is not so large, such that one should work in the original basisof photons and phonons. Results of these full calculations will be presented elsewhere. Here, like in the main text,we will focus on the resonant-interaction regime described in the main text, where the condition E + = 2 E − enhancescertain scattering processes. In this regime, the dominant self-energy processes for + and − polaritons are depictedin Fig. 4. Using the standard rules of the Keldysh technique [4], these diagrams correspond to:Σ R −− [ ω ] = 2 i (cid:101) g (cid:90) dω (cid:48) π (cid:0) G K −− [ ω (cid:48) ] G R ++ [ ω (cid:48) + ω ] + G A −− [ ω (cid:48) − ω ] G K ++ [ ω (cid:48) ] (cid:1) = 4 (cid:101) g n − − n + ω − ( E + − E − ) + i κ − + κ + , (S25)Σ R ++ [ ω ] = 2 i (cid:101) g (cid:90) dω (cid:48) π G K −− [ ω (cid:48) ] G R [ ω − ω (cid:48) ] = 2 (cid:101) g (1 + 2 n − ) ω − E − + iκ − . (S26)Note that because of the resonance condition, these self-energies scale as g /κ , whereas all other self-energy diagramsare suppressed by an additional small parameter κ/ω M . They can thus be neglected in the limit κ/ω M →
0. For asmall but realistic value of κ/ω M , the small-energy shifts associated with the non-resonant interaction terms in ˆ H can shift the condition needed for resonance. To describe these small shifts (which can be important for realisticparameters), it is sufficient to use standard second-order perturbation theory to treat the non-resonant terms. Thisis conveniently done via a Schrieffer-Wolff transformation, where a unitary transformation is performed to eliminatethe non-resonant terms to leading order in the Hamiltonian. The procedure generates new terms however at secondorder in g . Keeping only such terms which do not change the total number of polaritons, we obtain the general formgiven in the Hamiltonian ˆ H NR (Eq. (8) of the main text). Note that in this equation, the energy shifts δ σ includeboth the shifts arising from the non-resonant terms, as well as the energy shifts (cid:15) σ coming from the normal-orderingprocedure. Both such terms scale as g /ω M .0Including the energy shifts associated with the non-resonant processes (as described by ˆ H NR in Eq. (8) of the maintext), the self-energies above are modified to:Σ R −− [ ω ] = 4 (cid:101) g n − − n + ω − ˜ E + − + i κ − + κ + , ˜ E + − = E + − E − + δ + − δ − + U + − (¯ n − − ¯ n + ) , (S27)Σ R ++ [ ω ] = 2 (cid:101) g (1 + 2 n − ) ω − ˜ E −− + iκ − , ˜ E −− = 2 ( E − + δ − ) + 2 U −− (1 + 2¯ n − ) . (S28)The corrections due to the Kerr-type interaction constants U + − and U −− can be obtained by including ladder diagramsin the self-energies, as shown in Fig. 4b. OMIT REFLECTION COEFFICIENT
In an OMIT style-experiment, in addition to the main driving laser (which gives rise to the many-photon interaction G ), a second weak drive tone (the “probe”) is applied at a frequency ω pr to the cavity. This driving is described bya term in the Hamiltonian: ˆ H pr = − i √ κ cp (cid:16) ˆ d † ¯ d in , pr e − i ˜ ω pr t + h.c. (cid:17) (S29)where we work in the interaction picture determined by the main drive laser frequency, and hence ˜ ω pr = ω pr − ω L . κ cp parameterizes the coupling of the drive port to the cavity: for a one-port cavity, the total cavity damping rate κ = κ cp + κ int , where κ int describes internal cavity losses.As the amplitude ¯ d in probe drive is weak, we can use standard linear response theory to describe its effects. TheKubo formula thus tells us that to first order in ¯ d in , pr , the change in the cavity amplitude will be given by: δ (cid:68) ˆ d ( t ) (cid:69) = − i √ κ cp (cid:90) ∞−∞ dt (cid:48) (cid:16) ¯ d in , pr G Rdd ( t − t (cid:48) ) e − i ˜ ω pr t (cid:48) − ¯ d ∗ in , pr G Rd ¯ d ( t − t (cid:48) ) e i ˜ ω pr t (cid:48) (cid:17) (S30)= − i √ κ cp (cid:0) e − i ˜ ω pr t ¯ d in , pr G Rdd [˜ ω pr ] − e i ˜ ω pr t ¯ d ∗ in , pr G Rd ¯ d [ − ˜ ω pr ] (cid:1) . (S31)Here, G Rdd [ ω ] is the retarded Green function of the cavity as defined above, calculated to zeroth order in ˆ H pr , butincluding the effects of the nonlinear interaction g . G Rd ¯ d [ ω ] is the corresponding anomalous Green function defined as: G Rd ¯ d [ ω ] ≡ − i (cid:90) dtθ ( t ) (cid:68)(cid:104) ˆ d ( t ) , ˆ d (0) (cid:105)(cid:69) e iωt , (S32)Now, the standard input-output relation between input, output and cavity fields is [1]:ˆ d out ( t ) = ˆ d in ( t ) + √ κ cp ˆ d ( t ) (S33)Taking the average value of this equation, and defining the elastic amplitude reflection coefficient r [ ω pr ] as theamplitude of (cid:104) ˆ d out ( t ) (cid:105) at the probe frequency divided by ¯ d in , we obtain r [ ω pr ] = 1 − iκ cp G Rdd [ ω pr ] , (S34)as given in the main text. MASTER EQUATION SIMULATION
Starting from the system-bath Hamiltonians written in the polariton basis (Eqs. (S7) and (S9)), one can trace overthe dissipative baths and derive a master equation for the reduced density matrix ˆ ρ describing the polaritons usingstandard Born-Markov approximations [5]. One obtains: ∂ t ˆ ρ = − i (cid:104) ˆ H + ˆ H , ˆ ρ (cid:105) + L ˆ ρ (S35)1where the coherent system Hamiltonian ˆ H + ˆ H is written without any approximation, and the super-operator L describes the effects of the dissipative baths via standard Lindblad terms, L = κ − (1 + ¯ n − ) L [ˆ c − ] + κ − ¯ n − L [ˆ c †− ] + κ + (1 + ¯ n + ) L [ˆ c + ] + κ + ¯ n + L [ˆ c † + ] , (S36)with L [ˆ c ] · = ˆ c · ˆ c † − { ˆ c † ˆ c, ·} . (S37)Note that this master equation corresponds to each polariton seeing independent thermal baths, in direct analogy tothe form of the quantum Langevin equations in Eq. (S11).While Eq. (S35) is not a convenient starting point for deriving analytic results, it does allow us to numerically studythe system without having to assume a small value of g . Using Eq. (S35) and the quantum regression theorem [5],we have numerically calculated the cavity density of states ρ d [ ω ] = − Im G Rdd [ ω ] /π , finding good agreement with ouranalytic perturbative results even for g as large as κ (see Fig. 1 in main text). Note that to find agreement with thesenumerical results, it was crucial to include in the analytic theory the corrections associated with the non-resonantinteraction processes, ˆ H NR (c.f. Eqs.(S27) and (S28)).Finally, we have also numerically studied our system using a more conventional master equation, in which thedissipation is described by Lindblad superoperators which act in the photon and phonon basis, i.e. replace L inEq. (S35) with L , where: L = κL [ ˆ d ] + γ (1 + ¯ n M ) L [ˆ b ] + γ ¯ n M L [ˆ b † ] , (S38)and ¯ n M is a Bose-Einstein distribution evaluated at the mechanical frequency ω M and mechanical bath temperature.For the parameters studied in the paper, this conventional master equation yields results very similar to those obtainedfrom Eq. (S35). g CORRELATION FUNCTIONS