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 thepolariton 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

The rapidly growing field of cavity optomechanics seeks
to understand the interaction between photons and mechanicalmotion in driven electromagnetic cavities, hopefully in truly
quantum regime [1,2]. The past few years have seen many
breakthroughs, including laser cooling of mechanical motion
to the ground state [3,4], and the optomechanical generationof squeezed light leaving the cavity [5–7]. Although thebasic radiation pressure interaction between a mechanicalresonator and cavity photons is intrinsically nonlinear, almost
all the remarkable achievements in the field to date rely on
working in strongly driven regimes where the dynamics isessentially linear. To see truly nonlinear effects in the simplest
setting, one needs to achieve a single photon, single-phonon
optomechanical coupling g which exceeds both the mechanicalfrequency ωM and the cavity damping rate κ [8–11]. With
the exception of experiments using cold atoms [12,13], thisparameter regime remains challenging for experiments.
Nonlinear effects which only require g ∼ κ can be achievedin slightly more complex settings where the nonlinear interaction becomes resonant. This can occur in an optomechanicalsetup with two optical modes [14,15]. Alternatively, this canoccur in a standard single-mode optomechanical cavity which
is driven, such that the interaction becomes resonant in a basisof dressed states (so-called optomechanical polaritons). Thisoccurs both in regimes of weak driving [16,17] and strong
driving [17]; similar physics can also occur in membrane-inthe-middle–style optomechanical systems [18]. Note that suchstrong driving regimes, where the drive-enhanced optomechanical coupling exceeds dissipative rates, has been achievedin several experiments [19–21].
In this paper, we both expand upon and extend the results
previously reported in Ref. [17]. We consider a standard
single-cavity optomechanical system that is driven (possiblystrongly) in a regime where the nonlinear interaction isresonantly enhanced. We again address this system usinga Keldysh formulation [22,23] and perturbation theory; we
begin this paper by providing a more complete discussion ofthis approach to optomechanics, emphasizing the subtleties
involved in treating dissipation. Our work thus contributes
to a growing body of work using the Keldysh techniqueto address various quantum optics contexts [24–28]. Unlike
1050-2947/2015/91(3)/033836(19)our previous study, we go beyond a simple lowest-order
treatment and introduce a self-consistent perturbation theory.
This corresponds to an infinite partial resummation of selfenergy diagrams, and extends the range of couplings andtemperatures that we can address.
Our work also address a new set of physical phenomena.
Reference [17] focused on understanding the cavity densityof states (DOS), the quantity probed in optomechanicallyinduced transparency (OMIT) experiments [19,29–31]. Here,
we instead focus on the nonequilibrium state of this system:
What are the effective temperatures of the optomechanicalpolaritons? Of particular interest is how vacuum fluctuations
can result in nonzero effective temperatures. Such effects
are often termed “quantum heating” [32–36]; the simplest
example is the amplification of zero-point fluctuations by aparametric amplifier. Although often described using disparate
terms, this physics has been studied in a wide variety ofsystems ranging from driven nonlinear oscillators [32–35],
superconducting circuits [37,38], and circuit quantum electrodynamics [39] to phase transitions in driven-dissipative
many-body systems [27]. It also sets a limit to the minimum
mechanical temperature achievable using cavity cooling [40].
Here, we show that quantum heating effects in an optomechanical system lead to observable signatures in the output
spectrum of the cavity, i.e., the spectrum of output light thatwould be measured using a photomultiplier. Unlike an OMIT
experiment, such an experiment probes both the DOS ofthe cavity (as modified by the optomechanical interactions),
as well as the effective temperature of the optomechanicalpolaritons. This “quantum heating” is already present at thelevel of the linearized theory of the optomechanical cavity;
here, we investigate how it is modified by the nonlinear
interactions. Our work is timely, given that the generationof real photons from mechanical vacuum fluctuations wasrecently probed experimentally by Lecocq et al. [41].

A. Main findings

Our work has many technical aspects to it that will
hopefully be an aid in further studies of nonlinear quantum
optomechanics (e.g., optomechanical lattices [42–46]). It also
predicts several additional physical phenomena, in particular:

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MARC-ANTOINE LEMONDE AND AASHISH A. CLERKPHYSICAL REVIEW A 91, 033836 (2015)

1. Polariton thermalization

In the standard linearized theory of a driven optomechanicalcavity, quantum heating leads to very different effective temperatures for the two polariton modes (i.e., normal modes ofthe 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 thiscompetition in detail for two representative cases: a laser driveat 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 inthe cavity output spectrum by tuning the nonlinear interactions
into and out of resonance (see Fig. 9).

2. New instabilities

For a red-detuned laser, we find that a leading-order
treatment of the nonlinear interaction suggests a different kindof parametric instability not present in the linearized theory,
where one polariton mode acts as an incoherent pump mode
for the other (see Sec. V D). Including higher-order terms
stabilizes the system as expected (see Fig. 11).

3. Two-phonon cavity heating

We show that nonlinear heating of the cavity (as manifest
in the cavity spectrum) is greatly enhanced for red-detuned
laser drives near the second mechanical sideband (a detuning ≈ −2ωM ); this is a simple consequence of the lower optomechanical polariton being mostly phononic in this regime,
causing effects to be enhanced by the typically large phonon
lifetime (see Fig. 12).

B. Organization of the paper

The remainder of this paper is organized as follow. We
begin in Sec. II by introducing the basic driven optomechanicalsystem studied in this work, and reviewing the linearizedtheory. We also pay careful attention to the coupling todissipative baths, and to how quantum heating effects can ariseeven without nonlinearity. In Sec. III, we introduce the basic
aspects of the Keldysh technique as applied to optomechanics,
focusing first on the linearized system. In Sec. IV, wedevelop a Keldysh perturbation theory to treat the nonlinear
optomechanical interaction, and introduce our self-consistent
approach. In Sec. V, we discuss how the nonlinear interaction
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 parametricheating which arises with a red-detuned laser. In Sec. VI,
we discuss the observable consequences of our predictions
on the cavity output spectrum. We present our conclusions in
Sec. VII.

FIG. 1. (Color online) (a) A generic optomechanical cavity: acavity mode of frequency ωC and damping rate κ is driven by a laser
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
optomechanical polaritons; the resulting enhanced interaction shown
schematically and described in Eq. (8).
II. SYSTEM, TREATMENT OF DISSIPATION,
AND LINEARIZED THEORY
A. Hamiltonian in the polariton basis

We consider a standard optomechanical system where thefrequency 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]Ĥ = ωC â † â + ωM b̂† b̂ + g â † â(b̂ + b̂† )+ i(āin e−iωL t â † − H.c.) + Ĥdiss .

(1)

Here, â is the cavity mode, with frequency ωC and b̂ themechanical mode with frequency ωM . The parameter g is the
single-photon optomechanical coupling and āin is proportional
to the amplitude of a classical drive at frequency ωL . Finally,Ĥdiss describes dissipation due to the coupling to bosonicenvironments (both for the mechanics and the cavity). Going
into a rotating frame at the drive frequency and displacing thecavity field by its classical value, induced by the coherent drive{i.e., â(t) → e−iωL t [ā + d̂(t)]}, the Hamiltonian of Eq. (1) canbe expanded into a quadratic part, known as the linearized
optomechanical Hamiltonian, and a nonlinear interaction termsuch thatĤ = ĤL + ĤNL + Ĥdiss ,
†

†

(2a)
†

†

ĤL = −d̂ d̂ + ωM b̂ b̂ + G(d̂ + d̂ )(b̂ + b̂ ), (2b)ĤNL = g d̂ † d̂(b̂† + b̂).

(2c)

In the above, we have defined the laser detuning  = ωL −
ωC and the many-photon coupling constant G = āg; we takeā,g > 0 without loss of generality.1

1
The mean value of â (ā) is determined by solving the classicalequations of motion following from Eq. (1). We also shift themechanical lowering operator b̂ [i.e., b̂ = 0 in Eqs. (2)] to accountfor the static radiation pressure force. We also redefine the detuning
accordingly.

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PHYSICAL REVIEW A 91, 033836 (2015)

Our general approach is to diagonalize the quadratic partof the Hamiltonian ĤL (thus treating it exactly), and thentreat the terms due to nonlinear interaction (ĤNL ) as aperturbation. Unlike treatments based on polaron-transformed
Hamiltonians [8,9], we do not require the coherent drivingof the cavity to be so small that it too can be treated
perturbatively. We focus exclusively on a red-detuned laser( < 0), and only require the cavity drive to be weak enoughso that G2 < −ωM /4 ≡ G2crit . For drives stronger than thiscritical value (or for a blue-detuned laser), the linearized
coherent Hamiltonian ĤL is unstable, and corresponds to a
detuned parametric amplifier driven beyond threshold. This
critical value also coincides with the onset of the well-known
static optomechanical instability [1,47]. Operating near thisinstability has been investigated as an alternative promising
way to enhance the nonlinear interaction [18]. Note that thisinstability is also closely analogous to the superradiant phasetransition in the driven Dicke model studied in [27].
Focusing on  < 0 and G2 < −ωM /4, Eq. (2b) can be
diagonalized via a Bogoliubov transformation to yield
ĤL =



Eσ ĉσ† ĉσ .

(3)

σ =±

Here, ĉ± destroys an excitation in the eigenmode of ĤL withenergy E± > 0, given by1  2
E± = √ ωM+ 2 ±
2



 2
2
1/2
.
ωM − 2 − 16G2 ωM(4)Note that E− tends to zero as G approaches the critical value atthe instability Gcrit . As ĤL does not conserve the total numberof photons and phonons, the operators ĉ− ,ĉ+ mix photon andphonon annihilation and creation operators:††b̂ = αb,− ĉ− + αb,+ ĉ+ + ᾱb,− ĉ− + ᾱb,+ ĉ+ ,

(5a)

†
†
d̂ = αd,− ĉ− + αd,+ ĉ+ + ᾱd,− ĉ− + ᾱd,+ ĉ+ .

(5b)The coefficients αb/d,± and ᾱb/d,± are functions of /ωM
and G/ωM ; their explicit form in the case  = −ωM is given
in Eqs. (A1)–(A4) of Appendix A. As the excitations describedby ĉ− ,ĉ+ have both phononic and photonic components, werefer to them as polaritons in what follows.
Having diagonalized the linearized optomechanical Hamiltonian, we now express the nonlinear Hamiltonian ĤNL in thepolariton basis. We obtain interactions that do not conserve thetotal number of polaritons
ĤNL =



 A

† †
†
gσ σ σ ĉσ† ĉσ ĉσ + gσBσ σ ĉσ† ĉσ ĉσ + H.c.

σ,σ ,σ =±+ (A− ĉ− + A+ ĉ+ + H.c.).
A/B

(6)

Here, the constants gσ σ σ and Aσ are all proportional to g [seeEqs. (A7)–(A9) of Appendix A]. Note that the linear terms in
ĤNL (∝ Aσ ) arise from normal ordering ĤNL in the polariton
basis; physically, while the zero-polariton state is the vacuumof ĤL , this is no longer true when we include the nonlinear
interaction.

B. Resonant polariton interactions

As discussed in previous works [16,17] (and later in [48]),
we can enhance the effects of even a weak single-photon
coupling g by tuning G and  such that the nonlinear processes
in Eq. (6) that scatter a + polariton into two − polariB
tons (∝ g−−+
) become resonant. This requires E+ [,G] =2E− [,G]; for a given laser detuning in the range  ∈[−2ωM , − ωM /2], this can always be achieved by tuning
G = Gres [], where



2
4
− 4 4 + ωM
172 ωM.
(7)
Gres [] ≡√(10 −ωM )Once G is tuned to achieve this resonance condition,
one can show using standard perturbation theory that all
nonresonant nonlinear processes are suppressed by a factorof κ/(E+ − E− ) ∝ κ/ωM compared to the resonant process,where κ is the cavity damping rate (see [17] for more details).
In addition, in this regime, the relative modification of thepolariton energies and wave functions due to the linear terms
in Eq. (6) are strongly suppressed by a factor of (g/ωM )2 .
Thus, if we focus on parameters near this resonant regime,in the resolved sideband regime (κ/ωM1), and for weak
nonlinear interaction (g/ωM1), we can both ignore therenormalization of polariton energies and wave functions, andmake a rotating-wave approximation on Eq. (6), keeping only
the resonant interaction. The coherent system Hamiltonian inthis regime thus reduces to†Ĥeff =
Eσ ĉσ† ĉσ + g̃(ĉ+ ĉ− ĉ− + H.c.),
(8)
σ =±
B
g̃ = g−−+

whereis the effective nonlinear coupling [seeEq. (A8) in Appendix A]. Shown in Fig. 2(a) is the dependenceof Gres as a function of laser detuning , as well as the behaviorof g̃ on , when G is tuned to be Gres . For the rest of this paper,
we focus on the dynamics governed by the effective coherent
Hamiltonian Ĥeff .
C. Coupling to dissipative reservoirs

We now turn to a more careful consideration of the effects
of dissipation on our system. As is standard in optomechanics,
both the mechanics and the cavity are taken to be linearly
coupled to independent, Markovian bosonic baths (i.e., baths
with constant DOS over the frequency range of interest). Thisis analogous to the approach taken in input-output treatment ofdissipation (see, e.g. [49,50]). As we are interested in possibly
large many-photon optomechanical couplings G, care muststill be taken, as the eigenstates of our coherent Hamiltonian
are polaritons, not individual photons or phonons; similar
issues have recently been addressed in strongly-coupled circuit
QED systems [51]. Moreover, the strong driving of the cavity
can also lead directly to “quantum heating” effects, which
manifest themselves directly in the treatment of the cavity
dissipation. We describe these effects more in what follows.
1. Cavity dissipation in the presence of optomechanicalcoupling and driving

Consider first the interaction between the cavity andits dissipative reservoir. In the original laboratory frame,

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MARC-ANTOINE LEMONDE AND AASHISH A. CLERKPHYSICAL 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) whichconserve excitation number are time independent, whereas
the remaining excitation nonconserving terms are rapidly||). Since
oscillation at a frequency ±2ωL ≈ ±2ωC (ωCωC is much larger than all other energy scales in the rotating
frame (i.e., ωC
Eσ ,||,ωM ,g,κ), and since the bath oscillators necessarily have positive energies ωj > 0, these rapidly
oscillating terms can never become resonant. We can thussafely make a rotating-wave approximation, and drop them.
With this rotating-wave approximation, the cavity-bath
Hamiltonian then becomes time independent. Re-writing thecavity annihilation operator as per Eq. (2b), one obtains astandard rotating-wave system-bath interaction††(ωj − ωL )fˆj fˆj =ω̃j fˆj fˆj ,
(11)
Ĥκ =
j

j


Ĥκint = iκ  ˆ†(f d̂ − fˆj d̂ † ).
2πρC j j(12)Note crucially that in this final rotating frame, the transformed bath frequencies ω̃j can be negative (they extend downto −ωL ). While this may seem innocuous, things become
more interesting when we rewrite this interaction in termsof polariton operators [cf. Eqs. (5)]:κ  ˆ†
intĤκ = if (αd,σ ĉσ + ᾱd,σ ĉσ† ) + H.c.
(13)
2πρC j,σ j
FIG. 2. (Color online) (a) Gres , the value of the many-photon
coupling which leads a resonant nonlinear interaction, as a function oflaser detuning  [cf. Eq. (7)]; the corresponding effective nonlinear
coupling g̃ [cf. Eq. (8)] is also plotted. (b), (c) Coefficients relating theoptomechanical polaritons (eigenstates of the linearized Hamiltonian)
to original photon and phonon operators [cf. Eqs. (5)] as a functionof laser detuning  with G = Gres []. In (b), we plot the “normal”
coefficients which relate polariton destruction operators to photon
and phonon destruction operators, i.e., αb,σ = 0,0|ĉσ b̂† |0,0, with|0,0 being the polariton vacuum. In (c), we plot the “anomalous”
coefficients, i.e., ᾱb,σ = 0,0|ĉσ b̂|0,0; these coefficients are directly
related to the existence of “quantum heating” effects in the linearizedtheory [cf. Eq. (15)].

the cavity-bath interaction will have the following genericform [49]:κ  ˆ†† ˆintˆ
ωj fj fj ,Ĥκ = i(fj − fˆj )(â + â † ),
Ĥκ =
2πρ
C
j
j
(9)
where fˆj is the anihiliation operator for cavity-bath mode j(frequency ωj ), κ the damping rate of the photons inside thecavity, and ρC is the bath DOS. As we consider a Markovianbath, we take κ and ρC to be frequency independent.
We next transform to an interaction picture at the drive
frequency via the unitary
⎡
⎛
⎞⎤
 †
fˆj fˆj ⎠⎦ .
Û = exp ⎣−iωL t ⎝â † â +(10)

The anomalous terms which create or destroy two excitations
here should not be dropped; as the polariton energies EσωL ∼ ωC , these terms can be resonant in our interaction
picture, as bath modes having ω̃j < 0 can be involved.
Physically, such processes involve the creation of both apolariton and a bath excitation, while at the same time a(classical) drive photon is absorbed. Such heating processes
thus involve the interplay of the system drive and the bathzero-point fluctuations, and are at the heart of quantum
activation [32–35].
2. Quantum heating and effective temperature

The anomalous, excitation nonconserving terms in Eq. (13)can lead to polariton heating even if the cavity dissipation
is at zero temperature. We can naturally associate an effective
temperature to this heating by computing golden rule transition
rates [50]. Consider first the lower-energy − polaritons. Ĥκint
will cause transitions uphill in energy between an initial state
having N − 1 polaritons and a final state having N polaritons
at a rate N,N−1 ∝ N |ᾱd,− |2 . Similarly, it will cause transitions
downhill in energy from the N to N − 1 polariton state at a
rate N−1,N ∝ N |αd,− |2 . If these transitions were due to abath in true thermal equilibrium at temperature T , detailedbalance dictates that N,N−1 / N−1,N = exp(−E− /kB T ). In
our case, we can use the ratio of these rates to define theeffective temperature T−cav of the cavity dissipation as seen by
the − polaritons:je−E− /kB T− ≡
cav

N,N−1
N−1,N

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=

2
ᾱd,−
2
αd,−

.

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PHYSICAL REVIEW A 91, 033836 (2015)

If these transitions were the only dynamics of the − polaritons,
they would indeed cause them to reach a thermal state at
temperature T−cav , with a thermal occupancynB [E− ,T−cav ] =1eE− /kB T−cav

−1

=

2
ᾱd,−
2
2
αd,−
− ᾱd,−.

(15)

We thus see the two crucial ingredients needed to obtain anonzero effective temperature in a bosonic system where thephysical bath temperature is zero. We needed both coherent
drive (yielding effective negative energy bath modes in ourinteraction picture) and coherent parametric-amplifier typeinteractions (i.e., coherent interactions which do not conserve
particle number, and hence yield ᾱd,− = 0). In a completely
analogous fashion, one can also associate an effective temperature T+cav describing the quantum heating of the + polaritons.
We stress that these effective temperatures have nothing to dowith nonlinear interactions.
For a concrete example of this effective temperaturephysics, consider the special case of a cavity drive at the redmechanical sideband  = −ωM . One finds [cf. Eqs. (A1)–(A4)]√(1 − 1 ± 2G/ωM )2nB [E± ,T±cav ]|=−ωM =.
(16)√
4 1 ± 2G/ωM
Note that nB [E− ,T−cav ] diverges when G approaches theonset of parametric instability G → ωM /2. In this limit,
E− → 0, and the divergence of nB [E− ,T−cav ] is equivalent toa fixed effective temperature T−cav  ωM /4 = ||/4 near the
instability. Such behavior is generic for quantum heating nearparametric instabilities.
The type of quantum heating phenomena described here is
generic, and is found in a variety of related systems, though
the generic nature of the mechanism is often not appreciated.
Analogous effective temperatures also arise in mean-field
treatments of driven dissipative phase transitions. For example,
the study of the driven-dissipative Dicke model in Ref. [27]finds an effective temperature near the transition identical tothat quoted after Eq. (16). The physical origin is analogous
to that in our system: it arises from the interplay of coherent
parametric-amplifier interactions combined with a coherent
linear driving.
3. Mechanical dissipation

We now turn to the interaction between the mechanicalresonator and its dissipative bath. The starting interaction is
analogous to Eq. (9) for the cavity dissipation


γ  †
†
Ĥγ =
ωj ĝj ĝj , Ĥγint = i(ĝj − ĝj )(b̂ + b̂† ),
2πρ
M
j
j
(17)
where ĝj is a bath annihilation operator, γ the mechanicaldamping rate, and ρM is the constant DOS of the (Markovian)bath. In what follows, we consider the mechanical bath to beat temperature TM and define the mean number of excitations
inside the bath at ωM asn̄M
th ≡ nB [ωM ,TM ],with nB [ω,T ] being the Bose-Einstein distribution.

(18)

As the mechanics is undriven, the bath Hamiltonian andbath-system interaction are unchanged under the transformation of Eq. (10) to the rotating frame at the laser frequency. Thistime, we rewrite the interaction in terms of polariton operators
using Eqs. (5) before making further approximations. Thisdifference from the treatment of cavity dissipation stems fromthe fact that unlike ωC , ωM is comparable to Eσ . It is thus
crucial to go to the eigenstates basis of polaritons before
assessing which terms may be safely dropped. In the polariton
basis, we have †γĤγint = i(ĝ − ĝj )(αb,σ + ᾱb,σ )(ĉσ + ĉσ† ). (19)2πρM j,σ =± j
We can now consider the role of terms in Eq. (19) that do notconserve excitation number. In contrast to our treatment of thecavity dissipation, here such anomalous terms can be dropped
in a rotating-wave approximation. As there is no mechanicaldrive, there are no effective negative-energy mechanical bathmodes, and hence these terms can never be made resonant.
Thus, we finally obtain a simple rotating-wave interaction
between the mechanical bath and the polaritons:γ†Ĥγint = i[(ασ + ᾱσ )ĝj ĉσ + H.c.]. (20)2πρM j,σ =±As there are no excitation nonconserving terms in Eq. (20),
it is easy to confirm that the effective temperature of themechanical dissipation seen by the polaritons is simply equal
to the physical temperature of the mechanical dissipation.
As already emphasized, quantum heating requires both thepresence of coherent parametric-amplifier (paramp) type interactions and the presence of a coherent drive. Here, while thecoherent paramp interactions are present, there is no coherent
driving of the mechanics; as such, there is no quantum heating
effects involving the mechanical bath. We see that even if
both the cavity and mechanical baths have identical physicaltemperatures, the polaritons see them as having different
effective temperatures. The driven nature of the system thus
gives us interesting nonequilibrium physics even at the levelof the linearized (i.e., quadratic Hamiltonian) theory.
We end this section with a caveat on the validity of treating
dissipation via Markovian baths. For the cavity dissipation,
this is an excellent approximation, as we are always probing
the bath in a narrow interval of width ∼Eσ around ωC , aninterval over which the bath DOS can be treated as constant(recall that Eσ
ωC ). In contrast, a similar statement does
not hold for the mechanical dissipation: we will be probing themechanical bath at frequencies Eσ which could be significantly
different from the mechanical frequency ωM . As such, it is
not a priori obvious that the bath spectral density can betreated as flat. For simplicity, we will nonetheless use theMarkov bath approximation for the mechanics in what follows(consistent with the majority of works in optomechanics).
For the weak dissipation limit of interest, the main effects
of a nonflat bath spectral density could be easily incorporated
into our calculations. One would simply make the mechanicalcontribution κM to the intrinsic polariton decay rates [cf.
Eq. (23)] proportional to the mechanical bath density of statesat the relevant polariton energy.

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D. Lindblad master equation and effective polariton dissipation

For further insight, it is useful to use the form of the systembath 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 thestandard derivation (see, e.g., [49]) valid for weakly coupled
Markovian baths, we obtain
˙ = −i[Ĥeff ,ρ̂(t)]
ρ̂(t)
κ−
κ− 0
†
n̄ D[ĉ− ]ρ̂,+ (n̄0− + 1)D[ĉ− ]ρ̂ +
2
2 −
κ+ 0
κ+
†
n̄ D[ĉ+ ]ρ̂,+ (n̄0+ + 1)D[ĉ+ ]ρ̂ +
2
2 +

(21)

with the Lindblad superoperator D[ĉσ ]ρ̂ defined asD[ĉσ ]ρ̂ ≡ 2ĉσ ρ̂(t)ĉσ† − ĉσ† ĉσ ρ̂(t) − ρ̂(t)ĉσ† ĉσ .

(22)

The effective polariton damping rates appearing in Eq. (21)
are

 2
2
κσ = γ (αb,σ + ᾱb,σ )2 + κ αd,σ≡ κσmech + κσcav .
− ᾱd,σ(23)Here, we have introduced κσmech (κσcav ) as the contribution to thedamping rate of the σ polariton coming from the interaction
with the mechanical resonator (cavity) dissipative bath. The
corresponding effective bath thermal occupancies are12
γ (αb,σ + ᾱb,σ )2 nB [Eσ ,TM ] + κ ᾱd,σ
(24a)
κσ


κσmech nB [Eσ ,TM ] + κσcav nB Eσ ,Tσcav
.
(24b)
=
κσmech + κσcav

n̄0σ =

Here, TM is the (physical) temperature of the mechanical bath[cf. Eq. (18)], Tσcav is the temperature of the cavity bath as seenby the σ polariton [cf. Eq. (15)], and the different coefficients
α are given in Eq. (5). As expected, Eq. (24b) represents abosonic mode coupled independently to two dissipative baths.
An analogous expression holds for the effective mechanicaloccupancy used to describe cavity-cooling experiments [40].
Note that we have made a standard secular approximation,
allowing us to drop dissipative terms that do not conserve thenumber of each polariton independently in Eq. (21); this isvalid for the regime of interest Eσ ,|E+ − E− |κ,γ .
The thermal occupation number of the effective baths given
in Eqs. (24) and their corresponding temperatures, defined
as nB [Eσ ,Tσ0 ] ≡ n̄0σ , are plotted in Fig. 3 as a functionof the detuning . Anticipating our interest in nonlinear
interactions, for each  we adjust the control laser amplitude
so that G = Gres [i.e., the value that will make the nonlinear
interaction resonant, cf. Eq. (7)]; this can be done for any ∈ [−2ωM , − ωM /2]. One sees that even when the physicalbath temperature is zero, quantum heating effects can yield
effective polariton temperatures as large as ∼0.1 quanta (solid
curves). At nonzero physical temperature (dashed curves),
these quantum heating effects persist, but are swamped by thecontribution of mechanical noise at the edges of the detuning
range considered. This is simply because near the limits of thedetuning range, one polariton species is almost all phononic

FIG. 3. (Color online) (a) Damping rates κσ of the σ = ± polaritons [cf. Eq. (23)] in the linearized theory (i.e., g̃ = 0), as a functionof the detuning . For each , we tune the cavity control drive
to maintain G = Gres (). Near  = −2ωM , the − (+) polariton
damping tends to γ (κ) as it is mostly phononlike (photonlike);
the converse is true near  = −ωM /2. (b) Thermal occupation
numbers n̄0σ of the effective bath coupled to the σ polaritons [cf.
Eq. (24)] also when g̃ = 0 and G = Gres . Here, n̄M
th characterizes the(physical) temperature of the mechanical bath [cf. Eq. (18)]. (c) The
temperatures corresponding to the thermal occupation n̄0σ (kB = 1) inthe same regime as panels (a) and (b). For each curve, γ /κ = 10−4
and ωM /κ = 50.

(see Fig. 2) and becomes very sensitive to thermal fluctuationsof the mechanical bath.
Returning to zero physical bath temperature, anotherstriking feature in Fig. 3(c) is the sudden drop in effective
temperature for the more phononlike polariton branch at theedges of the detuning interval. As it will be of interest inwhat follows, we discuss this behavior for detunings near = −2ωM in more detail; here, one sees a sudden drop inn̄0− and T−0 . As  → −2ωM , Gres → 0, and the − polariton
becomes simply a phonon. Expanding n̄0− to lowest order inGres /ωM for TM = 0 and γ
κ, one gets
2

1 Gresn̄0− =

2 κ
κ 2
9 ωM
.
ᾱd,− ≈G2
κ−γ + 8 res
2 κ

(25)

9 ωM

We see there are two competing effects associated with
nonzero Gres . The numerator reflects the parametric heating

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PHYSICAL REVIEW A 91, 033836 (2015)

associated with the linearized optomechanical interaction.
The denominator in contrast reflects that the phononlike
polariton has its lifetime decrease as Gres increases and it
becomes more photonlike; this is just standard optomechanicaloptical damping. The result is that the net quantum heating is
maximized for [Gres /ωM ]2 ∼ γ /κ, corresponding to a laser
detuning  + 2ωM ∼ γ /κ; the maximum occupancy n̄0σ thatcan be obtained is 18 [as seen in Fig. 3(b)].
III. KELDYSH DESCRIPTION OF THE LINEARIZEDSYSTEMAs shown in the previous section, the driven nature of
the system leads to nonequilibrium physics even at the levelof the linearized theory (i.e., the two polariton species see
different effective temperatures). Consequently, we need touse the Keldysh formalism [23] in order to describe thedynamics and to properly construct a perturbation theory thattreats the nonlinear interaction present in Ĥeff [Eq. (8)]. In
this section, we quickly introduce this approach, considering
first the linearized Hamiltonian of Eq. (2b) and the couplings
with the environment given by Eqs. (12) and (20). We will
also map the resulting free Keldysh theory onto the simple
master equation previously derived [Eq. (21)], again working
in the regime where polaritons have a well-defined energy (i.e.,
Eσ ). We stress, however, that even without interactions,κσthe Keldysh approach is more general than a Lindblad-stylemaster equation, as it is not restricted to Markovian baths.
In the Keldysh formalism, we represent our linearized optomechanical system by a field theory which is general enoughto allow the system to be in an arbitrary, nonequilibrium state.
In this field theory, there are two time-dependent fields (classical and quantum) corresponding to each annihilation operator
in the original theory. Consequently, the quadratic action thatconserves the number of particles of our two independent
bosonic modes (polaritons) will have the following generalform: ∞ ∞∗∗SL =dt dt (cσ,cl(t),cσ,q(t))
σ =± −∞−∞× G−1
σ (t − t )cσ,cl (t ).
cσ,q (t )(26)Here, the cσ,q/cl (t) are complex functions of time and
G−1
σ (t − t ) is the (operator) inverse of the unperturbed (i.e.,g̃ = 0) Green’s function. The latter is given by a 2 × 2 matrixGKGRσ (t)σ (t)Gσ (t) =
.
(27)
GA
0
σ (t)
In terms of Heisenberg picture operators, each element isdefined as∗≡ −iθ (t)[ĉσ (t),ĉσ† (0)], (28a)GRσ (t) = GA
σ (t)
†
GK
σ (t) ≡ −i{ĉσ (t),ĉσ (0)},(28b)

where the expectations are taken with respect to the initial
density matrix without nonlinear interaction (g̃ = 0). Here,GRσ (t) and GA
σ (t) are the standard unperturbed retarded andadvanced Green’s functions, which govern the linear-response
properties of the unperturbed system. They are also related tothe unperturbed DOS of each polariton, given by
1 
ρσ0 [ω] = − Im GRσ [ω] .
π(29)Finally, GK
σ (t) is known as the (unperturbed) Keldysh Green’s
function. It encodes knowledge of the energy distributionfunction of each polariton (as we will see more clearly inthe following).
With these definitions in hand, one could follow thestandard approach used in input-output theory and derive theHeisenberg-Langevin equations from the coherent Hamiltonian of Eq. (8) and the particular form of the system baths
coupling given by Eqs. (12) and (20). From there, one candirectly get the bare Green’s functions by calculating Eqs. (28).
This approach has been used in our previous work [17]. Here,
we instead follow a different but equivalent route to obtain thebare Green’s functions and Keldysh action.
The goal here is to write the different Green’s functions
such that our description of the linearized theory in the Keldysh
formalism is completely equivalent to the master equation of
Eq. (21). As discussed before, Eq. (21) describes two independent bosonic modes with simple Markovian damping rates κσ[cf. Eq. (23)] and energies Eσ [cf. Eq. (4)]. Consequently,
the retarded (advanced) Green’s functions, which give theresponse functions of the system, have to adopt the simple
following form:∗GRσ [ω] = GA
=
σ [ω]

1
,
ω − Eσ + iκσ /2(30)

i.e., a polariton has an energy Eσ and a lifetime 1/κσ .
We now construct the Keldysh Green’s functions using thesame approach. As is standard [23], we define a distributionfunction f [ω] that relates the Keldysh and the retarded Green’s
functions, such that
 R 
GK
(31)
σ [ω] ≡ −2i(2f [ω] + 1)Im Gσ [ω] .
This function f [ω] parametrizes the occupation of different
polariton energy eigenstates. As an example, for a systemin thermal equilibrium at a certain temperature, f [ω] would
be the corresponding Bose-Einstein distribution and Eq. (31)
would be an exact statement of the fluctuation-dissipation
theorem [23]. In the particular case studied here, the freepolaritons have sharply peaked single-particle DOS ρσ [ω] [cf.
Eqs. (29) and (30) with κσEσ ], such that for the σ polariton,
the function f [ω] of Eq. (31) can be approximated as f [ω] f [Eσ ]. Finally, if we insist that the average occupancy of thepolariton matches that in the master-equation description, thenwe must have f [Eσ ] = n̄0σ . We thus have 0
  R GK(32)
σ [ω] = −2i 2n̄σ + 1 Im Gσ [ω] .
Describing the linearized theory in the Keldysh formalism
using the bare Green’s functions (30) and (32) is thus
completely equivalent to the Lindblad master equation (21).
We recall that the two assumptions underlying these two
equivalent descriptions are that the polaritons have sharply
peaked DOS (i.e., dissipation is weak) and that coupling
between the + and − polaritons due to dissipation is negligible[i.e., secular approximation made to derive Eq. (21), seeSec. II D].

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MARC-ANTOINE LEMONDE AND AASHISH A. CLERKPHYSICAL REVIEW A 91, 033836 (2015)

Finally, we note that it is possible to derive exact Langevinequations 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 introduces new fields ξσenv (t) which have a Gaussian action. One
can then exactly do the integrals over quantum and classicalfields. The resulting functional delta function corresponds to
the Langevin equationsκσcσ (t) − ξσenv (t),(33)
∂t cσ (t) = − iEσ +2where the noise ξσenv (t) is Gaussian with zero mean. The only
nonzero noise correlation functions are given by env  env ∗ 
ξσ (t) ξσ (t ) = κσ n̄0σ + 1/2 δ(t − t )δσ,σ . (34)
As expected, Eqs. (33) and (34) represent two uncoupleddamped harmonic oscillators each in contact with theirrespective finite-temperature Markovian baths.
IV. KELDYSH PERTURBATIVE TREATMENT
OF POLARITON INTERACTIONS
A. Self-energies and dressed Green’s functions

Having established the Keldysh formulation of the linearized optomechanical theory, we can now address the effectsof the nonlinear interaction as a perturbation. We assume
throughout this section that the drive laser has been tuned
to make the nonlinear interaction resonant, i.e., G = Gres , andthus consider only the resonant interaction process given in
Eq. (8).
First, the action generated by the nonlinear interaction inĤeff [Eq. (8)] in the cl-q basis is
 ∞
g̃
∗
∗
SNL = √
dt(c+,qc−,cl c−,cl + 2c+,cl
c−,q c−,cl2 −∞∗+ c+,qc−,q c−,q + c.c.),(35)
where the time dependence of the fields is implicit forclarity. Diagrammatically, each term in the nonlinear action
corresponds to a vertex shown in Fig. 4(a). The vertices witha single quantum field correct the classical saddle point ∞∂(SL + SNL ) =
dt [GR+ (t − t )]−1 c+,cl (t )∗∂c+,q(t) c±,q =0−∞∂(SL + SNL ) ∗∂c−,q(t) c±,q =0

g̃
(36a)
+ √ c−,cl (t)c−,cl (t),2 ∞=dt [GR− (t − t )]−1 c−,cl (t )FIG. 4. (Color online) Diagrams representing the self-energies
up to second order in g̃ in the Keldysh formalism. The classicalvertices are composed of only one quantum component, while thequantum vertex is composed of three.

versions of the Langevin equations in Eq. (33):κ−∗c− − 2i g̃c+c− − ξ−env , (37a)∂t c− = − iE− +2κ+c+ − i g̃c− c− − ξ+env , (37b)∂t c+ = − iE+ +2
where the autocorrelation functions of the ξσ (t) noise areunchanged by the nonlinear interactions. We have suppressed
the explicit time dependence of fields here for clarity.
We will not use these approximate quantum Langevinequations further, but proceed in a way that does not neglectterms that are cubic in the quantum fields. As we are interested
in weak nonlinear couplings g̃, we will compute the self-energy([ω]) of our Keldysh Green’s functions perturbatively toorder g̃ 2 . At this order, all relevant scattering processes (seeFig. 4) conserve the number of polaritons independently(i.e., the self-energies are diagonal in the +/− index).
Consequently, the Dyson equation that gives the Green’s
functions in presence of interactions can be separately written
for each polariton:

−1


σK [ω] σA [ω]GσK [ω] GσR [ω]−1= Gσ [ω] −.
0
0
GσA [ω]σR [ω](38)
Here, we used GσA,R,K [ω] to distinguish the full Green’s
functions (i.e., including the effects of g̃) from the unperturbed[ω].
ones GA,R,K
σ

−∞

+

√

∗
2g̃c+,cl (t)c−,cl(t)1. Retarded self-energies and interaction-induced
polariton damping(36b)

and thus correspond to a classical nonlinear potential. The
term with three quantum fields is more of a purely quantum
effect. It could be interpreted as an effective nonlinearity ofthe quantum noise.
One can still derive Langevin equations from the resulting
nonlinear action if one ignores the terms which are cubic inquantum fields. In this approximation, one obtains modified

The diagrams related to the second-order retarded selfenergies are shown in Fig. 4(b). From these, one straightforwardly calculates

033836-8

κ− + κ+κ−,+4 ω − (E+ − E− ) + i κ− +κ
2
κ−
κ+
,
+R [ω] = C+eff
2 ω − 2E− + iκ−−R [ω] = C−eff

(39a)
(39b)

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PHYSICAL REVIEW A 91, 033836 (2015)

equilibrium, the fact that E+ > E− guarantees this factor isintpositive, yielding −> 0. Our system, however, is not in true
thermal equilibrium: as discussed in the previous section, it ispossible to have (n̄0− − n̄0+ ) < 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. V D.
The retarded polariton self-energies presented here directly
lead to an interaction-induced modification of the polariton
DOS, ρσ [ω], given by

FIG. 5. (Color online) Solid lines: leading-order-in-g effective
cooperativities Cσeff associated with the nonlinear interaction [definedin Eq. (41)], as a function of the detuning in the resonant regime (G =Gres []). One sees that the Cσeff are enhanced near  = −2ωM . The
inset zooms on the region where C−eff goes below zero, which signals
the possibility of a new kind of instability, as discussed explicitly inSec. V D. Circles: results of the self-consistent perturbation theory
described in Sec. IV B, which includes diagrams at all orders in g.
For all values of Cσeff shown here, the approach converges and is ingood agreement with numerical simulations of the Lindblad masterequation [cf. Eq. (21)]. The parameters used are γ /κ = 10−4 , g = κ,ωM = 50κ, and n̄M
th = 0.

with σA [ω] = {σR [ω]}∗ . As discussed in detail in [17],the self-energies σR [ω] describe the hybridization between
the near-resonant |+ and |−,− polariton states. The resonant nonlinear processes underlying this hybridization [seeFig. 4(b)] are responsible for the sharply peaked self-energies
of Eqs. (39) (κσEσ ). We have introduced effective cooperativities Cσeff to parametrize how strong the decay rates
resulting from these processes are (i.e., imaginary part of theself-energy) on resonance, compared to the intrinsic polariton
linewidth. Defining
 R 
int
(40)
σ [ω] = −2 Im σ [ω] ,1 ρσ [ω] ≡ − Im GσR [ω]
π


1
1
.
= − Imπ
ω−Eσ +iκσ /2−σR [ω]Signatures of g in the DOS (and corresponding changes
to OMIT-style experiments) were the focus of our previous
work [17]. For strong enough nonlinear coupling g, theself-energies turn from simply describing an extra broadeningof the polaritons to describing the coherent hybridization of theresonant |+ and |−,− polariton states. The key consequenceof this is that the single peak in the + polariton DOS splits.
Further details about this splitting (and how it can be measured
via an OMIT-type experiment) can be found in Ref. [17].
2. Keldysh self-energies

We now turn to the Keldysh self-energies, which are also
directly calculated from the diagrams of Fig. 4(c). Here, weparametrize each Keldysh self-energy via a thermal occupancy
factor n̄int
σ associated with the interaction, defined such that  R σK [ω] ≡ −2i 2n̄int(43)
σ + 1 Im σ [ω] .
This parametrization is always possible if we let the n̄int
σ to befrequency dependent. However, we find them to be frequency
independent and given by
n̄int
− =

we haveC−eff ≡
C+eff ≡

int
− [E+ − E− ]κ−int
+ [2E− ]κ+=

The definitions of C−eff ,C+eff=

16g̃ 2 (n̄0− − n̄0+ ),κ− (κ− + κ+ )4g̃ 2 (2n̄0− + 1)
.
κ− κ+

(41a)

(42)

n̄0+ (n̄0− + 1)(n̄0− )2
, n̄int.
+ =00n̄− − n̄+2n̄0− + 1

(44)

We stress that these results (as well as the self-energy results
above) are based on only keeping the nonlinear polariton
interaction in Eq. (8), and thus assume that E+  2E− .

(41b)
B. Self-consistent calculation

are analogous to the definitionof the standard optomechanical cooperativity C = 4G2 /κγ as
the cavity-induced “optical damping” of the mechanics to theintrinsic mechanical damping. The effective cooperativities
are plotted in Fig. 5 as a function of the detuning (keeping
G = Gres for all detunings).
A crucial feature of the interaction-induced polariton
damping described by Eqs. (41) is their explicit temperaturedependence. This is a direct consequence of the multiparticle
nature of the relevant decay process. For the − polariton, we
int
have −∝ (n̄0− − n̄0+ ), as expected for a bosonic polarization
bubble; a similar damping rate is found for an oscillator
coupled quadratically to an oscillator bath [52]. In true thermal

The self-energy results discussed so far (and in Ref. [17])only retain diagrams to leading order in g. To capture higherorder effects and effectively resum diagrams at all orders in
perturbation theory, one can make the diagrams in Fig. 4
self-consistent. One simply replaces all internal propagatorsin the diagrams by full dressed propagators. The self-energy
thus becomes a functional of the full-dressed Green’s function,and the Dyson equation becomes a self-consistent equation
for the full Green’s function. Solving this self-consistent
Dyson equation allows us to capture a particular ensemble ofprocesses at all orders in g. Note that a related self-consistent
Keldysh approach was previously used to study a nonlinear
parametric amplifier near threshold [24–26].

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MARC-ANTOINE LEMONDE AND AASHISH A. CLERKPHYSICAL REVIEW A 91, 033836 (2015)

In practice, one solves such self-consistent equations
iteratively: in each step, one calculates the self-energy using
the current versions of the full Green’s functions, and then uses
these to update the Green’s functions which will be used forthe self-energy calculation in the next iteration. Applying thisiterative procedure until convergence solves the self-consistent
Dyson equation. To improve the accuracy of our approach, wehave applied this iterative strategy for the results shown in
Fig. 5 to Fig. 12. For each calculation, we have performed
20 iterations, which turn out to be more than enough to get
excellence convergence of the Green’s functions. Note that byusing this self-consistent approach, n̄int
σ , as defined in Eq. (43),becomes frequency dependent.
As expected, the self-consistent approach does not converge
if the nonlinear interactions become too strong. Convergence
is not solely controlled by the magnitude of g̃, but is primarily
determined by the effective cooperativities Cσeff defined in
Eqs. (41). These cooperativities involve both the magnitude ofg̃ and the polariton occupancies, reflecting the fact that large
temperature can also enhance the importance of nonlinearity.
As shown in the figures, we find that when the self-consistent
approach converges, it also is in excellent agreement withfull numerical simulations of the Linblad master equation
describing the system, Eq. (21).
V. INFLUENCE OF POLARITON INTERACTIONS ON
NONEQUILIBRIUM EFFECTS
A. Interaction-induced effective environment

From our previous discussion, we see that the nonlinear
interaction gives rise to self-energies which modify the singleparticle properties of polaritons. The imaginary part of theretarded self-energies describe interaction-induced polariton
damping rates σint [ω] [cf. Eq. (40)], whereas the Keldysh
self-energies describe interaction-induced heating effects, withassociated thermal occupancy factors n̄int
σ [cf. Eq. (44)]. Thissuggests that in terms of single-particle properties, the effects
of interactions are equivalent to having coupled the linearized
optomechanical system to new dissipative baths. In what
follows, we make this picture of an “interaction-induced
effective environment” explicit.
First, note that all single-particle polariton properties of oursystem are described by the effective quadratic action
Sσeff =

 ∞
−∞

∗
∗
dt dt (cσ,cl(t),cσ,q(t))Gσ−1 (t − t )cσ,cl (t ),cσ,q (t )(45)

where Gσ−1 (t − t ) is the Fourier transform of the inverse of the2 × 2 matrix of the full Green’s functions given in Eq. (38).
Further, as discussed in Sec. III, this quadratic action iscompletely equivalent to a set of linear Langevin equations.
Using the standard derivation [23], the effective action in
Eq. (45) is equivalent to the Langevin equations


 ∞
κσ
cσ (t) −dt σR (t − t )cσ (t )∂t cσ (t) = − iEσ +2−∞− ξσenv (t) − ξσint (t).

(46)

Here, σR (t − t ) is the Fourier transform of the retarded
self-energies given in Eqs. (39). The noise functions ξσenv (t)and ξσint (t) describe complex independent Gaussian noise
processes with zero mean. ξσenv (t) describes the intrinsic
polariton dissipation as discussed in Sec. III; its correlators
are given in Eq. (34). The only nonzero correlator of the newnoise ξσint (t) is int  int ∗ ξσ (t) ξσ (t ) = σint (t − t ) n̄int(47)
σ + 1/2 δσ,σ ,where σint (t − t ) is the Fourier transform of the frequencydependent interaction-induced polariton damping given in
Eq. (40).
These Langevin equations reproduce the intuitive picture
sketched above: each polariton species is now effectively
coupled to two independent dissipative environments, withcorresponding damping rates κσ and σint [ω], and corresponding thermal occupancies n̄0σ and n̄int
σ . The first bath corresponds
to intrinsic dissipation (i.e., the intrinsic mechanical and cavity
dissipation), whereas the second is due to polariton-polariton
interactions.
It is worth stressing that the induced damping rates σint [ω]are in general sharply peaked functions of frequency, due tothe resonant nature of the relevant scattering process. For someparameters, the width of σint [ω] can even be much smaller than
the width of the density of state ρσ [ω]; e.g., the + polaritonsfor  near −2ωM . In contrast, there are other cases where thedensity of state is much sharper than the interaction-induced
dissipation rate, as is the case for the + polariton for  near−ωM /2. As a result, the “interaction-induced” baths cannot
always be considered as Markovian.
The thermal occupancies n̄int
associated with theσinteraction-induced environments are plotted in Fig. 6 as afunction of the detuning , in the interesting case where all
intrinsic dissipation (i.e., mechanical bath, cavity bath) are atzero temperature.
B. Interaction-induced quantum heating

We now discuss in more detail the behavior of Eq. (44)which gives the thermal occupancies n̄int
σ of the effective
interaction-induced dissipative baths introduced in the previous subsection. For simplicity, we focus on the case of exact
resonance, where G = Gres and hence E+ = 2E− . Consider
first the case where the linear-theory polariton dissipation is inthermal equilibrium at temperature Teq , i.e., n̄0σ = nB [Eσ ,Teq ].
In this case, it is easy to confirm that for each polariton, n̄int
σ =n̄0σ , i.e., the interaction-induced dissipation also corresponds
to the same temperature Teq . Thus, if without interactions thepolaritons start in equilibrium at the same temperature, thenthe same is true with interactions.
The actual situation is, however, more complicated: dueto quantum heating effects, the effective temperatures of thetwo polariton species are different even without interactions.
n̄0− and n̄0+ are thus not related as they would be in thermal
equilibrium; this can be parametrized asn̄0+ ≡(n̄0− )2
+ δ n̄0+ .
2n̄0− + 1

(48)

Thermal equilibrium and the condition E+ = 2E− would imply δ n̄0+ = 0; δ n̄0+ = 0 means that even in the linearized theory,033836-10

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PHYSICAL REVIEW A 91, 033836 (2015)environments describe quantum heating physics. Together,
they will determine the total number of polaritons produced byquantum heating and, more specifically, the energy distributionfunction of the polaritons. We define this distribution functionin the standard manner, as an energy-dependent distributionfunction n̄eff
σ [ω]. This quantity is defined via the full Keldysh
and retarded polariton Green’s functions  R GσK [ω] ≡ −2i 2n̄eff(50)
σ [ω] + 1 Im Gσ [ω] .

FIG. 6. (Color online) Effective thermal occupancies for the various baths coupled to the two polariton species, as a function of with G = Gres []. Solid lines: occupancies associated with the“interaction-induced” baths to leading order in g [cf. Eqs. (44)].
Circles: same, but calculated using the all-orders self-consistent
approach of Sec. IV B. Dashed lines: occupancies of the intrinsic
polariton baths [cf. Eqs. (24)]. Note the leading-order occupancy
00
for the − polariton interaction bath (n̄int− ) diverges when n̄− = n̄+00(/ωM ≈ −0.8) and becomes negative when n̄+ > n̄− (/ωM −0.8), as shown in the inset. This divergence persists in the selfconsistent theory, but occurs at smaller-magnitude detunings. This
negative occupancy signals the possibility of a different kind of
instability, as discussed explicitly in Sec. V D. All curves are plotted−4
for n̄M
th = 0 [cf. Eq. (18)], γ /κ = 10 , ωM /κ = 50, and g = κ.

the two polaritons experience different effective temperatures(see Fig. 3).
Using this definition, the thermal occupancy of the −polariton interaction-induced bath becomes
0
0
n̄int
− = n̄− + δ n̄+(1 + 2n̄0− )2.
n̄0− (1 + n̄0− ) − δ n̄0+ (1 + 2n̄0− )If our polaritons were in thermal equilibrium at temperature
Teq , then the distribution function n̄eff
σ [ω] would simply be
the Bose-Einstein distribution nB [ω,Teq ]. In contrast, in oursystem this function will be determined by the thermal
occupancies of the two effective baths, and the strength of thecouplings (i.e., damping rates) to each. Using the expressionof the dressed Green’s functions G[ω] coming from theDyson equation [cf. Eq. (38)] and the relation between the
self-energies given in Eq. (43), one findsn̄eff
σ [ω] =int
int
0
σ [ω]n̄σ + κσ n̄σ.
int
σ [ω] + κσ(51)This is exactly the simple expression that would be expected
for a free bosonic mode coupled independently to two baths;
the same form holds for the linear theory [cf. Eqs. (24)].
In order to focus our attention on the contribution of thenonlinear interaction to n̄eff
σ [ω], we rewrite Eq. (51) using theexpressions for σint [ω] [cf. Eqs. (39) and (40)] and n̄int
σ [cf.
Eq. (44)]. Doing so, one gets0n̄eff
σ [ω] = n̄σ + Iσ

γσ2,(ω − ωσ )2 + γσ2

(52)

with
(49)

Thus, a deviation from true thermal equilibrium in the linear
theory (i.e., without polariton interactions) causes the occupancy of the interaction-induced bath n̄int
σ and the intrinsic bathn̄0σ (linear-theory dissipation) to deviate from one another. This
is not surprising: in this case, the nonlinear interaction between
the two polariton species tends to favor their thermalization,and hence transfers energy from the high-temperature species
to the low-temperature species.
Finally, we also stress that even in the case where theintrinsic mechanical and cavity dissipation is at zero temperature (i.e., the system only experiences vacuum noise),
the interaction-bath thermal occupancies n̄int
σ will be nonzero,and are in general different from n̄0σ . This is shown explicitly
in Fig. 6. We thus see that interactions change the effective
temperature associated with quantum heating effects.
C. Polariton energy distribution functions

The picture established so far is that our optomechanicalpolaritons are each effectively coupled to two independent
effective environments, one of which is self-generated anddue to the nonlinear optomechanical interaction. In the limit
where both the intrinsic cavity and mechanical dissipative
baths are at zero temperature (n̄Mth = 0), both these effective

 Cσeff0,(53a)
Iσ = n̄int
σ − n̄σ
1 + Cσeffκ− + κ+γ− =1 + C−eff , γ+ = κ− 1 + C+eff , (53b)2ω− = E+ − E− , ω+ = 2E− .
(53c)
The contribution from the interaction-induced environment
appears as a sharp Lorentzian in the polariton distributionfunctions. This is a direct consequence of the resonant natureof the relevant nonlinear scattering process.
For exact resonance (E+ = 2E− ), both n̄eff
σ [ω] and thesingle-particle DOS ρσ [ω] [cf. Eq. (42)] are peaked at Eσ ,so that the nonlinear interaction heating effects are maximal.
Even in this case, though, the frequency dependence of theinteraction contribution to n̄eff
σ [ω] can be very different thanthat of the polariton DOS. In this fully resonant case, thepolaritons distribution functions evaluated at ω = Eσ adopt
the following simple form:n̄eff
σ [Eσ ] =0
Cσeff n̄int
σ + n̄σ.
eff
Cσ + 1(54)

Equation (54) is plotted in Fig. 7 as a function of the laser
detuning  and is compared to n̄eff
σ [Eσ ] obtained using theself-consistent approach (cf. Sec. IV B).

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limit where γ
κ, one can simplify the instability condition,i.e., C−eff < −1, ton̄0+ > n̄0− +κ−2
κ− (κ− + κ+ )≈,16g̃ 216g̃ 2(56)

and the final mean number of − polaritons becomesn̄eff− [E− ] ≈FIG. 7. (Color online) (a) Solid lines: net polariton occupancies
in the presence of nonlinear interaction n̄eff
σ [Eσ ], calculated using
leading-order self-energies, as a function of  with G = Gres [].
Circles: same, but calculated using the self-consistent approach (cf.
Sec. IV B). Dashed curves: occupancies for g = 0, i.e., calculated
in the linearized theory. In each case, the mechanical temperature isMzero (n̄Mth = 0) and g = κ. (b) Same as (a), but now with n̄th = 100
and g = 0.1κ; main plot and inset show different ranges of . Near
−2ωM , one sees an important contribution to the mean number of+ polaritons due to nonlinear interaction; this contribution is greatly
enhanced by temperatures, as discussed in Sec. VI D. All the curves
are plotted for γ /κ = 10−4 and ωM /κ = 50.

VI. OBSERVABLE SIGNATURES OF QUANTUMHEATING EFFECTS

Among the more striking nonequilibrium behaviors possible in the linear theory is the possibility of having n̄0+ > n̄0− ,i.e., the thermal occupancy of the higher-energy + polariton
exceeds that of the − polariton. We discuss this regime inmore detail here, focusing on the exactly resonant case where
E+ = 2E− .
We start by recalling that the total damping of the −polariton isinteff− [E− ] = κ− (1 + C− ).

(57)

The surprising negative damping occurring here can beunderstood 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 findfor a degenerate parametric amplifier (DPA) pumped neardegeneracy [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 thesignal mode in a DPA, while the + polariton plays the roleof 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).
Note that nondegenerate parametric amplifier instability canbe realized in a linearized optomechanical system driven
with a blue-detuned laser (see, e.g., [53,54]). In contrast, theinstability described here occurs for a red-detuned drive.
Note that our discussion here is based solely on using
the leading-order results for the polariton self-energies. Including higher-order effects via our self-consistent approach
can dramatically change the onset and magnitude of theinteraction-induced negative damping. We discuss this morein Sec. VI C.

D. Nonlinear parametric heating

κ−tot [E− ] = κ− +16g̃ 2 n̄0+.
κ−2 − 16g̃ 2 n̄0+(55)

From Eqs. (41), we see that if we have the occupancy inversionn̄0+ > n̄0− , then C−eff < 0, and hence the contribution of thenonlinear interaction to the damping rate of the − polaritons
becomes negative. This is at first glance surprising: we haveopened a scattering process for the − polariton via thenonlinear interaction, and yet we get an increase in its lifetime.
We also have the possibility of an instability if C−eff < −1.
From Fig. 5, one sees that a negative C−eff occurs fordetunings near  = −0.5ωM . In that regime, the − polariton
is mostly photonic and the + polariton is mostly phononic.
Consequently, by having a high intrinsic mechanical bath
1, one naturally can achieve the inverted
temperature n̄Mthoccupancy regime where n̄0+n̄0− . In that case and in the

In the previous sections, we have demonstrated how
quantum heating effects can lead to a finite density of optomechanical polaritons at zero temperature; we also discussed
how these quantum heating effects can be modified by thenonlinear interaction. In this section, we discuss how theseeffects lead to observable signatures in the light leaving
the optomechanical cavity. We first relate the cavity output
spectrum to the polariton distribution functions, and thendiscuss specific parameter regimes where the heating effects
are most prevalent. In addition, we propose a way to effectively
control the strength of the nonlinear interaction in experiments
by tuning in and out the resonance condition (i.e., by varying
G at fixed detuning ). Doing so, one can explicitly isolate
and observe the nonlinear interaction signatures in the cavity
output spectrum.
A. Polariton energy distribution functions

To measure polariton occupancies, we consider a measurement of the flux of photons leaving our cavity (assuming asingle-sided cavity, and that the reflected classical drive tone isfiltered away). The spectrum of this flux is given in the standard
manner [49] by the normal-ordered cavity spectrum (alsoknown as the “lesser” Green’s function within the Keldysh

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technique),Sd [ω] ≡ ∞−∞dt eiωt d̂ † (0)d̂(t).

PHYSICAL REVIEW A 91, 033836 (2015)(58)

Rewriting Eq. (58) in terms of polariton Green’s functions
yields2
Sd [ω] = 2π
n̄eff
αd,σ
σ [ω]ρσ [ω]
σ =±

+ 2π



2
ᾱd,σ
(n̄effσ [−ω] + 1)ρσ [−ω].

(59)

σ =±

Here, the coefficients αd,σ and ᾱd,σ are the change-of-basis
coefficients introduced in Eq. (5a) and plotted in Fig. 2, andthe polariton DOS, ρσ [ω], is defined in Eq. (42). We are stillworking in a rotating frame with respect to the laser drive
frequency ωL , hence, ω = 0 implies output photons leaving
at the laser frequency. The negative frequency term meansthat removing a photon at frequency ω can involve creatinga polariton at frequency −ω, as expected from the presenceof “anomalous” terms in Eq. (5b). Finally, Eq. (59) reflects
the fact that the only nonzero polariton Green’s functions arethose that conserve the number of polaritons independently, as
discussed in Secs. III and IV.
Using the same assumptions, we also derive the cavity DOS
and its energy distribution function:


2
2
αd,σ
ρσ [ω] − ᾱd,σρσ [−ω] ,(60)
ρd [ω] =
σ =±

n̄eff
d [ω] =

1 Sd [ω].
2π ρd [ω](61)

From the cavity energy distribution, we can use the BoseEinstein distribution to define an effective cavity temperature(kB = 1)ω,
Tdeff [ω] ≡ (62)ln 1 + n̄eff1[ω]dwhich is always possible if we let the effective temperature tobe frequency dependent.
As discussed extensively in [17], the polariton DOSρσ [ω] can be directly measured in an OMIT-style experiment[19,29–31], where one measures the reflection of a weak
additional probe tone incident on the cavity. The cavity
spectrum in contrast also yields information on polariton
occupancies. As the polariton energies Eσ are well separated,the output spectrum will have a series of peaks corresponding
to the emission or absorption of a given polariton species.
The magnitude of these peaks is directly proportional to theoccupancy of the given polariton.
It is also useful to look at the total number of photons
due to a given polariton resonance, which we can obtainby integrating the output spectrum around the corresponding
resonance. We thus introduce ω0 +δωdω
tot
Sd [ω],n̄d [ω0 ,δω] ≡(63)
ω0 −δω 2π
where ω0 will be taken to be E± , and δω will be taken to belarger than the spectral width of the given polariton resonance.

In what follows, we consider signatures of quantum heatingin the spectrum for particularly interesting choices of the drive
laser detuning .
B. Limit of zero-temperature dissipation: Effects of nonlinear
interaction on quantum heating

The first studied limit is for a mechanical bath at zero temperature (n̄Mth = 0), where the finite number of polaritons inside
the optomechanical cavity exclusively comes from quantum
heating. We focus on two regimes: the red sideband drive,i.e.,  = −ωM , where both polaritons are equal mixture ofphotons and phonons, and the asymmetric polaritons regime,
where the polaritons are not equal combinations of photon
and phonon. For the latter regime, we chose  = −1.8ωM
as a representative laser detuning. The results predicted forthe cavity driven on the red sideband have the advantage
to be robust to temperatures since both polaritons have an
important photon part; this implies that even for finite n̄M
th ,quantum heating is still the prevalent source of polaritons.
In contrast, a laser detuned at  = −1.8ωM leads to morestriking modifications of the polaritons energy distribution
since Cσeff [ = −1.8ωM ] > Cσeff [ = −ωM ] (cf. Fig. 5). In
both cases, we show that the nonlinear interaction modifies theenergy distribution of the polaritons as it tends to thermalize
the two species.
1. Results for symmetric polaritons (red sideband drive)

In Fig. 8, we plot the cavity DOS [Eq. (60)], the cavity
spectrum [Eq. (59)], its energy distribution function [Eq. (61)],
and the corresponding effective temperature [Eq. (62)] for  =−ωM . The linearized theory (g = 0) is compared to the casewhere g = κ. For the latter interacting case, we present results
obtained from three different methods: the leading order in g̃
self-energies [Eqs. (42) and (52)], the self-consistent approachdescribed in Sec. IV B, and, finally, a numerical simulation of
the Lindblad master equation given in Eq. (21). By comparing
the three different approaches, one sees that for g = κ, higherorder corrections captured by the self-consistent approach play
an important role for the effective distribution functions andthe effective temperatures [Figs. 8(e)–8(h)]. In contrast, theDOS is already well described at the leading order in g̃, whichis in agreement with Ref. [17].
The splitting of the + polariton resonance in ρd [ω] andSd [ω] near ω = E+ arises from the hybridization between thestates |+ and |−,−; the resulting hybridized states become
spectrally resolved for g  κ (see [17] for more details). The
same hybridization phenomena give rise to a resonance in
eff
n̄eff
d [ω] and Td [ω] at the − polariton frequency ω = E− [seeFigs. 8(e) and 8(g)].
In Fig. 9, we plot the number of photons leaving the cavity
near each polariton resonance, as defined in Eq. (63), andshow that one can effectively isolate the effects of nonlinear
interaction. To do so, one varies G around Gres such that thenonlinear interaction gets amplified by a factor of ωM /κ when
the nonlinear process becomes resonant, i.e., for G = Gres ,compared to the off-resonant case, i.e., G − Gres  κ. Away
from resonance, the number of photons leaving the cavity isin good approximation given by the linearized theory (dashed1),
lines in Fig. 9). For the parameters here (n̄Mth = 0 and γ /κ033836-13

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FIG. 8. (Color online) (a), (b) Cavity DOS near the − polariton
resonance and the + polariton resonance, respectively, for a cavity
driven on the red sideband and when the nonlinear interaction isresonant (i.e., G = Gres ). We work in the frame rotating at the drive
frequency so that ω = 0 refers to the drive frequency while E+ =2E− = 63.24κ in this frame. The light dashed curves represent thelinearized theory, the green dotted ones represent the results to leading
orders in g̃ [Eq. (42) for the DOS], the full blue curves are the resultsof the self-consistent approach as described in Sec. IV B, and theblack curves are for the numerical simulation of the Lindblad masterequation shown in Eq. (21). (c), (d) Cavity spectrum in the same
conditions; the results to leading orders in g̃ are given in Eqs. (52)and (59). (e), (f) Cavity energy distribution function [Eq. (61)] and (g),(h) the corresponding (frequency-dependent) effective temperatures[Eq. (62)]. The parameters used for all the curves are γ /κ = 10−4 andωM /κ = 50, which leads to the leading-order effective cooperativities[i.e., Eq. (41)] C−eff = 0.18 and C+eff = 2.46.

the linearized theory leads to [see Eqs. (16) and (A4)]
2
0
n̄tot
d [E± ,5κ] ≈ αd,± n̄± ≈1 (G/ωM )2.
8 1 ± 2G/ωM(64)

Figure 9 clearly shows the thermalization between polaritons brought about the nonlinear interaction. Without
interactions, at  = −ωM the − polaritons have a lower
effective temperature than the + polaritons [cf. Fig. 3(c)].
When G is near Gres , the nonlinear interaction “turns on”
and allows the two polariton species to exchange energy andpartially thermalize (i.e., interactions heat up the − polaritons
while cooling down the + polaritons.)
2. Results for asymmetric polaritons

In Fig. 10, we plot the same functions as in Fig. 8, but inthe case where the laser detuning is  = −1.8ωM (and again,
G = Gres ). For this more negative detuning, the polaritons
are no longer an equal mixture of photons and phonons:

FIG. 9. (Color online) (a) Output photon flux in a bandwidth of5κ near the − polariton resonance E− [cf. Eq. (63)] as a function of the
many-photon coupling G, and for a control laser detuning  = −ωM .
(b) Same as (a) but near the + polariton resonance E+ . Note that G
can be varied by simply tuning the amplitude of the control laser.
On resonance, i.e., G = Gres , the nonlinear effects are enhanced bya factor of ωM /κ = 50 compared to the off-resonance case, i.e.,(G − Gres )  κ. The light dashed curves represent the linearizedtheory (g = 0), the solid blue curves are for the self-consistent
approach as described in Sec. IV B, and the black dashed curves are forthe numerical simulation of the Lindblad master equation shown (21),but this time, using the full nonlinear part of the Hamiltonian in
Eq. (6) that includes all the nonresonant nonlinear processes. For all
the curves, we used γ /κ = 10−4 and n̄M
th = 0.

the − polariton is more phononlike while the + is more
photonlike. This particular asymmetry leads to larger valuesof Cσeff than in the red sideband regime, mainly because of thelong lifetime of the phononlike polariton (cf. Fig. 5). Due tothese larger Cσeff , the results obtained to the leading order ing̃ are not sufficient to recover the numerical simulation of the
Lindblad master equation [cf. Eq. (21)] even for the DOS. It
is then crucial to use the self-consistent approach to properly
describe the effects of nonlinear interaction. Moreover, from
Figs. 10(g) and 10(h), one sees that unlike the case  = −ωM ,the nonlinear interaction cools down the − polaritons andheats up the + polaritons. This is also consistent with a partial
thermalization, as for  = −1.8ωM , without interactions theeffective temperature of the − polaritons is greater than thatof the + polaritons [cf. Fig. 3(c)].
C. Parametric amplification of the − polaritons

While our emphasis in this work has been on quantum
heating effects involving zero-temperature dissipation, ourapproach can also conveniently describe thermal nonlinear
phenomena. Perhaps the most striking example of this occurs
for detuning  ≈ −ωM /2, where leading-order perturbation
theory predicts the presence of a parametric instability at finite

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PHYSICAL REVIEW A 91, 033836 (2015)

FIG. 10. (Color online) Same as Fig. 8 but for laser detuning = −1.8ωM . In that case, E+ = 2E− = 92.08κ in the rotating
frame and the corresponding leading order [i.e., Eq. (41)] effective
cooperativities C−eff = 1.92 and C+eff = 5.40.

temperatures (see Sec. V D). In that regime, the nonlinear
interaction acts as an incoherently pumped degenerate parametric amplifier with the + polariton (mainly phonon) being
the (incoherent) pump and the − polariton (mainly photon)
being the signal mode.
In Fig. 11, we show the cavity DOS [cf. Eq. (60)], the cavity
spectrum Sd [ω] [cf. Eq. (59)], and the corresponding energy
distribution function n̄effd [ω] [cf. Eq. (61)] for a laser detuning = −0.65ωM and n̄M
th = 650. Combined with a damping
rate of the mechanical resonator γ = 10−3 κ, its resonant
frequency ωM = 50κ and g = κ, one obtains, to leading orderin g, an effective cooperativity C−eff = −0.97 [cf. Eq. (41)].
As discussed in Sec. V D, for C−eff = −1, the leading-order
perturbation theory predicts a parametric instability caused by
the nonlinear interaction. For C−eff  −1, one thus expects animportant narrowing of the cavity DOS as well as an importantheating of the cavity near the − polariton resonance. These
predictions from the leading-order self-energy are shown in
Fig. 11.
Not surprisingly, higher-order corrections (as captured by
the self-consistent self-energy) are especially important in thisregime and strongly contribute to prevent the system fromgoing unstable. More precisely, it is the hybridization between
the states |+ and |−− that competes with the parametricamplification of the − polaritons; the high number of −polaritons leads to an important modification of the energyof the hybridized states √12 (|+ ± |−−) so that the nonlinear
interaction ceases to be resonant. The result of this competitionis shown clearly in Fig. 11. While the leading-order theory

FIG. 11. (Color online) Same as Fig. 8 but for laser detuning  =−0.65ωM and finite temperature n̄M
th = 650. In these circumstances,leading-order perturbation theory predicts that the system is close
to a parametric instability (see Sec. V D). Note the mechanicaldamping rate here is larger than in previous plots (γ = 10−3 κ), as
this enhances the pumping effects of mechanical temperature. For
ωM = 50κ and g = κ, the leading-order effective cooperativities [cf.
Eq. (41)] are C−eff = −0.97 and C+eff = 0.75. In the frame that rotatesat the drive frequency, E+ = 2E− = 53.34κ. One sees from theseplots the crucial importance of higher-order-in-g terms.

predicts a photon occupancy at the − polariton resonanceof n̄effd [E− ] ≈ 50, in the self-consistent approach, one only
obtains a value 1.3. This is of course still much larger
than what would be obtained without interaction; in that case,n̄effd [E− ]  0.
D. Effective two-phonon absorption

Another striking example of temperature-enhanced nonlinear effects occurs for a laser detuning  ≈ −2ωM . In thisregime, the − polaritons are mostly phonons, which leads
to a thermal “stimulated emission” enhancement of −R [ω],leading to a strong modification of the cavity DOS. This
physics follows from Eqs. (39), and was discussed extensively
in Ref. [17]. However, as shown in Fig. 7, one also obtains
significant nonlinearity-induced heating of the cavity in thisregime, as we now describe.
Recall that if  = −2ωM , then Gres = 0 and the +(−) polaritons are exactly photons (phonons). A necessaryconsequence is that the amplitude for the resonant nonlinear
interaction vanishes, g̃ = 0. One thus ideally wants a detuning
close to, but not exactly equal to, −2ωM such that 0 < Gres
ωM . In the high-temperature limit (n̄M1) and for weakth
1 (i.e., gκ), the expression
nonlinear interaction C+eff
to lowest order in G/ωM for the + resonance in the cavity

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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; nooutput 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 thephenomenon tends to become purely classical. In order tosupport 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 singlephoton coupling constant g of the two-phonons absorption
signature in the cavity spectrum at the lowest order in g.
VII. CONCLUSIONS

FIG. 12. (Color online) (a) Signatures of the effective twophonon absorption in the cavity DOS near the + resonance; the inset
shows a larger range of frequency than the main plot. The light-bluedashed curve represents the linearized theory (g = 0), the green
dotted one represents the results to leading order in g [Eqs. (42), (52),and (59)], and the full dark-blue and black curves are the results ofthe self-consistent approach (S.C.) of Sec. IV B for n̄M
th = 100 and 0,
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
the + polariton population. The parameters used are γ /κ = 10−4 andωM /κ = 50.

spectrum simplifies to
 2
  

κ−2
κ G 2 g 2 16 n̄Mth
Sd [ω] ≈
.

3
2
γ ωM
κ
(ω − E+ )2 + κ−2
1 + 89 ωG2 γκ
M(65)
As discussed near Eq. (25), the competition between theparametric heating associated with the linearized optomechanical interaction and the standard optomechanical optical
damping leads to an optimal value of G/ωM where the9 γ
andheating is maximal. In the optimal case, (G/ωM )2 = 16κg 2 M 2
Sd [E+ ] ∼ ( κ ) (n̄th ) . Details of the calculations that lead toEq. (65) are presented in Appendix C.
In Fig. 12, the cavity DOS and the cavity spectrum near
the + polariton resonance is plotted for a laser detuning near
−2ωM and for finite mechanical bath temperatures n̄M
th . In
Fig. 12(a), we show the sharp dip in the density of state due
to nonlinear interaction, also described in Ref. [17]. Note,however, that effects of higher order in g, captured in the
self-consistent approach, considerably modify the predictions
made in Ref. [17], where only effects to leading order in g
were considered. This sharp feature is completely analogous
to the optomechanical induced transparency (OMIT) observedin the optomechanical cavity weekly driven on the red

In this work, we have described the effects of nonlinear
interaction on the nonequilibrium state of the optomechanicalcavity. We have shown the tendency of the nonlinear interaction to thermalize the polaritons, that it can lead to a newparametric instability for a red-detuned laser drive as well
as a temperature-enhanced effective two-phonon absorption.
In addition to these results, we have presented in details
many technical aspects with the aim to provide the proper
tools to investigate nonlinear effects in more complicatedoptomechanical systems. This work also opens the path toa more detailed characterization of this parametric instability
and further investigations of its consequences.
ACKNOWLEDGMENTS

We thank N. Didier and F. Beaudoin for helpful discussions.
This work was supported by NSERC and FQRNT.
APPENDIX A: CHANGE OF BASIS FOR A DETUNING
ON THE RED SIDEBAND ( = −ω M )In this appendix, we show particular examples of thetransformation used to go from the photon and phonon basis
to the polariton basis for  = −ωM .
We start with the change-of-basis coefficients αb/d,± andᾱb/d,± introduced in Eqs. (5):1 ωM + E±
,
√
E±
8ωM
1 ωM − E±,ᾱb,± = ± √√E±8ωM
1 ωM + E±,
αd,± = √√E±
8ωM
1 ωM − E±,ᾱd,± = √√E±8ωM
αb,± = ± √

(A1)
(A2)
(A3)
(A4)

with

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(A5)
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PHYSICAL REVIEW A 91, 033836 (2015)

From these coefficients, we can express the different effective
A/B
nonlinear coupling constants gσ σ σ and A± introduced in
Eq. (6). Here, we only show few examples:AA
A+ g+−++ g−++
g++−given by ∂− ĉ− = 0. In that case, we have
4i g̃ n̄0+ †2
ĉ− = −ĉ− − √ ξ̂− ,
κ−
κ−
2
†
⇒ ĉ− = √ (1 − |Q|2 )−1 (Qξ̂− − ξ̂− ),κ−= g[αd,+ ᾱd,+ (αb,− + ᾱb,− )+ (ᾱd,− αd,+ + ᾱd,+ αd,− )(αb,+ + ᾱb,+ )]
 


G
ωM 3/4 E− − E+ + ωM (E− + 2E+ )
g
,
= √
E+
4 2 E−
(A7)

(B4)
(B5)

with
Q≡


4i g̃ n̄0+
κ−

(B6)

.
†

B
g̃ ≡ g−−+

Imposing the right commutation relation [ĉ− ,ĉ− ] = 1, onefinds= g[αd,− ᾱd,− (αb,+ + ᾱb,+ )+ (ᾱd,− ᾱd,+ + αd,− αd,+ )(αb,− + ᾱb,− )]

 

 

G
G
ωM 3/4 1 + 2 ωM E− + 1 − ωM E+
g
,
=− √
E−
4 2 E+
(A8)


2
2
A− = g 2αd,− ᾱd,− + ᾱd,−(αb,− + ᾱb,− )+ ᾱd,++ (ᾱd,− αd,+ + ᾱd,+ αd,− )(αb,+ + ᾱb,+ )   E−G
ωM 3/4 2 ωM − 1 E+ + ωM (E+ − E− )
g
.
= √
E+
4 2 E−(A9)

APPENDIX B: PARAMETRIC AMPLIFIER DESCRIPTIONIn this appendix, we derive the condition to get an
instability from the parametric amplifier Hamiltonian and get
the corresponding mean number of signal excitations. These
calculations are based on the formalism introduced in [50].
Starting with the nonlinear interaction in the effective
Hamiltonian of Eq. (8) and considering the hypothetical casewhere the + mode is coherently pumped, such that ĉ+  = n̄0+ ,we get, in the mean-field approximation,Ĥeff ≈ g̃ n̄0+ (ĉ− ĉ− + H.c.).

(B1)

Using standard input-output theory, one derives the following
equation of motion:κ−√ĉ− − κ− ξ̂−∂t ĉ− = i[Ĥeff ,ĉ− ] −
2

κ−
√
†
ĉ− − κ− ξ̂− .
= −2g̃ n̄0+ ĉ− −2†[ξ̂− ,ξ̂− ] =Here, ξ̂− represents an incoming field, which includes noise
coming from the bath coupled to the − polaritons [as in
Eq. (33)]. Since we will focus only on quantities evaluated at
ω = E− and that Ĥeff of Eq. (B1) is written in the interaction
picture where E− = 0, we can seek for the particular solution

(B7)

†

which has the right units since ξ̂− ξ̂−  represents a rate at
which excitations are coming in. In the case of Gaussian noise†with zero mean (i.e., ξ̂− ξ̂−  = 0), as studied all along thiswork, we find that the mean number of − polaritons inside theoptomechanical cavity is†ĉ− ĉ−  =16g̃ 2 n̄0+
|Q|2
= 2
.
2
1 − |Q|κ− − 16g̃ 2 n̄0+

(B8)From this result, one immediately sees that for |Q|2 = 1, themean number of polaritons diverges. This condition sets thethreshold above which the parametric amplifier goes unstable.
More precisely, the system becomes unstable whenn̄0+ >

κ−2.
16g̃ 2(B9)These results have exactly the same form as Eqs. (56) and (57),but here, we have explicitly used the fact that the + polaritons
are coherently pumped such that ĉ+  = n̄0+ .
APPENDIX C: EXPANSION OF THE PHOTON SPECTRUM
FUNCTION FOR  ≈ −2ω M

In this appendix, we show details of the calculation that
leads to Eq. (65) starting from Eqs. (52) and (59). Here, thelimit of interest is for  ≈ −2ωM , G/ωM ∼ γ /κ1, n̄Mth
1, and C+eff
1. In these circumstances, photons are mainly +polaritons, such that Eq. (65) for frequencies near E+ reduces
to

(B2)
(B3)

κ− (1 − |Q|2 ),4

Sd [ω]|ω≈E+ ≈ 2π n̄eff+ [ω]ρ+ [ω].

(C1)

Also, for  ≈ −2ωM and G/ωM ∼ γ /κ1, the mean number of photons inside the cavity without nonlinear interactionn̄0+ is negligible compared to the nonlinear contribution. Thus,
from Eq. (52), we see that n̄eff+ [ω] reduces to a sharply
peaked Lorenztian with width γ+ ≈ κ− ≈ γ (1 + opt ) ∼ γ
2with the optical damping rate opt ≈ 89 ωG2 κ. Consequently,Mρ+ [ω] [Eq. (42)], which has a characteristic width of κ+ ≈ κ,

033836-17

MARC-ANTOINE LEMONDE AND AASHISH A. CLERKPHYSICAL REVIEW A 91, 033836 (2015)

can be evaluated at E+ . Doing so, one gets
Sd [ω]|ω≈E+ ≈ 2π n̄eff+ [ω]ρ+ [E+ ],

(C2)

1
2
2
.
≈
eff
π κ+ (1 + C+ )
πκ

(C3)

ρ+ [E+ ] =0Now, we can write n̄eff+ [ω] as (we have dropped n̄+ and
explicitly used E+ = 2E− )int effn̄eff+ [ω] ≈ n̄+ C+

κ−2,(ω − E+ )2 + κ−2

(C4)

with
eff
n̄int
+ C+ =

4g̃ (n̄0− )2a(t) = iain

2

κ− κ+

(C5)

.

Thus,
Sd [ω]|ω≈E+ ≈We solve these nonlinear coupled equations of motion via
a perturbation approach. Starting with a sinusoidal displacement x(t) = x0 sin(ωM t), we get the consequent evolutionof the optical mode and, then, the perturbed displacementof the mechanical resonator. This is a good approximation onlyin the weak-coupling regime (Gκ) where the eigenstates
have a well-defined number of photons and phonons. Thismethod is thus restricted to the extreme detuning case ( ≈−2ωM ) studied in Sec. VI D.
Following this method, one finds that the steady-state cavity
field is given byκ−2
16 g̃ 2 (n̄0− )2.
κ 2 κ− (ω − E+ )2 + κ−2

(C6)
γ n̄M

Again, for this particular limit, we can approximate n̄0− ≈ κ−thand g̃ 2 ≈ ( ωGM )2 g 2 , such that we get Eq. (65), i.e.,κSd [ω]|ω≈E+
 2

  
κ−2
κ G 2 g 2 16 n̄M
th
≈
.


2
γ ωM
κ
1 + opt (ω − E+ )2 + κ−

(C7)

γ

i (n−m) Jn ()Jm () i[(n+m)ωM −ωL ]t. (D3)eκ+ i(nωM − )n,m=−∞ 2

Here, Jn (x) are the Bessel functions of the first kind and where,
in the small displacement limit,  ≡ Lx00 ωωMC plays the role of theperturbation parameter.
Note that by computing the radiation pressure force acting
on the mechanical resonator in Eq. (D2) and by expanding
to the lowest order in , one recovers the standard optical
damping and optical spring constant coming from the linear
interaction [cf. Eq. (2b)] in the weak-coupling regime [40].
We focus here on the peak in the cavity spectrum thatcomes from the nonlinear interaction. According to Eq. (58),the proper definition of the classical counterpart of the photon
spectrum to use is

APPENDIX D: CLASSICAL TREATMENT OF THE
EFFECTIVE TWO-PHONON ABSORPTION

Scl [ω] ≡As shown in the Sec. VI D, the nonlinear process corresponding to the absorption of two phonons by a photon
from the classical drive is greatly enhanced by temperature.
In particular, even if the nonlinear interaction directly comesfrom single-photon dynamics, one can show that in the hightemperature limit, this phenomenon becomes purely classical.
In what follows, we present an accurate classical descriptionof this limit.
The classical Langevin equations for the cavity field
coupled to a mechanical resonator via radiation pressure force
in the limit of weak displacements are



κ
x(t)
ȧ(t) = − − iωC 1 −a(t) + i āin e−iωL t , (D1)
2
L0

+∞


1
ωC

 ∞
−∞

dt eiωt δa ∗ (t0 )δa(t0 + t)t0 .

(D4)Here, the fluctuations of the cavity field δa(t) are given bykeeping only the n = −m terms in Eq. (D3). The meanvalue . . .t0 means taking the average over all initial time
t0 and the coefficient 1/ωC is there to ensure that units matchthe definition of the photon spectrum used in the quantum
treatment [cf. Eq. (58)].
Using Eq. (D3) and focusing only on the frequency range
near 2ωM with  ≈ −2ωM , one finds that in the good cavity
limit (κωM ), the classical cavity spectrum reads as  G 2 g 2  M 2
Scl [ω] ≈ 16n̄th π δ(ω − 2ωM ),
ωM
κ

(D5)

Here, m is the mass of the mechanical oscillator, L0 is thelength of the cavity when x(t) = 0, and the damping rateκ has been introduced following the standard input-output
formalism [50]. The cavity field is normalized such that thesteady-state mean energy inside the cavity is Ūcav = |ā|2 =κ|āin |2 /[( κ2 )2 + 2 ], where ā is defined as the steady-statemean value of the cavity field a(t) = [ā + δa(t)]e−iωL t .

where we have used the definition of the single-photon
ωCcoupling constant g ≡ xZPF
, with xZPF being the zero-point
L0
motion of the mechanical resonator and the semiclassical2 2M
relation 12 mωM
x0 = ωM n̄M1.
th valid for n̄th
If one takes opt = 0 and γ → 0 in Eq. (65), Sd [ω ≈ E+ ]and Scl [ω ≈ 2ωM ] agree. The classical approach fails in getting
the width of the peak since the mechanical damping rate is
not taken into account while calculating the effects of themechanical motion on the cavity field. Moreover, the optical
damping does not come out in this result in the level ofapproximation we have used.

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

[2] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity
Optomechanics (Springer, Berlin, 2014).

ẍ(t) +

γ|a(t)|
2ẋ + ωM
x=
.
2
mL0
2

(D2)

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,
A. Sirois, J. Whittaker, K. Lehnert, and R. Simmonds, Nature(London) 475, 359 (2011).
[4] J. Chan, T. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause,S. Gröblacher, M. Aspelmeyer, and O. Painter, Nature (London)
478, 89 (2011).
[5] D. W. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms,
and D. M. Stamper-Kurn, Nature (London) 488, 476 (2012).
[6] A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M.
Aspelmeyer, and O. Painter, Nature (London) 500, 185 (2013).
[7] T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A.
Regal, Phys. Rev. X 3, 031012 (2013).
[8] A. Nunnenkamp, K. Børkje, and S. M. Girvin, Phys. Rev. Lett.
107, 063602 (2011).
[9] P. Rabl, Phys. Rev. Lett. 107, 063601 (2011).
[10] A. Kronwald, M. Ludwig, and F. Marquardt, Phys. Rev. A 87,
013847 (2013).
[11] A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, 133601(2013).
[12] K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn,
Nature (London) 4, 561 (2008).
[13] F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science
322, 235 (2008).
[14] M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt,
Phys. Rev. Lett. 109, 063601 (2012).
[15] P. Komar, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P.
Rabl, P. Zoller, and M. D. Lukin, Phys. Rev. A 87, 013839(2013).
[16] K. Borkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin,
Phys. Rev. Lett. 111, 053603 (2013).
[17] M.-A. Lemonde, N. Didier, and A. A. Clerk, Phys. Rev. Lett.
111, 053602 (2013).
[18] X. Xu, M. Gullans, and J. M. Taylor, Phys. Rev. A 91, 013818(2015).
[19] J. Teufel, D. Li, M. Allman, K. Cicak, A. Sirois, J. Whittaker,
and R. Simmonds, Nature (London) 471, 204 (2011).
[20] S. Groblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer,
Nature (London) 460, 724 (2009).
[21] E. Verhagen, S. Deleglise, S. Weis, A. Schliesser, and T.
Kippenderg, Nature (London) 482, 63 (2010).
[22] A. Kamenev and A. Levchenko, Adv. Phys. 58, 197 (2009).
[23] A. Kamenev, Field Theory of Non-Equilibrium Systems(Cambridge University Press, Cambridge, UK, 2011).
[24] C. J. Mertens, T. A. B. Kennedy, and S. Swain, Phys. Rev. A 48,
2374 (1993).
[25] C. J. Mertens, J. M. Hasty, H. H. Roark, D. Nowakowski, andT. A. B. Kennedy, Phys. Rev. A 52, 742 (1995).
[26] O. Veits and M. Fleischhauer, Phys. Rev. A 55, 3059 (1997).
[27] E. G. Dalla Torre, S. Diehl, M. D. Lukin, S. Sachdev, and P.
Strack, Phys. Rev. A 87, 023831 (2013).
[28] L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Phys. Rev.
B 89, 134310 (2014).

[29] G. S. Agarwal and S. Huang, Phys. Rev. A 81, 041803(2010).
[30] S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A.
Schliesser, and T. J. Kippenberg, Science 330, 1520 (2010).
[31] A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M.
Winger, Q. Lin, J. T. Hill, D. Chang, and O. Painter, Nature(London) 472, 69 (2011).
[32] M. Marthaler and M. I. Dykman, Phys. Rev. A 73, 042108(2006).
[33] M. I. Dykman, Phys. Rev. E 75, 011101 (2007).
[34] I. Serban and F. K. Wilhelm, Phys. Rev. Lett. 99, 137001 (2007).
[35] M. I. Dykman, M. Marthaler, and V. Peano, Phys. Rev. A 83,
052115 (2011).
[36] F. R. Ong, M. Boissonneault, F. Mallet, A. C. Doherty, A. Blais,
D. Vion, D. Esteve, and P. Bertet, Phys. Rev. Lett. 110, 047001(2013).
[37] J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, Phys.
Rev. Lett. 103, 147003 (2009).
[38] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R.
Johansson, T. Duty, F. Nori, and P. Delsing, Nature (London)
479, 376 (2011).
[39] V. Peano and M. Thorwart, Phys. Rev. B 82, 155129 (2010).
[40] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys.
Rev. Lett. 99, 093902 (2007).
[41] F. Lecocq, J. D. Teufel, J. Aumentado, and R. W. Simmonds,
arXiv:1409.0872.
[42] D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter,
New J. Phys. 13, 023003 (2011).
[43] M. Schmidt, V. Peano, and F. Marquardt, arXiv:1311.7095.
[44] W. Chen and A. A. Clerk, Phys. Rev. A. 89, 033854 (2014).
[45] A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S.
Gröblacher, and O. Painter, Phys. Rev. Lett. 112, 153603 (2014).
[46] V. Peano, C. Brendel, M. Schmidt, and F. Marquardt,
arXiv:1409.5375.
[47] A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H.
Walther, Phys. Rev. Lett. 51, 1550 (1983).
[48] Y.-C. Liu, Y.-F. Xiao, Y.-L. Chen, X.-C. Yu, and Q. Gong, Phys.
Rev. Lett. 111, 083601 (2013).
[49] C. Gardiner and P. Zoller, Quantum Noise: A Handbook ofMarkovian and Non-Markovian Quantum Stochastic Methods
with Applications to Quantum Optics, Springer Series in
Synergetics (Springer, Berlin, 2004).
[50] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and
R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[51] F. Beaudoin, J. M. Gambetta, and A. Blais, Phys. Rev. A 84,
043832 (2011).
[52] M. I. Dykman and M. A. Krivoglaz, Phys. Status Solidi 68, 111(1975).
[53] F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett.
96, 103901 (2006).
[54] M. Ludwig, B. Kubala, and F. Marquardt, New J. Phys. 10,095013 (2008).

033836-19