Basic Theory of Cavity Optomechanics Theory of Cavity Optomechanics Aashish A. Clerk and Florian Marquardt Abstract This chapter provides a brief basic introduction to the theory used

Basic Theory of Cavity Optomechanics

Aashish A. Clerk and Florian Marquardt

Abstract This chapter provides a brief basic introduction to the theory used to de-scribe cavity-optomechanical systems. This can serve as background information tounderstand the other chapters of the book.We first review the Hamiltonian and showhow it can be approximately brought into quadratic form. Then we discuss the clas-sical dynamics both in the linear regime (featuring optomechanical damping, opti-cal spring, strong coupling, and optomechanically induced transparency) and in thenonlinear regime (optomechanical self-oscillations and attractor diagram). Finally,we discuss the quantum theory of optomechanical cooling, using the powerful andversatile quantum noise approach.

1 The optomechanical Hamiltonian

Cavity optomechanical systems display a parametric coupling between the mechan-ical displacement x of a mechanical vibration mode and the energy stored inside aradiation mode. That is, the frequency of the radiation mode depends on x and can bewritten in the form ωopt(x). When this dependence is Taylor-expanded, it is usuallysufficient to keep the linear term, and we obtain the basic cavity-optomechanicalHamiltonian

H0 = h(ωopt(0)−Gx)a†a+ hΩMb†b+ . . . (1)

We have used ΩM to denote the mechanical frequency, a†a is the number of pho-tons circulating inside the optical cavity mode, and b†b is the number of phonons

Aashish ClerkDepartment of Physics, McGill University, Canada, e-mail: [email protected]

Florian MarquardtInstitute for Theoretical Physics, Universitat Erlangen-Nurnberg, Germany,e-mail: [email protected]

5

6 Aashish A. Clerk and Florian Marquardt

inside the mechanical mode of interest. Here G is the optomechanical frequencyshift per displacement, sometimes also called the “frequency pull parameter”, thatcharacterizes the particular system. For a simple Fabry-Perot cavity with an oscil-lating end-mirror (illustrated in Fig. 1), one easily finds G= ωopt/L, where L is thelength of the cavity. This already indicates that smaller cavities yield larger couplingstrengths. A detailed derivation of this Hamiltonian for a model of a wave field in-side a cavity with a moving mirror can be found in [1]. However, the Hamiltonian isfar more general than that derivation in a particular case might suggest: Whenevermechanical vibrations alter an optical cavity by leading to distortions of the bound-ary conditions or changes of the refractive index, we expect a coupling of the typeshown here. The only important generalization involves the treatment of more thanjust a single mechanical and optical mode (see the remarks below).

Fig. 1 A typical system incavity optomechanics con-sists of a laser-driven opticalcavity whose light field ex-erts a radiation pressure forceon a vibrating mechanicalresonator.

opticalcavity mechanical

modelaser

A coupling of the type shown here is called ’dispersive’ (in contrast to a ’dissi-pative’ coupling, which would make κ depend on the displacement). Note that wehave left out the terms responsible for the laser driving and the decay (of photonsand of phonons), which will be dealt with separately in the following.From this Hamiltonian, it follows that the radiation pressure force is

Frad = hGa†a . (2)

After switching to a frame rotating at the incoming laser frequencyωL, we intro-duce the detuning ∆ = ωL−ωopt(0), such that we get

H = −h∆ a†a− hGxa†a+ hΩMb†b+ . . . (3)

It is now possible to write the displacement x= xZPF(b+ b†) in terms of the phononcreation and annihilation operators, where xZPF = (h/2meffΩM)1/2 is the size of themechanical ground state wave function (“mechanical zero-point fluctuations”). Thisleads to

Basic Theory of Cavity Optomechanics 7

H = −h∆ a†a− hg0(b+ b†)a†a+ hΩMb†b+ . . . . (4)

Here g0 = GxZPF represents the coupling between a single photon and a singlephonon. Usually g0 is a rather small frequency, much smaller than the cavity de-cay rate κ or the mechanical frequencyΩM. However, the effective photon-phononcoupling can be boosted by increasing the laser drive, at the expense of introducinga coupling that is only quadratic (instead of cubic as in the original Hamiltonian).To see this, we set a = α + δ a, where α is the average light field amplitude pro-duced by the laser drive (i.e. α = ⟨a⟩ in the absence of optomechanical coupling),and δ a represents the small quantum fluctuations around that constant amplitude. Ifwe insert this into the Hamiltonian and only keep the terms that are linear in α , weobtain

H(lin) = −h∆δ a†δ a− hg(b+ b†)(δ a+ δ a†)+ hΩMb†b+ . . . (5)

This is the so-called “linearized” optomechanical Hamiltonian (where the equationsof motion for δ a and b are in fact linear). Here g = g0α is the enhanced, laser-tunable optomechanical coupling strength, and for simplicity we have assumed α tobe real-valued (otherwise a simple unitary transformation acting on δ a can bring theHamiltonian to the present form, which is always possible unless two laser-drivesare involved).We have thus arrived at a rather simple system: two coupled harmonicoscillators (Fig. 2).

Fig. 2 After linearization,the standard system in cavityoptomechanics represents twocoupled harmonic oscillators,one of them mechanical (at afrequency ΩM), the other op-tical (at an effective frequencygiven by the negative detuning−∆ = ωopt(0)−ωL).

(decay rate )

mechanical oscillator driven optical cavity

(decay rate )

Note that we have omitted the term −hg0 |α|2 (b+ b†), which represents a con-stant radiation pressure force acting on the mechanical resonator and would lead toa shift of the resonator’s equilibrium position. We can imagine (as is usually done inthese cases), that this shift has already been taken care of and x is measured from thenew equilibrium position, or that this leads to a slightly changed ”effective detun-ing” ∆ (which will be the notation we use further below when solving the classicalequations of motion). In addition, we have neglected the term −hg0δ a†δ a(b+ b†),under the assumption that this term is “small”. The question when exactly this termmay start to matter and lead to observable consequences is a subject of ongoingresearch (it seems that generally speaking g0/κ > 1 is required).As will be explained below, almost all of the elementary properties of cavity-

optomechanical systems can be explained in terms of the linearized Hamiltonian.


Of course, the Hamiltonian in Eq. (1) represents an approximation (usually, anextremely good one). In particular, we have omitted all the other mechanical normalmodes and all the other radiation modes. The justification for omitting the otheroptical modes would be that only one mode is driven (nearly) resonantly by the laser.With regard to the mechanical mode, optomechanical cooling or amplification in theresolved-sideband regime (κ < ΩM) usually affects only one mode, again selectedby the laser frequency. Nevertheless, these simplistic arguments can fail, e.g. whenκ is larger than the spacing between mechanical modes, when the distance betweentwo optical modes matches a mechanical frequency, or when the dynamics becomesnonlinear, with large amplitudes of mechanical oscillations.Cases where the other modes become important display an even richer dynamics

than the one we are going to investigate below for the standard system (one mechan-ical mode, one radiation mode). Interesting experimental examples for the case oftwo optical modes and one mechanical mode can be found in the chapter by Bahland Carmon (on Brillouin optomechanics), and in the contribution by Jack Sankey(on the membrane-in-the-middle setup).In the following sections, we give a brief, self-contained overview of the most im-

portant basic features of this system, both in the classical regime and in the quantumregime. A more detailed introduction to the basics of the theory of cavity optome-chanics can also be found in the recent review [2].

2 Classical dynamics

The most important properties of optomechanical systems can be understood al-ready in the classical regime. As far as current experiments are concerned, the onlysignificant exception would be the quantum limit to cooling, which will be treatedfurther below in the sections on the basics of quantum optomechanics.

2.1 Equations of motion

In the classical regime, we assume both the mechanical oscillation amplitudesand the optical amplitudes to be large, i.e. the system contains many photons andphonons. As a matter of fact, much of what we will say is also valid in the regime ofsmall amplitudes, when only a few photons and phonons are present. This is becausein that regime the equations of motion can be linearized, and the expectation valuesof a quantum system evolving according to linear Heisenberg equations of motionin fact follow precisely the classical dynamics. The only aspect missing from theclassical description in the linearized regime is the proper treatment of the quan-tum Langevin noise force, which is responsible for the quantum limit to coolingmentioned above.


We write down the classical equations for the position x(t) and for the complexlight field amplitude α(t) (normalized such that |α|2 would be the photon numberin the semiclassical regime):

x = −Ω 2M(x− x0)−ΓMx+(Frad+Fext(t))/meff (6)

α = [i(∆ +Gx)−κ/2]α+κ2αmax (7)

Here Frad = hG |α|2 is the radiation pressure force. The laser amplitude enters theterm αmax in the second equation, where we have chosen a notation such that α =αmax on resonance (∆ = 0) in the absence of the optomechanical interaction (G= 0).Note that the dependence on h in this equations vanishes once we express the photonnumber in terms of the total light energy E stored inside the cavity: |α|2 = E /hωL.This confirms that we are dealing with a completely classical problem, in which hwill not enter any end-results if they are expressed in terms of classical quantitieslike cavity and laser frequency, cavity length, stored light energy (or laser inputpower), cavity decay rate, mechanical decay rate, and mechanical frequency. Still,we keep the present notation in order to facilitate later comparison with the quantumexpressions.

2.2 Linear response of an optomechanical system

We have also added an external driving force Fext(t) to the equation of motion forx(t). This is because our goal now will be to evaluate the linear response of themechanical system to this force. The idea is that the linear response will displaya mechanical resonance that turns out to be modified due to the interaction withthe light field. It will be shifted in frequency (“optical spring effect”) and its widthwill be changed (“optomechanical damping or amplification”). These are the twomost important elementary effects of the optomechanical interaction. Optomechan-ical effects on the damping rate and on the effective spring constant have been firstanalyzed and observed (in a macroscopic microwave setup) by Braginsky and co-workers already at the end of the 1960s [3].First one has to find the static equilibrium position, by setting x = 0 and α = 0

and solving the resulting set of coupled nonlinear algebraic equations. If the lightintensity is large, there can be more than one stable solution. This ’static bistability’was already observed in the pioneering experiment on optomechanics with opticalforces by the Walther group in the 1980s [4]. We now assume that such a solutionhas been found, and we linearize around it: x(t) = x+δx(t) and α(t) = α+δα(t).Then the equations for δx and δα read:


δ x(t) = −Ω 2Mδx−ΓMδ x+

hGmeff

[α∗δα+ αδα∗]+Fext(t)meff

(8)

δα(t) = [i∆ −κ/2]δα+ iGαδx (9)

Note that we have introduced the effective detuning ∆ = ∆ +Gx, shifted due to thestatic mechanical displacement (this is often not made explicit in discussions of op-tomechanical systems, although it can become important for larger displacements).We are facing a linear set of equations, which in principle can be solved straight-forwardly by going to Fourier space and inverting a matrix. There is only one slightdifficulty involved here, which is that the equations also contain the complex con-jugate δα∗(t). If we were to enter with an ansatz δα(t) ∝ e−iωt , this automaticallygenerates terms∝ e+iωt at the negative frequency as well. In some cases, this may beneglected (i.e. dropping the term δα∗ from the equations), because the term δα∗(t)is not resonant (this is completely equivalent to the “rotating wave approximation”in the quantum treatment). However, here we want to display the full solution.We now introduce the Fourier transform of any quantity A(t) in the form A[ω ]≡!dt A(t)eiωt . Then, in calculating the response to a force given by Fext[ω ], we haveto consider the fact that (δα∗) [ω ] = (δα[−ω ])∗. The equation for δα[ω ] is easilysolved, yielding δα[ω ] = χc(ω)iGαδx[ω ], with χc(ω) = [−iω− i∆ +κ/2]−1 theresponse function of the cavity. When we insert this into the equation for δx[ω ],we exploit (δα∗) [ω ] = (δα[−ω ])∗ as well as (δx[−ω ])∗ = δx[ω ], since δx(t) isreal-valued. The result for the mechanical response is of the form

δx[ω ] =Fext[ω ]

meff(Ω 2M−ω2− iωΓM)+Σ(ω)

≡ χxx(ω)Fext[ω ] . (10)

Here we have combined all the terms that depend on the optomechanical interactioninto the quantity Σ(ω) in the denominator. It is equal to

Σ(ω) = −ihG2 |α|2 [χc(ω)− χ∗c (−ω)] . (11)

Note that the prefactor can also be rewritten as hG2 |α|2 = 2meffΩMg2, by insertingthe expression for xZPF = (h/2meffΩM)−1/2 and using (GxZPF |α |)2 = g2.One may call Σ the “optomechanical self-energy”[5]. This is in analogy to the

self-energy of an electron appearing in the expression for its Green’s function, whichsummarizes the effects of the interaction with the electron’s environment (photons,phonons, other electrons, ...).If the coupling is weak, the mechanical linear response will still have a single

resonance, whose properties are just modified by the presence of the optomechanicalinteraction. In that case, close inspection of the denominator in Eq. (10) reveals themeaning of both the imaginary and the real part of Σ , which we evaluate at theoriginal resonance frequencyω =Ω . The imaginary part describes some additionaloptomechanical damping, induced by the light field:


Γopt = −1

meffΩMImΣ(Ω) = g2κ

!1

(ΩM+ ∆)2+"κ2#2 −

1(ΩM− ∆)2+

"κ2#2

$

(12)The real part describes a shift of the mechanical frequency (“optical spring”):

δ (Ω 2) =1meff

ReΣ(Ω) = 2ΩMg2!

ΩM+ ∆

(ΩM+ ∆)2+"κ2#2 −

ΩM− ∆

(ΩM− ∆)2+"κ2#2

$

.

(13)Both of these are displayed in figure 3. They are the results of “dynamical backac-

tion”, where the (possibly retarded) response of the cavity to the mechanical motionacts back on this motion.

!2 0 2!4

!2

0

2

4

!2 0 2

!5

0

5

Detuning Detuning

Dam

ping

rat

e

Freq

uenc

y Sh

ift

-1 1 -1 1

Fig. 3 Optomechanical damping rate (left) and frequency shift (right), as a function of the effec-tive detuning ∆ . The different curves depict the results for varying cavity decay rate, running inthe interval κ/ΩM = 0.2, 0.4, ..., 5 (the largest values are shown as black lines). We keep the in-tracavity energy fixed (i.e. g is fixed). Note that the damping rate Γopt has been rescaled by g2/κ ,which represents the parametric dependence of Γopt in the resolved-sideband regime κ < ΩM. Inaddition, note that we chose to plot the frequency shift in terms of δ (Ω2) ≈ 2ΩMδΩ (for smallδΩ ≪ΩM).

2.3 Strong coupling regime

When the optomechanical coupling rate g becomes comparable to the cavity damp-ing rate κ , the system enters the strong coupling regime. The hallmark of this regimeis the appearance (for red detuning) of a clearly resolved double-peak structure inthe mechanical (or optical) susceptibility. This peak splitting in the strong couplingregimewas first predicted in [5], then analyzed further in [6] and finally observed ex-


perimentally for the first time in [7]. This comes about because the mechanical res-onance and the (driven) cavity resonance hybridize, like any two coupled harmonicoscillators, with a splitting 2g set by the coupling. In order to describe this correctly,we have to retain the full structure of the mechanical susceptibility, Eq. (10) at allfrequencies, without applying the previous approximation of evaluating Σ(ω) in thevicinity of the resonance.

1 1.20.8 1 1.20.8Frequency Frequency

Detuning

-1

-0.9

-1.1

Fig. 4 Optomechanical strong coupling regime, illustrated in terms of the mechanical suscepti-bility. The figures show the imaginary part of χxx(ω) = 1/(m(Ω2 −ω2 − iωΓ ) + Σ(ω)). Left:Imχxx(ω) as a function of varying coupling strength g, set by the laser drive, for red detuning onresonance, ∆ = −Ω . A clear splitting develops around g/κ = 0.5. Right: Imχxx(ω) as a functionof varying detuning ∆ between the laser drive and the cavity resonance, for fixed g/κ = 0.5.

2.4 Optomechanically induced transparency

We now turn to the cavity response to a weak additional probe beam, which can betreated in analogy to the mechanical response discussed above. However, an inter-esting new feature develops, due to the fact that usually Γ ≪ κ . Even for g≪ κ , thecavity response shows a spectrally sharp feature due to the optomechanical interac-tion, and its width is given by Γ = ΓM+Γopt. This phenomenon is called “optome-chanically induced transparency” [8, 9].We can obtain the modified cavity response by imagining that there is no me-

chanical force (Fext = 0), but instead there is an additional weak laser drive, whichenters as . . . + δαLe−iωt on the right-hand-side of Eq. (9). By solving the coupledset of equations, we arrive at a modified cavity response


δα(t) = χeffc (ω)δαLe−iωt , (14)

where we find

χeffc (ω) = χc(ω) ·!1+2imeffΩMg2χc(ω)χxx(ω)

". (15)

Note that in the present section ω has the physical meaning of the detuning betweenthe weak additional probe laser and the original (possibly strong) control beam atωL. That is: ω = ωprobe−ωL. The result is shown in figure 5. The sharp dip goesdown to zero when g2/(κΓM) ≫ 1. Ultimately, this result is an example of a verygeneric phenomenon: If two oscillators are coupled and they have very differentdamping rates, then driving the strongly damped oscillator (here: the cavity) can in-directly drive the weakly damped oscillator (here: the mechanics), leading to a sharpspectral feature on top of a broad resonance. In the context of atomic physics withthree-level atoms, this has been observed as “electromagnetically induced trans-parency”, and thus the feature discussed here came to be called “optomechanicallyinduced transparency”.We note that for a blue-detuned control beam, the dip turns into a peak, signalling

optomechanical amplification of incoming weak radiation.

1 1.50.5Probe detuning

1 1.50.5Probe detuning

Fig. 5 Optomechanically induced transparency: Modification of the cavity response due to the in-teraction with the mechanical degree of freedom. We show Reχeffc (ω) as a function of the detuningω between the weak probe beam and the strong(er) control beam, for variable coupling g of thecontrol beam (left) and for variable detuning ∆ of the control beam vs. the cavity resonance (right,at g/κ = 0.1). We have chosen κ/ΩM = 0.2. Note that in the left plot, for further increases in g,the curves shown here would smoothly evolve into the double-peak structure characteristic of thestrong-coupling regime.


2.5 Nonlinear dynamics

On the blue-detuned side (∆ > 0), where Γopt is negative, the overall damping rateΓM +Γopt diminishes upon increasing the laser intensity, until it finally becomesnegative. Then the system becomes unstable and any small initial perturbation (e.g.thermal fluctuations) will increase exponentially at first, until the system settles intoself-induced mechanical oscillations of a fixed amplitude: x(t) = x+Acos(ΩMt).This is the optomechanical dynamical instability (parametric instability), which hasbeen explored both theoretically [10, 11] and observed experimentally in varioussettings (e.g. [12, 13, 14] for radiation-pressure driven setups and [15, 16] for pho-tothermal light forces).

Fig. 6 When increasing acontrol parameter, such asthe laser power, an optome-chanical system can becomeunstable and settle into peri-odic mechanical oscillations.These correspond to a limitcycle in phase space of someamplitude A, as depicted here.The transition is called a Hopfbifurcation.

bifurcation

amplitude

phase

laserpower

In order to understand the saturation of the amplitude A at a fixed finite value, wehave to take into account that the mechanical ocillation changes the pattern of thelight amplitude’s evolution. In turn, the overall effective damping rate, as averagedover an oscillation period, changes as well. To capture this, we now introduce anamplitude-dependendoptomechanical damping rate. This can be done by noting thata fixed damping rate Γ would give rise to a power loss ⟨(meffΓ x) x⟩ = ΓmeffA2/2.Thus, we define

Γopt(A) ≡−2

meffA2⟨Frad(t)x(t)⟩ . (16)

This definition reproduces the damping rate Γopt calculated above in the limit A→ 0.The condition for the value of the amplitude on the limit cycle is then simply givenby

Γopt(A)+ΓM = 0 . (17)

The result for Eq. (16) can be expressed in terms of the exact analytical solutionfor the light field amplitude α(t) given the mechanical oscillations at amplitudeA. This solution is a Fourier series, |α(t)| =

!!! 2αLΩM ∑n αneinΩMt

!!!, involving Besselfunction coefficients. In the end, we find


Γopt(A) = 4!

κΩM

"2 g2

ΩMf (GAΩM

) , (18)

where

f (a) = −1a∑n

Imα∗n+1αn (19)

with

αn =12

Jn(−a)in+κ/(2ΩM)− i∆/ΩM

. (20)

Note that g denotes the enhanced optomechanical coupling at resonance (i.e. for∆ = 0), i.e. it characterizes the laser amplitude. Also note that ∆ includes anamplitude-dependent shift due to a displacement of the mean oscillator position xby the radiation pressure force ⟨Frad⟩. This has to be found self-consistently.

!4 !2 0 2 4

2

4

6

8

10

!4 !2 0 2 4

2

4

6

8

10

!4 !2 0 2 4

2

4

6

8

10

1-1 3-3 1-1 3-3 1-1 3-3

Ampl

itude

Effective detuning

Fig. 7 Optomechanical Attractor Diagram: The effective amplitude-dependent optomechanicaldamping rate Γopt(A), as a function of the oscillation amplitude A and the effective detuning∆ , for three different sidebands ratios κ/ΩM = 0.2, 1, 2, from left to right [Γopt in units ofγ0 ≡ 4(κ/ΩM)2 g2/ΩM, blue: positive/cooling; red: negative/amplification]. The optomechanicalattractor diagram of self-induced oscillations is determined via the condition Γopt(A) = −ΓM. Theattractors are shown for three different values of the incoming laser power (as parametrized by theenhanced optomechanical coupling g at resonance), with ΓM/γ0 = 0.1, 10−2, 10−3 (white, yellow,red).

The resulting attractor diagram is shown in figure 7. It shows the possible limitcycle amplitude(s) as a function of effective detuning ∆ , such that the self-consistentevaluation of x has been avoided.The self-induced mechanical oscillations in an optomechanical system are anal-

ogous to the behaviour of a laser above threshold. In the optomechanical case, theenergy provided by the incoming laser beam is converted, via the interaction, intocoherent mechanical oscillations. While the amplitude of these oscillations is fixed,the phase depends on random initial conditions and may diffuse due to noise (e.g.thermal mechanical noise or shot noise from the laser). Interesting features may


therefore arise when several such optomechanical oscillators are coupled, either me-chanically or optically. In that case, they may synchronize if the coupling is strongenough. Optomechanical synchronization has been predicted theoretically [17, 18]and then observed experimentally [19, 20]. At high driving powers, we note that thedynamics is no longer a simple limit cycle but may instead become chaotic [21].

3 Quantum theory

In the previous section, we have seen how a semiclassical description of the canoni-cal optomechanical cavity gives a simple, intuitive picture of optical spring and op-tical damping effects. The average cavity photon number ncav acts as a force on themechanical resonator; this force depends on the mechanical position x, as changesin x change the cavity frequency and hence the effective detuning of the cavity drivelaser. If ncav were able to respond instantaneously to changes in x, we would onlyhave an optical spring effect; however, the fact that ncav responds to changes in xwith a non-zero delay time implies that we also get an effective damping force fromthe cavity.In this section, we go beyond the semiclassical description and develop the full

quantum theory of our driven optomechanical system [22, 5, 23]. We will see thatthe semiclassical expressions derived above, while qualitatively useful, are not ingeneral quantitatively correct. In addition, the quantum theory captures an impor-tant effect missed in the semiclassical description, namely the effective heating ofthe mechanical resonator arising from the fluctuations of the cavity photon numberabout its mean value. These fluctuations play a crucial role, in that they set a limitto the lowest possible temperature one can achieve via cavity cooling.

3.1 Basics of the quantum noise approach to cavity backaction

We will focus here on the so-called “quantum noise” approach, where for a weakoptomechanical coupling, one can understand the the effects of the cavity backactioncompletely from the quantum noise spectral density of the radiation pressure forceoperator. This spectral density is defined as:

SFF [ω ] =! ∞

−∞dteiωt⟨F(t)F(0)⟩ (21)

where the average is taken over the state of the cavity at zero optomechanical cou-pling, and

F(t) ≡ hG"a†a−⟨a†a⟩

#(22)

is the noise part of the cavity’s backaction force operator (in the Heisenberg picture).


We start by considering the the quantum origin of optomechanical damping,treating the optomechanical interaction term in the Hamiltonian of Eq. 3 using per-turbation theory. Via the optomechanical interaction, the cavity will cause transi-tions between energy eigenstates of the mechanical oscillator, either upwards ordownwards in energy. Working to lowest order in the optomechanical coupling G,these rates are described by Fermi’s Golden rule. A straightforward calculation (seeSec.II B of Ref. [24]) shows that the Fermi’s Golden rule rate Γn,+ (Γn,−) for a tran-sition taking the oscillator from n→ n+1 quanta (n→ n−1 quanta) is given by:

Γn,± =

!n+

12∓12

"x2ZPFh2

SFF [∓ΩM] (23)

The optomechanical damping rate simply corresponds to the net rate at which en-ergy is lost from the mechanical resonator via these transitions:

Γopt = Γn,−−Γn,+ (24)

=x2ZPFh2

(SFF [ΩM]−SFF [−ΩM]) (25)

Note that Γopt is independent of n, the initial number of quanta in the oscillator,and thus describes simple linear damping (i.e. damping which is independent of theamplitude of the oscillator’s motion). Also note that our derivation has neglectedthe effects of the oscillator’s intrinsic damping ΓM, and thus is only valid if ΓM issufficiently small; we comment more on this at the end of the section.

Fig. 8 The noise spectrum ofthe radiation pressure forcein a driven optical cavity.This is a Lorentzian, peakedat the (negative) effectivedetuning. The transition ratesare proportional to the valueof the spectrum at +ΩM(emission of energy into thecavity bath) and at −ΩM(absorption of energy by themechanical resonator).

-3 -2 -1 0 1 2 3

There is a second way to derive Eq.(25) which is slightly more general, andwhich allows us to calculate the optical spring constant kopt; it also more closelymatches the heuristic reasoning that led to the semiclassical expressions of the pre-vious section. We start from the basic fact that both Γopt and δΩM,opt arise from thedependence of the average backaction force Frad on the mechanical position x. We


can calculate this dependence to lowest order in G using the standard equations ofquantum linear response (i.e. the Kubo formula):

δFrad(t) = −! ∞

−∞dt ′λFF(t− t ′)⟨x(t ′)⟩ (26)

where the causal force-force susceptibility λFF(t) is given by:

λFF(t) = −ihθ (t)⟨

"F(t), F(0)

#⟩ (27)

Next, assume that the oscillator is oscillating, and thus ⟨x(t)⟩ = x0 cosΩMt. Wethen have:

δFrad(t) = (−Re λFF [ΩM] · x0 cosΩMt)− (Im λFF [ΩM] · x0 sinΩMt) (28)

= −∆kopt⟨x(t)⟩−meffΓopt$dxdt

(t)%

(29)

Comparing the two lines above,we see immediately that the real and imaginary partsof the Fourier-transformed susceptibility λFF [ΩM] are respectively proportional tothe optical spring kopt and the optomechanical damping Γopt . The susceptibilitycan in turn be related to SFF [ω ]. In the case of the imaginary part of λFF [ω ], astraightforward calculation yields:

−Im λFF [ω ] =SFF [ω ]−SFF [−ω ]

2h(30)

As a result, the definition of Γopt emerging from Eq. (29) is identical to that inEq. (25). The real part of λFF [ω ] can also be related to SFF [ω ] using a standardKramers-Kronig identity. Defining δΩM,opt ≡

kopt2meffΩM

, one finds:

δΩM,opt =x2ZPFh2

! ∞

−∞

dω2π

SFF [ω ]

&1

ΩM−ω−

1ΩM+ω

'(31)

Thus, a knowledge of the quantum noise spectral density SFF [ω ] allows one to im-mediately extract both the optical spring coefficient, as well as the optical dampingrate.We now turn to the effects of the fluctuations in the radiation pressure force, and

the effective temperature Trad which characterizes them. This too can be directly re-lated to SFF [ω ]. Perhaps the most elegant manner to derive this is to perturbativelyintegrate out the dynamics of the cavity [24, 25]; this approach also has the ben-efit of going beyond simplest lowest-order-perturbation theory. One finds that themechanical resonator is described by a classical Langevin equation of the form:

mx(t) = −(k+ kopt)x(t)−mΓoptx(t)+ ξrad(t). (32)


The optomechanical damping Γopt and optical spring kopt are given respectively byEqs. (25) and (31), except that one should make the replacement ΩM → Ω ′

M ≡ΩM + δΩM,opt in these equations. The last term ξrad(t) above represents the fluc-tuating backaction force associated with photon number fluctuations in the cavity.Within our approximations of weak optomechanical coupling and weak intrinsicmechanical damping, this random force is Gaussian white noise, and is fully de-scribed by the spectral density:

Sξradξrad [ω ] =mΓopt coth!hΩ ′

M/2kBTrad"

= mΓopt (1+2nrad) . (33)

Here, Trad is the effective temperature of the cavity backaction , and nrad is the cor-responding number of thermal oscillator quanta. These quantities are determined bySFF [ω ] via:

1+2nrad ≡SFF [Ω ′

M]+SFF [−Ω ′M]

SFF [Ω ′M]−SFF [−Ω ′

M](34)

Note that as the driven cavity is not in thermal equilibrium, Trad will in generaldepend on the value ofΩM; a more detailed discussion of the concept of an effectivetemperature is given in Ref. [24].Turning to the stationary state of the oscillator, we note that Eq. (32) is identical

to the Langevin equation for an oscillator coupled to a thermal equilibrium bath attemperature Trad. It thus follows that the stationary state of the oscillator will bea thermal equilibrium state at a temperature Trad, and with an average number ofquanta nrad. As far as the oscillator is concerned, Trad is indistinguishable from atrue thermodynamic bath temperature, even though the driven cavity is not itself inthermal equilibrium.The more realistic case is of course where we include the intrinsic damping and

heating of the mechanical resonator; even here, a similar picture holds. The intrinsicdissipation can be simply accounted for by adding to the RHS of Eq. (32) a damp-ing term describing the intrinsic damping (rate. ΓM), as well as a stochastic forceterm corresponding to the fridge temperature T . The resulting Langevin equationstill continues to have the form of an oscillator coupled to a single equilibrium bath,where the total damping rate due to the bath is ΓM+Γopt, and the effective tempera-ture Teff of the bath is determined by:

neff =ΓMn0+Γoptnrad

ΓM+Γopt(35)

where n0 is the Bose-Einstein factor corresponding to the bath temperature T :

n0 =1

exp!hΩ ′

M/kBT"−1

(36)

We thus see that in the limit where Γopt ≫ ΓM, the effective mechanical tempera-ture tends to the backaction temperature Trad. This will be the lowest temperature


possible via cavity cooling . Note that similar results may be obtained by using theGolden rule transition rates in Eq. (23) to formulate a master equation describingthe probability pn(t) that the oscillator has n quanta at time t (see Sec. II B of Ref.[24]).Before proceeding, it is worth emphasizing that the above results all rely on the

total mechanical bandwidth ΓM+Γopt being sufficiently small that one can ignorethe variance of SFF [ω ] across the mechanical resonance. When this condition is notsatisfied, one can still describe backaction effects using the quantum noise approach,with a Langevin equation similar to Eq. (32). However, one now must include thevariation of SFF [ω ] with frequency; the result is that the optomechanical dampingwill not be purely local in time, and the stochastic part of the backaction force willnot be white.

3.2 Application to the standard cavity optomechanical setup

The quantum noise approach to backaction is easily applied to the standard optome-chanical cavity setup, where the backaction force operator F is proportional to thecavity photon number operator. To calculate its quantum noise spectrum in the ab-sence of any optomechanical coupling, we first write the equation of motion for thecavity annihilation operator a in the Heisenberg picture, using standard input-outputtheory [26, 27]:

ddta =

!−iωopt−κ/2

"a−

√κ ain. (37)

Here, ain describes the amplitude of drive laser, and can be decomposed as:

ain = e−iωLt!ain+ din

", (38)

where ain represents the classical amplitude of the drive laser (the input power isgiven by Pin = hωopt|ain|2), and din describes fluctuations in the laser drive. Weconsider the ideal case where these are vacuum noise, i.e. there is only shot noisein the incident laser drive, and no additional thermal or phase fluctuations. One thusfinds that din describes operator white noise:

#din(t)d

†in(t

′)$

=#[din(t), d

†in(t

′)]$

= δ (t− t ′) (39)

It is also useful to separate the cavity field operator into an average “classical” partand a quantum part,

a= e−iωLt eiφ!√

ncav+ d"

(40)

where eiφ√ncav is the classical amplitude of the cavity field, and d describes its

fluctuations.


It is now straightforward to solve Eq. (37) for d in terms of din. As we will beinterested in regimes where ncav≫ 1, we can focus on the leading-order-in-ncav termin the backaction force operator F:

F ≃ hG√ncav

!d+ d†

"(41)

Using this leading-order expression along with the solution for din and Eq. (39), wefind that the quantum noise spectral density SFF [ω ] (as defined in Eq.(21)) is givenby:

SFF [ω ] = h2G2ncavκ

(ω+∆)2+(κ/2)2(42)

SFF [ω ] is a simple Lorentzian, reflecting the cavity’s density of states, and is centredat ω = −∆ , precisely the energy required to bring a drive photon onto resonance.The form of SFF [ω ] describes the final density of states for a Raman process wherean incident drive photon gains (ω > 0, anti-Stokes) or loses (ω < 0, Stokes) a quantah|ω | of energy before attempting to enter the cavity. From Eq. (25), we can imme-diately obtain an expression for the optomechanical damping rate; it will be largeif can make the density states associated with the anti-Stokes process at frequencyΩ ′M much larger than that of the Stokes process at the same frequency. The optical

spring coefficient also follows from Eq. (31).We finally turn to nrad, the effective temperature of the backaction (expressed as

a number of oscillator quanta). Using Eq. (34) and (42), we find:

nrad = −(Ω ′

M+∆)2+(κ/2)2

4Ω ′M∆

(43)

As discussed, nrad represents the lowest possible temperature we can cool our me-chanical resonator to. As a function of drive detuning ∆ , nrad achieves a minimumvalue of

nrad###min

=

$κ4Ω ′

M

%2 2

1+&1+(κ2/4Ω ′2

M)(44)

for an optimal detuning of

∆ = −&Ω ′2M+κ2/4. (45)

We thus see that if one is in the so-called good cavity limit ΩM ≫ κ , and if thedetuning is optimized, one can potentially cool the mechanical resonator close to itsground state. In this limit, the anti-Stokes process is on-resonance, while the Stokesprocess is far off-resonance and hence greatly suppressed. The fact that the effec-tive temperature is small but non-zero in this limit reflects the small but non-zeroprobability for the Stokes process, due to the Lorentzian tail of the cavity density ofstates. In the opposite, “bad cavity” limit whereΩM ≪ κ , we see that the minimum


of nrad tends to κ/ΩM≫ 1, while the optimal detuning tends to κ/2 (as anticipatedin the semiclassical approach).Note that the above results are easily extended to the case where the cavity is

driven by thermal noise corresponding to a thermal number of cavity photons ncav,T .For a drive detuning of ∆ = −ΩM (which is optimal in the good cavity limit), onenow finds that that the nrad is given by [6]:

nrad =

!κ4ΩM

"2+ncav,T

#

1+2!

κ4ΩM

"2$

(46)

As expected, one cannot backaction-cool a mechanical resonator to a temperaturelower than that of the cavity.

3.3 Results for a dissipative optomechanical coupling

A key advantage of the quantum noise approach is that it can be easily applied toalternate forms of optomechanical coupling. For example, it is possible have sys-tems where the mechanical resonator modulates both the cavity frequency as wellas the damping rate κ of the optical cavity [28, 29]. The position of the mechani-cal resonator will now couple to both the cavity photon number (as in the standardsetup), as well as to the “photon tunnelling” term which describes the coupling ofthe cavity mode to the extra-cavity modes that damp and drive it. Because of thesetwo couplings, the form of the effective backaction force operator F is now modi-fied from the standard setup. Nonetheless, one can still go ahead and calculate theoptomechanical backaction using the quantum noise approach. In the simple casewhere the cavity is overcoupled (and hence its κ is due entirely to the coupling tothe port used to drive it), one finds that the cavity’s backaction quantum force noisespectrum is given by [30, 31]:

SFF [ω ] =

!G2κ ncav4κ

"%ω+2∆ − 2G

Gκ κ&2

(ω+∆)2+κ2/4(47)

Here, G= −dωopt/dx is the standard optomechanical coupling, while Gκ = dκ/dxrepresents the dissipative optomechanical coupling. For Gκ = 0, we recover theLorentzian spectrum of the standard optomechanical setup given in Eq. (42) whereasfor Gκ = 0, SFF [ω ] has the general form of a Fano resonance. Such lineshapes ariseas the result of interference between resonant and non-resonant processes; here,the resonant channel corresponds to fluctuations in the cavity amplitude, whereasthe non-resonant channel corresponds to the incident shot noise fluctuations on thecavity. These fluctuations can interfere destructively, resulting in SFF [ω ] = 0 at thespecial frequency ω = −2∆ + 2G/Gκ . If one tunes ∆ such that this frequency co-incides with −ΩM, it follows immediately from Eq. (34) that the cavity backaction


has an effective temperature of zero, and can be used to cool the mechanical res-onator to its ground state. This special detuning causes the destructive interferenceto completely suppress the probability of the cavity backaction exciting the me-chanical resonator, whereas the opposite process of absorption is not suppressed.This “interference cooling” does not require one to be in the good cavity limit, andthus could be potentially useful for the cooling of low-frequency (relative to κ) me-chanical modes. However, the presence of internal loss in the cavity places limits onthis technique, as it suppresses the perfect destructive interference between resonantand non-resonant fluctuations [30, 31].

References

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Basic Theory of Cavity Optomechanics Theory of Cavity Optomechanics Aashish A. Clerk and Florian Marquardt Abstract This chapter provides a brief basic introduction to the theory used