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Quantum Noise Interference and Back-action Cooling in Cavity Nanomechanics

Florian Elste,1S. M. Girvin,2and A. A. Clerk1

1Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8

2Department of Physics, Yale University, New Haven, Connecticut 06520, USA

(Dated: March 12, 2009)

We present a theoretical analysis of a novel cavity electromechanical system where a mechanical

resonator directly modulates the damping rate κ of a driven electromagnetic cavity.

that via a destructive interference of quantum noise, the driven cavity can effectively act like a zero-

temperature bath irrespective of the ratio κ/ωM, where ωM is the mechanical frequency. This scheme

thus allows one to cool the mechanical resonator to its ground state without requiring the cavity to

be in the so-called ‘good cavity’ limit κ ? ωM. The system described here could be implemented

directly using setups similar to those used in recent experiments in cavity electromechanics.

We show

PACS numbers:42.79.Gn, 07.10.Cm, 42.50.Lc.

Introduction.— Much of the rapid progress in fabri-

cating and controlling nanomechanical devices has been

fueled by numerous promising technological applications.

However, progress has also been motivated by the real-

ization that such systems are ideally poised to allow the

exploration of several fundamental quantum mechanical

effects.Various studies have addressed such issues as

entanglement, quantum-limited measurement and Fock

state detection in systems containing nanoscale mechan-

ical resonators [1, 2, 3, 4]. The issue of quantum back-

action has also received considerable attention in these

systems [5, 6]. Understanding the back-action properties

of a detector in a quantum electromechanical or optome-

chanical system is necessary if one wishes to do quantum-

limited position detection. Further, this back-action can

in some cases be exploited to achieve significant non-

equilibrium cooling of the mechanical resonator. This is

of particular importance, since a pre-requisite to seeing

truly quantum behaviour in these systems is the ability

to cool the mechanical resonator to near its ground state.

A particularly effective back-action cooling scheme is

so-called “cavity cooling” [7, 8, 9, 10]. Here, a mechan-

ical resonator is dispersively coupled to a driven elec-

tromagnetic cavity (i.e. the cavity’s resonance frequency

depends on the mechanical displacement x). By moni-

toring the frequency of the cavity, one can make a sen-

sitive measurement of x. In addition, the cavity photon

number necessarily acts as a noisy force on the mechan-

ics; for a suitably chosen cavity drive, this force can be

used to effectively cool the mechanical resonator. This

approach has been used in a number of recent experi-

ments, both with optical cavities coupled to mechanical

resonators [11, 12], as well as in systems using microwave

cavities [13, 14]. Theoretically, it has been shown that

such schemes could in principle allow ground state cool-

ing of the mechanical resonator [9, 10]. Similar physics

even occurs in seemingly very different systems, e.g. a

superconducting single-electron transistor coupled to a

mechanical resonator [15, 16].

In this Letter, we present a theoretical analysis of

a generic electromechanical (or optomechanical) system

which is the dual of the dispersive coupling discussed

above. We again consider a driven cavity coupled to a

mechanical resonator: now, however, the mechanical dis-

placement x does not change the cavity frequency, but

rather changes its damping rate κ. As we will discuss,

such a coupling could be achieved in a microwave cav-

ity system, or in an optical cavity containing a moveable

‘mebrane-in-the-middle’ [12, 17]. We show that for such

a dissipative cavity-mechanical resonator coupling, inter-

ference effects are important in determining the quantum

back-action effects on the mechanical system; this is not

the case for a purely dispersive coupling. In particular,

we show that one can use destructive interference to allow

the cavity to act as an effective zero-temperature bath,

irrespective of the ratio of the mechanical frequency ωM

to the cavity linewidth κ; as such, ground state cool-

ing is possible without requiring the ‘good cavity’ limit

ωM? κ. This is in sharp contrast to the case of a disper-

sive coupling, where ground state cooling is only possible

if ωM ? κ. From a practical perspective, being able

to deviate from the good cavity limit is advantageous,

as it allows one to use small drive detunings and hence

achieve much larger effective cavity-mechanical resonator

couplings. We show that this destructive interference ef-

fect persists in the case where one has both a dissipative

and dispersive coupling; we also show that the dissipative

coupling can also allow for a quantum-limited position

measurement.

Model.— We consider a mechanical oscillator (fre-

quency ωM, displacement x) whose motion weakly mod-

ulates the damping rate κ and resonant frequency ωRof

a driven electromagnetic cavity. For small displacements,

both ωRand κ will have a linear dependence on x, and

we can describe the system via the Hamiltonian (? = 1):

ˆH = ωRˆ a†ˆ a + ωMˆ c†ˆ c +?

Hamiltonians, while Hγdescribes the intrinsic mechani-

cal damping by an equilibrium bath at temperature Teq.

The third term inˆH describes the bosonic bath responsi-

qωqˆb†

qˆbq+ˆHdamp+ˆHint+ˆHγ.

The first two terms describe the cavity and mechanical

arXiv:0903.2242v2 [cond-mat.mes-hall] 13 Mar 2009

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2

(b)

00.20.4 0.6

0.01

0.1

1

SFF(ω) / SFF,max

nosc

ω/κ

Δ/ωM

(a)

-3

0123

0.0

0.2

0.4

0.6

0.8

1.0

-2

-1

FIG. 1:

∆ = κ/2, for different ratios of the dispersive to dissipa-

tive optomechanical couplings.

responds to

˜ A/˜B = 10, the long-dashed green curve to

˜ A/˜B = 0.75 and the short-dashed red curve to˜ A/˜B = 0.

Each curve has been scaled by its maximum value. (b) Num-

ber of oscillator quanta nosc versus drive detuning ∆, ob-

tained by solving the full equations of motion. The dotted

blue curve is the Bose-Einstein factor neff corresponding to

the back-action effective temperature Teff when κ = ωM (i.e.

neff = SFF(−ωM)/(SFF(ωM) − SFF(−ωM)). The remaining

curves correspond to˜B¯ a = 0.2, neq = 50, γ = 10−6ωM, and

κ/ωM = 0.2 (green solid curve), κ/ωM = 1.0 (purple long-

dashed curve), κ/ωM = 5.0 (red short-dashed curve).

(a) Back-action force noise spectral density for

The solid blue curve cor-

ble for the dissipation and driving of the cavity. Working

in the usual Markovian limit where κ ? ωR, and where

the bath density of states ρ can be treated as energy-

independent over relevant frequencies, the cavity-bath

interaction takes the form [18, 19]:

?

We stress that the Markovian limit is well justified both

for cavities used in optomechanical and microwave elec-

tromechanical systems. The coupling between the cavity

and mechanics is then described by:

?1

The dimensionless coupling strengths ˜A,˜B above are

given by˜Bκ = (dκ/dx)xzpt,˜Aκ = (dωR/dx)xzpt, where

xzptis the zero-point motion amplitude of the mechani-

cal oscillator. Setting˜B = 0 recovers a purely dispersive

coupling, while setting˜A = 0 corresponds to a purely

dissipative optomechanical coupling.

Quantum noise.— We may now identify the quantum

back-action force operatorˆF ≡ −(d/dx)ˆHint. For a suf-

ficiently weak optomechanical coupling, linear-response

theory applies, and the unperturbed quantum noise spec-

trum ofˆF will determine both the back-action damp-

ing and back-action heating of the mechanical res-

onator by the driven cavity [15, 19].

spectrum is SFF(ω) =

that SFF(ωM) is proportional to the Fermi’s golden

rule rate for the absorption of a mechanical quan-

tum by the driven cavity, while SFF(−ωM) is pro-

portional to the corresponding emission rate.

ˆHdamp = −i

κ

2πρ

?

q

?

ˆ a†ˆbq−ˆb†

qˆ a

?

.

(1)

ˆHint = (ˆ x/xzpt)

2

˜BˆHdamp+˜Aκˆ a†ˆ a

?

.

(2)

The relevant

?dτeiωτ?ˆF(τ)ˆF(0)?0. Recall

The

effective temperature associated with the back-action

noise as seen by the mechanical oscillator is then

given by kBTeff ≡ ωM(log[SFF(ωM)/SFF(−ωM)])−1,

while the back-action damping is given by γBA

x2

the total mechanical damping rate to be small enough

that the resonator is only sensitive to noise at ω = ±ωM.

We calculate SFF(ω) by linearizing the full quantum

dynamics about the solutions of the classical equations

of motion (see below), and by making use of the input-

output formalism of quantum optics [18, 19]. Letting

∆ = ωdrive−ωRbe the detuning of the cavity drive, and

¯ a = ?ˆ a?, one finds:

?

=

zpt(SFF(ωM) − SFF(−ωM)).These results presume

SFF(ω) = κ

˜B|¯ a|

2xzpt

?2?

ω + 2∆ −2˜

(ω + ∆)2+ κ2/4.

A

˜ Bκ

?2

(3)

In the limit˜B → 0 of a purely dispersive coupling,

Eq. (3) reduces to a simple Lorentzian, in agreement with

Refs. [9, 10]. This form has a simple interpretation: it

corresponds to the Lorentzian density of final states rele-

vant to a Raman process where an incident drive photon

gains an energy ?ω while attempting to enter the cavity.

The optimal back-action cooling discussed in Refs. [9, 10]

involves choosing ∆ = −ωMand κ ? ωM. For these pa-

rameters, the drive photons are initially far from being

on resonance with the cavity. The absorption of energy

from the oscillator is resonantly enhanced, as it moves

an incident drive photon onto the cavity resonance. In

contrast, emission of energy to the oscillator is greatly

suppressed, as the drive photon is moved even further

from the cavity resonance. Thus, the˜B = 0 form of

SFF(ω) and the resulting cooling are simply explained

as a density of states effect.

In the more general case where the dissipative op-

tomechanical coupling˜B is also non-zero, we see that

SFF(ω) is not a simple Lorentzian; as such, the back-

action physics cannot be interpreted solely in terms of a

density of states effect. In general, SFF(ω) has an asym-

metric Fano lineshape (see Fig. 1); in particular, the cav-

ity emission noise SFF(−ωM) is equal to zero whenever

the detuning ∆ = ωM/2 + (˜A/˜B)κ ≡ ∆opt. Thus, for

this value of the detuning, one finds that the cavity acts

as an effective zero-temperature bath, irrespective of the

ratio κ/ωM. For ∆ = ∆opt, one has:

γBA,opt=˜B2|¯ a|2κ

ω2

M

?

3ωM/2 + (˜A/˜B)κ

?2

+ κ2/4

.

(4)

For ∆ = ∆opt, the weak-coupling, quantum noise ap-

proach yields the equilibrium number of quanta in the

oscillator to be nosc = γneq/(γBA,opt+ γ), where γ is

the intrinsic damping rate of the oscillator, and neq is

the Bose-Einstein factor associated with the bath tem-

perature Teq. One can thus cool to the ground state for

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3

a sufficiently large coupling and/or cavity drive. This

possibility of ground state cooling for an arbitrary κ/ωM

ratio is a main result of this paper. Note that the optimal

detuning ∆optis greater than 0: ground state cooling is

possible even though the drive photons would seemingly

need to “burn off” energy to enter the cavity. This is in

stark contrast to the purely dispersive case where a posi-

tive detuning leads to heating and a negative-damping

instability; it highlights the fact that the back-action

physics here is not simply a density of states effect. Note

finally that there is also an “optical spring” effect asso-

ciated with the back-action; it can be obtained directly

from SFF(ω) [9].

A heuristic explanation of the Fano form of SFF(ω)

can be given. Fano lineshapes arise generically as a re-

sult of interference between resonant and non-resonant

processes; the situation is no different here. In the usual

˜B = 0 case, the only source of back-action force noise

is the number fluctuations of the cavity field ˆ a. How-

ever, when˜B ?= 0, the mechanical oscillator mediates

the coupling between the cavity and the cavity’s dissi-

pative bath. As a result, it is subject to two sources of

noise: the shot noise associated with the driving of the

cavity, as well as the fluctuations of ˆ a. The first of these

noise processes is white, whereas the second is not: it

is simply the shot noise incident on the cavity filtered

by the ω-dependent cavity susceptibility. The interfer-

ence between these two noises yields a Fano lineshape for

SFF(ω), and the destructive interference at ∆ = ∆opt

which causes Teff = 0. Note that Fano interference in

quantum electromechanical systems has recently been

discussed in Ref. [20], albeit in a completely different

context.

Equations of motion.— To address whether the de-

structive noise interference effect persists beyond the sim-

plest weak-coupling regime, we now examine the full so-

lutions of the linearized Heisenberg equations of motion

for our system. For simplicity, we focus on the case˜A = 0

in what follows. Working in frame rotating at the drive

frequency, and writing ˆ a = ¯ a+ˆd, the linearized equations

take the form:

?

˙ˆ c = −

2

Here¯binis the amplitude of the coherent cavity drive,ˆξ

describes vacuum noise entering the cavity from the drive

port, and ˆ η describes thermal noise associated with the

mechanical damping γ; bothˆξ and ˆ η are operator-valued

white noise [18, 19]. The back-action force noise operator

in Eq. (6) takes the form:

˙ˆd =

i∆ −κ

?

2

?

ˆd −

˜B

2κ

ˆ c −√γˆ η + iˆf.

?

¯ a +

¯bin

√κ

??ˆ c + ˆ c†?−√κˆξ,(5)

iωM+γ

?

(6)

ˆf = i

˜B

2

√κ(¯ a∗ˆξ −¯b∗

inˆd) + h.c.

(7)

We see the two contributions to the back-action discussed

earlier: the first term corresponds to the direct contribu-

tion of the drive shot noise, while the second term rep-

resents the contribution from fluctuations in the cavity

field. As discussed, the interference of these terms yields

a Fano lineshape.

Upon solving the equations of motion, the oscillator

spectrum Scc=?dt?ˆ c†(t)ˆ c(0)e−iωt? is found to be:

Scc(ω) = |˜ χM(−ω)|2[σeq(−ω) + SFF(ω)]

where:

(8)

˜ χM(ω) = χM(ω)/[1 + iχM(ω)Σ(ω)],

Σ(ω) = −i˜B2|¯ a|2χ∗

ωM∆

?

(9)

(10)

M(−ω)χR(ω)χ∗

∆2−3

?

R(−ω) ×

,

?

4κ2+ iωκ

?

σeq(ω) = γneq

1 +ReΣ(ω)

ωM

?

+

????

Σ(ω)

2ωM

????

2

(1 + 2neq)

?

.

(11)

Here, we denote the bare mechanical and cavity sus-

ceptibilties by χM(ω) = (−i(ω − ωM) + γ/2)−1and

χR(ω) = (−i(ω + ∆) + κ/2)−1. Eq. (8) has a simple

interpretation: the oscillator responds with a modified

susceptibility ˜ χM to two independent fluctuating forces,

corresponding to the two terms in the equation. The

first, described by σeq, is the fluctuating thermal force

associated with the intrinsic oscillator damping γ. The

second is the back-action from the driven cavity. We see

that its form is not affected by the coupling strength:

the same spectral density SFF(ω) (evaluated at˜A = 0)

found earlier in the weak-coupling regime (c.f. Eq. (3))

appears here. Thus, the quantum noise interference dis-

cussed above continues to play a role even for moderate

coupling strengths. A strong cavity-mechanical resonator

coupling will nonetheless modify the physics, as the de-

structive interference occurring when ∆ = ∆optonly oc-

curs at the single frequency ωM. For a sufficiently strong

coupling, the oscillator’s total damping will become large

enough that the oscillator will be sensitive to noise at fre-

quencies away from ωM, frequencies where the destruc-

tive interference is not complete. As a result, the cavity

will no longer appear to the oscillator as an effective zero-

temperature bath. Note that when˜A ?= 0, the spectrum

Scc(ω) still has the general form given by Eqs. (8, 9, 11):

one now simply uses the full form of SFF(ω), as well as

a self-energy that contains extra terms ∝˜A.

To see whether this resonance-broadening effect as well

as other strong coupling effects preclude ground state

cooling at the optimal detuning ∆ = ∆opt, we calcu-

late the average number of oscillator quanta noscdirectly

by integrating Scc(ω) in Eq. (8). In Fig. 2 we show the

expected cooling for realistic parameter values, as a func-

tion of the cavity drive strength. Strong-coupling effects

lead to an optimal drive strength, beyond which nosc

Page 4

4

0.0.2 0.4

0.001

0.1

10

B|a|

nosc

C

L

C1(x)

(a)

(b)

Z0

γ/ωM = 10-5

γ/ωM = 10-6

~

FIG. 2: (a) Number of mechanical quanta nosc versus cou-

pling strength˜B|¯ a|, for˜ A = 0, neq = 50, κ/ωM = 1 and

for an optimal detuning ∆ = ωM/2; the mechanical damping

γ is as marked. Dashed curves are results from the linear-

response, quantum noise approach, whereas solid curves are

obtained from the full solutions of the equations of motion.

(b) Schematic of a cavity (modeled as an LC resonator) cou-

pled to a transmission line (impedance Z0) via an x-dependent

capacitance C1(x).

starts to increase. Nonetheless, the minimum value of

nosccan still be significantly less than 1. Again, we stress

that the dissipative coupling discussed here is in many

ways advantageous to a system with a purely dispersive

optomechanical coupling, as one does not need to use a

large detuning to achieve a low effective temperature.

Physical realization.— The dissipative optomechani-

cal coupling analyzed here could be realized in a mi-

crowave electromechanical system similar to those stud-

ied in Refs. [13, 14]. In such systems, a capacitor C1

couples the cavity to a transmission line (impedance Z0)

which both drives and damps the cavity. One would now

need to make C1 mechanically compliant (see Fig. 2b).

In general, such a setup will have both dissipative and

dispersive optomechanical couplings (i.e.˜A ?= 0,˜B ?= 0).

A careful analysis shows that in the physically relevant

regime C1 ? C, one can have ˜A ? ˜B if the cavity

impedance ZR =

Z0? 50Ω. For example, taking experimentally achiev-

able parameters ωR = 2π × 10 GHz, C = 3.2 pF and

C1 = 0.01C results in κ/ωR ? 10−3and |˜A/˜B| ? 2.5.

Eq. (3) then implies that ground-state cooling via de-

structive noise interference is possible if one uses a drive

detuning ∆ = ∆opt= ωM/2 + 2.5κ. We stress this con-

clusion is independent of the magnitude of ωM/κ; in con-

trast, if one had a purely dispersive cavity - mechanical

resonator coupling, ground state cooling is only possi-

ble if ωM ? κ. A second setup where the dissipative

optomechanical coupling could be realized is an optical

Fabry-Perot cavity containing a thin, moveable dielectric

slab. If the cavity is constructed with mirrors having dif-

ferent reflectivities, the motion of the membrane will nat-

urally modulate the damping rate of the optical modes.

The resulting value of˜B can be easily derived using the

approach of Ref. [17].

Position detection.— A straightforward calculation

shows that the power spectrum ofˆbout, the output field

?L/C is made slightly smaller than

from the cavity, is modified in a simple way by the

coupling to the oscillator: the oscillator adds a term

Sxx(−ω)SFF(ω) (where ω is measured from the drive fre-

quency); this holds for both˜A,˜B ?= 0. Here, Sxx(ω) =

?dτeiωτ?ˆ x(τ)ˆ x(0)?.

ture of the oscillator. Note that for an optimal detuning

∆ = ∆opt, the destructive interference effect means that

the peak at ω = ωMin the output spectrum will vanish.

This could serve as an interesting test of our predictions.

Finally, specializing to the case where ωM ? κ, one

can also use the dissipative optomechanical coupling to

make a quantum-limited, weak continuous position mea-

surement of the oscillator. One drives the cavity off res-

onance, and performs a homodyne detection of the ap-

propriate quadrature of the output field from the cavity.

One finds from Eqs. (5),(6) that for any non-zero detun-

ing ∆, the corresponding back-action noise is as small

as is allowed by the uncertainty principle, meaning that

one can reach the quantum limit on the total added noise

[19].

Conclusions.— We have considered a generic system

where a mechanical resonator modulates the damping of

a driven quantum electromagnetic resonator, and demon-

strated how Fano interference is important to the back-

action physics. In particular, one can have a perfect

destructive interference which allows the back-action to

mimic a zero-temperature environment; as such, such a

system should allow near ground-state cooling of the me-

chanics.

We thank J. Harris for useful comments. This work

was supported by NSERC, CIFAR and the NSF under

grants DMR-0653377 and DMR-0603369.

One can thus use the peaks at

ω = ±ωM in the output spectrum to infer the tempera-

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