PHYSICAL REVIEW A 88, 063833 (2013)

Arbitrarily large steady-state bosonic squeezing via dissipation
Andreas Kronwald,1,* Florian Marquardt,1,2 and Aashish A. Clerk3
1

2

Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstrasse 7, D-91058 Erlangen, Germany
Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany
3
Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8(Received 23 July 2013; revised manuscript received 11 November 2013; published 19 December 2013)
We discuss how large amounts of steady-state quantum squeezing (beyond 3 dB) of a mechanical resonatorcan be obtained by driving an optomechanical cavity with two control lasers with differing amplitudes. The
scheme does not rely on any explicit measurement or feedback, nor does it simply involve a modulation of an
optical spring constant. Instead, it uses a dissipative mechanism with the driven cavity acting as an engineered
reservoir. It can equivalently be viewed as a coherent feedback process, obtained by minimally perturbing thequantum nondemolition measurement of a single mechanical quadrature. This shows that in general the conceptsof coherent feedback schemes and reservoir engineering are closely related. We analyze how to optimize thescheme, how the squeezing scales with system parameters, and how it may be directly detected from the cavity
output. Our scheme is extremely general, and could also be implemented with, e.g., superconducting circuits.
DOI: 10.1103/PhysRevA.88.063833

PACS number(s): 42.50.Pq, 42.50.Nn, 07.10.Cm, 85.85.+j

I. INTRODUCTION

Among the simplest nonclassical states of a harmonic
oscillator are quantum squeezed states, where the uncertainty
of a single motional quadrature is suppressed below thezero-point level [1]. Such states are of interest for a variety of
applications in ultrasensitive force detection [3]; they are also a
general resource for continuous variable quantum-information
processing [4]. It has long been known that coherent parametricdriving can be used to generate squeezing of a bosonicmode; for a mechanical resonator, this simply amounts tomodulating the spring constant at twice the mechanicalresonance frequency [5]. Such a simple parametric interaction
can yield at best steady-state squeezing by a factor of 1/2
below the zero-point level (the so-called 3 dB limit) [6]; if one
further increases the interaction strength, the system becomes
unstable and starts to self-oscillate.
The rapid progress in quantum optomechanics [7–10],
where a driven electromagnetic cavity is used to detect andcontrol a mechanical resonator, has led to a renewed interestin the generation of squeezing. One could simply use radiationpressure forces to define an oscillating spring constant [11–15],
though this cannot surpass the usual 3 dB limit on stationary
squeezing. One can do better by combining continuous
quantum measurements and feedback, either by making aquantum nondemolition (QND) measurement of a single
motional quadrature [3,16–18], or by combining detuned
parametric driving with position measurement [19,20]. While
such schemes can generate quantum squeezing well past the
3 dB limit, they are difficult to implement, as they requirenear-ideal measurements and feedback. The 3 dB limit could
also be surpassed by continuously injecting squeezed light
directly into the cavity [21], but this is also difficult as one
needs to start with a source of highly squeezed light. We note*

andreas.kronwald@physik.uni-erlangen.de1050-2947/2013/88(6)/063833(10)that a mechanical resonator could also be squeezed via a pulsed
optomechanical scheme [22].
In this work, we discuss a remarkably simple scheme forgenerating steady-state mechanical squeezing well beyond the
3 dB level. We use a two-tone driving of an optomechanicalcavity, without any explicit measurement or feedback[cf. Fig. 1(a)]. As described in Refs. [3,16,18], if one drives
the cavity with equal amplitude at both ωcav ±  (where ωcav isthe cavity frequency and  is the mechanical frequency), thenthe cavity only couples to a single mechanical quadrature X1 ,allowing for a QND measurement and preventing any backaction disturbance of X1 . In contrast, we consider a situation inwhich the two drive tones have different amplitudes. One thus
no longer has a QND situation, and there will be a backaction
disturbance of X1 . However, this disturbance acts to suppress

FIG. 1. (Color online) (a) An optomechanical cavity is driven on
the red and blue mechanical sideband with different laser amplitudes.
This leads to steady-state mechanical squeezing beyond 3 dB. (b)Steady-state quadrature fluctuations X̂12  [in units of the zero-point
fluctuations (ZPF)] as a function of the blue laser driving strength
G+ for different cooperativities C = 4G2− /(κM ). The dark blue
region indicates squeezing beyond 3 dB. The red dashed line isthe variance of a squeezed vacuum state with squeeze parameterr = arctan G− /G+ . An optimal choice of G+ /G− exists (orange
circles) maximizing the amount of squeezing for each C. (Parameters:M /κ = 10−4 , nth = 10, C = 10,25,50,102 ,103 ,104 .)063833-1©2013 American Physical Society

KRONWALD, MARQUARDT, AND CLERKPHYSICAL REVIEW A 88, 063833 (2013)

the fluctuations of X1 , to a level even below the zero-pointlevel. Our scheme thus realizes a coherent feedback operation,
where the driven cavity both measures the X1 quadrature andautonomously applies the corresponding feedback operationnecessary to “cool” X1 (cf. Appendix A and Refs. [23–25] forother examples of coherent feedback in optomechanics). This,
hence, suggests the general recipe to construct a coherent
feedback scheme by perturbing a QND measurement setup
minimally. Equivalently, one can think of the scheme as an
example of reservoir engineering [26]: the driven cavity acts
effectively as a bath whose force noise is squeezed. Thissuggests that in general the concepts of coherent feedback
and reservoir engineering are closely related.
In what follows, we provide a thorough analysis of the
optimal steady-state squeezing generated by our scheme,
showing that the squeezing is a sensitive function of theratio of the cavity drive amplitudes. We also show that forrealistic parameters, one can obtain mechanical squeezing
well beyond the usual 3 dB limit associated with a coherent
parametric driving. While we focus here on optomechanics,
our scheme could also be realized in other implementations ofparametrically coupled bosonic modes, e.g., superconducting
circuits [27,28]. Note that related dissipative mechanisms
can be utilized to prepare the motion of a trapped ion ina squeezed state [29], to squeeze a mechanical resonator
which is coupled to a two-level system [30], and to produce
spin squeezing of atoms in a cavity [31,32]. Unlike thoseworks, our analysis does not rely on describing the engineered
reservoir (the driven cavity) via a simple Lindblad masterequation; in fact, we explicitly discuss corrections to suchan approximation, which we show to become significant for
current experiments. We also note that reservoir engineering
approaches to optomechanics have been previously considered
for generating entanglement (two-mode squeezing) [33,34], as
well as coherence in arrays [35].
II. MODEL

We consider a standard optomechanical system, wherea single cavity mode couples to a mechanical resonator
via radiation pressure; cf. Fig. 1(a). It is described by the
optomechanical Hamiltonian [36]Ĥ = ωcav â † â + b̂† b̂ − g0 â † â(b̂† + b̂) + Ĥdr ,(1)

where the two-tone laser driving Hamiltonian reads
Ĥdr = (α+ e−iω+ t + α− e−iω− t )â † + H.c.
â (b̂) is the photon (phonon) annihilation operator, and g0 is
the optomechanical coupling. ω± and α± are the frequency
and amplitude of the two lasers, respectively. We apply
the displacement transformation â = ā+ e−iω+ t + ā− e−iω− t +d̂ to (1) and go into an interaction picture with respect tothe free cavity and mechanical resonator Hamiltonian. Here,ā± is the coherent light field amplitude due to the twolasers. If  is strongly temperature-dependent, oscillationsin the average cavity intensity can yield spurious parametricinstabilities [37]; these could be suppressed by adding anappropriate third drive tone without strongly degrading thegeneration of squeezing [38] (cf. Appendix G).

We next take the two lasers to drive the mechanicalsidebands of a common mean frequency ω̄, i.e., ω± = ω̄ ± ,
and assume |ā± |  1. For ω̄ far detuned from ωcav , one caneliminate the cavity to obtain an “optical spring” which ismodulated at 2 [14,15]; Ref. [11] obtains an analogous
effect by weakly amplitude-modulating a single strong driveat ωcav − . In contrast, we take ω̄ = ωcav as well as ā+ = ā−[in contrast to backaction-evasion (BAE) schemes [3,16,18]].
Applying a standard linearization to Eq. (1), we find that thelinearized Hamiltonian in our interaction picture isĤ = −d̂ † (G+ b̂† + G− b̂) + H.c.
− d̂ † (G+ b̂e−2it + G− b̂† e2it ) + H.c.

(2)

Here, G± = g0 ā± are the enhanced optomechanical coupling
rates; without loss of generality, we assume G+ ,G− > 0.
The quantum Langevin equations describing the dissipative
dynamics read [39]√iκd̂˙ = [Ĥ ,d̂] − d̂ + κ d̂in .
2
A similar equation holds for b̂, where the cavity decayrate κ is replaced by the mechanical decay rate M . The†
nonzero input noise correlators read d̂in (t)d̂in (t  ) = δ(t −††t  ), b̂in (t)b̂in (t  ) = (nth + 1)δ(t − t  ), and b̂in (t)b̂in (t  ) =nth δ(t − t  ), where nth is the thermal occupancy of themechanical bath.
III. SQUEEZING GENERATION

We now present two intuitive ways of understanding thegeneration of steady-state squeezing in our scheme. For
physical transparency, we focus on the good cavity limitκ  , and thus ignore counterrotating terms in Eq. (2) until
the last section.
Note first that if G+ = G− (i.e., equal drive amplitudes),the cavity√only couples to the mechanical quadrature X̂1 =(b̂† + b̂)/ 2; cf. Eq. (2). As discussed earlier, this allows a
QND measurement of X̂1 [16,18]. If G+ = G− , the cavity still
couples to a single mechanical operator, a Bogoliubov-mode
annihilation operatorβ̂ = b̂ cosh r + b̂† sinh r,
where the squeezing parameter r is defined via tanh r =G+ /G− . We also assume G+ < G− , which ensures stability.
The Hamiltonian (2) becomesĤ = −G d̂ † β̂ + H.c.,(3)√where the coupling G = G2− − G2+ . This is a beam-splitter
Hamiltonian well known from optomechanical sideband
cooling [40,41]. However, instead of allowing the cavity tocool the mechanical mode, it can now cool the mode β̂.
As the vacuum of β̂ is the squeezed state Ŝ(r)|0 (whereŜ(r) = exp[r(b̂b̂ + b̂† b̂† )/2]) [5], this cooling directly yields
steady-state squeezing. In general, 
2 X̂12 = e−2r [1 + 2β̂ † β̂ + β̂ β̂ + β̂ † β̂ † ].
(4)
If β̂ is in its ground state, 2X̂12  = e−2r . Thus, the cavity
acts as an engineered reservoir that can cool the mechanical resonator into a squeezed state. We note that related

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PHYSICAL REVIEW A 88, 063833 (2013)

IV. SQUEEZING VERSUS DRIVING STRENGTHS

We solve the quantum Langevin equations (first in therotating-wave approximation) and consider the steady-statemechanical squeezing as a function of G+ /G− , cf. Fig. 1(b),holding constant both the ratio of damping rates M /κ andthe red-laser amplitude [parametrized via the cooperativity
C = 4G2− /(κM )]. Without the blue-detuned laser, i.e., G+ =0, we have standard optomechanical sideband cooling: both
quadrature variances are reduced as compared to the thermalcase [40,41]. Turning on G+ , the quadrature variance X̂12  firstdecreases with increasing G+ /G− . In general, X̂12  exhibitsa minimum as a function of G+ /G− which becomes sharperwith increasing cooperativity C. For large C, the minimum
variance is well below 1/2 the zero-point value, i.e., the3 dB limit. This minimum results from the competition of
two opposing tendencies. On one hand, increasing G+ /G−
increases the squeezing parameter r and thus the squeezing
associated with the vacuum of β. On the other hand, increasing
G+ /G− reduces G, and hence suppresses the ability of thecavity to cool β. The optimum squeezing is thus a tradeoff
between these tendencies.

(a)quadrature squeezing

40
30
squeezing
20
10
below ZPT

0
1

100

1x104

1x106

1x108

1x1010cooperativity(b)eff. thermal quantaentanglement-via-dissipation schemes [31,33,34] are basedon cooling a delocalized Bogoliubov mode. In contrast, westudy a localized mode which directly leads to a (single-mode)squeezed mechanical steady state.
As mentioned, one can also interpret the squeezing generation without invoking a Bogoliubov mode, but rather as acoherent feedback operation where the cavity both measuresand perturbs X̂1 . In the simplest large-κ
limit, the feedback√
causes both X̂1 and X̂2 = i(b̂† − b̂)/ 2 to be damped at a rate
opt = 4G 2 /κ, but adds negligible fluctuations to X̂1 (smaller
than the zero-point fluctuations that one would associatewith opt ; cf. Appendix A). Thus, despite being driven withclassical light, the cavity acts as a squeezed reservoir leading
to mechanical squeezing.

10
g

lin

up

1

0.1

ng

co

ro

st

1

100

1x104

1x106

1x108

1x1010cooperativity

FIG. 2. (Color online) Maximized steady-state squeezing andstate purity. (a) Squeezing for fixed M /κ and optimized driving
strength G+ [cf. (5)], as a function of G− (parametrized by thecooperativity C). Squeezing beyond 3 dB is apparent even formoderate C. Black lines represent the full theory. These curves
are well described by (6) for C  100. Yellow dashed lines show
the prediction of a Lindblad master equation [LME, Eq. (D1)].
(b) Effective thermal occupancy neff in the mechanical steady state,
for optimized parameters, as a function of C. The colors are thesame as in (a). The full theory and the LME differ drasticallyin the strong-coupling regime where G  κ (gray shaded region,
beginning first for nth = 100). (Parameters: M /κ = 10−4 , nth = 0,
and nth = 100.)
VI. STATE PURITY

V. OPTIMAL SQUEEZING

Consider a fixed red-laser amplitude (i.e., G− ) large
enough that the cooperativity C  1. The value of G+ which
maximizes squeezing is then


G+ 
1 + 2nth
1 1 + 2nth
−2r
≈1−
≈
, i.e., e.
G− optimal
C
2
C
(5)
The corresponding minimum value of X̂12  is
 2  M
1 + 2nth(1 + 2nth ) +;(6)
2 X̂1 ≈
κC
cf. Fig. 2(a). We see that even for moderate values of C andnonzero nth , quantum squeezing beyond 3 dB is achieved;
cf. Fig. 2(a). As C is increased further, the amount of squeezing
saturates to a level set by the ratio of the mechanical heating
rate to κ. Note that if one attempts to describe the effect of thecavity on the mechanical resonator via an effective Lindblad
master equation, one misses this saturation; cf. Fig. 2(a) (seeAppendix D). An analogous Lindblad approach was recently
analyzed in the context of spin squeezing in Ref. [32].

It follows from Eqs. (4)–(6) that the maximal squeezingof our scheme (at fixed cooperativity C) corresponds toβ̂ † β̂ > 0. The steady state is thus a squeezed thermal state. To
quantify the purity of this state, we define an effective thermal
occupancy from the determinant of the mechanical covariancematrix, i.e.,  
(7)(1 + 2neff )2 = 4 X̂12 X̂22 − 4{X̂1 ,X̂2 }2 .
The mechanical state is a pure squeezed vacuum state if
neff = 0, while the mixedness of the state increases with neff .
As shown in Fig. 2(b), for moderately strong C, one can bothachieve √squeezing beyond 3 dB and a low-entropy state, withneff ∼ ( 2 − 1)/2 ≈ 0.2 (independent of all parameters; see
Appendix B). This again is in marked contrast to coherent
parametric driving, where the maximal squeezing of 3 dB is
associated with a diverging neff . This is also in contrast to thesqueezing generated by a BAE measurement and feedback,where strong squeezing is also associated with neff  1(cf. [18] and Appendix C).
As we increase C further, we enter the strong-coupling
regime where G  κ,M and the cavity and the Bogoliubov

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KRONWALD, MARQUARDT, AND CLERKPHYSICAL REVIEW A 88, 063833 (2013)

mode hybridize. The squeezing saturates in this regime [cf.
Eq. (6)], whereas neff increases without√ bound. For optimized
couplings, we find n2eff ∼ (M /2κ) (1 + 2nth )C in the large-C
limit. Thus, while one can get enhanced squeezing in thestrong-coupling regime, it comes at the price of strongly
reducing the purity of the squeezed state. It is also worthemphasizing that, as shown in Fig. 2(b), the Lindblad masterequation approximation fails to describe accurately both thequadrature squeezing and neff at strong coupling; this is not
surprising as the Lindblad approach cannot describe the hybridization physics important in this regime; cf. Appendix D.
VII. SQUEEZING DETECTION

Squeezing of X1 can be detected by making a singlequadrature backaction-evading measurement. One generates
the squeezed state as described above by having G+ < G− .
To then measure the squeezing, one simply increases G+
so that G+ = G− , thus allowing a QND measurement of
X1 [16,18,38]. The measurement must be fast compared to
the rate at which the mechanical dissipation rethermalizes X1 ,leading to the condition C  2nth + 1 [18].
A simpler method for verifying squeezing is to keep G+ <G− and use the cavity output spectrum to extract the occupancyof the β̂ mode: photons in the output are a measure of how “hot”
this mode is. This is analogous to how mechanical temperature
is obtained in sideband cooling setups [40,41]. The output
spectrum is given by†S[ω] = dt eiωt δ âout (t)δ âout (0).
√Here, δ âout = âout − âout  and âout + âin = κ â [39], where
we have assumed an ideal, single-sided cavity. We find in thegood cavity limit (cf. Appendix E)G2β̂ † β̂.
dω S[ω] = 8π κ 2(8)
4G + κ(κ + M )Thus, knowing the red and blue driving strength G± as well as
the cavity decay rate κ  M is sufficient to measure β̂ † β̂.
The knowledge of β̂ † β̂ is enough to find a rigorous upperbound of the squeezing, sinceX̂12   e−2r [1 + 2β̂ † β̂](see Appendix E). For large C, this upper bound coincides
with√the actual value of X̂12  (up to corrections ∼ 1/ C).
VIII. EFFECTS OF COUNTERROTATING TERMS

We now turn to the effects of the counterrotating terms
in Eq. (2) which can play a role when one deviates fromthe extreme good cavity limit κ  . These additional terms
cause the cavity to nonresonantly heat the Bogoliubov modeβ̂; as a result, the coupling-optimized quadrature squeezing
becomes a nonmonotonic function of the cooperativity C.
This is shown in Fig. 3, where we have solved the fullquantum Langevin equations, and we took parameters froma recent experiment in microwave cavity optomechanics [42].
Steady-state squeezing beyond 3 dB still exists even for only
moderately resolved sidebands (/κ ∼ 5). To estimate theonset of the nonresonant heating, we can calculate the leading

FIG. 3. (Color online) Squeezing vs C for optimized driving
strengths, using realistic experimental parameters [42], and the fulltheory. Yellow and white dashed lines show the theory including
effects due to a nonzero sideband parameter [i.e., no rotating-wave
approximation, in contrast to Fig. 2(a)]. The black dashed line
shows the expectation for κ/  = 0. Thus, squeezing beyond 3 dB is
expected for state-of-the-art experiments. (Parameters: yellow lines:M /κ = 10−4 , κ/  = 1/50; white dashed lines: M /κ = 10−5 ,κ/  = 1/5.)

O[(κ/ )2 ] correction to X̂12 . Insisting it be much smaller
than the smallest variance possible in the extreme good cavity
limit (and taking C  1) leads to the following condition on
the cooperativity:


 2
κ
3/2
.
(9)
C  1 + 2nth
M κThis condition also ensures that the previous results for the
optimized coupling strengths remain valid. Further discussionof bad cavity effects (as well as parameters relevant to recent
optical-frequency optomechanics experiments) are presented
in Appendix F.
IX. CONCLUSION

We have shown that large steady-state squeezing of amechanical resonator can be achieved by driving an optomechanical cavity at both the red and blue mechanicalsideband, with different amplitudes. For realistic parameters,
steady-state quantum squeezing well beyond the 3 dB limit canbe generated. By adding a final-state transfer pulse, our scheme
could also be used to generate strong optical squeezing. It isalso general enough to be realized in other implementations ofparametrically coupled bosonic modes (e.g., superconducting
circuits).
Note added: Very recent work by Didier, Qassemi, and Blais
analyzes an alternative dissipative squeezing mechanism [43].
ACKNOWLEDGMENTS

We thank N. Didier and A. Blais for drawing our attention to
the possibility of dissipative squeezing, and we acknowledge
support from the DARPA ORCHID program through a grant063833-4

ARBITRARILY LARGE STEADY-STATE BOSONIC . . .

PHYSICAL REVIEW A 88, 063833 (2013)

from AFOSR, the Emmy-Noether program, the European
Research Council, and the ITN cQOM. A.K. thanks A.A.C.
for his hospitality at McGill.
APPENDIX A: SEMICLASSICAL PICTURE OFSQUEEZING GENERATIONAs described in the main text, one can obtain a semiclassicalunderstanding of the squeezing generation of our schemeby formally eliminating the cavity from the dynamics; weprovide more details on this approach here. In addition to themechanical resonator quadratures X̂1 and X̂2 , we introduce thecavity quadratures by√√Û1 = (d̂ † + d̂)/ 2 and Û2 = i(d̂ † − d̂)/ 2.
Using the Hamiltonian (2), the Heisenberg-Langevin equations take the formMX̂˙ 1 = −(G− − G+ )Û2 −X̂1 + M X̂1,in ,2√
κÛ˙ 2 = (G− + G+ )X̂1 − Û2 + κ Û2,in ,
(A1)
2
and
√
κ
Û˙ 1 = − (G− − G+ ) X̂2 − Û1 + κ Û1,in ,2M˙X̂2 = (G− + G+ ) Û1 −X̂2 + M X̂2,in , (A2)2where we have also introduced quadratures of the input noise
operators d̂in and b̂in . Note that these are two decoupledsets of equations. They immediately let us understand the
backaction-evading limit G+ = G− [16,18]: the cavity Û2quadrature measures the mechanical resonator’s quadratureX̂1 without disturbing its time evolution. The generation ofsteady-state squeezing, however, requires that G+ be slightly
smaller than G− . In this case, the cavity Û2 quadrature stillmeasures X̂1 . However, the cavity Û2 quadrature also acts as
a force on X̂1 ; we can view this as a weak, coherent feedback
force [24] [cf. the first line of Eq. (A1)]. As we now show, thiseffective feedback leads directly to steady-state squeezing.
It is convenient to work in the Fourier domain, where1f [ω] = √dt eiωt f (t).
2πThese equations also imply that the correlations X̂1 X̂2 +X̂2 X̂1  are zero.
The cavity also introduces new noise terms driving eachmechanical quadrature. We can parametrize them by aneffective temperature in the standard way by considering
their magnitude compared to the corresponding cavity-induced
damping:√ i [ω]M+ i [ω] X̂1 [ω] = − κÛ2,in [ω]2G− + G++ M X̂1,in [ω]
(A3)

and
−iω +

√ i [ω]M+ i [ω] X̂2 [ω] = κÛ1,in [ω]2G− − G++ M X̂2,in [ω], (A4)

where the self-energy [ω] = −i(G2− − G2+ )/(κ/2 − iω) is
the same for both quadratures. The imaginary part ofdescribes damping of the mechanical quadratures by the cavity.

κ| [ω]/(G− + G+ )|2,−2 Im [ω](A5)

1 + 2neff,X2 [ω] ≡κ| [ω]/(G− − G+ )|2.
−2 Im [ω](A6)

Taking the low frequency limit, we have
1 + 2neff,X1 [0] =G− − G+,
G− + G+

G− + G+
1 + 2neff,X2 [0] =.
G− − G+(A7)

Thus, while the cavity damps both mechanical quadraturesin the same way, the noise added to the X̂1 quadrature ismuch smaller than the noise added to the X̂2 quadrature(this is different from coherent parametric driving, where thesqueezed X̂1 quadrature experiences extra damping whereasX̂2 experiences extra negative damping). Moreover, the magnitude of the noise added to the X̂1 quadrature is smaller than
the the zero-point noise one would associate with the optical
damping opt ≡ −2 Im [0], i.e., neff,X1 [0] < 0. We thus seethat the cavity effectively acts as a squeezed reservoir, i.e.,a reservoir whose force noise is quadrature squeezed. If thiscavity-induced dissipation dominates the intrinsic mechanicaldissipation, this directly yields squeezing of the mechanicalresonator.
Finally, it is interesting to note that for a fixed C  1, theoptimal ratio of G+ /G− given in Eq. (5) of the main text
can be given a simple interpretation in terms of the effective
optical damping opt = 4G 2 /κ introduced above (with G 2 =G2− − G2+ ). Using the result of Eq. (5), we have
opt 

optimalM (1 + 2nth )4C
1 + 2nthM (1 + 2nth )e2r .

Eliminating the cavity quadratures from the mechanicalequations of motion, we find−iω +1 + 2neff,X1 [ω] ≡(A8)One can easily confirm that this is exactly the rate at which
the β mode is heated by the mechanical bath (in the larger limit). We thus see that the optimal coupling condition
represents a simple impedance matching: the rate at which
the engineered reservoir (the cavity) extracts quanta from theβ mode should match the rate at which it is “heated” by the
intrinsic mechanical dissipation.
APPENDIX B: SQUEEZED-STATE PURITY

In general, the effective thermal occupancy neff (quantifying
the purity of the mechanical state) is defined by  
4 X̂12 X̂22 = (1 + 2neff )2 ,(B1)063833-5

KRONWALD, MARQUARDT, AND CLERKPHYSICAL REVIEW A 88, 063833 (2013)

cf. Eq. (7), since X̂1 X̂2 + X̂2 X̂1  = 0. The two variances interms of the Bogoliubov mode β̂ read 
2 X̂12 = e−2r (1 + 2β̂ † β̂ + β̂ β̂ + β̂ † β̂ † ), 
2 X̂22 = e2r (1 + 2β̂ † β̂ − β̂ β̂ − β̂ † β̂ † ).
(B2)
As discussed in the main text, optimal squeezing involves a
tradeoff between maximizing the squeeze parameter r (which
requires large G+ /G− ) and maximizing the effective coupling
G to the Bogoliubov mode (which requires small G+ /G− ).
The maximum squeezing at fixed C thus corresponds to β notbeing in its vacuum state. Equations (B1) and (B2) thus imply
that neff = 0; cf. Fig. 2. The optimally squeezed state is thus
in general a thermal squeezed state.
As is also discussed in the main text, there is a generalregime where one has large optimized squeezing while at thesame time having an almost pure state. This occurs for the“moderately strong coupling” regime where C  1 while atthe same time G < κ (no strong-coupling hybridization of β̂and the cavity). The latter condition is satisfied for optimized
couplings as long as C is small enough to satisfy

κ 2
1
1
.
(B3)
C
16 M
2nth + 1
In this regime, one can achieve optimized steady-statesqueez√ing well beyond 3 dB and a low entropy: neff ∼ ( 2 − 1)/2 ≈
0.2 (cf. Fig. 2).
To see this, we now focus on the limit of large cooperativities C = 4G2− /(κM ) while keeping away from thestrong-coupling regime. Formally, one can perform this limit
by sending M → 0 (in contrast to the previous discussions)while keeping all other parameters fixed. Then, for G < κ andfor large C we find4G2− (1 + 2nth ) + κ 2 nth 1(1 + 2neff )2 ≈ 2 1 +→2√√
κ 2 1 + 2nthC
(B4)√or neff ≈ ( 2 − 1)/2. Thus, independent of the choice ofparameters, one √always finds that the effective thermal occupancy neff ∼ ( 2 − 1)/2 in the moderately strong-coupling
limit (i.e., C  1 while G < κ). Note that in this regime, theBogoliubov mode β is characterized by β̂ † β̂,β̂ β̂ → 1/4.
This follows from the fact that β̂ β̂ → β̂ † β̂ as C → ∞ andEqs. (B1)–(B4).

also provides specific examples where coherent feedback
control schemes can outperform Gaussian measurement-based
schemes.
For simplicity, we focus on the regime of large cooperatives
C = 4G2− /(κM ) where large squeezing is possible both in ourscheme and the BAE measurement scheme. Since the latter
scheme was only analyzed in the limit of no strong-coupling
effects, we consider the same limit here: we keep G+ ,G− κ while having C → ∞ by taking M → 0. If we simply
focus on the maximum possible squeezing achievable at a
fixed cooperativity, there is no fundamental advantage of ourdissipative scheme over the BAE scheme, as both predict a
scaling: 
2 X̂12 ≈

1 + 2nth.
C

Of course in practice, achieving this value using BAE
measurement and feedback could be very challenging, as it
requires near-ideal measurements and feedback.
However, the advantage of our coherent feedback scheme(even on an ideal, fundamental level) becomes apparent whenstudying the purity of the generated squeezed state. As alreadydiscussed, if we stay out of the strong-coupling regime, themechanical squeezed state is almost in a pure state, with theeffective number of thermal quanta neff tending to ∼ 0.2 forlarge C as per Eq. (B4). In contrast, the BAE measurementplus-feedback scheme yields(1 + 2neff )2 =

√
1 + 2nth C → ∞in the same limit. Thus, our scheme (an example of coherent
feedback) yields a far more pure state than the measurementplus-feedback approach. This represents a significant advantage over the measurement-based approach.
We note that one could improve the state purity achieved in
the BAE measurement scheme by measuring both quadraturesof the cavity output (instead of measuring only the quadraturethat contains information on the coupled mechanical quadrature X̂1 ). One would thus also learn something about thebackaction noise driving the unmeasured quadrature X̂2 . This
would reduce the conditional variance, and thus improve thestate purity. The analogous situation involving the dispersive
measurement of a qubit is well studied; see, e.g., [44].

APPENDIX C: COMPARISON AGAINSTMEASUREMENT-BASED SCHEMES

2. Comparison against stroboscopic measurements

1. Comparison against measurement-based feedback squeezing

As discussed in the main text, the ability of our scheme
to generate large amounts of stationary quantum squeezing
with low entropy indicates that it outperforms what is possible
with a simple coherent parametric driving (i.e., spring constantmodulation). Here, we also suggest that it has significant
advantages compared to schemes for squeezing based ona backaction-evading (BAE) single-quadrature measurement
plus feedback [18]. As our scheme can be viewed as akind of coherent feedback operation, this apparent advantage
is reminiscent of claims made in Ref. [24]. That work

An alternative way of generating mechanical resonatorsqueezing is to perform a stroboscopic QND position measurement as suggested in Ref. [17]. In this scheme, themeasurement rate is modulated in time periodically. The
backaction-evading scheme is basically a stroboscopic schemewith a particular choice for how the measurement rate ismodulated (i.e., sinusoidally). Instead of the cooperativity,
the crucial parameter determining the amount of squeezing
generated is now given by the measurement rate. Based on
the analysis of Ref. [17], the scaling of squeezing and state
purity of the stroboscopic measurement scheme is essentiallythe same as in the backaction-evading scheme.

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3. Comparison against pulsed optomechanics schemes

Let us now compare our dissipative scheme to a pulsed
optomechanical scheme, where large-amplitude pulses of light
driving an optomechanical cavity are used to realize effective
strong position measurements, which, hence, can generatesqueezing [22]. We note that this scheme requires κ  .
Since this scheme effectively realizes a strong position measurement, the amount of√squeezing scales with the parameter(κ/G)2 , where G = g0 NP , and NP is the mean numberof photons per light pulse. This is in marked contrast to our
dissipative scheme, where the cooperativity C determines theamount of squeezing. In addition, the pulsed scheme does
not generate truly stationary squeezing, which is again in
sharp contrast to our dissipative scheme. Finally, we note thatusing the pulsed scheme, it is currently challenging to getsqueezing from this scheme experimentally [45]. However,
our scheme should generate large amounts of squeezing evenfor state-of-the-art experiments; cf. Fig. 3 of our manuscript.
APPENDIX D: EFFECTIVE LINDBLAD
MASTER EQUATIONIn this appendix, we derive an effective Lindblad masterequation which describes the effects of the cavity (theengineered reservoir) on the mechanical resonator. Such an
approach is common in studies of reservoir engineering;in contrast, the approach we use in the main text goesbeyond this approximation. We start by considering the
Hamiltonian (3)Ĥ = −G d̂ † β̂ + H.c.,
where G 2 = G2− − G2+ , β̂ = b̂ cosh r + b̂† sinh r, cosh r =G− /G, and sinh r = G+ /G. Taking the limit of a large
cavity damping rate κ, one can use standard techniques [46]to eliminate the cavity and derive a Lindblad-form masterequation for the reduced density matrix of the mechanicalresonator ρ̂. This takes the formρ̂˙ = M (nth + 1) D[b̂]ρ̂ + M nth D b̂† ρ̂ + opt D β̂ ρ̂,

expression which we have already found using the full theory,
Eq. (5) in the main text. For this optimized coupling and in thelimit of large cooperativity C, the Lindblad approach predicts 2
1 + 2nth.
2 X̂1 ≈C
Comparing with Eq. (6) in the main text, we see that the
Lindblad approach misses the saturation of squeezing toM (1 + 2nth )/κ in the large-C limit. The approximations
used to derive the Lindblad master equation tacitly assume
κ → ∞, and thus neglect the finite rate at which the cavity isable to expel energy extracted from the mechanical resonator.
When focusing on the purity of the squeezed state, we findthat the Lindblad master equation predicts
12nth(1 + 2neff )2 ≈ 2 + √√ .
2nth + 1 C√
This is in strong contrast to the prediction (1 + 2neff )2 ∼ Cof the full theory. This is because the Lindblad master equation
cannot capture strong-coupling effects.
APPENDIX E: CAVITY OUTPUT SPECTRA ANDSQUEEZING DETECTION

Let us consider the cavity output spectrum,†S[ω] = dt eiωt δ âout (t)δ âout (0),where δ âout = âout − âout  and âout + âin =single-sided cavity [39]. We find thatS[ω − ωcav ] =√κ â for an ideal,

16κM [G2+ (nth + 1) + G2− nth ],|N [ω]|2

whereN [ω] = 4G 2 + (M − 2iω)(κ − 2iω).
As shown in Fig. 4, the coupling to the β mode gives rise
to weight in the output spectrum near the cavity resonance

where D[Â] = Âρ̂ Â† − Â† Âρ̂/2 − ρ̂ Â† Â/2. Expressing the
Bogoliubov mode in terms of the original operators b̂ and b̂†yieldsρ̂˙ = [M (nth + 1) + opt cosh2 r]D[b̂]ρ̂+ [M nth + opt sinh2 r]D[b̂† ]ρ̂+ opt cosh r sinh r DS [b̂]ρ̂+ opt cosh r sinh r DS [b̂† ]ρ̂,

(D1)

where opt = 4G 2 /κ and DS [b̂]ρ̂ = b̂ρ̂ b̂ − b̂b̂ρ̂/2 − ρ̂ b̂b̂/2.
The last two terms on the right-hand side (RHS) of thisequation do not conserve the number of mechanical quanta,
and are directly responsible for the generation of squeezing.
This Lindblad master equation is similar to the one discussed in
the context of dissipative preparation of spin squeezed atomic
ensembles [32].
Using Eq. (D1), we can again calculate the optimal value
of G+ which maximizes the squeezing (with other parameters
and G− held fixed). We find that this approach leads to the same

FIG. 4. (Color online) Cavity output spectra. (a) Cavity output
spectrum for weak coupling. The spectrum at ω − ωcav as well as
the area of the spectrum are directly proportional to the occupancyβ̂ † β̂. (b) If we increase the cooperativity, we observe a normal
mode splitting. This is a signature of the “strong-coupling regime,”
where the Bogoliubov mode and the photons hybridize. [Parameters:(a) M /κ = 10−4 , nth = 10, and C = 104 . (b) Same as (a)but C = 106 .]063833-7

KRONWALD, MARQUARDT, AND CLERKPHYSICAL REVIEW A 88, 063833 (2013)

frequency. For weak coupling (G < κ), one has a simple
Lorentzian peak, whereas for strong coupling (G > κ) a
double-peak structure emerges. The condition for this strong
coupling to occur was given in Eq. (B3).
As discussed in the main text [cf. Eq. (8)], one can detect
the squeezing of the mechanical resonator by first measuringβ̂ † β̂ from the integrated output spectrum, and then using
this to bound the variance of the X̂1 quadrature. The general
expression for the X̂1 variance in terms of the Bogoliubov
mode β̂ is given in Eq. (B2). Since in general|β̂ β̂|  β̂ † β̂ + 12
(which can be shown by using the Cauchy-Schwarz inequality), one finds a general upper bound for the squeezing,
2X̂12   2e−2r [1 + 2β̂ † β̂].
In the limit of large cooperativity, and hence large r, one
can also find a lower bound by making use of the theoreticalpredictions (for κ/  = 0). For any value of the squeezing
parameter r, we find that β̂ β̂ ∼ β̂ † β̂. In the limit of large
cooperativities (and hence large r),
nth + 1− 2 e−2r β̂ † β̂.
β̂ β̂ ≈ 1 + 4(E1)2nth + 1
Thus,

where

 
2 X̂12 ≈ e−2r [1 + 4ζ β̂ † β̂],
1 + nthζ =1+ 2
− 1 e−2r  1.
1 + 2nth

FIG. 5. (Color online) (a) Quadrature variance X̂12  as a functionof the blue driving strength G+ /G− including finite sideband
parameter effects. The black curve shows the analytical result whereas
the circles represent findings due to a numerical simulation of thefull Hamiltonian (2). (b) Zoom of (a). We see that the mechanicalresonator can be squeezed beyond the 3 dB limit. (Parameters:M /κ = 10−4 , κ/  = 1/50, nth = 100, and C = 5 × 106 .)the counterrotating terms is small. As we increase the cooperativity, the squeezing parameter r also increases [cf. Eq. (5) forthe case κ/  = 0], such that the counterrotating terms become
more and more important. Since r ≈ ln[4C/(1 + 2nth )]/4 forκ/  = 0 and large C, we find that the smaller nth , the earlier
these corrections become important. When increasing thecooperativity further, maximum squeezing is assumed first,
after which squeezing gets lost again.

(E2)
APPENDIX G: AVOIDING THE
PARAMETRIC INSTABILITY

Using this estimate, we finally find the lower bound, 
2 X̂12  e−2r (1 + 4β̂ † β̂).

It turns out that due to the two-tone driving, the radiation pressure force F ∝ |ā+ e−iω+ t + ā− e−iω− t |2 oscillates attwice the mechanical frequency, since ω± = ωcav ± . In an

APPENDIX F: EFFECTS OF FINITESIDEBAND PARAMETERS

To consider effects due to finite sideband parameters,
we perturbatively solve the quantum Langevin equations forthe cavity and mechanical resonator operators by using thefull, time-dependent Hamiltonian (2), keeping the leading
corrections in κ/ . We also compare with a full numericalsolution of the equations.
In Fig. 5, the quadrature variance X̂12  is shown as afunction of the blue laser driving strength G+ /G− including
bad cavity effects. The black curve depicts our analytical
perturbative expression (which is too lengthy to be reportedhere), whereas the orange circles show the result of a numericalsimulation of (2). Note that X̂12  is strongly nonmonotonic.
We find that a unique optimum of the driving strength G+ /G−
maximizing steady-state squeezing still exists [cf. Fig. 5(b)].
A plot showing the maximized squeezing as a function of
the cooperativity C for fixed sideband parameter κ/  anddecay rates M /κ is shown in Fig. 3, where parameters of astate-of-the-art experiment [42] have been assumed. Figure 6
also shows maximized squeezing as a function of C forexperimental parameters of state-of-the-art photonic crystal
experiments [47].
Let us now discuss the influence of the counterrotating
terms in more detail. For small cooperativities, the effect of

FIG. 6. (Color online) Squeezing as a function of the cooperativity for optimized driving strengths where realistic experimental
parameters are assumed [47]. The yellow line and the white dashedlines show the theory including effects due to a finite sideband
parameter. The black dashed line shows the expectation for κ/  =0. Thus, squeezing beyond 3 dB is expected for state-of-the-art
experiments. (Parameters: M /κ = 10−4 , κ/  = 1/10.)

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PHYSICAL REVIEW A 88, 063833 (2013)

to the Hamiltonian (1). One then finds that the radiationpressure force reads
F ∝ |ā+ e−iω+ t + ā− e−iω− t + ā3 e−iω3 t+iϕ |2 .
When choosing ω3 = ωcav − 3 [38] we find that the component of F oscillating at 2 vanishes if ϕ = π and ā3 = ā+ .
Linearizing the resulting Hamiltonian again and going into
an interaction picture with respect to the free cavity andmechanical resonator Hamiltonian, we find that the third drive
tone gives rise to an additional term,
Ĥnew = G3 e3it [b̂e−it + b̂† eit ]d̂ † + H.c.,(G1)

Ĥdrive,add = α3 (e−iω3 t+iϕ â † + H.c.)to the linearized Hamiltonian (2), where G3 = g0 ā3 = G+ .
The impact of this additional counterrotating term on thegeneration of steady-state squeezing is shown in Fig. 7. In thisfigure, the quadrature variance X̂12  is shown as a functionof the blue laser driving strength G+ /G− including all counterrotating terms. For medium cooperativity, the influence ofthe third tone is negligible. For larger cooperativity, deviations
become visible as G+ → G− . However, the minimum valueof X̂12  is changed little and lies still well beyond 3 dB.
Let us now briefly discuss why the deviations become
apparent as we increase the cooperativity and as we approach
G+ → G− . An increase of the cooperativity C leads toan increase of the squeezing parameter r, such that the
influence of counterrotating terms becomes larger; cf. thediscussion in the previous section. Thus, the influence of
the additional, counterrotating terms (G1) increases withincreasing C. Additionally, to cancel the unwanted frequency
component 2 of the radiation pressure force, we have tochoose G3 = G+ . Thus, as we increase G+ , the magnitude of
the additional Hamiltonian (G1) also increases, leading to alarger perturbation of the steady-state quadrature variance.
Thus, to conclude, we can avoid the parametric instability
by adding a third driving tone while still generating squeezing
well beyond 3 dB.

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FIG. 7. (Color online) Quadrature variance X̂12  as a function of
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