Dissipation-driven two-mode mechanical squeezed states in optomechanical systems

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by  Huatang Tan

Dissipation-driven two-mode mechanical squeezed states in optomechanical systems
Huatang Tan,1,2 Gaoxiang Li,1 and P. Meystre2
1Department of physics, Huazhong Normal University, Wuhan 430079, China
2B2 Institute, Department of Physics and College of Optical Sciences, The University of Arizona, Tucson, AZ 85721
Inthispaper,weproposetwoquantumoptomechanicalarrangementsthatpermitthedissipation-
enabledgenerationofsteadytwo-modemechanicalsqueezedstates. Inthefirstsetup,themechanical
oscillatorsareplacedinatwo-modeopticalresonatorwhileinthesecondsetupthemechanicaloscil-
latorsarelocatedintwocoupledsingle-modecavities. Weshowanalyticallythatforanappropriate
choice of the pump parameters, the two mechanical oscillators can be driven by cavity dissipation
intoastationary two-modesqueezedvacuum,providedthatmechanical dampingisnegligible. The
effect of thermal fluctuations is also investigated in detail and show that ground state pre-cooling
3 oftheoscillators in notnecessary forthetwo-modesqueezing. Theseproposals can berealized ina
1 numberof optomechanical systems with current state-of-the-art experimentaltechniques.
0
2 PACSnumbers:

n
a
J I. INTRODUCTION double advantage of being independent of specific initial
states and of leading to steady states robust to decoher-
4
2 Quantum squeezing and entanglement have been ob- ence.
served in a number of atomic and photonic systems and Here,weproposetoexploitcavitydissipationtogener-
h] are expected to play an increasing role in applications ate the steady-statetwo-mode squeezing oftwo spatially
separated mechanical oscillators. These states are also
p ranging from measurements of feeble forces and fields
- to quantum information science [1–4]. For example, it entangledstatesofcontinuousvariables,abasicresource
t in quantum information precessing. We propose specifi-
n has been know for over three decades that squeezed vi-
a brational states are of importance for the measurement callytwodifferentsetups. Inthefirstonetwomechanical
u oscillatorsareplacedinsideatwo-modeopticalresonator,
beyond the standard quantum limit of the weak signals
q expected to be produced e.g. in gravitational wave an- while inthe secondone the mechanicaloscillatorsarelo-
[
cated in two separate single-mode cavities coupled by
tennas [5].

1 Although achieving such effects in macroscopic sys- photon tunneling. The cavities are driven in both cases
v tems remains a major challenge due in particular to the by amplitude-modulated lasers. In the first model we
8 show analytically that for appropriate mechanical oscil-
increasing rate of environment-induced decoherence [6],
9 lator positions and pump laser parameters the mechani-
recent progress toward the ground-state cooling of mi-
6
cal oscillators can be driven into a stationary two-mode
5 cromechanical systems [7–11] may change the situation
squeezedvacuumbycavitydissipation,providedthatme-
. significantly in the near future. In particular, the char-
1 chanicaldamping is negligible. Inthe secondcase atwo-
acterizationofquantumground-statemechanicalmotion
0
step driving sequence can likewise give rise to a steady
[12],thequantumcontrolofamechanicalresonatordeep
3
two-modemechanicalsqueezedvacuum state. The effect
1 in the quantum regime and coupled to a quantum qubit
of thermal fluctuations on the resulting squeezed states
: [13], and the demonstration of optomechanical pondero-
v is also investigated in detail.
motive squeezing [14] are important steps toward the
i
X broad exploration of quantum effects in truly macro- The remainder of this paper is organized as follows.
Section II introduces our first model of two mechanical
r scopic systems [15–23]. Several proposals have been put
a oscillators coupled by radiation pressure to the field of a
forward to generate mechanical squeezing in an optome-
common two-mode optical resonator. It then discusses
chanical oscillator, including the injection of nonclassi-
how to prepare a two-mode mechanical squeezed state.
cal light [24], conditional quantum measurements [25],
Section III shows that cavity damping can likewise be
and parametric amplification [26–29]. In all cases, deco-
exploited to create a squeezed mechanical state of two
herence and losses are a dominant limiting factor in the
mechanical oscillators in separated single-mode cavities.
amount of squeezing that can be achieved.
Finally, Section IV is a conclusion and outlook.
A new paradigm in quantum state preparation and
controlhas recently receivedincreasedattention. Its key
aspectisthatitexploitsquantumdissipationinthe gen-
II. MECHANICAL OSCILLATORS IN A
eration of specific quantum states. Quantum reservoir
SINGLE TWO-MODE CAVITY
engineeringhasbeenproposedtopreparedesirablequan-
tumstates[30–33]andperformquantumoperations[34],
and the creation of steady-state entanglement between A. Model and equations
two atomic ensembles by quantum reservoir engineering
hasbeenexperimentallydemonstrated[35]. Thisdissipa- We first consider an extension of the “membrane-in-
tive approachto quantumstate preparationpresentsthe the-middle” arrangement of cavity optomechanics [36]
2
where two mechanical oscillators, modeled as vibrating
dielectric membranes of identical frequencies ω , are lo-
m
cated inside a driven two-mode optical resonator, as il-
lustrated in Fig. 1. The mechanical modes are charac- A2
terized by the bosonic annihilationoperatorsCˆ and the
j
ctiaovnitoypmeroadtoerss, Aaˆt.frTeqhueesnecmieosdωescjar(ejd=riv1e,n2)b,ybyamapnlnitihuidlae-- C1 A1 C2
j
modulated lasers of frequencies ω . The Hamiltonian of
lj
the driven cavity-oscillatorssystem reads (~=1) FIG. 1: Schematic plot of two vibrating membranes (Cj)
placed at appropriately-chosen positions in a driven cavity
H = ωcjAˆ†jAˆj +ωmCˆj†Cˆj +gjkAˆ†jAˆj(Cˆk+Cˆk†) with two frequency-nondegenerateresonant modes (Aj).
j,kX=1,2n
+ iEj(t)e−iωljtAˆ†j −iEj∗(t)eiωljtAˆj , (1) where
o
where (t) are the time-dependent amplitudes of the
Ej ∆j =δj +gj(βj +βj∗),
pump lasers. Their specific forms will be given later.
The single-photon optomechanical coupling constants with δ = ω ω , and the effective optomechanical
j cj lj
gjk = ωLcj√fjmkω(x¯mk) [36], where L is the length of the cav- coupling strengt−hs are given by
ity, m are the identical masses of the membranes, and
fjk(x¯k) = √12−RRk2ksicno(s22k(c2jkx¯ckj)x¯k). Here Rk and x¯k are the χj(t)=gjαj(t). (7)
reflection coefficients and equilibrium positions of the The density matrix ρ˜ of the sub-system composed of
j
two membranes and kcj the wave numbers of the cav- the cavity mode aj and the normal mode bj satisfies the
ity modes. master equation
For an appropriate combination of cavity length and
mpoesmsibbrleanteo pfionsditaiosnistuoaftitohnesmucehmtbhraatneths,es“eseymFimg.e1t,riictali”s ρ˜˙j(t) = −i[H˜jlin,ρ˜j]+ κ2j(2aˆjρ˜jaˆ†j −aˆ†jaˆjρ˜j −ρ˜jaˆ†jaˆj)
γ
asantdis“fyantthiesyemqumaelittriiecs” optomechanical coupling strengths + 2m(n¯th+1)(2ˆbjρ˜jˆb†j −ˆb†jˆbjρ˜j −ρ˜jˆb†jˆbj)
γ
g = g =g , + 2mn¯th(2ˆb†jρ˜jˆbj −ˆbjˆb†jρ˜j −ρ˜jˆbjˆb†j), (8)
11 12 1
g = g =g , (2)
21 22 2 where κ is the cavity dissipation rate and γ the me-
− j m
chanical damping rate, taken to be the same for both
as discussed in Ref. [37]. Introducing then the new
oscillators. The mean thermal phonon number is n¯ =
bosonic annihilation operators (e~ωm/kBT 1)−1, with kB the Boltzmann constantthand
−
Bˆ =(Cˆ +Cˆ )/√2, (3a) T the temperature.
1 1 2
Bˆ =(Cˆ Cˆ )/√2, (3b)
2 1 2
−
B. Stationary two-mode mechanical squeezed
the Hamiltonian H becomes H = jH˜j, where vacuum via cavity dissipation
H˜j = ωcjAˆ†jAˆj +ωmBj†Bˆj +PgjAˆ†jAˆj(Bˆj +Bˆj†) We now show how cavity dissipation can be exploited
+ iEj(t)e−iωljtAˆ†j −iEj∗(t)eiωljtAˆj. (4) tvoacpuruemparoefththeestvaitbiorantairnygtmwoe-mmbordaenems.echTaonicthailssqeunedezwede
IntermsofthenormaloperatorsBˆ ,thesystemisthere- consider an effective optomechanical coupling strength
j
χ (t) of the form
fore decoupled and reduces to two independent single- j
membrane optomechanical systems.
Further decomposing the operators Aˆj and Bˆj as χj(t)=χj1e−i(Ωjt−φj)+χj2, (9)
Aˆ = Aˆ (t) +aˆ α (t)+aˆ , where χj1 and χj2 are constants. This situation can be
j j j j j
h i ≡ realized by a pump laser of the form discussed in Sec-
Bˆ = Bˆ (t) +ˆb β (t)+ˆb , (5)
j j j j j tion IIC.
h i ≡
Forweakoptomechanicalcouplingwehave∆ δ ,so
andwith |αj(t)|2 ≫haˆ†jaˆji and|βj(t)|2 ≫hˆb†jˆbji one can that if the cavity-laser detuning δj and the mojd≃ulajtion
linearize the Hamiltonians H˜j to get frequency Ωj are
H˜jlin =∆jaˆ†jaˆj+ωmˆb†jˆbj+[χ∗j(t)aˆj+χj(t)aˆ†j](ˆbj+ˆb†j), (6) δj =ωm, Ωj =2ωm, (10)
3
the transformation aˆj aˆje−i∆jt and ˆbj ˆbje−iωmt describes quantum-state transfer between the cavity
→ →
reduces the Hamiltonian H˜jlin to mode aj and the mechanicalmode bj in the transformed
picture. It can be inferred from that master equation
H˜jlin = (χj1e−iφjˆbj +χj2ˆb†j)aj (11) that in the transformed picture and for γm =0 the cav-
ity mode a and the mechanical mode b asymptotically
+ (χj1e2iωmt−iφjˆbj +χj2e−2iωmtˆb†j)aˆj +H.c. decay to thje ground state j
This form can be further simplified for a mechanical fre-
̺ ( )= 0 0 0 0 . (20)
quency ω χ , in which case the rapid oscillating j ∞ | aj bjih aj bj|
m jk
terms e±2iωm≫t can be neglected and Simplyreversingtheunitarytransformation(13)wehave
then that in the steady-stateregime the normalmode b
H˜jlin ≃(χj1e−iφjˆbj +χj2ˆb†j)aˆj +H.c. (12) is indeed in the squeezed vacuum state j
This Hamiltonian describes the coupling of the cavity ρ˜ ( )=Sˆ(ξ )0 0 Sˆ (ξ ). (21)
field a to the normal mode b simultaneously via para- bj ∞ j | bjih bj| † j
j j
metric amplification – through the term proportional to Itispossibletoadjusttheamplitudeχ andphaseφ
jk j
χj1–andviaabeamsplitter-typecoupling–throughthe of the optomechanical coupling coefficient (9) in such a
constantcontributionχj2. Itisknownthatthefirstterm waythatthe squeezingparameterssatisfythe conditions
in the Hamiltonian (12) leads to photon-phonon entan-
glement and phononic heating of the normal mode bj, r ≡rj, (22a)
while the second term results in quantum state transfer φ φ =φ π, (22b)
1 2
≡ −
between the cavity mode a and the mechanical mode
j
bj as well as to mechanical cooling (cold damping). To in which case the two normal modes b1 and b2 exhibit
ensurethe stabilityofthesystem,the couplingstrengths the same amount of steady-state squeezing, but in per-
must satisfy the inequality χ > χ , that is, cooling pendicular directions. With Eqs. (3) and (21) we then
j2 j1
should dominate over anti-damping. have
When absorbing a laser photon the parametric am-
plification term results in the simultaneous emission of ρc1c2(∞)=Sˆ12(ξ12)|0c1,0c2ih0c1,0c2|Sˆ1†2(ξ12) (23)
a photon into the cavity mode a and a phonon in the
j where we have introduced the two-mode squeezing oper-
normalmode b , while the beam-splitter interactioncor-
j ator
respondstotheannihilationofaphotonandtheemission
of a phonon. We now show that the combined effect of Sˆ12(ξ12)=exp(−ξ1∗2cˆ†1cˆ†2+ξ12cˆ1cˆ2), (24)
these processes is the generation of steady-state single-
mode squeezing of the normal mechanical mode bj, or and ξ12 = re−iφ. This two-mode squeezed vacuum of
equivalentlyoftwo-modesqueezingoftheoriginalmodes the mechanical oscillators can be thought of as the out-
c and c . put from a 50:50 beam splitter characterizedby the uni-
1 2
We proceed by first performing the unitary transfor- tary transformation (3), the two inputs being the nor-
mation malmodes bj squeezedin perpendicular directions. This
shows that for γ 0 the dissipation of the intracavity
m
̺j(t)=Sˆj†(ξj)ρ˜j(t)Sˆj(ξj), (13) fieldcanbeexploit→edtopreparepuretwo-modemechan-
ical squeezed vacuum state. The minimum time t re-
min
where the squeezing operator
quired for preparing such states can be evaluated from
the eigenvalues of Eq (17),
Sˆ(ξj)=exp −ξj∗ˆb†j2/2+ξjˆb2j/2 , (14)
with h i η = κj κ2j G2.
j± − 2 ±s 4 − j
ξj = rje−iφj, (15)
rj = tanh−1(χj1/χj2). (16) Fonoersfiynmdms tetric=pa4r/aκmfeotrerGsκ=κ/κ2j.,χj =χjk,andGj =G
min
≥
For γ =0 the master equation (8) becomes then
m
κ
̺˙j(t)=−i[Hjlin,̺j]+ 2j(2aˆj̺jaˆ†j −aˆ†jaˆj̺j −̺jaˆ†jaˆj), C. The choice of pump lasers
(17)
Next we consider the choice of pump laser amplitudes
where
(t) that result in the time-dependent optomechanical
j
Hjlin =Gj(aˆ†jˆbj +ˆb†jaˆj), (18) Ecoupling(9),andthespecificform(22)thatresultsinthe
pure two-mode mechanicalsqueezing (23). FromEq. (7)
with
an obvious starting ansatz is
Gj =χj2 1−(χj1/χj2)2, (19) Ej(t)=Ej1e−i(Ωjt−ϕj1)+Ej2eiϕj2, (25)
q
4
where ϕ are the initial phases of two components with D. Thermal fluctuations
jk
amplitudes and Ω are modulation frequencies. The
jk j
E
corresponding amplitudes αj(t) and βj(t) of the cavity Forfinitemechanicaldamping,γm =0,themechanical
and mechanical modes are modesc andc arenolongerbeinap6 uresqueezedstate,
1 2
d but rather in a two-mode squeezedthermal state. In the
dtαj(t) = −[κ+iδj +igj(βj +βj∗)]αj +Ej1e−i(Ωjt−ϕ1j) state, we find from Eqs. (3) and (8)
d + Ej2eiϕj2, hcˆ†jcˆji∞ =d0d1cosh2r−d0d2sinh2r+sinh2r, (31a)
dtβj(t) = −(γm+iωm)βj −igj|αj|2. (26) hcˆ1cˆ2i∞ =[−(d0d1+1/2)sinh2r+d0d2cosh2r]e(i3φ1,b)
It is difficult to find exactsolutions of these equations in
general. For the case of the weak optomechanical cou- where d = n¯ cosh2r+sinh2r, d = (n¯ + 1)sinh2r,
1 th 2 th 2
plingstrengthsg ,however,approximateanalyticalsolu- and
j
tions can be found by expanding the amplitudes α and
j 4κG2
βj in powers of gj as αj = α(j0) +α(j1) +α(j2) +··· and d0 =1− (κ+γm)(κγm+4G2). (32)
β = β(0) + β(1) + β(2) + . Substituting these into
j j j j ··· The quantum correlations between the mechanical os-
Eqs. (26) and for the time t 1/κ, ω γ , δ =ω ,
m m j m
≫ ≫ cillatorscanbequantifiedbythesumofvariances[3,38]
and Ω = 2ω one then finds (higher-order corrections
j m
can be derived straightforwardly) ∆ = (Xˆθ1 +Xˆθ2)2 + (Xˆθ1+π2 Xˆθ2+π2)2 , (33)
ε ε EPR h 1 2 i h 1 − 2 i
α(j0) = κ2j+1 ωm2 e−i(Ωjt−φj)+ κ2j+2 ωm2 , (27a) whereXˆjθj =(cˆje−iθj+cˆ†jeiθj)/√2arequadratureopera-
torswithlocalphaseθ . Avalue of∆ <2 isasigna-
α(j1) =0p, βj(0) =0, βj(2) =0, p (27b) tureofEinstein-Podolsjky-Rosen(EPRE)P-tRypecorrelations
α(j2) = 2ig3j2ωEj2((κ22Ej+21ω+23)E2j22)e−i(φj−ϕj1) breestpwoenedninthgetotwthoemidecehalanqiucaanltmumodmese,cwhiatnhic∆alElPimRi=t[03]c.oIrn-
m m ourcasewefindthat∆ isminimumforthechoice
2ig2 (3 2 +2 2 ) EPR,min
+ jEj1 Ej1 Ej2 e−i[Ωjt−(2φj−ϕj1)] of local phases θ1+θ2 = φ. In the long-time limit it is
3ωm(κ2+ωm2 )2 equal to
2ig2 2
− 3ω (κj2Ej+1Eωj22 )2ei(Ωjt−ϕj1) ∆EPR,min =2e−2r(1−d0)+2(2n¯th+1)d0. (34)
m m
2ig2 2 We first observe that for κ = 0 we have d0 = 1 and
− 3ω (κ2+ω2jE)j31/E2(jκ2 3iω )e−2i(Ωjt−φj), (27c) hence ∆EPR(∞)=4n¯th+2,showingthat inthe absence
m m − m of cavity dissipation the oscillators are just prepared in
βj(1) =−ωgj((Eκj212++Eωj222)) − 3ωgj(Eκj21+Ej2ω2 )ei(Ωjt−φj) tthoethteheernmviarlonstmateenti.mTphoissedcobnyfirtmhesitrhmatecchavanitiycadliscsoipuaptliionng
m m m m
is an essential component of this realization of steady-
g
+ jEj1Ej2 e−i(Ωjt−φj), (27d) state two-mode mechanical squeezing. From Eq. (34),
ω (κ2+ω2 )
m m steady-state squeezing requires that
with the phases
1 d
n¯th <n¯th,max = − 0 1 e−2r . (35)
φ = ϕ +ϕ , 2d −
j j1 j2 0
(cid:0) (cid:1)
ϕj2 = arctan(ωm/κ). (28) Inpractice,itisimportanttobeabletooperateatashigh
anumberofmeanthermalphononsaspossible. Thiscan
For coupling strengths g κ,ε ,ω , for example
for gj ∼10−6ωm, κ∼0.05ωjm≪, a{ndEjjkk ∼m1}04ωm, we find bateivaechpiaevraemdebtyerd”ec4rGea2sing1d.0Fwrhoimle Ekeqe.p(in3g2)thaend“cfooorptehre-
|1α0(j2)4|ω∼.1W≪e|cαa(jn0)|th∼er1e0fo4raensdetδj∆∼=ωmδ ≫+ggj((ββj ++ββj∗)) ∼ realistic case κ≫γκγmm, d≫0 reduces approximately to
− m j j j j j∗ ≃ γ κγ
δ , and the effective optomechanical coupling strength m m
j d + , (36)
χ (t) g α(0) takes the required form (9) for 0 ≃ κ 4G2
j ≃ j j
indicating that by increasing the coupling frequency G
g
j jk while keeping the ratio χ /χ fixed it is possible to in-
χ = E . (29) 1 2
jk
κ2+ω2 crease the value of n¯ which is approximately given
m th,max
by
Finally, Eqs. (28) and (22bp) give
4κχ (χ χ )
1 2 1
ϕ21−ϕ11 =π, ϕj2 =arctan(ωm/κ). (30) n¯th,max ≃ γm[κ2+4(χ−22−χ21)]. (37)
5
Hamiltonian is
Hlin = ∆aˆ†jaˆj +ωmcˆ†jcˆj +[χ∗(t)aˆj +χ(t)aˆ†j](cˆj +cˆ†j)
A1 A2 j
X
C1 C2 + J12(aˆ1aˆ†2+aˆ†1aˆ2), (40)
(a) where
χ(t)=gα(t) (41)
(b)
FIG.2: (a)Twovibratingmembranes(Cj)inseparatesingle- andthedetuning∆≃(ωc−ωl)fortheweakoptomechan-
mode cavities (Aj) are optically coupled by either an optical ical coupling. By introducing the new bosonic operators
fiber. (b)Twoevanescent-wave-coupledoptomechanicalcrys-
tal nanocavities.
ˆb =(cˆ cˆ )/√2, (42a)
1,2 1 2
±
dˆ =(aˆ aˆ )/√2, (42b)
1,2 1 2
As a concrete example, for a cavity dissipation rate of ±
κ 0.05ω , a mechanical damping γ 10 4ω , and the Hamiltonian separates as before into the sum of two
m m − m
co∼upling strengths χ 0.01ω and χ∼ 0.03ω , we uncoupled Hamiltonians, H = H˜lin, where
1 ∼ m 2 ∼ m lin j=1,2 j
have n¯ 70. This indicates that two-mode me-
th,max
chanicalsquee∼zing is robustagainstthermal fluctuations H˜jlin =∆jdˆ†jdˆj +ωmˆb†jˆbj +[χ∗(t)Pdˆj +χ(t)dˆ†j](ˆbj +ˆb†j),
and ground-state precooling of the mechanical modes
(43)
may not be necessary.
witheffectivedetunings∆ =∆+ and∆ =∆ .
1 12 2 12
J −J
The dynamics of the two independent optomechanical
sub-systems are governed by the same master equations
III. MECHANICAL OSCILLATORS IN
as before, see Eq. (8).
SEPARATE SINGLE-MODE CAVITIES
In contrast to the preceding case, though, in the two-
cavity setup it is not possible to simultaneously achieve
In this section we show that the same dissipative
thesingle-modesqueezingofthenormalmodesb andb
1 2
approach can also be utilized to generate a two-mode
since there is only one driving laser field and one set of
squeezed state of two mechanical oscillators in sepa-
control parameters (ω ,Ω,ϕ ). However, for sufficiently
l j
rate single-mode cavities. The specific example that
weakmechanicaldecoherenceitispossibleinprincipleto
we consider consists of two identical single-mode cavi-
implement a two-step process that can still achieve that
ties of frequency ω , optically coupled to via fiber (a)
c goal.
or evanescent-wave coupling (b), see Fig. 4. Each cavity
For the first step we choose the frequencies ω and Ω
l
field is driven by a modulated laser and optomechani-
such that
cally coupledto amechanicaloscillatoroffrequency ω .
m
AdoptingthesamesymbolsasbeforetheHamiltonianof ∆ = ω ω + =ω ,
1 c l 12 m
that system reads − J
Ω = 2ω , (44)
m
H = ωcAˆ†jAˆj +ωmCˆj†Cˆj +gAˆ†jAˆj(Cˆj +Cˆj†) and the phases
jX=1,2n ϕ = φ arctan(ω /κ)
1 1 m
+ iE(t)e−iωltAˆ†j −iE∗(t)eiωltAˆj ϕ2 = arc−tan(ωm/κ), (45)
+ J12(Aˆ1Aˆ†2+Aˆ†1Aˆ2), o (38) where φ1 is arbitrary. With the transformation aˆj
→
where g account for the optomechanical coupling aˆje−i∆jt and ˆbj → ˆbje−iωmt, the Hamiltonians H˜jlin re-
duce to
strengths,assumedtobeidentical,and describesthe
12
J
cpouumpplinfigelbdestwoefeenqtuhaeltfwreoqucaevnictyiesω. Haenrde wtiemceo-dnesipdeenrdtewnot H˜1lin =(χ1e−iφ1ˆb1+χ2ˆb†1)dˆ1
l
amplitude +(χ2e−2iωmtˆb1+χ1ei(2ωmt−φ1)ˆb†1)dˆ1+H.c.
(46a)
E(t)=E1e−iΩt+iϕ1 +E2eiϕ2. (39) H˜2lin =(χ1ei(2J12t−φ1)ˆb1+χ2e−2iJ12tˆb†1)dˆ1
For that symmetrical situation, both cavity fields have +(χ1e2i(J12−ωm)tˆb1+χ2ei[2(J12+ωm)t−iφ1]ˆb†1)dˆ1
+H.c. (46b)
the same amplitude α (t) = α(t) and the linearized
j
6
For χ ω , , ω , the non-resonantterms with
j m 12 12 m
≪{ J |J − |}
in these Hamiltonians can be neglected and they reduce
tsroietsupHa˜et1cliitnoinv≃ealys.(χeF1noec−roiutφhn1etˆbe1mre+oddχien2ˆbb†11S)edtˆc1htii+soniHsI.fIcoB.rm.anNadlelygHl˜et2hclitneins≃gama0es, ̺th,bi = nXb∞i=0(n¯i+n¯in1b)inbi+1|nbiihnbi|, i=1,2
ξ = (r r˜)e iφ,
before the mechanical damping, γ = 0, cavity dissipa- 2 −
m −
tionbringslikewisethatnormalmodeintoasteady-state 1 2d0(d1+d2)+1
r˜ = ln , (53)
single-mode squeezed vacuum for long enough time, and 4 2d (d d )+1
0 1 2
−
atthesametimemodeb simplydecaysintothevacuum,
2
i.e., with the mean thermal excitation number n¯1 = n¯th and
n¯ = (d d +1/2)2 d2d2 1/2. Note however that
2 0 1 − 0 2 −
ρ˜ (t t )=Sˆ(ξ )0 0 Sˆ (ξ ), (47a) a two-step procedure is not required in that situation
ˆb1 ≥ min 1 | b1ih b1| † 1 sinceapsinglestepwithlaserparameterssatisfyingeither
ρ˜ (t t )= 0 0 . (47b)
ˆb2 ≥ min | b2ih b2| Eq. (44) [(48)] will result in the preparation of mode b1
[or b ] in a squeezed thermal state while the other mode
where t , as defined before, is a time long enough that 2
min in the thermal state (52a), with the degree of EPR cor-
steady state has been reached. After the first step the
relations between the mechanical modes c and c
mechanical oscillators c and c are therefore prepared 1 2
1 2
in a pure two-mode squeezed state, although not a stan- ∆ ( )=e 2r(1 d )+(2n¯ +1)(1+d ).
dard two-mode squeezed vacuum. That latter goal can EPR,min ∞ − − 0 th 0
(54)
be achieved in a second step by changing the frequency
and phase of the pump laser at time tmin so that Hence, the maximum mean number of thermal phonons
n¯ for which squeezing can be achieved is
th,max
∆ = ω ω =ω ,
2 c l 12 m
− −J
1 d
Ω = 2ωm, (48) n¯th,max = − 0 (1 e−2r), (55)
2(1+d ) −
0
and the phases
which is obviously smaller than that in Eq.(35) because
just one normal mode is now in a squeezed state and it
ϕ = φ arctan(ω /κ)+π,
1 1 m
− is therefore harder to maintain quantum correlations.
ϕ = arctan(ω /κ). (49)
2 m
Fortheweakoptomechanicalcoupling,wethenhavethat
IV. CONCLUSION
H˜lin 0, which means that the mode b evolves freely
1 ≃ 1
in the state of Eq. (47a), while H˜lin (χ e iφ2ˆb +
2 ≃ 1 − 2 Inconclusion,wehaveproposedtwopossiblequantum
χ2ˆb†2)dˆ2 +H.c. After a time t ≥ 2tmin both the normal optomechanical setups to generate two-mode mechani-
modesb1andb2arethereforeinsteady-statesingle-mode cal squeezed states of mechanical oscillators in optical
squeezed vacua, cavities driven by modulated lasers. We showed analyti-
callythatforappropriatelaserpumpparametersthetwo
ρ˜b1(t≥2tmin)=Sˆ(ξ1)|0b1ih0b1|Sˆ†(ξ1), (50a) oscillators can be prepared into a stationary two-mode
ρ˜ (t 2t )=Sˆ(ξ )0 0 Sˆ (ξ ), (50b) mechanical squeezed vacuum with the aid of the cavity
b2 ≥ min 2 | b2ih b2| † 2 dissipation when choosing appropriate positions of the
andthemechanicalmodesc andc areinthetwo-mode membranes in the optical cavities. The effect of ther-
1 2
squeezed vacuum state, mal fluctuations on the two-mode mechanical squeezing
was also investigated in detail, and we showed that me-
ρc1c2(t≥2tmin)=Sˆ12(ξ12)|0c1,0c2ih0c1,0c2|Sˆ1†2(ξ12). cmheacnhiacanlicsaqluoesecziilnlagtoisrsacthoietvhaebirleqwuiatnhtouumt pgrreo-uconodlinstgattehse.
(51)
The present schemes are deterministic and can be im-
When accounting for mechanical damping the two-
plemented in a variety of optomechanical systems with
step preparationscheme remains efficient for mean ther-
current state-of-the-artexperimental techniques.
mal phonon number such that γ n¯ κ so that
m th
[γ n¯ ] 1 t κ 1 and thermal ≪effects can be
m th − min −
≫ ≫
neglected during the state preparation. In case this con-
Acknowledgments
dition is not satisfied, γ n¯ κ, following the second
m th
≥
step the mode b is thermalized while the mode b is in
1 2
a squeezed thermal state, This work is supported by the National Natural
Science Foundation of China (Grant Nos. 11274134,
ρ˜ (t 2t )=ρ˜ , (52a) 11074087 and 61275123), the National Basic Research
b1 ≥ min th,b1 Program of China (Grant No. 2012CB921602), the
ρ˜ (t 2t )=Sˆ(ξ )ρ˜ Sˆ (ξ ), (52b)
b2 ≥ min 2 th,b2 † 2 DARPA QuASAR and ORCHID programs through
7
grants from AFOSR and ARO, the US Army Research also supported by the CSC.
Office,andthe USNationalScienceFoundation. THT is
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