Light-to-matter entanglement transfer in optomechanics

0.4 MB·

by  Eyob A. Sete

Light-to-matter entanglement transfer in optomechanics
Eyob A. Sete,1 H. Eleuch,2 and C.H. Raymond Ooi3
1Department of Electrical Engineering, University of California, Riverside, California 92521, USA∗
2Department of Physics, McGill University, Montreal, Canada H3A 2T8
3Department of Physics, University of Malaya, Kuala Lumpur 50603, Malaysia
We analyze a scheme to entangle the movable mirrors of two spatially separated nanoresonators
via a broadband squeezed light. We show that it is possible to transfer the EPR-type continuous-
variableentanglementfromthesqueezedlighttothemechanicalmotionofthemovablemirrors. An
optimalentanglementtransferisachievedwhenthenanoresonatorsaretunedatresonancewiththe
vibrationalfrequenciesofthemovablemirrorsandwhenstrongoptomechanicalcouplingisattained.
4 Stationaryentanglementofthestatesofthemovablemirrorsasstrongasthatoftheinputsqueezed
1 light can be obtained for sufficiently large optomechanical cooperativity, achievable in currently
0 available optomechanical systems. The scheme can be used to implement long distance quantum
2 state transfer provided that thesqueezed light interacts with the nanoresonators.

v
o
I. INTRODUCTION pendent movable mirrors can be entangled in the steady
N
state as a result of entanglement transfer from the two-

3 Quantum state transfer between two distant parties is mode squeezed light. More interestingly, the entangle-
1 an important and a rewarding task in quantum infor- ment in the two-mode light can be totally transferredto
mation processing and quantum communications. Sev- the relativeposition andthe totalmomentum of the two
]
h eralproposalshavebeen put forwardemployingschemes movable mirrors when the following conditions are met:
p based on cavity quantum electrodynamics (QED) [1–3]. 1) the nanoresonators are resonant with the mechanical
- More recently quantum state transfer in quantum op- modes, 2) the resonator field adiabatically follows the
t
n tomechanics,wheremechanicalmodesarecoupledtothe motion of the mirrors, and 3) the optomechanical cou-
a optical modes via radiation pressure, has become a sub- pling is sufficiently strong. We alsoshow that the entan-
u
ject of interest [4–9]. In particular, entanglement trans- glement transfer is possible in the nonadiabatic regime
q
[ fer between two spatially separated cavities is appealing (low mechanical quality factor), which is more closer to
in quantum information. Entangling two movable mir- experimental reality. Unlike previous schemes [12, 13],
3 rors of an optical ring cavity [10], two mirrors of two wheredouble-orringcavityisconsidered,ourschemecan
v different cavities illuminated by entangled light beams be used, in principle, for practical test of entanglement
5
[11], and two mirrors of a double-cavity set up coupled between two distant movable mirrors, for example, by
0
2 totwoindependentsqueezedvacua[12]havebeenconsid- connectingthe squeezedsourceto the nanoresonatorsby
5 ered. Recently,entanglingtwomirrorsofaringcavityfed anopticalfibercable. Giventherecentsuccessfulexperi-
. by two independent squeezed vacua has been proposed mentalrealizationofstrongoptomechanicalcoupling[14]
1
0 [13]. This, however, cannot be used to implement long andavailabilityofstrongsqueezingupto 10dB[15],our
4 distance entanglement transfer because the two movable proposal of efficient light-to-matter entanglement trans-
1 mirrors belong to the same cavity. fer may be realized experimentally.
: In this work, we propose a simple model to entangle
v
i the states of two movable mirrors of spatially separated
X
nanoresonatorscoupledtoacommontwo-modesqueezed
r vacuum. Thetwo-modesqueezedlight,whichcanbegen-
a
erated by spontaneous parametric down-conversion, is II. MODEL
injected into the nanoresonators as biased noise fluctua-
tions with nonclassicalcorrelations. The nanoresonators
are also driven by two independent coherent lasers (see Weconsidertwonanoresonatorseachhavingamovable
Fig. 1). The modes of the movable mirrors are coupled mirrorandcoupledtoacommontwo-modesqueezedvac-
to their respective optical modes and to their local envi- uumreservoir,for example,fromthe outputofthe para-
ronments. Ouranalysisgoesbeyondtheadiabaticregime metric down converter. One mode of the output of the
[11]byconsideringthemoregeneralcaseofnonadiabatic squeezed vacuum is sent to the first nanoresonator and
regime and asymmetries between the laser drives as well the other mode to the second nanoresonator. The mov-
as mechanical frequencies of the movable mirrors. Us- able mirror Mj oscillates at frequency ωMj and interacts
ing parameters from a recent optomechanics experiment with the the jth optical mode. The jth nanoresonator
[14], we show that the states of the two initially inde- is also pumped by external coherent drive of amplitude
ε = 2κ P /~ω , where κ is the jth nanoresonator
j j j Lj j
dampipng rate, Pj the drive pump power of the jth laser
and ω is its frequency. The schematic of our model
∗ Correspondingauthor: [email protected] systemLjis depicted in Fig. 1. The system Hamiltonian
2
Squeezed
light source

(cid:1839)(cid:2869) + + (cid:1839)(cid:2870)
F F
(cid:1832)(cid:2869) (cid:2013)(cid:2869) (cid:1832)(cid:2870) (cid:2013)(cid:2870)
, Coherent ,
laser sources
(cid:1853)(cid:2869)(cid:2925)(cid:2931)(cid:2930) (cid:1853)(cid:2870)(cid:2925)(cid:2931)(cid:2930)
FIG.1. Schematicsoftwonanoresonatorscoupledtoatwo-modesqueezedlightfromspontaneousparametricdown-conversion.
The output of the squeezed source is incident on the resonators as noise operators F1 and F2 (see text for their correlation
properties). The first (second) nanoresonator movable mirror M1(M2) is coupled to the nanoresonator mode of frequency
ωr1(ωr2) viaradiation pressure. Thenanoresonator are also drivenbyan externalcoherent laser lasers of amplitudeεj. Inthe
strong optomechanical coupling regime, the states of the two movable mirrors can be entangled due to the squeezed light. A
Faraday isolator F is used to facilitate unidirectional coupling. The output fields a1,out and a2,out can be measured using the
standard homodynedetection method to determine theentanglement between themirrors.
has the form (~=1) where the movable mirrors are damped by the ther-
mal baths of mean number of photons n =
th,j
2 [exp(~ω /k T ) 1]−1. Thesqueezedvacuumoperators
H = [ωMjb†jbj +ωrja†jaj +gja†jaj(b†j +bj) F andMFj† hBavejt−he following non vanishing correlation
Xj=1 j j
properties [13]:
+(a†jεjeiϕje−iωLjt+ajεje−iϕjeiωLjt)], (1)
F (ω)F†(ω′) =2π(N +1)δ(ω+ω′), (6)
h j j i
where ω is the jth nanoresonator frequency, ϕ is the
rj j F†(ω)F (ω′) =2πNδ(ω+ω′), (7)
ipshtahseeosifntghleejpthhoitnopnutopfiteoldmaenchdagnjic=al(cωoruj/pLlijn)gp, w~/hMichjωdMej- hhFj1(ω)F2j(ω′)ii=2πMδ(ω+ω′−ωM1 −ωM2), (8)
scribes the coupling of the mechanical mode with the F†(ω)F†(ω′) =2πMδ(ω+ω′ ω ω ), (9)
intensity of the opticalmode [16], whereL is the length h 1 2 i − M1 − M2
j
of the jth nanoresonator and M is the mass of the jth where N = sinh2r and M = sinhrcoshr with r being
j
movable mirror; ω is the frequency of the jth coher- the squeeze parameter for the squeezed vacuum light.
Lj
ent pump laser; b is the annihilation operator for the
j
jth mechanical mode while a is the annihilation oper-
j
ator for the jth optical mode. Using the Hamiltonian III. LINEARIZATION OF QUANTUM
(1), the nonlinear quantum Langevin equations for the LANGEVIN EQUATIONS
optical and mechanical mode variables read [4, 5, 12]
The coupled nonlinear quantum Langevin equations
γ [Eqs. (2)and(3)]areingeneralnotsolvableanalytically.
b˙j =−(iωMj + 2j)bj −igja†jaj +√γjfj, (2) Toobtainanalyticalsolutiontotheseequations,weadopt
the following linearization scheme [17]. We decompose
the mode operators as a sum of the steady state average
κ
a˙j =−( 2j−i∆j)aj−igjaj(b†j+bj)−iεjeiϕj+√κjFj, (3) aanndd ba fl=uβctu+atδiobn, qwuhaenrteuδma oapnedraδtboraarseaojpe=ratαojrs+. Tδhaej
j j j j j
mean values α and β are obtained by solving Eqs. (2)
where γ is the jth movable mirror damping rate, ∆ = j j
j j
and (3) in the steady state
ω ω is the laser detuning, f is noise operator de-
Lj − rj j
scribing the coupling of the jth movable mirror with its
own environment while Fj is the squeezed vacuum noise αj ≡haji= κ−/i2εjeiiϕ∆j′ , (10)
operator. Note that Eq. (3) is written in a frame rotat- j − j
ing with ω . We assume that the mechanicalbaths are
Lj
Markovian and have the following non zero correlation
ig α 2
j j
properties between their noise operators [17, 18]: β b = − | | , (11)
j j
≡h i γ /2+iω
j Mj
f (ω)f†(ω′) =2π(n +1)δ(ω+ω′), (4)
h j j i th,j where ∆′ = ∆ g (β +β∗) is the effective detuning,
f†(ω)f (ω′) =2πn δ(ω+ω′), (5) whichincjlude tjhe−disjplajcemejntofthe mirrorsdue to the
h j j i th,j
3
radiation pressure force. The contribution from the dis- IV. ENTANGLEMENT ANALYSIS
placement of the movable mirrors is proportional to the
intensity of the nanoresonator field, n¯j αj 2. In prin- In order to investigate the entanglement between
ciple,wecanarbitrarilychoosethedetun≡in|gs|∆′j provide the states of the movable mirrors of the two spatially
that we are away from the unstable regime [18]. separated nanoresonators, we introduce two EPR-type
Using aj = αj +δaj and bj = βj +δbj, Eqs. (2) and quadrature operators for the mirrors, namely their rela-
(3) can be written as tivepositionX andthetotalmomentumY: X =X X
1 2
and Y = Y + Y , where X = (δ˜b + δ˜b†)/√2−and
γ 1 2 l l l
δb˙j =−(iωMj + 2j)δbj +Gj(δaj −δa†j)+√γjfj,(12) Yl = i(δ˜b†l −δ˜bl)/√2. We apply entanglement criterion
[21] for continuous variables which is sufficient for non-
κ
δa˙j =−( 2j −i∆′j)δaj −Gj(δb†j +δbj)+√κjFj,(13) Gaussian states, and sufficient and necessary for Gaus-
sian states. According to this criterion, the states of the
where j gj αj =gj√n¯j is the many-photon optome- movable mirrors are entangled if
G ≡ | |
chanicalcoupling. Since the phase ofthe coherentdrives
can be arbitrary, for convenience we have chosen the ∆X2+∆Y2 <2. (17)
phase of the input field to be ϕ = arctan(2∆′/κ )
j − j j Thusformaximallyentangledstates(EPRorBellstates)
sothatα = iα . Noticethatthelinearizedequations
j j
− | | the total variance becomes 0, while for separable states
(12) and (13) can be described by an effective Hamilto-
nian (~=1) the sum of the variances will be equal or greater than 2.
2
H=Xj=1hωMjδb†jδbj −∆′jδa†jδaj A. Adiabatic regime
+i (δa δa†)(δb +δb†) (14) An optimal quantum state transfer (in this case from
Gj j − j j j i the two-mode squeezed vacuum to the mechanical mo-
with a new effective many-photon optomechanical cou- tion of the mirrors) is achieved when the nanoresonator
pling Gj, which is stronger than the single photon cou- fields adiabatically follow the mirrors, κj ≫ γj,Gj [12],
pling gj by a factor of √n¯j. The effective Hamiltonian which is the case for mirrors with high-Q mechanical
(14) describes two different processes depending on the factor and weak effective optomechanical coupling. (In
choice of the laser detuning ∆′ [16]. Here we want em- fact the condition κ γ can also be expressed as
j j j
≫
phasizethatωMj ≫γj and∆j ≫κj sothatwecanapply ωrj ≫ ωMj(Qrj/QMj).) Inserting the steady state solu-
the rotating wave approximation. The latter condition tionof (16)into(15),weobtainequationsdescribingthe
is the case when the resonators are strongly off-resonant dynamics of the movable mirrors
withthelaserfields. When∆′ = ω ,withintherotat-
idnugcewsatvoeapp=roxiimat2ion,t(hδeajinδtbe†r−acδtMaio†jδnbH)a,mwhilitcohniiasnrerlee-- δ˜b˙j =−Γ2jδ˜bj +qΓajF˜j +√γjf˜j, (18)
HI − j=1Gj j j− j j
vantforquantumPstatetransfer[4,5]andcooling(trans- where Γj = Γaj +γj with Γaj = 4Gj2/κj being the ef-
ferring of all thermal phonons into cold photon mode) fective damping rate induced by the radiation pressure
[20]. In quantum optics, it is referred to as a ’beam- [22].
splitter’ interaction. Whereas, when ∆′ = +ω (in ro- First, let us consider the variance of the relative posi-
j Mj
tatingwaveapproximation),theinteractionHamiltonian tion of the two mirrors ∆X2, which can be expressed
takes a simple form = i 2 (δa δb δa†δb†), as ∆X2 = X2 X 2. Since the noise operators
whichdescribesparamHeItric−amPpljifi=c1aGtjionijntejra−ctionj anjd correspondingh toith−ehtwoi-mode squeezed vacuum Fj as
can be used for efficient generation of optomechanical well as the movable mirrors baths fj have zero mean
squeezing and entanglement. In this work, we are in- values, it is easy to show that X = 0. Therefore,
terested in quantum state transfer and hence choose ∆X2 = hX12i + hX22i − hX1X2ih− hiX2X1i. To evalu-
∆′ = ω . Thus, for ∆′ = ω and in a frame ro- ate these correlations, it is more convenient to work in
tajting−witMhjfrequencyω j(neg−lectMinjgthefastoscillating frequency domain. To this end, the Fourier transform of
Mj
terms), one gets Eqs. (18) yields
δ˜b˙j =−γ2jδ˜bj +Gjδa˜j +√γjf˜j, (15) δ˜b (ω)= ΓajF˜j(ω)+√γjf˜j(ω). (19)
δa˜˙j = κjδa˜j jδ˜bj +√κjF˜j, (16) j p Γj/2+iω
− 2 −G
TheexpectationvalueofthepositionX ofthefirstmov-
1
where we have introduced a notation for operators: o˜= able mirror can be expressed as
oexp(iω t).
IntheMfojllowingsectionwe use these equationsto ana- X2 = 1 ∞ ∞ dωdω′ei(ω+ω′)t X (ω)X (ω′) .
lyzetheentanglementofthestatesofthemovablemirrors h 1i 4π2 Z Z h 1 1 i
−∞ −∞
via entanglement transfer. (20)
4
2.0 tical nanoresonators coupled to two-mode squeezed vac-
uum. Wealsoassumetheexternallaserdrivestohavethe
2 1.5 samestrengthandthe thermalbathsofthe twomovable
Y
2D+DX 1.0 r0.5 mntihr)r.orTsotothbiseeantdt,hseetstainmgeΓt1em=pΓer2a=tuΓre, Γ(nat1h,=1 =Γa2nt=h,2Γ=a,
0.5 1.0 M1 = M2, ωr = ωr1 = ωr2, ωM = ωM1 = ωM2,
κ = κ = κ , and γ = γ = γ, and using the rela-
2.0 1 2 1 2
0.0 tion N = sinh2r,M = sinhrcoshr, the variance of the
0 200 400 600 800
relative position (25) takes a simple form
THΜKL
2Γ 2γ
FIG. 2. Plots of the sum of variances ∆X2+∆Y2 vs bath ∆X2+∆Y2 = a e−2r+ (2nth+1)
γ+Γ γ+Γ
a a
temperature T of the movable mirrors for drive laser power
P = 10 mW and frequency ωL = 2π×2.82×1014 Hz(λ = = 8e−2rG2/γκ+2+4nth
1064 nm), mass of the movable mirrors M1 = M2 = 145 ng, 4 2/γκ+1
frequency of the nanoresonator ωr = 2π ×5.26×1014 Hz, 2 G 2(1+2n )
length of the cavity L = 125 mm, the mechanical motion = C e−2r+ th , (26)
damping rate γ = 2π ×140 Hz, ωM = 2π ×947×103 Hz, C+1 C+1
nanoresonator damping rate κ=2π×215×103 Hz, and for where = 4 2/γκ = 4n¯g2/γκ is the optomechani-
different values of the squeezing parameter r: 0.5 (blue solid C G
cal cooperativity [23]. In the absence of the two-mode
curve), 1.0 (red dashed curve), and 2.0 (green dotted curve).
The bluedashed line represents ∆X2+∆Y2 =2. squeezed vacuum reservoir r = 0, Eq. (26) reduces to
∆X2+∆Y2 =2+4n /( +1), which is always greater
th
C
than 2, indicating the mechanical motion of the two
Using the correlation properties of the noise operators mirrors cannot be entangled without the squeezed vac-
[Eqs. (4)-(9)], we obtain uum. This is because the motion of the mirrors are ini-
tiallyuncorrelatedandtheirinteractionviavacuumdoes
1 Γ γ
X2 = (2N +1) a1 + 1 (2n +1). (21) not create correlations. In the limit 1 (a weaker
h 1i 2 Γ1 2Γ1 th,1 condition [24] for strong coupling regCim≫e), the sum of
the variances can be approximated by ∆X2 + ∆Y2
Similarly, it is easy to show that
≈
2exp( 2r)+4n / . Therefore,when4n / <1,which
th th
1 Γ γ − C C
X2 = (2N +1) a2 + 2 (2n +1). (22) can be achieved for sufficiently large number of photons
h 2i 2 Γ2 2Γ2 th,2 in the nanoresonator, the sum of the variances can be
2 Γ Γ less than 2 when
X X = X X = a1 a2M. (23)
h 1 2i h 2 1i pΓ1+Γ2 1
r> ln[1/(1 2n / )], (27)
th
Therefore,using Eqs. (21)-(23), the variance of the rela- 2 − C
tive position of the movable mirrors becomes
indicatingtransferofthe quantumfluctuationsofthein-
put fields to the motion of the movable mirrors. This
∆X2 = 1(2N +1) Γa1 + Γa2 4 Γa1Γa2M canbeinterpretedasentanglementtransferfromlightto
2 (cid:18) Γ Γ (cid:19)− pΓ +Γ
1 2 1 2 mechanical motion. The interesting aspect is that this
γ γ
+ 1 (2n +1)+ 2 (2n +1). (24) quantum state transfer scheme can, in principle, be ex-
th,1 th,2
2Γ 2Γ
1 2 tendedtolongdistance statetransferifthe twonanores-
It is easy to show that the variance of the total momen- onators are kept far apart but connected by, for exam-
tumofthemovablemirrorsisthesameasthatofX,i.e., ple, an opticalfiber cable to the output of the two-mode
∆X2 =∆Y2. Thus,thesumofthevariancesofthe rela- squeezed vacuum. Obviously, the entanglement between
tivepositionandtotalmomentumofthemovablemirrors the mirrors would degrade when the distance between
is given by the resonators is increased owing to the decrease in de-
gree of squeezing as a result of environmental couplings.
Recently, similar transfer scheme from light to matter
γ γ
∆X2+∆Y2 = 1(2nth,1+1)+ 2(2nth,2+1) has been proposed [1, 25, 26].
Γ Γ
1 2 For realistic estimation of the entanglement between
+(2N +1) Γa1 + Γa2 8 Γa1Γa2M. (25) the movable mirrors, we use parameters from recent
(cid:18) Γ Γ (cid:19)− pΓ +Γ experiment [14]: laser frequency ω = 2π 2.82
1 2 1 2 L
× ×
1014 Hz(λ = 1064 nm), ω =2π 5.64 1014 Hz (ω =
r r
× ×
2ω ), M = M = 145 ng, L = 25 mm, κ = 2π 215
L 1 2
1. Identical nanoresonators 103Hz, γ = 2π 140 Hz, ω = 2π 947 ×103 H×z.
M
× × ×
In Fig. 2, we plot the sum of the variances of X and
To elucidate the physics of light-to-matter entangle- Y as a function the temperature of the thermal bath of
ment transfer, we first consider a simplified case of iden- themovablemirrors. Thisfigureshowsthatthe movable
5
4
2.0 r nth
1
0.5
2Y 1.5 1.0 2 3 C=23 5
D Y C=11.6
2+X 1.0 2.0 2+D 2 10
D X
0.5 D C=2.3
1
0.0
0 2 4 6 8 10 0
0 10 20 30 40 50
PHmWL
C
FIG. 3. Plots of the sum of variances ∆X2+∆Y2 vs drive
FIG. 4. Plots of the sum of variances ∆X2 +∆Y2 vs the
pumppowerforthermalbathtemperatureT =50µK ofthe
movable mirrors, ωr =2π×2.82×1014 Hz, and for different optomechanical cooperativity C for squeeze parameter r = 1
and for various values of the thermal bath photon numbers:
valuesofthesqueezingparameterr: 0.5(bluesolidcurve),1.0
(red dashed curve), and 2.0 (green dotted curve). All other nth = 1(T = 62.2µK) (green dotted curve), nth = 5 (T =
parameters as the same as in Fig. 2. The blue dashed line 236 µK) (red dashed curve), and nth = 10 (T = 452 µK)
represents ∆X2+∆Y2 =2. (bluesolid curve).
mirrors are entangled when the nanoresonators are fed greater than 2 independent of the degree of squeezing
with squeezed light. Notice that based on the definition of the input field, indicating no quantum state transfer
of the quadrature operators X and Y, an optomechan- from the squeezed light to the mechanical motion of the
ical quadrature squeezing [18, 19, 23] is achieved when movablemirrors,andhence the mirrorsremainunentan-
∆X2 <1 or ∆Y2 <1. This implies that whenever there gled. Figure 4 shows the plot of the entanglement mea-
is optomechanical squeezing, the two movable mirrors sure vs the optomechanical cooperativity as a function
are always entangled. This shows a direct relationship of the thermal bath photon numbers. For r = 1.0 and
between optomechanical squeezing and entanglement of n=1.0 (62.2 µK) the motion of the two mirrors are not
the mechanical modes of the movable mirrors. entangled up to =2n [1 exp( 2r)]−1 2.3.
th
C − − ≈
It is also interesting to see the dependence of the
mirror-mirror entanglement on the pump laser power
strength. Figure 3 shows that for a given squeeze pa-
2. Effect of asymmetric coherent drives and mechanical
rameter r and the thermal bath temperature T of the frequencies
movable mirrors, there exists a minimum pump power
strength for which the movable mirrors are entangled.
We next analyze the effect of the asymmetries in the
The minimum power required to observe mirror-mirror
strength of coherent drives and in the vibrational fre-
entanglement can be derived from (26) by imposing the
quencies of the movable mirrors. Figure 5a illustrates
condition that ∆X2+∆Y2 <2, which yields
thatforaconstantthermalbathtemperaturesT =T =
1 2
2n 0.25 mK of the movable mirrors and squeeze parame-
th
> . (28)
C 1 exp( 2r) ter r = 2.0, there exist input laser powers P1 and P2,
− − where ∆X2 +∆Y2 is minimum or the entanglement is
Using the explicit form of in = 4 2/γκ, we then the strongest. It turns out that for identical nanores-
G C G
obtain (r =0) onators, strong entanglement is achieved when P =P .
6 1 2
α Notice also that the width of the entanglement region is
P > (1 e−2r)(exp[~ω /k T] 1), (29) mainlydeterminedbytheinputpower: thehigherthein-
M B
− − put powers, the wider the width of entanglement region
where α γωM L2ω [(κ/2)2 + ω2 ]/2ω2 is a factor becomes.Figure5bshowsoptimized∆X2+∆Y2 overthe
≡ 1 M M r
which can be fixed at the beginning of the experiment input power P2 for a given P1 for different values of the
(note here that M = M ). It is easy to see from (29) thermal bath temperatures T and T . As expected the
1 2 1 2
thatforagiventhermalbathtemperatureT ofthemov- entanglementdegradesasthethermalbathtemperatures
able mirrors, increasing r decreases the minimum power of the mirrors increase and the entanglement persists at
required to achieve entanglement between the mirrors. higher temperatures for sufficiently strong pump power
When the number of thermal bath photons increases, strength (see green-dotted curve for T1 =T2 =0.5mK.)
the minimum value of the cooperativity parameter for Tuning the frequencies of the movable mirrors also af-
which the entanglement occurs increases. In the weak fects the degree of the mirror-mirror entanglement. As
couplingregime,wheretheoptomechanicalcooperatively shown in Fig. 6a, for a fixed temperatures of the ther-
is much less than one, 1, the sum of the variances mal bath of the movable mirrors T =T =0.25mK and
1 2
C ≪
(26) that characterize the entanglement can be approxi- squeeze parameter r = 2.0 and drive powers P = P =
1 2
matedby∆X2+∆Y2 2+2 e−2r+4n. Thisisalways 11mWandthefrequencyω ofthefirstmovablemirror,
≈ C M1
6
2.0
2.0
HaL HaL
2Y 1.5 2Y 1.5
D D
2+X 1.0 P1Hm5WL 2+X 1.0 ΩM1(cid:144)2Π7H4k7HzL
D D
0.5 10 0.5 947
15 T1=T2=0.25mK T1=T2=0.25mK 1147
0.0 0.0
0 5 10 15 20 25 0.6 0.8 1.0 1.2 1.4
P2HmWL ΩM2(cid:144)2ΠHMHzL
2.0 2.0
HbL T1=T2HmKL HbL
2 1.5 0.5mK 1.5 0.05
Y
D
2+ 1.0 1.0 0.25
X 0.25mK 0.50
D
0.5 0.5
T1=T2=0.05mK
0.0 0.0
0 2 4 6 8 10 12 0.2 0.4 0.6 0.8 1.0 1.2
P1HmWL ΩM1(cid:144)2ΠHMHzL
FIG. 5. (a)∆X2 +∆Y2 vs the input drive power P2 of the FIG. 6. (a)∆X2+∆Y2 vs the vibrational frequency ωM of
2
secondnanoresonatorandforvariousvaluesoftheinputdrive thesecond nanoresonatorandforvariousvaluesofthevibra-
power P1 of the first nanoresonator and assuming the same tionalfrequencyωM ofthefirstnanoresonatorandassuming
1
thermalbathtemperaturesofthemovablemirrorsT1 =T2 = theinputlaserpowersP1 =P2 =11mWandsqueezeparam-
0.25 mK and squeeze parameter r = 2.0. (b) ∆X2 +∆Y2 eter r = 2.0. (b) ∆X2 +∆Y2 vs the vibrational frequency
vstheinputdrivepowerofthefirstnanoresonator optimized ωM of the first nanoresonator optimized over ωM and for
1 2
over the input power of the second nanoresonator and for different values of T1 and T2 and squeeze parameter r =2.0.
different values of T1 and T2 and squeeze parameter r =2.0. All other parameters are the same as in Fig. 2. The blue
All other parameters are the same as in Fig. 2. The blue dashed line in both figures represents ∆X2+∆Y2 =2.
dashed line in both figures represents ∆X2+∆Y2 =2.
(30) and the properties of the noise operators (4)-(9),
there exists a frequency ω of the second movable mir- the sum of the variances of the relative position X and
M2
rorforwhichtheentanglementismaximum. Thesmaller total momentum Y of the movable mirrors (for identical
ω is, the stronger the entanglement becomes. The op- nanoresonators)is found to be
M1
timumentanglementdecreaseswithincreasingfrequency
2 κ e−2r 2(2n +1) γ
ωM1 ofthefirstmovablemirrorandeventuallydisappears ∆X2+∆Y2 = C + th 1+ C .
atsufficientlylargeω andrelativelyhightemperatures +1 κ+γ +1 κ+γ
M1 C C (cid:2) (cid:3)
(see Fig. 6b.) (31)
We immediately see that for κ γ, , Eq. (31) reduces
≫ G
to the expression (26) derived in the adiabatic approxi-
B. Nonadiabatic regime mation. In general, for the dissipation rate of the mov-
ablemirrorsγ comparabletotheresonatordecayκ ,the
j j
expression (31) can be significantly different from (26).
So far we have discussed the mirror-mirror entangle-
In Fig. 7 we present a comparison showing the en-
ment induced by the squeezed light in the adiabatic
tanglement transfer in the adiabatic and nonadiabatic
regime (κ γ , ). We next derive a condition for
j j j
≫ G regimes. The main difference comes from the mechan-
entanglement valid for both adiabatic and nonadiabatic
ical dissipation rate γ. Since the adiabatic approxima-
regimes. We also study the field-field entanglement in
tion assumes negligible mechanical dissipation rate, the
the regime where the two mirrors are entangled.
transfer is more efficient than the non adiabatic case.
The dynamics of the movable mirrors in the nonadia-
This however is an ideal situation, which requires very
batic regime is described by the coupled equations (15)
high mechanical quality factor. In general, for low me-
and (16). Solving the Fourier transforms of these equa-
chanicalqualityfactorthe mechanicaldissipationcanbe
tions yields
significant,leadingto a less efficiententanglementtrans-
fer. As can be noted from Fig. 7, the mirror-mirror
δ˜bj = κjd/2(+ω)iω√γjf˜j + dG(jω)√κjF˜j, (30) ecnaltadnisgsliepmaetinotndriamtienγis/hκesinwchreeansetshefrnoomrm0.a0l1izetdo 0m.0e5ch.aWnie-
j j
note that when the dissipation rate increases, large co-
where d (ω)= 2+(γ /2+iω)(κ /2+iω). Thus using operativity (strong coupling) is required to observe the
j Gj j j
7
2.0 333...555 33..55
333...000 n =5 33..00
2 1.5 (cid:87)(cid:75)
Y
D 222...555 22..55
2+ 1.0
DX Adiabatic 2y222...000 22..00(cid:19)(cid:58)
0.5 Nonadiabatic,Γ(cid:144)Κ=0.05 (cid:36)(cid:12)111...555 (cid:38)(cid:3)(cid:32)(cid:3)(cid:20)(cid:24) 11..55 (cid:36)(cid:1)(cid:12)
Nonadiabatic,Γ(cid:144)Κ=0.01 2x (cid:19)(cid:1)(cid:57)
0.0 (cid:36)111...000 (cid:38)(cid:3)(cid:32)(cid:3)(cid:22)(cid:19) 11..00 (cid:36)
0 10 20 30 40 50
C 000...555 (cid:38)(cid:3)(cid:32)(cid:3)(cid:28)(cid:19) 00..55
000...000 00..00
FIG. 7. Plots of the sum of the variance of the quadra- 000...000 000...555 111...000 111...555 222...000
ture operators X, Y for the mirror versus the optomechan- rrr
ical cooperativity parameter in the adiabatic regime [(26)]
(red solid curve) and in the nonadiabatic regime [Eq. (31)] FIG. 8. Plots of the sum of the variance of the quadrature
for γ/κ = 0.01 (black dashed curve) and 0.05 (black dot- operatorsX andY forthemirror∆X2+∆Y2[Eq. (31)](red
dashed curve). Here we used nth = 5 and squeeze parame- curves with different C values) and the quadrature operators
ter r = 2. The blue dashed line in both figures represents x,yforthefield∆x2+∆y2[Eq. (34)](bluesolidcurve)vsthe
∆X2+∆Y2 =2. squeezeparameterrfordifferentvaluesoftheoptomechanical
cooperativity parameter C = 15 (red solid curve), 30 (red
dashedcurve),and90(reddottedcurve). Hereweusedγ/κ=
mirror-mirrorentanglement. 6.5 × 10−4, nth = 5. The blue dashed line shows ∆x2 +
To gain insight into the transfer of entanglement from ∆y2=∆X2+∆Y2=2 below which thestationary states of
the squeezed light to the motion of the mirrors, it is the movable mirrors as well as the nanoresonator modes are
important to study the entanglement between the op- entangled.
tical modes of the nanoresonators. This can be ana-
lyzedbyintroducingtwoEPR-typequadratureoperators
x=x x and y =y +y , where x =(δa˜ +δa˜†)/√2 while the entanglementof the states ofthe movable mir-
1− 2 1 2 l l l
and y = i(δa˜† δa˜ )/√2. The optical modes of the rors increases as the optomechanical coupling becomes
l l − l stronger or increases (Fig. 8). It is interesting to see
nanoresonatorsare entangled if
C
thatforsufficientlystrongcoupling(largevaluesofcoop-
∆x2+∆y2 <2. (32) erativity, ), the entanglement between the states of the
C
movablemirrorscanbe asstrongasthatofthe squeezed
Solving the Fourier transforms of Eqs. (2) and (3), we light. Therefore, in addition to choosing the mechanical
obtain frequency to be ∆′ = ω and adiabatic approximation
M
−
(κ γ, ), it is imperative to attain strong coupling
δa˜j(ω)= Gj √γ1f˜j(ω)+ γj/2+iω√κjF˜j, (33) regi≫me toGachieve the maximum entanglement between
−d (ω) d (ω)
j j the states of the movable mirrors.
Experimentally, the entanglement between the states
where d (ω) = 2 +(κ /2+iω)(γ /2+iω). The sum
j Gj j j of the movable mirrors can be measured by monitoring
of the variances of x and y for identical nanoresonators
the phase andamplitude [10] ofthe transmittedfield via
reads
the method of homodyne detection, in which the signal
∆x2+∆y2 = 2C(2nth+1) γ is brought into interference with a local oscillator that
+1 γ+κ serves as phase reference. For other variants of optical
C
measurementschemesseeRef. [16]. Withtheavailability
κ 1 γ
+2 + e−2r (34) [15] of strong squeezing sources up to 10 dB squeezing
(cid:18)κ+γ 1+ γ+κ(cid:19)
C (90%) below the standard quantum limit, our proposal
which for the case γ/κ 1 and strong coupling regime can be realized experimentally.
≪
( 1) reduces to
C ≫
γ
∆x2+∆y2 2(2n +1) +2e−2r. (35) V. CONCLUSION
th
≈ γ+κ
We note from (35) that in the strong coupling regime, In summary, we have analyzed a scheme to entan-
the field-field entanglement is mainly determined by the gle the vibrational modes of two independent movable
thermal bath temperature and squeeze parameter, not mirrors and spatially separated nanoresonators via two-
on the value of . For experimental parameter in Ref. mode squeezed light. We showed that in the regime of
[10]we haveγ/κC=6.5 10−4 andassumingthe thermal strongcoupling 1(4 2 κγ)andwhenthenanores-
× C ≫ G ≫
bath mean photon number n =5, the field-field entan- onator field adiabatically follows the motion of the mir-
th
glementisinsensitivetothe increaseofthe cooperativity rors,thequantumfluctuationsofthetwo-modesqueezed
8
light is transferred to the motion of the movable mir- ACKNOWLEDGMENTS
rors, creating stationary entanglement between the vi-
brational modes of the movable mirrors. It turns out
thatanentanglementofthestatesofthemovablemirrors
as strong as the entanglement of the two-mode squeezed
light can be achieved for sufficiently large optomechani-
cal cooperativity or equivalently for sufficiently strong EAS acknowledges financial support from the Office
C
optomechanicalcoupling. Wealsoconsideredalessstrin- of the Director of National Intelligence (ODNI), Intel-
gent condition–nonadiabatic regime which is more real- ligence Advanced Research Projects Activity (IARPA),
istic thanthe adiabaticapproximationandstillobtained through the Army Research Office Grant No. W911NF-
entanglement transfer from the two-mode light to the 10-1-0334. All statements of fact, opinion or conclusions
movablemirrors. Giventhe recentsuccessfulexperimen- containedhereinarethose ofthe authorsandshouldnot
talrealizationofstrongoptomechanicalcoupling[14]and be construedas representingthe officialviews orpolicies
well-developed method of homodyne measurement, our of IARPA, the ODNI, or the U.S. Government. He also
proposalforefficientlight-to-matterentanglementtrans- acknowledge support from the ARO MURI Grant No.
fer may be realized experimentally. W911NF-11-1-0268.
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