Generating EPR beams in a cavity optomechanical system

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by  Zhang-qi Yin

Generating EPR beams in a cavity optomechanical system
Zhang-qi Yin1,2 and Y.-J Han2
1Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China
2FOCUS center and MCTP, Department of Physics,
University of Michigan, Ann Arbor, Michigan 48109, USA
We propose a scheme to produce continuous variable entanglement between phase-quadrature
amplitudes of two light modes in an optomechanical system. For proper driving power and detun-
ing, the entanglement is insensitive with bath temperature and Q of mechanical oscillator. Under
realistic experimental conditions, we find that the entanglement could be very large even at room
temperature.
9 PACSnumbers: 03.67.Bg,42.50.Wk,07.10.Cm
0
0
2 Entanglement is the key resource of the field of quan- scheme,otherthanpulseinRef. [9]. Themostattractive

n tum information. Light is the perfect medium to dis- feature of our scheme is that the entanglement between
a tribute entanglement among distant parties. Entangled output beams is nearlynotchangedunder differentbath
J light with continuous variable (CV) entanglement be- temperature and Q of the mechanical oscillator. Within

9 tweenphase-quadratureamplitudesoftwolightmodesis the experimentally available parameters[13, 14], we find
2 widely used in teleportation, entanglement swap, dense the maximum two-mode squeezing could be higher than

coding,etc. [1]. Thistypeofentangledstateisalsocalled 16 dB under room temperature. The entanglement of
]
h Einstein-Podolsky-Rosen (EPR) state. The EPR beams formation (EOF) between two modes is larger than 5
p have been generated experimentally by a nondegenerate [15]. Since the coupling efficiency between cavity and
- optical parameter amplifier [2], or Kerr nonlinearity in fiber could be larger than 99% in the WGM cavity sys-
t
n an optical fiber [3]. The later one is simpler and more tem [16], we neglect the coupling induced noises in this
a reliable. TheKerrnonlinearityisusedtogeneratetwoin- paper.
u
dependentsqueezedbeams. With interferenceatabeam
q
splitter, the EPRentanglementis obtainedbetweenout-
[
put beams. However, Kerr nonlinearity in fiber is very

3 weak,whichlimits entanglementbetween output beams.
v
4 It was found that strong Kerr nonlinearity appeared
2 in an optomechanical system consisting of a cavity with
4 a movable boundary [4, 5, 6]. Besides, the single-mode
0
squeezing could be made insensitive with thermal noise
.
1 [5], which makes the scheme very attractive. However,
1 the frequencyofoutputsqueezedbeamscannotbe made
8
identical, which makes interference difficult. Then it
0
was generalized to two-mode schemes in order to gen-
:
v erate EPR beams without interference [7, 8, 9, 10, 11].
i However,they are either verysensitive to thermalnoises
X FIG. 1: (Color online) Experimental setup. Cavity
[7,8,10,11],orrequiringultrahighmechanicaloscillator
r modes a and b, which are driven by four lasers, couple
a Q ∼ 108 to suppress thermal noise effects [9], which is to the mechanical mode am.
2 to 3 orders higher than the present available parame-
ters [12]. The practical scheme to generate EPR beams
in an optomechanical system needs to overcome these
problems.
AsshowninFig. 1,weconsideranoptomechanicalsys-
Inthispaperweproposeapracticalschemetoproduce tem consisting of a WGM cavity with a movable bound-
EPRbeamsinanoptomechanicalsystem,whichconsists ary. There are two cavity modes a and b with the same
of a whispering-gallery mode(WGM) cavity with a mov- frequency but the opposite momentum. They are cou-
ableboundary. Wefindthat,similarlyasthesingle-mode pling with the same mechanicaloscillationmode a and
m
scheme[5],thethermalnoiseinthetwo-modeschemecan driven by four lasers, two from the right-hand side with
be greatlysuppressedbyadiabaticallyeliminating anos- frequenciesω andω ,the othertwofromthe left-hand
L L′
cillator mode. By precisely tuning the laser power and side with frequencies ωL and ωL′. ain and bin are input
detuning,theoscillationmodeisadiabaticallyeliminated lights. aoutandaoutareoutputlights. Twolowermirrors
1 2
and two output sideband modes are entangled. Unlike haveveryhighprobability(>99%)toreflectthe driving
thecavity-freescheme[9],ourschemerequiresmodestos- lasers. Soweneglectthe reflectinginducednoisefora
out
cillator Q. Besides, the output light is continuous in our andb . ThesystemHamiltonianisH =H +H +H ,
out 0 d I
2
where [17, 18, 19, 20] i∆ α +2iη2ω α (α 2+ α 2) γα iΩj =0. Because
j j m j | 1| | 2| −2 j− 2
γ ω , the imaginary part of β can be neglected.
m m
H0 =~ωp(a†a+b†b)+~ωma†mam In th≪e limit ∆j ≫ 2η2wm(|α1|2 +|α2|2), we find αj ≃
Hd =~(Ω2aa+ Ω2bb)e−iωLt+~(Ω2′aa+ Ω2′bb)e−iωL′t+H.c. eΩqju/aqtiγon2s+a4re∆l2ji.neIanritzheedlaims it |α| ≫ |hapi|, the Langevin
HI =~ν(a†b+ab†)+~ηωm(a†a+b†b)(am+a†m). (1) a˙j = −iηωmαj(am+a†m)+(i∆′j − γ2)aj +√γaijn,(5)
Here a, b and a are the annihilation operators for the
m 2
γ
loaprtifcraelqaunendcmy.ecΩhjanwicitahlmjo=deas,,bωipsatnhdeωdmrivairnegtahmeirplaintugdue- a˙m = −iηωmXp=1(α∗pap+αpa†p)−(iωm+ 2m)am
and defined as Ωj = 2 Pj/~ωLτ, where Pj is the input +√γ ain, (6)
laser power and τ =1/pγ is the photon loss rate into the m m
output modes. ν is the couplingstrengthbetween cavity where j =1,2 and ∆′j =∆j +2η2ωm(|α1|2+|α2|2). We
modes a and b. For the WGM cavity system, it ranges suppose ∆′1 < 0 and ∆′2 > 0. We define δ = (∆′2 −
from 100 MHz to 10 GHz [19, 20]. The dimensionless ∆′1)/2−ωm and d=−(∆′1+∆′2)/2.
parameter η = (ωp/ωm)(xm/R) is used to characterize In the limit ωm ≫ δ,d,γ,γm, the Langevin equations
optomechanicalcoupling,withx = ~/mω the zero- (5) and (6) can be simplified as
m m
point motion of the mechanical resonpator mode [21], m a˙ = ida iηω α a γa +√γain,
itseffectivemass,andRacavityradius. IntypicalWGM 1 − 1− m 1 m− 2 1 1
cavWiteydseyfisnteemthsewneofirmndalηm∼od1e0s−a4. =(a+b)/√2anda = a˙2 =−ida2−iηωmα2a†m− γ2a2+√γai2n,
1 2
γ
(aand−Ωb)/√+2Ω. W=e0suaprpeossaettishfieecdo.nTdihteioHnsamthialttoΩniaa−nΩcabn=b0e a˙m =iδam−iηωm(α∗1a1+α2a†2)− 2mam+√γmaimn.
′a ′b (7)
written as
With proper detuning and input power, we can always
H =~(ωp+ν)a†1a1+~(ωp−ν)a†2a2+~ωma†mam tfiunneeththeeFcoauvriiteyr cmoomdpeonamenptslitoufdethαe1in=traαca2vi=ty αfi.eldDbey-
Ω Ω
+~( 21a1e−iωLt+ 22a2e−iωL′t+H.c.) (2) ωa(,tγ) =, we√12cπanR−∞a∞diaeb−aiωti(ct−altl0y)ae(lωim)dinωa.teItnhethae lmimoidte.δW≫e
+~ηωm(a†1a1+a†2a2)(a†m+am), getmam(ω)≃ηωδm(α∗a1+αa†2)− √iγδmaimn. Tmhen we have
dwehteurneinΩg1∆= Ω=aω+Ωbωand Ων2a=ndΩ∆′a−=Ωω′b. Weωde+finνe.tAhes quantum Langevin equations for a1(ω) andγa†2(−ω)
showninFig1. 1,wLit−hbepa−msplitters2andFL′ar−adapyrotator, −iωa1(ω)=−ig′a1(ω)−iga†2(−ω)− 2a1(ω)
we can get the output mode of a1 and a2. We assume +√γai1n(ω)+ a˜maimn(ω),
both cavity and oscillator modes are weakly dissipating (8)
p γ
atratesγ andγm,respectively,whereγm ≪ωm. Wecan −iωa†2(−ω)=ig′a†2(−ω)+iga1(ω)− 2a†2(−ω)
get quantum Langevin equations [22]
+√γa†2in(−ω)− γ˜maimn(ω),
Ω γ p
a˙j = i∆jaj −iηωmaj(am+a†m)−i 2jaj − 2aj(t) where g = η2|α|2ωm2 /δ, γ˜m = (η|α|ωm/δ)2γm, and g′ =
g+d. In Eq. (8), we neglect the phase of α because it is
+√γaijn for j =1,2, (3) not important.
a˙m = −iηωmXj=21a†jaj −(iωm+ γ2m)am+√γmaimn,(4) ~aimnD(ωen)o=te ~aa(aimωnin()ω(ω))=. (cid:0)Waa†21e((−ωgω)e)t(cid:1),th~aeinf(oωll)ow=ing(cid:0)ami2ani1a†n(t(−ωriω)x)(cid:1)eqaunad-
tion (cid:0)− m (cid:1)
where thermal noise inputs are defined as corre-
hlaatminio†n(t),fauimnn†c(tti′o)nis ha=imn†(t),ahimanim(nt(′t))i,aimn=(t′)inmδ(=t − t′0),, where A~a(ω)=√γ~ain(ω)+pγ˜m~aimn, (9)
b0ha,ajitnwh†i(ttfh)o,ranijmno(stct′i)hlileat=tohrehrammijno†ad(lte)o.,cacijuWn†p(eatn′)sciuyp=pnuohsmaeijbnec(tra)v,oiatfyijnt(hmte′)roimde=asl A=(cid:18)−iω+−iγ2g+ig′ −iω+igγ2 −ig′(cid:19).
couple with vacuum bath. Usingboundaryconditionsaojut(ω)=−aijn(ω)+√γaj(ω)
TosimplifyEqs. (3)and(4),weapplyashifttonormal for j =1,2, we can calculate output field as
cnouomrdbienras,tew,haijch→aarej +choαsje,namto→canacmel+alβl.cαnjumanbderβtaerrme cs ao1ut(ω)=G(ω)ai1n(ω)−H(ω)ai2n†(−ω)+I(ω)aimn(ω),
in the transformedequations. We find they should fulfill ao2ut†(ω)=G(ω)∗ai2n†(ω)−H(ω)∗ai1n(−ω)−I(ω)∗aimn(−ω),
the following requirements: β η(α 2 + α 2), and (10)
1 2
≃ − | | | |
3
where G(ω)=(ω2+γ2 +g2 g2 ig γ)/∆(ω), H(ω)= cavity radius is R=38 µm. The dimensionless coupling
∆ig(γω/)∆=(ω(),iIω(ω+)=γ)2(−+i4ωg+2 γ2g−−2.ig′′+−ig)′√γγ˜m/∆(ω), and pnaorisaemdeoteesrnisoηtd≃ec1r0e−a4s.eWtheefimnadxiifm0u<mden≪tanγg,ltehmeetnhterwmitahl
− 2 ′ − strong enough input power. This is because we adiabat-
Let us define the dimensionless position and momen-
tum operators of fields Xjout(ω)= ao1ut(ω)+ao1ut†(−ω) iecffaelclytseolifmthinearmteatlhneoimsee.chAasnsihcaolwminodFeiga.n2d, tshuepptermespsetrhae-
and Pjout(ω)= aojut(ω)−aojut†(−ω(cid:2)) /i, for j =1,2. We(cid:3) turechangedoesnotchangethemaximumentanglement
define the corre(cid:2)lationmatrix of the o(cid:3)utput field as Vij = with proper driving and detuning. But the higher the
(ξ ξ +ξ ξ )/2 ,where ξ =(Xout,Pout,Xout,Pout). We
h i j j i i 1 1 2 2 temperature, the less the entanglement spectrum width.
calculate the correlation matrix with Eq. (10). Up to
The embeded figure shows that the maximum squeezing
local unitary transformation, the standard form of it is
could be larger than 16 dB at room temperature.
n 0 k 0
x 7
0 n 0 k α=1000
VS =kx 0 n −0x , (11) 6 αα==11350000
0 k 0 n 
 − x 
5
where n= (ω2+ γ2 +g2 g2)2+(g2+g2)γ2+[(ω+
g g)2+ γ{2]γγ˜ (24n +1−) /′∆(ω)2,′k = V2 +V2, F4
w′h−ere V14 4= −2mgγ(wm2 + γ4}+|g2 −|g′2)x/|∆(pω)|21,4 V242=4 EO3
2g gγ2 + [(ω +g g)2 + γ2]γγ˜ (2n + 1) /∆(ω)2.
{ ′ ′ − 4 m m } | | 2
This is the symmetric Gaussian state. The EOF for the
symmetric Gaussian states is defined as [15]
1
E =C (n k )log [C (n k )]
F + − x 2 + − x (12) 0
C (n k )log [C (n k )] −2 −1.5 −1 −0.5 0 0.5 1 1.5 2
− − − x 2 − − x ω/γ
where C (x) = (x 1/2 x1/2)2/4. V describes an en- FIG. 3: (Color online) EOF for different cavity mode
−
tangleme±nt state if and±only if n k < 1. Based amplitude α. Here we adopt δ/2π=10 MHz, T =300
x
− K, γ /2π =3.2MHz, and d=0.07γ.
on the standard form of matrix (11), we also find that p
δ2(X +X ) = δ2(P P ) = n k . We define the .
1 2 1 2 x
h i h − i −
two-mode squeezing as S = 10log (n k ).
− 10 − x As shown in Fig. 3, the bigger the cavity mode am-
6 plitude α, the larger the output entanglement. Because
20
α Ω / γ2+4∆2, the output entanglement is pro-
5 33300K0KK Squeeze(dB)10 ptaojnrgt≃iloenmajelnqttoisdsrpivliinttgejdaminptolitutwdoe.syBmumtettrhiec ppeeaakksowfheenn-
4 driving is very strong. The splitting distance is pro-
0
−2 0 2
ω/γ portional to driving power. Increasing driving power
F
O3 can decrease the entanglement too. This is because
E
adiabatical elimination condition ω δ are not valid
≪
2 around peaks for very strong driving. So the driving
power should be neither too big nor too small. For
the specific α and δ, we find there is an optimum d
1
which makes entanglement maximum and the entangle-
ment peaks appear near ω = 0. The optimum d is
−02 −1.5 −1 −0.5 0 0.5 1 1.5 2 d = (η2ω2α2/δ)2+γ2/4 (ηωα)2/δ, corresponding
ω/γ too squpeezing So = −10log10(−4d2o/γ2) and entanglement
FIG. 2: (Color online) EOF for different temperature. whichis obtainedfromEq. (12)with n kx=4(d /γ)2.
o
−
α is 1000, δ/2π=10 MHz, d=0.07γ. The maxima Itisobviousthatthehighertheinputpower,thesmaller
| |
squeezing is larger than 16 dB when T =300 K. the optimum d. In the mean time, we find that decreas-
ing the mechanical Q factor nearly does not change the
We now estimate the bath noise influence in experi- entanglement spectrum if d is around its optimum value
mentally accessible conditions [13]. The cavity resonant and the condition ω /Q δ is fulfilled. Leaving other
m
≪
frequency ω = 2π 300 THz. The oscillator frequency parametersunchanged,Qcouldbeaslowas300. Consid-
p
×
ω =2π 73.5 MHz. The mechanical Q factor is about eringthedifficultyofincreasingthemechanicaloscillator
m
×
30 000. The cavity and oscillator modes decay rates are Q, the above finding makes our scheme more practical.
γ = 2π 3.2 MHz and γ = ω /Q, respectively. The We also test the stability of our scheme. As shown
m m
×
4
in Fig. 4, the optimum d is around 0.07γ if α = 1000, isting, which requires ha†jaji ≪ |α|2. During numerical
δ/2π = 10 MHz. To maintain such high entanglement, calculation, a a is in the order of 103, which is much
†
weneedtopreciselycontroltheddownto0.02γ 2π 60 less than αh2 1i06. The other two approximations are
p4kηHre2zcω.imsde|αids|e2dt.eufiTnnhienedghaiasgnhdde=rder−nivt(a∆inn′1gg+lep∆mow′2e)ne/tr2isi=snr−eee(qd∆ueid1r,+e∼td∆hea2t)m/×to2hr−ee rbpoaetntaidcteainnlgtellywim.|aFiv|noear∼atαipopnrδo1x≫0i3m,ωatt,hiγeomnd,rωiwvmhini≫cghacδm,adnp,lbiγte,uγfdumelfiΩallneisddinaindtdihaee--
sametime. TomaintaintheentanglementashighasFig. order of 1011 Hz,∼which is much lower than the distance
4, the laser spectrum width should be less than 60 kHz betweenadjacentcavitymodes∆ω =c/(Rn ) 5 1012
0
andthedrivingpowerfluctuationshouldbelessthan1%. Hz,wherecisthelightspeedinavacuum,n t∼her×efrac-
0
The lowerentanglementbetweentwobeams isneededto tiveindexofsilica. Thereforetheapproximationthatone
maintain, the larger the optimum d is. Therefore higher laseronlydrivesonecavitymodeisvalid. Laserpoweris
fluctuations of detuning and driving power are allowed. needed in the order of 10 mW, which is available in the
laboratory.
6
d=0.07γ In conclusion, we have proposed a scheme to gener-
5 d=0.09γ ate EPR lights in an optomechanical system. Two side-
d=0.05γ
band modes, which couple with the mechanical mode,
4 are driven by lasers. After adiabatically eliminating the
the mechanical mode, we find that the output sideband
F
O3 modes are highly entangled. The higher power of the
E
drivinglaser,thelargerentanglementoftheoutputlight.
2 To maintainthe entanglement, we need to precisely con-
trol the driving power and laser frequency at the same
time. With proper parameters, the entanglement is in-
1
sensitive to the thermal noise and mechanical Q factor.
We test the scheme by experimental available parame-
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 ters. ThoughinthispaperwefucusonWGMcavitysys-
ω/γ
tems, our scheme can be realized in other optomechani-
FIG. 4: (Color online) EOF for different d. Here we calsystems,aslongasthe mechanicalmodefrequencyis
adopt ωm/2π=73.5 MHz, T =300K,γm =ωm/30 000, much larger than the cavity decay rate.
γ/2π=3.2 MHz, and α =1000.
| |
. We thank Lu-ming Duan for helpful discussions. We
thankYun-fengXiaoandQingAiforvaluablecomments
Before conclusion, we briefly discuss the approxima- on the paper. ZY was supported by the Government of
tions we used. Our scheme needs the steady states ex- China through CSC (Contact No.2007102530).
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