Table Of ContentEntangling two macroscopic mechanical mirrors in a two-cavity optomechanical system
Jie-Qiao Liao,1 Qin-Qin Wu,2 and Franco Nori1,3
1CEMS, RIKEN, Saitama 351-0198, Japan
2Department of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414000, China
3Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
(Dated: January 8, 2014)
We propose a simple method to generate quantum entanglement between two macroscopic me-
chanical resonators in a two-cavity optomechanical system. This entanglement is induced by the
radiation pressure of a single photon hopping between the two cavities. Our results are analyt-
ical, so that the entangled states are explicitly shown. Up to local operations, these states are
two-mode three-component states, and hence thedegree of entanglement can be well quantifiedby
the concurrence. By analyzing the system parameters, we find that, to achieve a maximum av-
4
erage entanglement, the system should work in the single-photon strong-coupling regime and the
1
deep-resolved-sidebandregime.
0
2
PACSnumbers: 03.67.Bg,42.50.Wk,42.50.Pq
n
a
J
Quantumentanglement[1,2],asacornerstoneofquan- Inparticular,wewillfocusonthesingle-photonstrong-
7 tum physics, plays an important role in the foundation coupling regime [33–41], in which the radiation pressure
ofquantumtheoryandalsohaspotentialapplicationsin ofasinglephotoncanproduceobservableeffects. Inthis
]
h quantum technology, such as quantum information sci- regime,people havefound that strongphoton nonlinear-
p ence[3]andquantummetrology[4]. Inparticular,howto ity at the few-photon level (e.g., photon blockade) can
- prepare macroscopic mechanical entanglement is of high be induced by the radiation-pressure coupling [33, 38–
t
n interest and significance, because such macroscopic en- 40]; moreover,resolved phonon sidebands and frequency
a tanglement might provide explicit evidence for quantum shifts can be observed in the photon emission and scat-
u
phenomena[5]andevenmightpossiblyhelpustoclarify tering spectra [36]. So, a natural question is whether
q
thequantum-to-classicaltransition,aswellasthebound- a single photon can also induce a considerable entan-
[
ary between classical and quantum worlds [6]. Recently, glement between the two mechanical resonators in the
1 muchattentionhasbeenpaidtothecreationofquantum single-photonstrong-couplingregime. Below,wewillad-
v
entanglement in macroscopic mechanical systems. Some dress this question by analytically solving the dynamics
4
proposals have been brought forward to generate quan- of the system.
8
tum entanglement in various mechanical resonators [7– Specifically, we consider a two-cavity optomechanical
3
1 22]. system, which consists of two optomechanical cavities
. (Fig. 1). Each cavity is formed by a fixed end mirror
1 Ingeneral,wecanclassifythesestate-preparationpro-
and a moving one. We focus on a single-mode electro-
0 posals into two categories, according to the coupling
4 channels: either direct coupling or indirect coupling. In magnetic field in each cavity. This field couples to the
1 the former case, the two mechanical resonators are cou- mechanical motion of the moving mirror via the radia-
: tion pressure coupling. In addition, the fields in the two
v pled to each other directly. In the latter case, some kind
cavitiescouple to eachother via a photon-hoppinginter-
i of an intermediate is needed to induce an effective inter-
X action. Without loss of generality, we assume that the
action between the two mechanical resonators. There-
r fore, the intermediate link should be able to couple with two optomechanical cavities are identical. The Hamilto-
a nian of the system is (~=1)
the mechanical resonators. In this sense, cavity optome-
chanical systems [23–25] can provide a natural platform
H = ω a†a +ω b†b g a†a (b†+b )
tboecianudsuectehaerneiinstaenraicnttiroinnsbicetcwoueepnlinmgemcheachnaicnailsmresboentawteoerns S jX=1,2h c j j M j j − 0 j j j j i
optical and mechanical degrees of freedom. Motivated ξ(a†a +a†a ), (1)
− 1 2 2 1
by this feature, in this paper we propose to study the
generation of macroscopic mechanical entanglement in a where aj (a†j) and bj (b†j) are, respectively, the anni-
two-cavity optomechanical system. This system is com- hilation (creation) operators of the single-mode cavity
posed of two coupled optomechanical cavities. In each field and the mechanical motion of the moving mirror in
cavity, the electromagnetic fields couple to the mechan- the jth (j = 1,2) optomechanical cavity, with respec-
ical motion of one moving end mirror. The connection tive resonant frequencies ωc and ωM. The parameter
between the two cavity fields is built through a photon- g0 = ωcxzpf/L is the single-photon optomechanical cou-
hopping interaction. This photon connection will induce pling strength, where x = 1/(2Mω ) is the zero-
zpf M
an entanglement between the two mechanical mirrors. pointfluctuationofthemovingpmirrorwithmassM,and
We note that somepreviousstudies haveconsideredvar- L is the restlength ofthe cavity. The parameterξ is the
ious entanglements in optomechanical systems [26–32]. photon-hopping coupling strength between the two cav-
2
b a a b the Jaynes-Cummings Hamiltonian under the rotating-
1 1 2 2 wave approximation,
w M w cc x w cc w M HII = ω2M(|0ia1|1ia2 a2h1|a1h0|−|1ia1|0ia2 a2h0|a1h1|),
+g(1 0 1 0b†+ 0 1 0 1b )
FIG. 1. (Color online) Schematic diagram of the two-cavity | ia1| ia2 a2h |a1h | 2 | ia1| ia2 a2h |a1h | 2
+ω b†b . (7)
optomechanical system. Each Fabry-Perot cavity is formed M 2 2
byafixedendmirrorandamovingone. Theelectromagnetic Now, we can solve the dynamics of the system Hamil-
fields in the cavity couple to the mechanical motion of the
tonianH inthesingle-photonsubspace,becausethetwo
S
movingmirrorthroughtheradiation-pressureinteraction,and
Hamiltonians H and H are solvable. The unitary evo-
thefieldsinthetwocavitiescoupletoeachotherviaaphoton- 3 II
lution operator associated with the Hamiltonian H in
hoppinginteraction. S
the single-photon subspace can be expressed as
U(t)=e−iHSt =V†V†V†e−iH3tV e−iHIItV V . (8)
ities. We note that some previous studies have consid- 1 2 3 3 2 1
ered multicavity optomechanical systems with only one Based on this unitary evolution operator U(t), we can
mechanical resonator [42–48]. calculate the state of the system at any time t once the
To solve the Hamiltonian HS, we first introduce the initialstateisgiven. Toinduceanentanglementbetween
transformation V = exp[π(b†b b†b )], and the trans- thetwomechanicalresonators,weintroduceasinglepho-
1 4 1 2− 2 1
formed Hamiltonian then becomes tonas anintermediate. The initialstate ofthe systemis
assumed to be ψ(0) = 1 0 0 0 . After some
H =V H V† | i | ia1| ia2| ib1| ib2
1 1 S 1 tediouscalculations,thestateofthesystemattimetcan
=ω (a†a +a†a )+ω (b†b +b†b ) be obtained as
c 1 1 2 2 M 1 1 2 2
+g(a†a +a†a )(b† +b )+g(a†a a†a )(b† +b ) ψ(t) =U(t)ψ(0)
1 1 2 2 1 1 2 2− 1 1 2 2 | i | i
−ξ(a†1a2+a†2a1), (2) = 12e−iΘ(t)Db1[β(t)/√2]Db2[β(t)/√2]
dwehcearye,tghe≡to−tagl0c/a√v2it.ypIhnotthone naubmsebnecreinofthciasvsityystpemhoitsona ×(|1ia1|0ia2{[exp(iωMt)+cos(gt)]|0ib1|0ib2
i
conservedquantity because of[a†a +a†a ,H ]=0, and + sin(gt)(0 1 1 0 )
1 1 2 2 1 √2 | ib1| ib2 −| ib1| ib2 (cid:27)
hence we can restrict the system within a subspace with
a definite total photon number. In this work, we will +|0ia1|1ia2{[exp(iωMt)−cos(gt)]|0ib1|0ib2
consider the single-photon case, i.e., a†a + a†a = 1, i
1 1 2 2 + sin(gt)(0 1 1 0 ) , (9)
then the Hamiltonian (2) is reduced to √2 | ib1| ib2 −| ib1| ib2 (cid:27)(cid:19)
H1 =ωc+ωMb†1b1+g(b†1+b1)−ξ(a†1a2+a†2a1) w(gh2e/rωe t)h[seinp(ωhaste)/fωactor ist],Θ(ta)nd= D(ωc[β+(t)ω/M√/22])t =+
+ωMb†2b2+g(a†2a2−a†1a1)(b†2+b2). (3) exp[(βM(t)b† βM∗(t)b M)/√−2] is the displacbejment operator
j − j
ntNrioaawnnstHfohr1emcmaaotnidobenebV1fu2dret=hcoeeurxptplr[eaπ4sn(fsarfo†1omarm2o−etdhaet†2roam1)o],deths.eUHsainmgiltthoe- winigtGhtewntheoeracplalavyr,itatymhefietesetldrastβea(|tnψ)d(=tt)wi−oi(sgma/nωecMehn)at(na1nic−galleed−misoωtMdaettes)..inHvoolwv--
H =V H V† =H +H , (4) ever, we can obtain macroscopic mechanical entangle-
2 2 1 2 I II ment by measuring the state of the two cavity fields:
where H and H are defined by (i)Ifthecavity-fieldstateisdetectedin 1 0 ,i.e.,
I II | ia1| ia2
the single photon is in the first cavity, then the state of
HI =ωc+ωMb†1b1+g(b†1+b1), (5a) the two mechanical modes becomes
HII =ξ(a†2a2−a†1a1)+ωMb†2b2 |ϕ1(t)i=Db1[β(t)/√2]Db2[β(t)/√2]|φ1(t)i, (10)
+g(a†1a2+a†2a1)(b†2+b2). (5b) with
These two parts HI and HII are commutative, i.e., |φ1(t)i=N1{[exp(iωMt)+cos(gt)]|0ib1|0ib2
[H ,H ] = 0. In particular, the Hamiltonian H can
I II I +(i/√2)sin(gt)(0 1 1 0 ) (,11)
be diagonalized with the displacement transformation | ib1| ib2 −| ib1| ib2 o
V =exp[ g (b† b )] as follows:
3 ωM 1− 1 where N1 =1/ 2+2cos(gt)cos(ωMt) is the normaliza-
tion constant. pWe can evaluate the probability for de-
g2
H3 =V3HIV3† =ωMb†1b1+ωc− ω . (6) tecting this componentstate |1ia1|0ia2 basedonEq.(9).
M This is
For the Hamiltonian HII, if we consider the case ωM = 1
2ξ g,andinthesingle-photoncase,itcanbewrittenas P1(t)= [1+cos(gt)cos(ωMt)]. (12)
2
≫
3
(a)
1 1 0.5
renceC () 10.5 0.5 bility P ()1 C ()ave
oncur Proba 00 0.5 1 1.5 2
C Time
0 0 FIG. 3. (Color online) Dynamics of the average concurrence
0 0.5 1 1.5 2
(b) Cave(t) given in Eq. (17). The parameters are the same as
1 1 those in Fig. 2.
)
( )
C 2 (2
urrence 0.5 0.5 ability P |φ1(t)i and |φ2(t)i are sin2(gt)
Conc Prob C1(t)= 2[1+cos(gt)cos(ωMt)], (16a)
sin2(gt)
C (t)= . (16b)
0 0 2 2[1 cos(gt)cos(ω t)]
M
0 0.5 1 1.5 2 −
Time To evaluate the performance of the entanglement gen-
eration, in Fig. 2, we show the evolution of the concur-
FIG. 2. (Color online) Dynamics of the concurrence (red
rences and the corresponding probabilities for the two
solid curves) and the probability (blue short-dashed curves)
forstates(a)|φ1(t)iand(b)|φ2(t)i. Here,τ ≡gtisthescaled states |φ1(t)i and |φ2(t)i. Here we use the scaled time
time and ωM/g=15. τ = gt, and for satisfying the condition ωM g, we
≫
choose ω = 15g. We can see from Fig. 2 that the
M
concurrences exhibit some oscillations with a sine-cosine
envelope, and that the probabilities show inverse behav-
(ii)Ifthecavity-fieldstateisdetectedin 0 1 ,i.e.,
| ia1| ia2 ior with respect to the concurrence. When the concur-
the single photon is in the second cavity, then the state
renceislarge,theprobabilityissmall,andviceversa. So
of the two mechanical modes is
there is a competition between achieving either a large
entanglementoralargeprobability. Atcertaintimes,the
ϕ (t) =D [β(t)/√2]D [β(t)/√2]φ (t) , (13)
| 2 i b1 b2 | 2 i concurrence could be very close to 1. These times can
be determined based on Eq. (16). For the state φ (t) ,
1
where we choose the time t such that cos(ω t) = 1|, thein
M
C (t) = sin2(gt) . If the time t also satisfies t−he condi-
φ (t) = [exp(iω t) cos(gt)]0 0 1 4sin2(gt/2)
| 2 i N2{ M − | ib1| ib2 tionsin2(gt) 1,thenwehavetheapproximaterelations
+(i/√2)sin(gt)(0 1 1 0 ) (,14) sin2(gt) (g≪t)2 andsin2(gt/2) (gt)2/4,andC (t) 1.
| ib1| ib2 −| ib1| ib2 o However≈,theprobabilityforthis≈maximumentan1glem≈ent
state is very small because P = sin2(gt/2) 1. Sim-
1
with = 1/ 2 2cos(gt)cos(ω t). The probability ≪
2 M ilarly, we can determine the times for obtaining a large
N −
for this compopnent state is entanglement state φ (t) . As a compromise, we can
2
| i
choosethetimes τ/π =n+1/2(n=0,1,2,...). Atthese
1
times, the concurrences of the states φ (t) and φ (t)
P (t)= [1 cos(gt)cos(ω t)]. (15) 1 2
2 M | i | i
2 − are 1/2, and the two probabilities are also 1/2.
Based on the above analyses, we now introduce an
So far, we have obtained the quantum entangled average concurrence to evaluate the entanglement effi-
states of the two mechanical modes. Below, we give a ciency:
proper description and quantification of the entangle-
1
ment. Since the displacement operators Db1[β(t)/√2] Cave(t)=P1(t)C1(t)+P2(t)C2(t)= sin2(gt).(17)
2
and D [β(t)/√2] are local operations, the states ϕ (t)
b2 | l i
and φ (t) (l = 1,2) have the same degree of entangle- In Fig. 3, we illustrate the dynamics of the average con-
l
| i
ment. The states φ (t) and φ (t) are actually two- currence. We can see that the maximum average entan-
1 2
| i | i
mode three-component pure states, so we can employ glement can be achieved at times τ = (n + 1/2)π for
the concurrence [49] as the measurement of the degree n = 0,1,2.... So the shortest time is τ = π/2. In
min
of entanglement. The respective concurrences for states this case, β(τ ) = 2g/ω = 2/15, and the states
min M
− −
4
become be realized using atoms. We assume that two atoms in
theirgroundstatesareflyingthroughthetwocavities,re-
1 spectively. Ifoneatomisexcited,thenthecorresponding
φ (τ ) = φ (τ ) = [ i0 0
| 1 min i | 2 min i √2 − | ib1| ib2 cavity is in a single-photonstate, and the other cavity is
i in its groundstate. Since the single photonhas been ab-
+√2(|0ib1|1ib2 −|1ib1|0ib2)(cid:21). (18) sorbedbythe atominthemeasurement,the twocavities
afterthe measurementareintheirgroundstates. There-
fore,in their subsequent dynamics,both mechanicaland
This state is a typical two-mode three-component state.
opticalmodes only experience free evolution, so the gen-
Itsconcurrenceis1/2,whichisinthe sameorderasthat
erated entanglement is preserved. We should point out
for the Bell states. Since φ (τ ) = φ (τ ) , the
| 1 min i | 2 min i that, in the single-photon preparation and measurement
two mechanical resonator and the two cavity fields are
processes, the photon-atom coupling strength should be
decoupled at these times.
much larger than other coupling scales, so that other
We now give some remarks on the possible realization
physicalprocessesarenegligibleduringtheπ/2-rotation.
of this proposal.
Insummary,wehaveanalyticallystudiedtheentangle-
(i) In the present method, we consider an ideal case
mentdynamicsbetweentwomechanicalmirrorsinatwo-
in which the photon dissipation is neglected. So the
cavity optomechanical system. This entanglement is in-
time needed for the state generation should be much
ducedbytheradiationpressureofasinglephoton,which
shorter than the cavity photon lifetime. If we denote
ishoppingbetweenthetwocavities. Theexplicitformof
the cavity-field decay rate as γ , then the lifetime of a
c these states was obtained by analytically solving the dy-
single photon in the cavity should be 1/γ . Therefore,
c namical evolution of the system. Up to local operations,
∼
thethresholdconditionforentanglementgenerationtime
these states are two-mode three-component states, and
is t = π/(2g) 1/γ , which leads to the system param-
c hence we can quantify the entanglement by the concur-
≪
eter condition g γ (the mechanical decay rate γ is
c m rence. Itwasfoundthataconsiderableentanglementcan
≫
much smaller than γ ). This is the single-photonstrong-
c be created when the system works in the single-photon
coupling condition in optomechanics. In addition, since
strong-coupling regime and the deep-resolved-sideband
we used the condition ω g , the system should also
M 0 regime. Here, considerableentanglementmeans that the
≫
work in the deep-resolved-sidebandregime ω γ .
M c amount of entanglement of these states is comparableto
≫
(ii) Topreparethe initial state|1ia1|0ia2|0ib1|0ib2, the that of the Bell states.
twomechanicalmirrorsshouldbe cooledto their ground J.Q.L.wouldliketothankZhi-JiaoDengandXin-You
states, which can be realized by the ground-state cool- Lu¨ for discussions. J.Q.L.is supported by the JapanSo-
ing method [50, 51]. For the cavity-field state prepara- ciety for the Promotion of Science (JSPS) Foreign Post-
tion and measurement, it would be necessary to use the doctoral Fellowship No. P12503. Q.Q.W. is supported
techniques developed for cavity-QED [52]. The single bytheScientificResearchFundoftheEducationDepart-
cavity photon could be loaded by a two-level atom. An ment of Hunan Province No.12C0716. F.N. is partially
excited atom is assumed to fly through the cavity. By supported by the RIKEN iTHES Project, MURI Cen-
carefullycontrollingthevelocityoftheatom,suchthata ter for Dynamic Magneto-Optics, JSPS-RFBR Contract
π/2-rotation transition occurs, a single photon could be No. 12-02-92100, Grant-in-Aid for Scientific Research
emitted into the cavity. (S), MEXT Kakenhi on Quantum Cybernetics, and the
(iii)Themeasurementofthecavity-fieldstatescanalso JSPS via its FIRST program.
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