P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21

Entangling Distant Atoms by Interference of Polarized Photons

Xun-Li Feng,1 Zhi-Ming Zhang,2 Xiang-Dong Li,1 Shang-Qing Gong,1 and Zhi-Zhan Xu1

1Laboratory for High Intensity Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences,Shanghai 201800, People’s Republic of China

2Department of Applied Physics, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China(Received 15 January 2003; published 28 May 2003)

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We propose a scheme to generate the entangled state of two �-type three-level atoms trapped indistant cavities by using interference of polarized photons. Two possible spontaneous emission channelsof each excited atom result in a coherent superposition of the states of two atoms. The subsequentdetection of the different polarized photons reveals that both atoms are in different ground states, but aninterference effect prevents us from distinguishing which atom is in which ground state; the atomsare thus entangled. In comparison with the original proposal of interference-induced entanglement[C. Cabrillo, J. Cirac, P. Garcia-Fernandez, and P. Zoller, Phys. Rev. A 59, 1025 (1999)], in our schemethe weakly driven condition is not required, and the influence of atomic excitement and atomic recoil onthe entanglement fidelity can be eliminated.

DOI: 10.1103/PhysRevLett.90.217902 PACS numbers: 03.67.Mn, 03.65.Ud, 42.50.Ct

sequently, then the state of the system is projected into anentangled state of two ground states. The idea of interfer-

in which ground state with certainty, so we arrive at asuperposition of two product states jgLi1 � jgRi2 and

Entanglement shared by distant parties is not only akey ingredient for the tests of quantum nonlocality [1],but also a basic resource in achieving tasks of quantumcommunication and quantum computation [2]. Recently,various quantum systems have been suggested as pos-sible candidates for engineering of quantum entangle-ment; among them the cavity-quantum-electrodynamics(CQED) systems are always paid more attention. This isdue to the fact that cold and localized atoms are not onlythe source of local entanglement, but also well suited forstoring quantum information in long-lived internalstates; photons are the natural source for fast and reliabletransport of quantum information over long distances; inparticular, CQED also provides mechanisms for commu-nicating between cavities [3]. Numerous proposals inCQED have been made for entangling atoms couplingto the mode(s) of a single cavity [4,5] and atoms in twoor more cavities [3,6–10].

Most of the schemes for the generation of the entangledstates of atoms are based on direct or indirect interactionsbetween the atoms that are to be entangled. Very recently,Cabrillo et al. presented a novel scheme to entangledistant atoms by using a single-photon interference effectrather than by using an effective interaction between them[11]. The main idea of their scheme is as follows. Twoidentical �-type three-level atoms A and B, each with anexcited state jei and two ground states jg1iand jg2i andinitially in a ground state, say jg1i, are driven by weaklaser pulses simultaneously; only one of them is driven tothe excited state jei. Because one cannot determine whichatom is excited, a superposition of the states of jg1iA �jg1iB, jeiA � jg1iB, and jg1iA � jeiB is thus prepared. Ifthe transition of spontaneous emission from the excitedstate jei to the other ground state jg2i is detected sub-

0031-9007=03=90(21)=217902(4)$20.00

ence for producing entanglement has also been used toentangle atomic ensembles [12].

In this Letter, we propose a scheme for preparing theentanglement between distant atoms by using the inter-ference of polarized photons. The model we are consid-ering consists of two identical �-type three-level atoms,with the atoms 1 and 2 trapped in optical cavities A and B,respectively, as shown in Fig. 1. Each atom has an excitedstate jei and two degenerate ground states jgLi and jgRi.The transitions jei ! jgLi and jei ! jgRi are stronglycoupled to, respectively, left- and right-circularly polar-ized cavity modes. Such an atomic level structure can beachieved using Zeeman sublevels [13] and has been pro-posed to entangle two atoms in a single cavity veryrecently [5]. The photons leaking out from the cavitiesA and B first transmit a quarter wave plate (QWP),respectively, then are focused on a polarizing beam split-ter (PBS), and finally detected by two photodetectors D1

and D2. We assume that the photons coming from cavity Aand cavity B pass through equal distance to PBS, and bothcavities are one sided so that the only leakage of photonsoccurs through the sides of the cavities facing QWP.

Suppose both atoms are initially prepared in theirexcited state jei, which can be achieved, for example,by illuminating a left-circularly polarized � laser pulseon each atom if both atoms are in the ground state jgLi.Every excited atom has two possible transitions jei !jgLi or jei ! jgRi, corresponding to an emission of a left-or right-circularly polarized photon. If both photodetec-tors D1 and D2 click simultaneously after a finite waitingtime, we detect two photons with different polarizations,so we can make sure that the two atoms are now in thedifferent ground states: one in state jgLi and the other instate jgRi. However, we cannot determine which atom

2003 The American Physical Society 217902-1

||e⟩

|g ⟩ |gR⟩

(a)

1

D2

(b)

L

b

D

FIG. 1. (a) Atomic level structure. (b) Experimental setup.Two distant atoms 1 and 2 trapped in cavities A and B,respectively, and are initially driven to their excited state jei.The photons leaking out from the cavities A and B are detectedby photodetectors D1 and D2 after transmitting quarter waveplates (QWP) and a polarizing beam splitter (PBS).

P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21

jgRi1 � jgLi2, that is, an entangled state of two atoms.Otherwise, if none of the photodetectors click or only oneclicks during the waiting time, we fail to generate thedesired entangled state, and should repeat the processagain until we find both photodetectors D1 and D2 clicksimultaneously.

In order to illustrate our approach explicitly, let us firstexamine the resonant interaction between a �-type three-level atom, whose level structure is depicted in Fig. 1(a),and two cavity modes in an ideal cavity. The cavitymodes have left- and right-circular polarization, respec-tively. For simplicity, we neglect the atomic spontaneousemission to other modes. The interaction Hamiltoniantakes the following form:

HI � �hX

k�L;R

�k�akjeihgkj � ayk jgkihej; (1)

where the subscripts k � L;R present the left- and right-circularly polarized cavity modes, respectively. ayk and ak

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are the creation and annihilation operators of photons ofthe k mode, and �k is the coupling constant between the kmode and the atom. We suppose �k to be real, and for thesake of generality we allow the coupling between theatom and the cavity modes to be different, i.e., �L �

�R. If the atom is prepared initially in its excited statejei and cavity modes in their vacuum states j0L; 0Ri,where j0L; 0Ri � j0Li � j0Ri, one can find the evolutionof the state of the atom-field system after the interactingtime t,

j��ti � cos�tjeij0L; 0Ri

� i1

�sin�t��LjgLij1L; 0Ri � �RjgRij0L; 1Ri;

(2)

where � ��������������������2L � �2

R

q.

Now we show how to generate a desired two-atomentangled state. Suppose the two atoms trapped in cavitiesA and B (see Fig. 1) are driven to their excited states bytwo laser pulses simultaneously. After time t, the jointstate of the two subsystems, cavity A� atom 1 and cavityB� atom 2, is described by j��tiA � j��tiB; herej��tiA [j��tiB] denotes the state of cavity A and atom1 (cavity B and atom 2) and has the form of Eq. (2). Afterpassing through the QWP, the left- and the right-circularly polarized photons become linearly polarizedwith their polarizing directions perpendicular to eachother. Since the polarizing directions of the photons arerelative to the PBS and symmetrical in the joint statej��tiA � j��tiB, without loss of generality, we assumethe left-circularly polarized photons become verticallypolarized, and the right-circularly polarized photons be-come horizontally polarized, with the following trans-formation:

j1L; 0Ri ! jVi; (3a)

j0L; 1Ri ! jHi: (3b)

Here, jVi and jHi denote vertically and horizontallypolarized photons, respectively. For the sake of simplic-ity, we have ignored the vacuum modes in the abovenotation because they have no contribution to the clickof the photodetectors D1 and D2. For the same reason, wewill neglect the terms such as jeij0L; 0Ri in the joint statej��tiA � j��tiB in the following analysis. After thephotons pass through the QWP, the joint state j��tiA �j��tiB evolves into

��LjgLi1jViA � �RjgRi1jHiA � ��LjgLi2jViB � �RjgRi2jHiB; (4)

where the subscripts 1 and 2 denote atoms 1 and 2, and A and B denote the photons from cavities A and B, respectively.The PBS transmits horizontally polarized photons and reflects vertically polarized ones [14]. Therefore, the photonswill be split into mode a and mode b at the PBS according to their polarization; after the photons pass through the PBS,Eq. (4) becomes

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P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21

��LjgLi1jViA;a � �RjgRi1jHiA;b � ��LjgLi2jViB;a � �RjgRi2jHiB;b; (5)

where subscripts i; j (i � A;B; j � a; b) denote the transformation from input mode i to output mode j. Since the photoninterference effect at the PBS, one cannot distinguish which cavity the output photons come from, therefore we canneglect the subscripts i in the above equation and rewrite Eq. (5) as

��LjgLi1jVia � �RjgRi1jHib � ��LjgLi2jVia � �RjgRi2jHib � �2LjgLi1jgLi2jViajVia � �2

RjgRi1jgRi2jHibjHib

� �L�R�jgLi1jgRi2 � jgRi1jgLi2jViajHib: (6)

If we find both detectors D1 and D2 click simultaneously,the joint state j��tiA � j��tiB will project into a maxi-mally entangled state:

1���2

p �jgLi1jgRi2 � jgRi1jgLi2; (7)

the probability of getting such a state is2�2

L�2Rsin

4��t=��2L � �2

R2. In the case of certain inter-

acting time t, the probability is determined by the cou-pling constants �L and �R and reaches its maximum valuesin4��t=2 when both coupling constants �L and �R areequal. It is interesting to note that the fidelity of the finalentangled states is not affected by the coupling constants�L and �R. Considering jgLi and jgRi are stable groundstates (or metastable states), the entangled state we haveobtained is free from the cavity loss as well as sponta-neous emission.

The scheme given here is similar in spirit to that ofCabrillo et al. [11]. Both schemes are based on the inter-ference effect of light to generate a coherent superposi-tion of atomic states rather than on the effectiveinteraction between the two atoms. But physically thetwo schemes are different. In the scheme of Cabrilloet al., the coherent superposition of atomic states is gen-erated in the photon absorption process, while in ourscheme it is generated in the photon emission process.In addition, our scheme has several advantages.

First of all, in the scheme of Cabrillo et al., the laserpulse has to be sufficiently weak or short to ensure theprobability of exciting both atoms is much smaller thanthe probability of exciting the relevant coherent super-position. In such a case, it will be a time-consuming taskto detect one spontaneously emitted photon, while in ourscheme the weakly driven condition is not required asboth atoms are excited simultaneously.

Second, in the scheme of Cabrillo et al., the probabilitythat both atoms are excited to their upper states is notabsolutely zero even under weakly driven conditions. Thiswill influence the fidelity of the final entangled states [11],while in our scheme the driving condition does not influ-ence the fidelity of the final entangled states, in fact, if thecondition that both atoms are driven to their excited statesis not met, that is, only one atom or neither of the atoms isdriven to the excited state, corresponding to at most onephotodetector click. According to our proposal, suchexperimental results will be thrown out. Therefore thedriving condition only influences the efficiency in ourscheme.

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Third, in our scheme, the influence of atomic recoil onthe entanglement fidelity could be eliminated. When anatom absorbs or emits photons, it is always accompaniedby a recoil. In the single emission case of Ref. [11], theatomic recoil leaves a trace of which atom has emitted thephoton, thus also destroying the entanglement. But in ourscheme both atoms absorb and emit photons with thesame energy simultaneously if both photodetectors D1

and D2 click at the same time; the influence of the atomicrecoil on the entanglement fidelity can thus be erased.

Considering the cavity decay and the photon observa-tion in our model, it is necessary for us to adopt thequantum trajectory theory [15,16] for a further explana-tion. The evolution of the quantum system under continu-ous detection, conditional to observing a particulartrajectory of counts, can be described by a pure statewave function j�c�ti. During the time interval whenno photon is detected, the wave function evolves accord-ing to a non-Hermitian effective Hamiltonian due to itscoupling with the environment. Because in our modelthere is no coupling between the modes of cavity A andthose of cavity B, the wave function can thus be written asj�c�ti � j�c�tiA � j�c�tiB. Here j�c�tiA and j�c�tiBstand for the wave function of cavity A� atom 1 and thatof cavity B� atom 2, and are governed by the followingeffective Hamiltonian, respectively,

Hjeff � Hj

I � i �hkX

k�L;R

aykjakj; �j � A;B; (8)

where we have assumed that both cavities have the sameloss rate k for all modes; Hj

I (j � A;B) takes the form ofEq. (1). A direct calculation gives the following result forj�c�tij (j � A;B) in the good cavity case �L; �R > k:

j�c�tij ���

cos�t�k2�

sin�t�jeij0L; 0Ri

� i1

�sin�t��LjgLij1L; 0Ri

� �RjgRij0L; 1Ri�� exp

��1

2kt�; (9)

where � ������������������������������������2L � �2

R � k2=4q

.The detection of photons is accompanied by the wave

function collapse j�c�ti ! Cj�c�ti, and the probabilitydensity for such a detection to occur at time t is P �h�c�tjCyCj�c�ti. If both photodetectors D1 and D2

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.00

0.05

0.10

0.15

0.20

: λL=λ

R=10κ

: λL=λ

R=5κ

: λL=2λ

R=10κP

κt

FIG. 2. Probability density P for detecting two photons withperpendicular polarizations as a function of kt with differentvalues of coupling constants �L and �R.

P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21

click at the same time t, two photons with perpendicularpolarizations are annihilated with C taking the followingform C � �aLA � aLB�aRA � aRB. In this case, the wavefunction j�c�ti collapses into the desired entangled state(7) with the probability density

P ��2L�

2R

�4 exp��2ktsin4�t: (10)

In Fig. 2, we plot the probability density P as a function ofkt with the different values of coupling constant �L and�R. From Fig. 2, we find the waiting time for the effectivedetection can be chosen to be a few times of cavity life-time 1=k.

In the above discussion, we have neglected the influ-ence of the spontaneous emissions. Let us take account ofit now. On the one hand, the photons from the spontaneousemissions to the free modes (rather than the cavitymodes) run with random directions and cannot be de-tected by photodetectors D1 and D2, therefore the effi-ciency of successful preparation of the desired entangledstate is reduced. On the other hand, because the time whenan atomic spontaneous emission occurs is stochastic anduncontrollable, one cannot assure that both atoms deex-cite at the same time. Correspondingly, we might obtainthese detecting results that both photodetectors D1 and D2

click at different times. In this case, the sources of thedetected photons may be identified according to the earlyor the late occurrence of atomic recoil; the entanglementis thus destroyed. In comparison, with the single emissioncase of Ref. [11], the influence of atomic recoil on theentanglement fidelity can be eliminated by sacrificing theefficiency. We simply eliminate such experimental resultsthat both D1 and D2 click at different times and keeponly those that both photodetectors D1 and D2 click at thesame time.

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In summary, we have proposed a scheme to generatethe entangled state of two �-type three-level atomstrapped in distant cavities by using interference of polar-ized photons. Compared with the original scheme ofinterference-induced entanglement [11], our scheme hasseveral advantages: It does not require weakly drivenconditions and can eliminate the influence of atomicexcitement and atomic recoil on the entanglement fidelity.

This work was supported by the Natural ScienceFoundation of China (Grants No. 60178014 andNo. 10274089).

Note added.—After the submission of this work, webecame aware of several schemes for the generation ofdistant atoms (ions) by means of photon interferencebeing proposed independently very recently [17].

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