Output of multiphoton entangled states with cavity QED

Jie-Hui Huang1 and Shi-Yao Zhu1,2

1Department of Physics, Hong Kong Baptist University, Hong Kong, China2Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China

�Received 18 May 2006; published 22 September 2006�

A scheme for the output of an Einstein-Podolsky-Rosen entangled photon pair is proposed in a cavity QEDsystem composed of two high Q cavities and a moving atom passing through them and with a final detectionon the atomic ground states. The theoretical derivation shows that the production of multientangled photonscan be achieved by passing the atom through additional cavities along the atom’s path.

DOI: 10.1103/PhysRevA.74.032329 PACS number�s�: 03.67.Mn, 42.50.Dv, 03.65.Ud, 42.50.Pq

I. INTRODUCTION

Since the arguments concerning the incompleteness ofquantum mechanics described first by Einstein, Podolsky,and Rosen in 1935 �1� and the Bell’s theorem proposed byBell in 1965 �2�, quantum entanglement has attracted muchattention from physicists for several decades. It not onlygives the possibility for testing the nonlocality, one of themost extraordinary properties of quantum mechanics, butalso has many potential applications in quantum cryptogra-phy �3�, quantum dense coding �4�, and quantum teleporta-tion �5�. Some kinds of entanglement can be classified byconsidering the equivalence of entangled states under localoperations assisted with classical communication �LOCC�,e.g., the Einstein-Podolsky-Rosen �EPR� state �1� of two qu-bits, the Greenberger-Horne-Zeilinger �GHZ� state �6�, andthe W state of three qubits �7�. The classification for four ormore qubits systems is much more complicated �8�.

Due to decoherence, which is almost unavoidable becauseof the interaction of the system under consideration with theenvironment, the entangled state is so fragile that a strictrequirement for experimental technology is needed to pro-duce and detect it. Early work on the experimental realiza-tion of the entangled states is focused on a bipartite system,such as two photons from a nonlinear crystal �9�, two ions ina trap �10�, and two atoms in a cavity QED system �11�.Multipartite entanglement is also achieved in experiments,such as the entangled states among three �12� or four photons�13�, two atoms and one cavity mode �14�, four trapped ions�15�, and five photons recently �16�. The storage and ma-nipulation of the entanglement, related to quantum memoryin quantum information processing, is another challenge inexperiments. Kuzmich and van der Wal et al. reported thegeneration of delayed and correlated photon pairs in theatomic ensemble of cesium �17� and rubidium atoms �18�,respectively in 2003, and the observed time delay betweenStokes and anti-Stokes photons in each pair is no longer than1 �s. Longer storage time in similar atomic systems wasreported by different groups in the last three years �19�.

How to effectively generate and control entangled pho-tons, the promising information carrier for quantum commu-nication, is always an interesting and challenging problem.Up to the present time, the most commonly used method of

producing entangled photon pairs is through parametricdown conversion, where each single pump photon is splitinto two entangled photons via the nonlinear effect of somespecial crystals �20�. However, the photon pair generationrate per pump pulse is very low in spontaneous parametricdown conversion �21�. Moreover, to ensure the desired re-sult, a two- or more fold coincidence measurement is neces-sary for this stochastic process, which also decreases the ef-ficiency.

The developing technology of microwaves and opticalcavities provides another way to coherently exchange quan-tum information between material qubits �atoms or ions� andphotonic qubits �22–24�. Some experimental progress on theentanglement or the nonclassical correlations in cavity QEDhas been reported �11,23–25�. Besides the generation of en-tangled atoms, such as atomic GHZ �26�, cluster �27�, andBell �28� states, some proposals on the atom-photon en-tanglement in a cavity QED system have also been put for-ward �29�. Here we would like to focus our attention on thegeneration of entangled photon pairs based on the model ofadiabatic passage, which has been introduced and studied tomap atomic coherence to photon state in an optical cavity�30�. Some schemes aiming at the creation of GHZ photonmultiplets �31� and EPR entangled atoms �32� have beenproposed based on this model. More recently, Zhou et al.proposed a scheme where the EPR entangled photon pairscan be created through the coherent control on a movingsix-level atom in a high Q cavity, assisted by a classical�-polarized pumping field �33�. However, on the generationof entangled photon pairs outside the cavity they only madea qualitative discussion with a non-Hermitian effectiveHamiltonian to represent the interaction between the internalcavity field and the outside field. Here we present a detailedcalculation, showing how the entanglement between photonsin free space can be achieved through the cavity dissipation,and this scheme could be generalized to produce multiphotonentanglement. In Ref. �31�, Lang and Kimble investigatedthe generation of GHZ photon multiplets in a similar model.However a 2�n+1�-level atomic structure is required to pro-duce n entangled photons in their scheme, which is hard tofind in real atoms when n is large. Our scheme provides apossible way to overcome this problem by using a four-levelatom assisted with a final detection on it, which is easier torealize in experiments.

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II. GENERATION OF ENTANGLED PHOTONS OUTSIDETHE CAVITIES

In our scheme, the multipartite entanglement is achievedthrough the coherent coupling between a four-level atom �in-set in Fig. 1� pumped by classical lasers and two orthogonalcircularly polarized cavity modes, and the exchange of in-cavity photons with out-cavity photons. The proposed ex-perimental setup is mainly composed of four cavities ar-ranged in a straight line and two detectors for the atomiclevels �g1� and �g2� �see Fig. 1�. Two strong linearly polarizedpump lasers, driving the atomic �-transition �g1�↔ �e1� and�g2�↔ �e2� simultaneously, are injected into the two identicalhigh Q cavities C1 and C2 in the direction perpendicular tothe cavity axis, respectively. Each one of the two microwavecavities R1 and R2, sandwiching the cavities C1 and C2, arefilled with a classical microwave field, which couples thetransition between the two hyperfine ground states �g1� and�g2� �34�. For simplicity, we assume that the amplitude,phase, and coupling time of the classical microwave in R1and R2 are suitably adjusted �� /2 pulse� so that the four-level atom prepared in the ground state �g1� will stay in thesuperposition ���0��= 1

�2��g1�+ �g2�� when it exits the cavity

R1.When the moving atom in state ���0�� enters the high Q

cavity C1, it is first driven by a �-polarized pump laser. Weassume the classical pump pulse is so strong ��1����.where �1 is the Rabi frequency of the laser, � is the one sidedecay rate of the cavity modes, and � is the couple strengthbetween the atomic transitions and the corresponding cavitymode� that other processes such as the interaction betweenthe atom and the cavity modes and the spontaneous decay ofthe cavity field, can be ignored during pumping. Under thisassumption, the duration Tp=� /2�1 is required to transformthe atomic state to ���Tp��= 1

�2��e1�+ �e2�� in the resonant case

�p=�a ��p is the frequency of the pump laser; �a is theatomic transition frequency�. We have set the origin of timet=0 when the atom leaves the cavity R1 and reaches thecavity C1.

After crossing the pump area in the cavity, the excitedatom will interact with the two orthogonal �left- and right-handed polarized� cavity modes in the following, whichcouple the two dipole-allowed transitions �e1�↔ �g2� and�e2�↔ �g1�, denoted by − and +, respectively. Moreover,the dissipation of the cavity field is taken into account, that is

to say, the emitted photon by the atom will leak out thecavity soon, which makes it possible to verify the entangle-ment of the photons outside the cavity or to export the en-tangled photons for other interested purposes. The interac-tions could be described by the Hamiltonian under rotating-wave approximation in the interaction picture

HI = − ���e1�e1� + �e2�e2��

+ ���e1�g2�aL + aL†�g2�e1� + �e2�g1�aR + aR

† �g1�e2��

+ k

�gke−i��0−�k�t�bk

�L�+aL + bk�R�+aR�

+ gk*ei��0−�k�t�aL

†bk�L� + aR

†bk�R��� , �1�

where � is the detuning between the atomic transition andthe cavity mode �the same detuning for the two decay chan-nels �e1�↔ �g2� and �e2�↔ �g1� is supposed�; aL and aL

† �or aRand aR

†� are the annihilation and creation operators for theinternal left �or right�-handed polarized cavity mode. The lastterm represents the interaction of the internal cavity modeswith the reservoir outside �35�, where bk

�L� and bk�L�+ �or bk

�R�

and bk�R�+� are the annihilation and creation operators for the

kth vacuum mode with frequency �k and left �or right� handpolarization, and � and gk are the corresponding couplingconstants. The evolution of the system obeys the Schrödingerequation. It is not difficult to derive the time- dependent statevector of this system based on the initial vacuum state forboth the internal cavity field and the reservoir outside, andthe atomic initial state ���Tp��= 1

�2��e1�+ �e2�� �see the Appen-

dix for detailed calculation�

���t�� = A1�t��e1��0��0� + A2�t��e2��0��0� + B1�t��g2��1L��0�

+ B2�t��g1��1R��0� + k

C1,k�t��g2��0��11,kL �

+ k

C2,k�t��g1��0��11,kR � , �2�

where

A1�t� = A2�t�

=1

�2��1 − �2����

2+ �1 e�1�t−Tp� − ��

2+ �2 e�2�t−Tp�� ,

FIG. 1. �Color online� The ex-perimental setup; the external lin-early polarized pump lasercouples the two dipole-forbidden�-transitions �g1�↔ �e1� and�g2�↔ �e2�. The two cavity modes�left- and right-handed polarized�couple the atomic dipole-allowed− and + transitions �e1�↔ �g2�and �e2�↔ �g1�, respectively.

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B1�t� = B2�t� =i�

�2��1 − �2��e�2�t−Tp� − e�1�t−Tp�� ,

C1,k�t� = C2,k�t�

=�gk

�2��1 − �2��

Tp

t

e−i��0−�k�t��e�2�t�−Tp� − e�1�t�−Tp��dt�

=�gke

−i��0−�k�Tp

�2��1 − �2��

0

t−Tp

e−i��0−�k�t��e�2t� − e�1t��dt�. �3�

The basis for the Hilbert space of the whole system is formedby the tensor products of three substates, representing theatomic state and the number states for each polarization ofthe internal cavity field and the emission field outside, e.g.,�g2��1L��0� means the atom in �g2�, one left-handed polarizedphoton in the cavity and the vacuum field outside; and�g2��0��11,k

L � means the atom in �g2�, the vacuum field in thecavity, and one left-handed polarized photon with frequency�k outside. The subscript “1” in �11,k

L � or �11,kR � means the 1st

emitted photon. �1 and �2 are the two roots of equation�2+ � �

2 − i���+�2− i2��=0, and �=2��g�0

�2D��0� is one sidedecay rate of the cavity modes with D��0� being the modedensity around frequency �0 in the reservoir. In deriving Eq.�3�, the Markov approximation �35–37� has been applied.

A typical atom-cavity interaction time is approximately afew tens of �s in the relevant experiments, which is slightlylonger than the photon storage time in a high Q optical cav-ity. So if the atom’s passage time in cavity C1 is very long�T0�

1� �Tp, determined by the atom’s velocity and cavity

size�, the excited atom in ���Tp��= 1�2

��e1�+ �e2�� will decayback to its ground states, and the emitted photon �either left-or right-handed polarized� will leak out the cavity com-pletely, leaving the cavity in the vacuum again, correspond-ing to A1�T0�=A2�T0��B1�T0�=B2�T0��0. Then the sys-tem’s state has a simpler form when the atom exits the highQ cavity C1,

���T0�� ��

�2��1 − �2�

k

gke−i��0−�k�Tp

�0

T0−Tp

e−i��0−�k�t��e�2t� − e�1t��dt���g1��0��11,kR �

+ �g2��0��11,kL �� . �4�

The dynamics in the second high Q cavity C2 is almostthe same as that in cavity C1 with a different “initial” state,because one photon has been emitted, which is entangledwith the moving atom �4�. When the atom enters the secondcavity C2 the �-polarized pump laser performs a local opera-tion on the atom subsystem and simultaneously excites thetransitions �g1�→ �e1� and �g2�→ �e2� again �other processesare ignored because of the large Rabi frequency: �2=�1�. Inthe ideal situation, another pump duration Tp=� /2�2 is re-quired to change the system to

���T0 + Tp�� ��

�2��1 − �2�

k

gke−i��0−�k�Tp

�0

T0−Tp

e−i��0−�k�t��e�2t� − e�1t��dt���e1��0�

�11,kR � + �e2��0��11,k

L �� . �5�

Then the second photon is emitted through the interaction ofthe excited atom with the cavity modes in cavity C2, and itwill also leak outside due to the dissipation of the cavity. TheHamiltonian in this process is the same as the one describedin �1�. Following the similar calculation above, we can ob-tain the time-dependent state vector of this stage as

���t�� = k

A1,k�t��e1��0��11,kR 0� +

k

A2,k�t��e2��0��11,kL 0�

+ k

B1,k�t��g2��1L��11,kR 0� +

k

B2,k�t��g1��1R��11,kL 0�

+ k�

k

C1,k,k��t��g2��0��11,kR 12,k�

L � + k�

k

C2,k,k��t�

�g1��0��11,kL 12,k�

R � , �6�

where

A1,k�t� = A2,k�t�

= A1,k�T0 + Tp�1

�1 − �2���

2+ �1 e�1�t−T0−Tp�

− ��

2+ �2 e�2�t−T0−Tp�� ,

B1,k�t� = B2,k�t�

= A1,k�T0 + Tp�i�

�1 − �2�e�2�t−T0−Tp� − e�1�t−T0−Tp�� ,

C1,k,k��t� = C2,k,k��t�

= A1,k�T0 + Tp��gk�

�1 − �2�

T0+Tp

t

e−i��0−�k��t�

�e�2�t�−T0−Tp� − e�1�t�−T0−Tp��dt�

= A1,k�T0 + Tp��gk�e

−i��0−�k���T0+Tp�

�1 − �2

�0

t−T0−Tp

e−i��0−�k��t��e�2t� − e�1t��dt�,

A1,k�T0 + Tp� =�gke

−i��0−�k�Tp

�2��1 − �2�

�0

T0−Tp

e−i��0−�k�t��e�2t� − e�1t��dt�. �7�

Under the same approximation made in cavity C1 �the samepassage time T0�

1� �Tp is assumed for the atom in the cavi-

ties C1 and C2�, the second photon will also leak out of the

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cavity C2 completely. When the atom leaves this cavity, thestate of the whole system becomes

���2T0�� � � �

�1 − �2 2

k

gke−i��0−�k�Tp

�0

T0−Tp

e−i��0−�k�t��e�2t� − e�1t��dt�k�

gk�

e−i��0−�k���T0+Tp��0

T0−Tp

e−i��0−�k��t�

�e�2t� − e�1t��dt�1�2

��g1��11,kL 12,k�

R �

+ �g2��11,kR 12,k�

L �� � �0�

=1�2

��g1��11L12

R� + �g2��11R12

L�� � �0� , �8�

where the vacuum state of the internal cavity field �0� isseparable from the rest of the system. The two polarizedemission fields composed of continuous modes outside thecavity are defined as two one-photon states,

�1 jR� =

�

�1 − �2

k

gke−i��0−�k���j−1�T0+Tp�

�0

T0−Tp

e−i��0−�k�t��e�2t� − e�1t��dt��1 j,kR � , �9a�

and

�1 jL� =

�

�1 − �2

k

gke−i��0−�k���j−1�T0+Tp�

�0

T0−Tp

e−i��0−�k�t��e�2t� − e�1t��dt��1 j,kL � . �9b�

Here �1 jR� or �1 j

L� means the jth photon with right-or left-handpolarization in the reservoir, which is emitted in the timerange ��j−1�T0+Tp , jT0�. As we mentioned above, the atom-cavity interaction time and the photon storage time in high Qoptical cavities are assumed much shorter than the atom’spassage time in each cavity, which ensures one photon instate �1 j

R� or �1 jL�. The phase factor e−i��0−�k���j−1�T0+Tp� is a

function of the sequence number of emitted photons “j.”This time delay �phase difference� is related to the memoryof quantum states and has quite an important significance inquantum information processing. It can be concluded that theatom with two coherent ground states plays a key role for thememory, which acts as an intermediate, not only recordingthe information of the former photons but also conveying itto the later ones. The time delay between emitted photonsalso makes it possible to spatially separate them in theirpropagation direction.

The result �8� means the successful generation of a tripar-tite entangled state including two emitted photons outsidethe cavity and the moving atom. The two photons areemitted one after another with a controllable time delay ap-proximating the atom’s passage time T0 in each high Q cav-

ity. The two-photon entanglement states could be obtainedfrom the tripartite entangled state �8� by a measurementon the atom in the rotated basis �↑ �= 1

�2��g1�+ �g2�� and

�↓ �= 1�2

��g1�− �g2��. The state Eq. �8� can be rewritten as

1�2

��g1��11L12

R� + �g2��11R12

L��

=1

2��↑���11

L12R� + �11

R12L�� + �↓���11

L12R� − �11

R12L��� .

�10�

Both measurement results on the atom, �↑� and �↓�, induce thecollapse from the tripartite entangled state to the EPR typetwo-photon entangled state

��EPR� =1�2

��11L12

R� + �11R12

L�� , �11a�

or

��EPR� � =1�2

��11L12

R� − �11R12

L�� . �11b�

The state Eq. �8� can also be transformed to the followingstate,

���� =1

2��g1���11

L12R� − �11

R12L�� + �g2���11

L12R� + �11

R12L��� � �0� ,

�12�

by letting the atom pass through a microwave cavity R2. Inthis cavity, the interaction Hamiltonian is expressed as,

HI� = i�0ei�0�g2�g1� − i�0e−i�0�g1�g2� . �13�

By solving the Schrödinger equation, it is easy to find thatthe state �12� is achieved from Eq. �8� under the initial phase�0=0 and the interaction duration of � /4�0. The detectionof the atom on the state �g1� or �g2� by the state-selective fieldionization in detectors Dg1

and Dg2also results in the gen-

eration of the entangled photon pair of Eq. �11a� or Eq.�11b�.

In fact, we can achieve multiphoton entanglement in thiscavity QED system by repeating the operations on the four-level atom in additional cavities arranged along the atom’spath �see Fig. 2�. It could be deduced from Eqs. �4�, �8�, and�12� that after the atom passes N identical high Q cavitiesand leaves the cavity R2, the system’s final state will be

��� =1

2��g1���11

L12R¯ 1N

L� − �11R12

L¯ 1N

R�� + �g2���11L12

R¯ 1N

L�

+ �11R12

L¯ 1N

R��� � �0�, �N is odd�; �14a�

or

��� =1

2��g1���11

L12R¯ 1N

R� − �11R12

L¯ 1N

L�� + �g2���11L12

R¯ 1N

R�

+ �11R12

L¯ 1N

L��� � �0�, �N is even� . �14b�

The detection on the atom projects the N photons into two

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generalized n-party GHZ states �38� with 50% possibility foreach.

Finally, we give a brief discussion about the experimentalfeasibility of this scheme. First, the realistic atomic four-level structure with two upper Zeeman sublevels and twolower Zeeman sublevels involved in this scheme could befound in real atoms, such as a J=1/2 to J=1/2 transition of198Hg+ ions �39�. If possible, a very strong coupling betweenthe atomic transitions and the corresponding cavity modesshould be chosen to enhance the generation of the cavityphotons and to weaken the influence of other possible noiseinteractions, such as spontaneous decay of the atom, whichhas been ignored in the above analysis. Second, the atom’spassage time in each cavity can be controlled by the atom’svelocity through Doppler-resolved and time-of-flight tech-niques �24�. Within the coherence time of the atom’s twoground states, which should be very long compared with thestorage time of the photon in the cavity, a very low velocityought to be selected to prolong the atom’s interaction time ineach cavity; thus a more pure EPR or generalized GHZ statewill be achieved for the photons outside the cavities, after themeasurement on the atom. According to our calculation, toachieve a two-photon entangled state with higher than 95%fidelity �40� �defined by F= �EPR����EPR� or F�= �EPR� �����EPR� �, �EPR and �EPR� are maximally entangledstates described as Eqs. �11a� and �11b�, and � ���� is thedensity matrix of the field outside the cavities, correspondingto the detection result �g2� ��g1�� on the atom�, a shortestinteraction time T0�8/� is required for the atom in eachcavity �the conditions �=0 and �=� are assumed in the cal-culation for simplicity�. Third, the inevitable dissipation inoptical cavities leads to the output of entangled photons inthis scheme and the large decay rate � helps to shorten thephoton’s emission time from cavity to reservoir outside. Asmentioned above, all the photons emitted from the cavitiesare separable in time domain, so we can measure their polar-ization correlations through the time-resolved coincidence�17,41�. Because we have ignored the spontaneous decay ofthe atom in this theoretical analysis, the success probabilityof achieving entangled photon multiplets in this scheme willreach unity accordingly.

III. CONCLUSION

To summarize, the entanglement between an atom andphotons or among photons can be achieved through the co-

herent coupling among an atom, the cavity modes, and anexternal classical pump laser. Different from the usual para-metric down-conversion processes �20�, the generation of themultiphoton entangled state is deterministic in our scheme.Owing to the long time atomic coherence of the two groundstates, the characteristic of quantum memory is displayed inthis system, which is quite useful in quantum informationprocessing. Since each passing atom produces a pair of en-tangled photons, we can enhance the signal by injecting alarge number of atoms into the experimental setup one byone, with a fixed interval longer than the atom’s transit timein it. All the requirements in this scheme should be well inaccord with current experimental technology.

ACKNOWLEDGMENTS

This research was supported by RGC Grant No.�NSFC05-06.01� of HK Government and FRG of HK BaptistUniversity.

APPENDIX

After the moving atom leaves the pump area in the high Qcavity, it will interact with the two orthogonal cavity modesand emit one photon �either left- or right-handed polarized�.This photon will leak outside due to the dissipation of thecavity. The Hamiltonian could be expressed in theSchrödinger picture

H = HF + HA + HAF + HR + HFR, �A1�

where

HF = �0�aL†aL + aR

†aR� ,

HA = �g��g1�g1� + �g2�g2�� + �e��e1�e1� + �e2�e2��

=1

2�A��e1�e1� − �g2�g2� + �e2�e2� − �g1�g1��

+1

2�2�g + �A���e1�e1� + �e2�e2� + �g1�g1� + �g2�g2�� ,

HAF = ���e1�g2�aL + aL†�g2�e1� + �e2�g1�aR + aR

† �g1�e2�� ,

FIG. 2. �Color online� The ex-perimental setup to generate mul-tiphoton entanglement. A series ofidentical cavities are arrangedalong the atom’s path.

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HR = k

�k�bk�L�+bk

�L� + bk�R�+bk

�R�� ,

HFR = k

�gk�bk�L�+aL + bk

�R�+aR� + gk*�aL

†bk�L� + aR

†bk�R��� .

�A2�

If we choose

H0 = �0�aL†aL + aR

†aR� + ��e + ����e1�e1� + �e2�e2��

+ �g��g1�g1� + �g2�g2�� + k

�k�bk�L�+bk

�L� + bk�R�+bk

�R��

= �0�aL†aL + aR

†aR� +1

2�2�g + �0���e1�e1� + �e2�e2�

+ �g1�g1� + �g2�g2�� +1

2�0��e1�e1� − �g2�g2� + �e2�e2�

− �g1�g1�� + k

�k�bk�L�+bk

�L� + bk�R�+bk

�R�� , �A3�

the Hamiltonian �A1� turns to a simpler expression in theinteraction picture

HI = iU˙

U† + UHU†

= − ���e1�e1� + �e2�e2�� + ���e1�g2�aL + aL†�g2�e1�

+ �e2�g1�aR + aR† �g1�e2�� +

k

�gke−i��0−�k�t�bk

�L�+aL

+ bk�R�+aR� + gk

*ei��0−�k�t�aL†bk

�L� + aR†bk

�R��� , �A4�

where the unitary transformation is defined by U=eiH0t/, and�=�0−�A is the detuning between atomic transition andcavity mode. The state vector of the whole system in thisstage has the form like this

���t�� = A1�t��e1��0��0� + A2�t��e2��0��0� + B1�t��g2��1L��0�

+ B2�t��g1��1R��0� + k

C1,k�t��g2��0��11,kL �

+ k

C2,k�t��g1��0��11,kR � , �A5�

The evolution of the above state vector obeys theSchrödinger equation i�=HI�, then we can obtain the re-lations between these coefficients

A1�t� = i�A1�t� − i�B1�t� ,

B1�t� = − i�A1�t� − ik

gk*ei��0−�k�tC1,k�t� ,

C1,k�t� = − igke−i��0−�k�tB1�t� , �A6a�

and

A2�t� = i�A2�t� − i�B2�t� ,

B2�t� = − i�A2�t� − ik

gk*ei��0−�k�tC2,k�t� ,

C2,k�t� = − igke−i��0−�k�tB2�t� . �A6b�

By applying the Markov approximation �35–37�, we have

B1�t� = − i�A1�t� − ik

gk*ei��0−�k�tC1,k�t�

= − i�A1�t� − ik

gk*ei��0−�k�t

�Tp

t

− igke−i��0−�k�t�B1�t��dt�

= − i�A1�t� − k

�gk�2�Tp

t

ei��0−�k��t−t��B1�t��dt�

= − i�A1�t� −�

2B1�t� , �A7a�

and

B2�t� = − i�A2�t� −�

2B2�t� . �A7b�

�=2�g�0D��0� is the damping constant of the cavity mode

and D��0�=V�02 /�2c2 �with V being the quantization vol-

ume� is the mode density around frequency �0 in the reser-voir. The equations in �A6� have the solution based on the“initial”state of this stage ��Tp�= 1

�2��e1�+ �e2�� � �0��0� as

follows:

A1�t� = A2�t�

=1

�2��1 − �2����

2+ �1 e�1�t−Tp� − ��

2+ �2 e�2�t−Tp��

B1�t� = B2�t� =i�

�2��1 − �2��e�2�t−Tp� − e�1�t−Tp��

C1,k�t� = C2,k�t�

=�gk

�2��1 − �2��

Tp

t

e−i��0−�k�t��e�2�t�−Tp� − e�1�t�−Tp��dt�

=�gke

−i��0−�k�Tp

�2��1 − �2��

0

t−Tp

e−i��0−�k�t��e�2t� − e�1t��dt�.

�A8�

JIE-HUI HUANG AND SHI-YAO ZHU PHYSICAL REVIEW A 74, 032329 �2006�

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�1 and �2 are the two roots of equation �2+ � �2 − i���+�2

− i2��=0; The duration T0 is assumed so long that the ex-

cited atom in ���Tp�� completely decays back to groundstates �A1�T0�=A2�T0��0�. In addition, the first photon �ei-ther left-or right-handed polarized� emitted by the atom leaksout the cavity entirely �B1�T0�=B2�T0��0�. Then the state ofthis system is simplified to be

���T0�� ��

�2��1 − �2�

k

gke−i��0−�k�Tp

�0

T0−Tp

e−i��0−�k�t��e�2t� − e�1t��dt���g1��0��11,kR �

+ �g2��0��11,kL �� . �A9�

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