Long-distance quantum communication with individual atoms and atomic ensembles

Gong-Wei Lin,1 Xu-Bo Zou,1,* Xiu-Min Lin,2 and Guang-Can Guo1

1Department of Physics, Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026,People’s Republic of China

2School of Physics and Optoelectronics Technology, Fujian Normal University, Fuzhou 350007, People’s Republic of China�Received 10 November 2008; published 24 April 2009�

A scheme is presented for realizing long-distance quantum communication with individual atoms and atomicensembles. In the scheme, deterministic and storable atom-photon entanglement sources can be obtained byadiabatic evolution of dark states, which makes this scheme largely enhance the efficiency for long-distanceentanglement distribution. Furthermore, due to many-atom interference effect, requirement of the strong-coupling condition can be greatly relaxed. Thus this scheme opens an alternate avenue for a scalable quantumnetwork.

DOI: 10.1103/PhysRevA.79.042332 PACS number�s�: 03.67.Hk, 42.50.Dv, 32.80.�t

I. INTRODUCTION

Quantum communication over long distance via noisychannel is a challenging task due to the exponential scalingof the error probability with the length of the channel. Toovercome this problem, Briegel et al. �1� introduced the con-cept of quantum repeater, combining entanglement swapping�2� and quantum memory to efficiently extend the achievabledistances. With atomic ensembles and linear optics, Duan etal. �3� presented a realistic scheme for a quantum-repeaterarchitecture. Fundamental to this protocol, which is knownas the Duan-Lukin-Cirac-Zoller �DLCZ� protocol, is thatatomic ensembles are used as quantum memories, whichhave the ability to generate and retrieve single “spin” exci-tations. In the efforts of realizing this protocol, significanttheory and experimental advances have been achieved re-cently �4–14�, including the creation of entanglement be-tween separate quantum nodes �4� and the realization of tele-portation between photonics and atomic qubits �11�. A high-brightness source of narrowband identical-photon pairs wasalso demonstrated �6�. In these protocols �4–14�, atomic en-sembles are used as memory qubits to avoid the challengingrequest for strong coupling between individual atoms andphotons. But to avoid “multispin” excitations, they use weakoff-resonant “write” pulses to excite atomic ensembles. Thisresults in a small distribution rate of entangled states �15�. Itseems that the protocols based on cavity quantum electrody-namics �QED� with individual atoms allow the implementa-tion of deterministic protocols in quantum information sci-ence �16�, such as realizing single-photon source �17�,entangling an atom with a photon �18�, and mapping quan-tum states between an atom and a photon �19�. However,they generally require the strong atom-cavity coupling in ahigh-finesse cavity, which is still a challenging pursuit �20�.

In this paper, a scheme is presented for realizing long-distance quantum communication with individual atoms andatomic ensembles. In the scheme, the individual atoms,atomic ensembles, and photons act as the control qubits,memory qubits, and flying qubits, respectively. Generation

and retrieval of atom-photon entanglement sources can beachieved by adiabatic evolution of dark states. The schemeproposed in this paper has the following significant advan-tages: �i� our scheme largely enhances the efficiency of en-tanglement distribution since creation of single spin excita-tions of atomic ensembles in our scheme can work in adeterministic way by control of individual atoms. �ii� Due tocollective interference effect of atomic ensembles, the effec-tive coupling strength between atoms and cavity modesscales up with �N �N is the effective number of atoms in theensemble�. Thus our scheme can greatly relax the require-ment of the strong-coupling condition. �iii� Our scheme isbased on two-photon Hong-Ou-Mandel-type interferencewhich relaxes the long-distance stability requirements�13–15�. �iv� In our scheme, atom-photon entanglement canbe directly extended to multimode case, which not onlypromises a speedup in entanglement generation but also al-lows long-distance distribution of multiparticle Greenberger-Horne-Zeilinger �GHZ� state.

II. GENERATION AND RETRIEVAL OF ATOM-PHOTONENTANGLEMENT SOURCE

The basic element of our system is illustrated schemati-cally in Fig. 1. Two local optical cavities 1 and 2 are con-nected by a short optical fiber �21,22�. Some individual at-oms trapped in cavity 1 act as the control qubits. Twospatially overlapped atomic ensembles A and B with thenumbers of atoms Na and Nb trapped in cavity 2 act as thememory qubits �23�. The cavity 1 �2� supports two degener-ate cavity modes a1�2�h and a1�2�v with different polarizationsh and v, respectively. For generation of storable atom-photonentanglement, an individual atom is prepared in the state��0�+��1�, where � and � are the superposition coefficientswith relation ���2+ ���2=1. The ensemble A �B� is initiallypumped into �Ga�= � i=1

Na �ga�i ��Gb�= � i=1Nb �gb� j�. First, we let

switch K be on the transmitted state and apply two classicalfields to drive the transitions �0�↔ �e� and �sb�↔ �eb� withtime-dependent Rabi frequency �1�t� and �2�t�, respec-tively. The interaction between the atoms and cavity modescan be written as Ha−c=�1�t��e��0�+gv�e��r�a1v+�NbgvEb

†a2v+�2�t�Eb†Sb+H.c. �24�, where Mb

†*xbz@ustc.edu.cn

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=1 /�Nbi=1Nb �mb� j�gb� �M =S, m=s or M =E, m=e� denotes

the collective atomic excitation operator. The coupling be-tween the cavity fields and the fiber modes bh and bv, in“short fiber limit,” can be described by �21,22,25,26� Hfib=�bh�a1h

† +a2h† �+�bv�a1v

† +a2v† �+H.c., where � is the coupling

strength between the cavity modes and the fiber modes.The interaction Hamiltonian for atom-cavity-fiber systemHI=Ha−c+Hfib has a particular zero-energy eigenstate,i.e., dark state: �D1�t��� �gv�2�t��0�I−�1�t��2�t��r�a1v

†

+�2�t��1�t��r�a2v† −�Nbgv�1�t��r�Sb

†��vac�, where I is iden-tity operator, �vac� denotes that atomic ensemble A �B� is inthe initial state �Ga� ��Gb��, and the cavity modes are invacuum states. Under the conditions gv��1�t� and �Nbgv��2�t�, we have

�D1��t�� �cos �1�0�I − sin �1�r�Sb†��vac� , �1�

where tan �1=�Nb�1�t� /�2�t�. We note that there is a factorof �Nb over the single atom case �21,22,25,26� due to themany-atom interference effect, which greatly relaxes the re-quirements of the strong-coupling condition. By adiabati-cally changing �1�t� and �2�t�, we can coherently map thequantum state ���0�+��1���vac� onto �−��r�Sb

†+��1���vac�.Then transferring �r� back into −�0�, we can obtain ���0�Sb

†

+��1���vac�. In a similar way, we apply two classical fieldsto drive the transitions �1�↔ �e� and �sa�↔ �ea� with time-dependent Rabi frequency �3�t� and �4�t�, respectively. Un-der the conditions gh��3�t� and �Nagh��4�t�, we have adark state

�D2��t�� �cos �2�1�I − sin �2�f�Sa†��vac� , �2�

where tan �2=�Na�3�t� /�4�t� and Sa†=1 /�Nai=1

Na �sa�i�ga�.By adiabatically changing �3�t� and �4�t�, we can map thequantum state ���0�Sb

†+��1���vac� onto ���0�Sb†

−��f�Sa†��vac�. Then transferring �f� back into −�1�, we have

���0�Sb† + ��1�Sa

†��vac� . �3�

We can extend the method above to realize multimode-memory-based entanglement source �27,28�. Suppose thatthere are n �nNa , Nb� accessional states �ska� ��skb�� inatomic ensembles A �B� and the quantum states can be co-

herently mapped between �sa� ��sb�� and �ska� ��skb��, with k=1,2 , . . . ,n. We generate the entangled state in Eq. �3�, thentransfer Sa

†�vac� �Sb†�vac�� to one of n accessional states �ska�

��skb��. By repeating these two processes, we can obtain amultiparticle entanglement source,

���0��k=1

n

Skb† + ��1��

k=1

n

Ska† �vac� . �4�

If we have n individual atoms, in a similar way, let n indi-vidual atoms sequentially entangle with atomic ensembles Aand B in Eq. �3�, then Sa

†�vac� �Sb†�vac�� is stored in one of n

accessional states �ska� ��skb��. We will obtain

k=1

n

��k�0�kSkb† + �k�1�kSka

† ��vac� . �5�

For retrieval of the stored states Sa†�vac� and Sb

†�vac� ascorresponding photon polarization states, we let the switch Kbe on a reflected state and simultaneously apply the classicalfields to drive the transitions �sb�↔ �eb� and �sa�↔ �ea� withtime-dependent Rabi frequency �2��t� and �4��t�, respec-tively. We have two dark states,

�D3�t�� = �cos �3Sb† + sin �3�v�I��vac� , �6�

�D4�t�� = �cos �4Sa† + sin �4�h�I��vac� , �7�

where tan �3=�2��t� /�Nbgv, tan �4=�4��t� /�Nbgh, and �v���h�� denotes a photon in polarization v �h�. By adiabaticallychanging �2��t� and �4��t�, we can coherently map the quan-tum state ���0�Sb

†+��1�Sa†��vac� onto ���0��v�+��1��h���vac�.

Other storable state Ska† �vac� and Skb

† �vac� can be, respec-tively, mapped back onto Sa

†�vac� and Sb†�vac�, then be re-

trieved. By adiabatic evolution, Simon et al. �29� experimen-tally demonstrated retrieval of the stored states in a cavitywith a high efficiency r84%.

Next we briefly discuss the feasibility of generation andretrieval of atom-photon entanglement source. The condi-tions �I�t�=gv�h� /�1�3��t��1 and �II�t�=�Nb�a�gv�h� /�2�4��t��1 are required for Eqs. �1� and �2�. Suppose that minimal�I

min=�IImin=�10, where �I�II�

min is the minimal �I�II��t�. Underthe adiabatic condition, the time for preparation and retrievalof entanglement source tp and tr are on the order of 100

�Ngand

10�Ng

, respectively. Here, no loss of generality, we have as-sumed that gh=gv=g and Na=Nb=N. In experiment �29� therequired time tr100 ns. Thus tp10tr1 �s can beachieved. We note that dark states in Eqs. �1� and �2� do notinvolve the occupation of cavity modes. The efficient deco-herence time of the cavity is greatly prolonged by a factor ofabout �2=100 �30�. Thus the efficiency for preparation ofentanglement source p, in principle, can be up to r84% under current experiment technique.

III. ENTANGLEMENT DISTRIBUTION

As shown in Fig. 2, there are two nodes LA and LB in anetwork. The distance between them L0 Latt, with Latt as thechannel attenuation length. Suppose that we have prepared

Flying qubit

K

3�1�

1

f r

vh

0

e

Control qubit

as

ae be

4�h

ag

'4�

bgbs

2�v

'2�

Memory qubit

Cavity 2Cavity 1

FIG. 1. �Color online� Setup for generation and retrieval ofatom-photon entanglement source. Two local optical cavities 1 and2 are connected by a short optical fiber. The individual atoms,atomic ensembles, and photons act as the control qubits, memoryqubits, and flying qubits, respectively. K denotes a switch.

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two multimode-memory-based entanglement sources in Eq.�5�, i.e., ��k

A�0�kASkb

† +�kA�1�k

ASka† ��vacA� and ��k

B�0�kBSkb

†

+�kB�1�k

BSka† ��vacB�, where the superscript A �B� denotes the

node LA �LB�. The stored states in both nodes are simulta-neously retrieved and then interfere at middle side of twonodes. In the ideal case, detection of two photons in either oftwo output ports for kth time bin would result in a measure-ment of the state Ska

†2�Skb†2. Then we know that an entangled

pair,

�kA�k

B�0�kA�0�k

B � �kA�k

B�1�kA�1�k

B, �8�

has been successfully created. The success probability P0�n /2�p

2r2d

2e−L0/�2Latt� �d is the single-photon detectionefficiency�, and the average generation time is given by T0Tcc / P0, where Tcc is the classical communication time. Wenote that the entangled state in Eq. �8� is somewhat differentfrom that in previous quantum-repeater protocol �3–14� sincethe entangled state in Eq. �8� keeps track of the information�k

A�B� and �kA�B� of the initial qubit state. Thus, the scheme

here is not just an entangling protocol but is instead an en-tangling gate with the final quantum state depending on theinitial state. Duan et al. �31� showed that such probabilistic,but heralded, quantum entangling gate can be used for scal-able quantum computation by a combination of the quantumrepeater and the cluster state approach.

After the successful generation of entanglement withinthe attenuation length, we want to extend the quantum com-munication distance. This is done through entanglementswapping, As shown in Fig. 3, there are four nodes �LA , LB�and �LC , LD� and suppose that we have created two pairs ofmaximal entanglement, i.e., ��0�A�0�B+ �1�A�1�B� /�2 and

��0�C�0�D+ �1�C�1�D� /�2, between the nodes �LA , LB� and�LC , LD�, respectively. First, the quantum states of indi-vidual atoms in nodes LB and LC are transferred to corre-sponding atomic ensembles. The process for transferring issimilar to that for preparation of entanglement source. Afterthe state �0��vac� is coherently mapped onto −�r�Sb

†�vac� withadiabatically changing the dark state in Eq. �1�, we transfer�r� to −�f� instead of −�0�, then coherently map �1��vac� onto−�f�Sa

†�vac� with adiabatically changing the dark state in Eq.�2�. Thus we can obtain the state ��0�ASb

†+ �1�ASa†��vacB�

� ��0�DSb†+ �1�DSa

†��vacC� /2. Second, the stored states innodes LB and LC are simultaneously retrieved as single pho-tons to performed Bell-state measurement. This leads thestate of individual atoms in nodes LA and LD to collapse intoa entangled state ��0�A�0�D� �1�A�1�D� /�2.

The above method for connecting entanglement can becascaded to arbitrarily extend the communication distance.For the mth �m=1,2 , . . . ,N� entanglement connection, wefirst prepare in parallel two pairs of entangled state with thesame communication length Lm−1 and then perform entangle-ment swapping as shown in Fig. 3, which now succeeds witha probability Pm. After a successful two detectors click, thecommunication length is extended to Lm=2Lm−1. As the mthentanglement connection needs to be repeated on average1 / Pm times, the total time needed to establish an entangledstate over the distance Lm=2mL0 is given by

Tm T0�L/L0�log2�3/2� �

m=0

N

�1/Pm� L0

ncp2 eL0/Latt�L/L0�log2

3/�2p�,

�9�

where the factor of �L /L0�log2�3/2�

arises because entanglementhas to be generated for two links before every entanglementconnection and we have assumed that P=r

2d2e−L0/�2Latt� is a

constant �28�. Thus the total time Tm only scales polynomi-ally with the communication distance L. The total time Tm tocreate entanglement across two communication nodes at adistance of 1280 km can be numerically estimated. In ourcalculation, we assume that L0=10 km, n=3, prd0.95, Latt=22 km, and c=2.0�108 m /s in optical fibers.Our numerical result gives a total time of about 9.6 ms.However, in previous protocols typical times to create en-tanglement at a distance up to 1000 km are about a few tensof seconds with single-photon interference �3,15� and severalhours with two-photon interference �13�.

Suppose that entanglement has been established betweentwo distant nodes �LI , LII�, i.e., ��0�I�0�II+ �1�I�1�II� /�2, usingthe method for generation of GHZ state in Eq. �5�, we canobtain

��0�I�0�II�k=1

n

Skb�I�† �

k�=1

n

Sk�b�II�† + �1�I�1�II�

k=1

n

Ska�I�†

� �k�=1

n

Sk�a�II�† �vacI�

��vacII�/�2, �10�

which is a long-distance multiparticle GHZ state. Since theatomic states can be effectively retrieved as correspondingphoton polarization states, the state in Eq. �10� is equivalentto a nonlocal GHZ photon state, which can be used for many

K

K

Node AL

Node BL

DD

HWP

D

PBS

D

FIG. 2. �Color online� Schematic setup to entanglement genera-tion. Where PBS, K, HWP, and D denote polarizing beam splitter,switch, half-wave plate, and single-photon detector, respectively.

Node BL

Node AL

Node CL

Node DL

FIG. 3. �Color online� Schematic setup to entanglement connec-tion through swapping. First, the quantum states of individual atomsin nodes LB and LC are transferred to corresponding atomic en-sembles. Second, the stored states in nodes LB and LC are simulta-neously retrieved as a single photon to performed Bell-state mea-surement. Other notations are the same as in Fig. 2.

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entanglement-based communication protocols, such as quan-tum teleportation, cryptography, and Bell inequality detec-tion �3�.

IV. BRIEF DISCUSSION AND CONCLUSION

We have supposed that perfect coupling between the cavi-ties and optical fibers can be achieved. In Ref. �32�, thestrong coupling ��95%� of fiber-cavity system has been ob-served. In our scheme, we assume that the polarization andfrequency of the classical fields is different from the cavitymodes h and v, and the photon scattering from the classicalfields into the cavity modes is so small that can be ignored.We note that the switches, the fibers, and the double-cavitysystem are not prerequisite to our scheme. If we assume thatthe control qubits and the corresponding memory qubits aretrapped in a same cavity and they can be addressed individu-ally by the classical fields, under the same conditions we canalso obtain the dark states in Eqs. �1� and �2�. However, itrequires that the control qubits and memory qubits are sepa-rated by at least one optical wavelength �33�. All of opera-tions in the cavities can be accomplished with adiabatic evo-lution of dark states. Even if the experimental parameterscannot be fully controlled, the operations still have a veryhigh fidelity �34�. Our scheme is based on two-photon Hong-Ou-Mandel-type interference which relaxes the long-distancestability requirements �13�. The dominant noise in ourscheme is photon loss, which includes atomic spontaneousemissions, coupling inefficiency of photons into and out ofthe cavity, contributions from channel attenuation, and inef-ficiency of the single-photon detectors. The photon loss onlydecreases the success probability but has no contribution tothe gate infidelity if the operations succeed �i.e., if two-photon count is registered�.

For a potential experimental system, we consider lasercooled individual and ensemble of 133Cs atoms �29,23� thatare trapped in cavities 1 and 2, respectively. The requiredatomic level configuration can be chosen from the hyperfinestates of 133Cs. For individual atoms, the states �0�, �1�, �f�,and �r� correspond to �5S1/2 ;F=2,m=−1�, �5S1/2 ;2 ,1�,

�5S1/2 ;3 ,−1�, and �5S1/2 ;3 ,1�, while �e� corresponds to�5P3/2 ;3 ,0�. For atomic ensembles A �B�, �ga� ��gb�� and �sa���sb�� correspond to �5S1/2 ;3 ,−3� ��5S1/2 ;3 ,3�� and�5S1/2 ;2 ,−2� ��5S1/2 ;2 ,2��, while �ea� ��eb�� corresponds to�5P3/2 ;3 ,2� ��5P3/2 ;3 ,2��. Other hyperfine levels in theground-state manifold can be used as accessional states �ska���skb��. We assume that the quantum states can be coherentlymapped between �sa� ��sb�� and �ska� ��skb��. This may be donethrough applying external fields to lift their degeneracy orapplying techniques from control theory �35� or a “holo-graphic quantum register” with hundreds of qubits is en-coded in collective excitations with definite spatial phasevariations �36�. We note that recently two spatially over-lapped atomic ensembles inside an optical resonator acted asquantum memory for arbitrary polarization states have beenrealized in experiment �23�, and a fast high-fidelity many-particle gate by entangling a control atom and an atomicensemble was also proposed and analyzed �37�.

In summary, we have proposed a scheme for the realizinglong-distance quantum communication with individual atomsand atomic ensembles by the way of quantum-repeater archi-tecture. In the scheme, two local optical cavities are con-nected by a short optical fiber. Some individual atoms andatomic ensembles, respectively, trapped in the cavities act asthe control qubits and memory qubits. Deterministic andstorable atom-photon entanglement sources can be obtainedby adiabatic evolution of dark states. No strong-couplingcondition for the cavities is required in our scheme due tocollective interference effects of atomic ensembles. Thus ourscheme opens an alternate avenue for long-distance quantumcommunication.

ACKNOWLEDGMENTS

This work was funded by National Natural Science Foun-dation of China �Grants No. 10574022 and No. 60878059�,the Natural Science Foundation of Fujian Province of China�Grant No. 2007J0002�, the Foundation for Universities inFujian Province �Grant No. 2007F5041�, and “Hundreds ofTalents ” program of the Chinese Academy of Sciences.

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