Annals of Physics 325 (2010) 1018–1025

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Annals of Physics

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Long-distance quantum communication with ‘‘polarization”maximally entangled states

Fang-Yu Hong a,*, Shi-Jie Xiong b, W.H. Tang a

a Department of Physics, Center for Optoelectronics Materials and Devices, Zhejiang Sci-Tech University, Xiasha College Park,Hangzhou, Zhejiang 310018, Chinab National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China

a r t i c l e i n f o

Article history:Received 10 July 2009Accepted 1 February 2010Available online 6 February 2010

Keywords:Quantum entanglementQuantum repeaterAtomic ensemble

0003-4916/$ - see front matter � 2010 Elsevier Indoi:10.1016/j.aop.2010.02.001

* Corresponding author.E-mail address: honghfy@163.com (F.-Y. Hong).

a b s t r a c t

We propose a scheme for long-distance quantum communicationwhere the elementary entanglement is generated through two-photon interference and quantum swapping is performed throughone-photon interference. Local ‘‘polarization” maximally entangledstates of atomic ensembles are generated by absorbing a singlephoton from on-demand single-photon sources. This scheme isrobust against phase fluctuations in the quantum channels, more-over speeds up long-distance high-fidelity entanglement genera-tion rate.

� 2010 Elsevier Inc. All rights reserved.

Entanglement plays a fundamental role in quantum information science [1] because it is a crucialrequisite for quantum metrology [2], quantum computation [3,4], and quantum communication [3,5].Quantum communication opens a way for completely secure transmission of keys with the Ekert pro-tocol [6] and exact transfer of quantum states by quantum teleportation [7]. Because of losses andother noises in quantum channels, the communication fidelity falls exponentially with the channellength. In principle, this problem can be circumvented by applying quantum repeaters [5,8–11], ofwhich the basic principle is to separate the full distance into shorter elementary links and to entanglethe links with quantum swaps [7,12]. A protocol of special importance for long-distance quantumcommunication with collective excitations in atomic ensembles has been proposed in a seminal paperof Duan et al. [13]. After that considerable efforts have been devoted along this line [14–20].

In Duan–Lukin–Cirac–Zoller (DLCZ) protocol, entanglement in the elementary links is created bydetecting a single photon from one of two ensembles. The probability p of generating one excitationin two ensembles is related to the fidelity of the entanglement, leading to the condition p� 1 to

c. All rights reserved.

F.-Y. Hong et al. / Annals of Physics 325 (2010) 1018–1025 1019

guaranty an acceptable quality of the entanglement. But when p! 0, some experimental imperfec-tions such as stray light scattering and detector dark counts will contaminate the entangled stateincreasingly [20], and subsequent processes including quantum swap and quantum communicationbecome more challenging for finite coherent time of quantum memory [16]. To solve this problem,protocols based on single photon source [16,17] and photon pair source [18] were suggested. How-ever, for the scheme proposed in Ref. [16] the ‘‘vacuum” coefficient c0 [13] of the state of the elemen-tary link is near 1, which causes the probability pi ði ¼ 1;2; . . . ;nÞ of successful quantum swap to bevery low and thus the capability of the scheme in increasing quantum communication rate to be weak,where n is the nesting level of swap. For the schemes suggested in Refs. [17,18], the same problemexists owing to the fact that the efficiency of storage of a single photon in a quantum memory isfar from ideal. Furthermore, all schemes based on measuring a single-photon via single-photon detec-tors suffer from the imperfections from the detector dark counts and its incapability of distinguishingone photon from two photons.

Here we present a protocol for long-distance quantum communication using linear optics andatomic ensembles. To overcome the low probability p in DLCZ protocol, we generate the entanglementin every node with on-demand single photon source. To solve the problem of the large ‘‘vacuum” coef-ficient c0 in Refs. [16–18], the quantum swapping is performed based on ‘‘polarization” maximallyentangled states [13]. Our scheme can automatically eliminate the imperfection arising from the inca-pability of the single-photon detectors in distinguishing one photon from two photons and can ex-clude partially the imperfection due to the detector dark counts, which is the major imperfectionon the quality of the entanglement for the previous schemes [17]. With this scheme the quantumcommunication rate can be significantly increased by several orders of magnitude with higher quan-tum communication fidelity for a distance 2500 km compared with the DLCZ protocol. To be insensi-tive to the phase fluctuation in the quantum channel [19,21], our previous propose for quantumcommunication [22] employs two-photon Hong–Ou–Mandel-type (HOMT) interferences to generatelocal entanglement, to distribute basic entanglement between distance L0, and to connect entangle-ment with quantum swap. Because the phase instability in the local quantum channel is easy to con-trol, this scheme uses single-photon Mach–Zehnder-type interferences to generate local entanglementand to connect entanglement, and uses two-photon HOMT interferences only to distribute basicentanglement to simplify the physical set-up.

The quantum memory in our scheme can be a cloud of Na identical atoms with pertinent levelstructure shown in Fig. 1b. One ground state jgi and two metastable states jsi and jtimay be providedby, for instance, hyperfine or Zeeman sublevels of the electronic ground state of alkali-metal atoms,where long relevant coherent lifetime has been observed [23–25]. The atomic ensemble is opticallythick along one direction to enhance the coupling to light [13]. State je1i is an excited state. A singlephoton emitted with a repetition rate r from an on-demand single-photon source [16,26] located half-way between quantum memories L and R in every node is split into an entangled state of opticalmodes Lin and Rin (Fig. 1a) described by

Fig. 1.single-p

jwinð/Þi ¼1ffiffiffi2p ðj0Lin

j1Rini þ ei/j1Lin

ij0RiniÞ; ð1Þ

where / denotes an unknown difference of the phase shifts in the L and R side channels. This statethen is coherently mapped onto the state of atomic ensembles L and R:

a b

(a) Schematic illustration of entanglement establishment between two atomic ensembles L and R through on-demandhoton sources. (b) The relevant level configuration of atoms in the ensembles and the coupling with pulses.

Fig. 2.effectiv

1020 F.-Y. Hong et al. / Annals of Physics 325 (2010) 1018–1025

jwð/ÞiLR ¼1ffiffiffi2p TyL þ ei/TyR

� �j0aiLj0aiR ð2Þ

by applying techniques such as adiabatic passage based on dynamic electromagnetically inducedtransparency [16], where T � 1=

ffiffiffiffiffiffiNap PNa

i¼1jgiihtj is the annihilation operator for the symmetric collec-tive atomic mode T [13] and j0ai � �ijgii is the ensemble ground state. Considering photon loss, whichincludes the optical absorption in the quantum channel and the inefficiency of the excitation transferfrom the optical mode to quantum memory mode, the state of ensembles R and L can be described byan effective maximally entangled (EME) state [13]

qLRðc0;/Þ ¼1

c0 þ 1ðc0j0a0aiLRh0a0aj þ jwð/ÞiLRhwð/ÞjÞ; ð3Þ

where c0 is the vacuum coefficient.Before proceeding we discuss the conversion of the collective atomic excitation T into the atomic

excitation S given by S � 1=ffiffiffiffiffiffiNap PNa

i¼1jgiihsj. Consider the atoms have an excited state je2i satisfying thecondition that the dipole moments of the atomic transitions er1 ¼ ehgjrje2i ¼ 0; er2 ¼ ehsjrje2i – 0, ander3 ¼ ehtjrje2i– 0 [27]. The transition jsi ! je2i of each of these atoms is coupled to a quantized radi-ation mode described by an annihilation operator a with a coupling constant g; the transitions fromje2i ! jti are resonantly driven by a classical control field of Rabi frequency Xc2 (Fig. 2). The interac-tion Hamiltonian of this systems is in the form [28]

Hin ¼ �hgaXN

i¼1

rie2s þ �hXc2ðtÞ

XN

i¼1

rie2t þ H:c:; ð4Þ

where rilm ¼ jliiihmj is the flip operator of the ith atom between states jli and jmi. This interaction

Hamiltonian has the dark state with zero adiabatic eigenvalue [28–30],

jDi ¼ cos hðtÞSyjgij1i � sin hðtÞTyjgij0i; ð5Þ

where tan h ¼ g=Xc2ðtÞ and jmi denotes the radiation state with m photon. Thus with this dark state,by applying a retrieval pulse of suitable polarization that is resonant with the atomic transitionjti ! je2i, the atomic excitation T in an atom ensemble can be converted into the atomic excitationS while a photon which has polarization and frequency different from the retrieval pulse is emitted[13,22,24,27,28,31]. Because this conversion process does not involve the collective enhancement,its efficiency is low.

Now we discuss the generation of local entanglement. Two pairs of ensembles are prepared in thesame EME state qLiRi

ði ¼ 1;2Þ at every node with the vacuum coefficient c0 (Fig. 2). The / parametersin qL1R1

and qL1R1are equal assuming that the two EME states are generated through the same station-

ary channels. The state of the two pairs of ensembles can be described with qL1R1� qL2R2

. By applyingretrieval pulses on resonance with the atomic transition jti ! je2i, the atomic excitations T are trans-formed simultaneously into excitations S while photons are emitted. After the conversion, the stimu-lated photons overlap at a 50%–50% beam split (BS), and then are recorded by the single-photondetectors DL1 ;DL2 ðDR1 ;DR2 Þ which measures the combined radiation from two samples, ayþLaþL oray�La�L (ayþRaþR or ay�Ra�R), with a�i ¼ a1i � ei/i a2i; i ¼ L;R [13]. In the following discussion, we assume/L ¼ /R, which is easy to control for the local transformation [15,21]. Only the coincidences of the

a b

(a) The relevant level configuration of the atoms in the ensemble and coupling pulses. (b) Configuration for projecting ane maximally entangled (EME) state to a ‘‘polarization” maximally entangled (PME) state.

F.-Y. Hong et al. / Annals of Physics 325 (2010) 1018–1025 1021

two-side detectors are recorded, so the protocol succeeds with a probability pr only if both of thedetectors on the left and right sides have a click. Under this circumstance, the vacuum componentsin the EME states, the state components TyL1

TyL2jvaci, and TyR1

TyR2jvaci have no effect on the experimen-

tal results, where jvaci is the ground state of the ensemble j0a0a0a0aiL1R1L2R2. Thus, after the conversion,

the state of system of four ensembles can be written as the following polarization maximally entan-gled (PME) state

Fig. 3.count band BRi

where

jwi�PME ¼ ðSyL1SyR2 � SyL2SyR1Þjvaci=

ffiffiffi2p

: ð6Þ

Without loss of generality, we assume that the generated PME is jwiþPME in the following discussion. Thesuccess probability for entanglement generation at every node is pr ¼ g2

pg2s g2

e1g2

d=2, where we denotethe probability of emitting one photon by the single-photon source with gp, the efficiency for theatomic ensemble storing a photon by gs, the efficiency for the atomic ensemble emitting a photon dur-ing the process Tyj0ai ! Syj0ai by ge1

, and the single-photon detection efficiency by gd. The averagewaiting time for successful generating a local entanglement state is Tl ¼ 1

rpr.

Then we show how to distribute basic entanglement between neighboring nodes at a distance L0.The atomic ensembles at neighboring nodes A and B are prepared in the state jwiþPME, then illuminatedsimultaneously by retrieval laser pulses on resonance of the atomic transition jsi ! je3i, where je3i anexcited state, the atomic excitations S are transformed simultaneously into anti-Stokes photons. Weassume the anti-Stokes photons are in an orthogonal polarization state jHi from ensemble AR1;BL1

and jVi from ensemble AR2;BL2, which represent horizontal and vertical linear polarization,respectively.

After the conversion, the stokes photons from site A and B at every node are directed to the polar-ization beam splitter (PBS) and experience two-photon Bell-state measurement (BSM) (shown inFig. 3) at the middle point to generate an entanglement between the atomic ensembles ALi andBRi ði ¼ 1;2Þ. Only the coincidences of the two single-photon detectors D1 and D4 (D1 and D3) or D2

and D3 (D2 and D4) are recorded, so the protocol is successful only if each of the paired detectors havea click. Under this circumstance, the vacuum components in the EME states, one-excitation compo-nents like SyLu

jvaci, and the two-excitation components SyAL1SyBR1jvaci and SyAL2

SyBR2jvaci have no effect

on the experimental results [33]. A coincidence click between single-photon detectors, for example,D1 and D4 will project the four atomic ensembles into PME state [21,32,33]

jWiþAB ¼1ffiffiffi2p SyAL1

SyBR2þ SyAL2

SyBR1

� �jvaci: ð7Þ

Schematic illustration of basic entanglement generation with length L0. Up to a local unitary phase shift the coincidenceetween single-photon detectors D1 and D4 (D1 and D3) or D2 and D3 (D2 and D4) will project the atomic ensembles at ALi

ði ¼ 1;2Þ into a PME state in the form of Eq. (7). PBS ðPBS�Þ transmits jHi ðjþiÞ photons and reflects jVi ðj�iÞ photons,j�i ¼ 1ffiffi

2p ðjHi � jViÞ.

1022 F.-Y. Hong et al. / Annals of Physics 325 (2010) 1018–1025

The success probability for entanglement generation within the attenuation length is pb ¼ g2e2g2

dg2t =2,

where ge2 denotes the efficiency for the atomic ensemble emitting a photon during the processSyj0ai ! j0ai and gt ¼ exp½�L0=ð2LattÞ� is the fiber transmission efficiency with the attenuation lengthLatt .

After successful generation of PME states within the basic link, we can extend the quantum com-munication distance through entanglement swapping with the configuration shown in Fig. 4. We havetwo pairs of ensembles—A1;A2;BL1 and BL2 , and BR1 ;BR2 ;C1 and C2—located at three sites A, B, and C.Each pair of ensembles is prepared in the PME state (Eq. (7)). The stored atomic excitations of fourensembles BL1 and BL2 , and BR1 ; BR2 are transferred into light at the same time with near unity effi-ciency. The stimulated optical excitations interfere at a 50%–50% beam splitter, and then are detectedby single-photon detectors D1;D2;D3, and D4. Only if each pair of detectors ðD1;D2Þ, and ðD3;D4Þ, has aclick, the protocol is successful with a probability p1 ¼ g2

e2g2d=2 and a PME state in the form of Eq. (7) is

established among the ensembles A1;A2;C1, and C2 with a doubled communication distance. Other-wise, we need to repeat the previous processes .

The scheme for entanglement swapping can be applied to arbitrarily extend the communicationdistance. For the ith ði ¼ 1;2; . . . ;nÞ entanglement swapping, we first prepare simultaneously two pairsof ensembles in the PME states (Eq. (7)) with the same communication length Li�1, and then makeentanglement swapping as shown by Fig. 4 with a success probability pi ¼ g2

e2g2d=2. After a successful

entanglement swapping, a new PME state is established and the communication length is extended toLi ¼ 2Li�1. Since the ith entanglement swapping needs to be repeated on average 1=pi times, the aver-age total time needed to generating a PME state over the distance Ln ¼ 2nL0 is given by [18]

Ttot ¼L0

cþ 1

rpr

� �1

pb

Qmi¼1pi

32

� �n

ð8Þ

with c being the light speed in the optical fiber.After a PME state has been generated between two remote sites, quantum communication proto-

cols, such as cryptography and Bell inequality detection, can be performed with that PME state like theDLCZ scheme [13]. The established long-distance PME state can be used to faithfully transfer unknownstate through quantum teleportation with the configuration shown in Fig. 5. Two pairs of atomicensembles L1;R1 and L2;R2 are prepared in the PME state. The unknown state which is to be trans-ferred is described by aSyI1

þ bSyI2

� �j0a0aiI1I2

with unknown coefficient a and b, where SyI1and SyI1

arethe collective atomic operators for the two ensembles I1 and I2. The collective atomic excitations inthe ensembles I1; L1 and I2; L2 are transferred into optical excitations simultaneously. After a 50%–50% beam splitter, the optical excitations are measured by detectors DI1 ;DL1 and DI2 ;DL2 . Only if thereis one click in DI1 ;DL1 and one click in DI2 ;DL2 , the state transfer is successful, and the unknown stateaSyI1þ bSyI2

� �j0a0aiR1R2

appears in the ensembles R1 and R2 up to a local p-phase rotation. Unlike theDLCZ protocol, this scheme does not need posterior confirmation of the presence of the excitationto teleportation unknown state.

Fig. 4. Configuration for entanglement swapping.

Fig. 5. Configuration for probabilistic quantum teleportation of an unknown atomic ‘‘polarization” state.

F.-Y. Hong et al. / Annals of Physics 325 (2010) 1018–1025 1023

Now we evaluate the perform of the scheme numerically. The conversion efficiency ge1may be low,

assuming to be 0.01. If we assume that r ¼ 50 MHz;gp ¼ 1;gs ¼ ge2¼ 0:9;gd ¼ 0:9; Ln ¼ 2500 km;

Latt ¼ 22 km for photons with wavelength of 1:5 lm [17], c ¼ 2:0� 105 km=s, and n ¼ 4, Eq. (8) givesthe average total time Ttot ¼ 2251 s, in contrast to the average total time Ttot ¼ 65;0000 s for the DLCZprotocol and Ttot ¼ 15;300 s for single-photon source (SPS) protocol [17] with the above parameters.Thus, compared with the SPS protocol, this scheme can significantly reduce the average total time forsuccessful quantum communication. Note that e2 can be enhanced by putting the atomic ensembles ina low-finesse ring cavity [13] and one can exploited many kinds of on-demand single-photon sources,such as molecule-based sources with max rate 100 MHz and quantum-dot-based sources with maxrate 1 GHz [34].

To enhance the conversion efficiency ge1, we can use a cavity with a quality factor Q. According tothe literature [13], in the free-space limit the signal-to-noise ratio Rsn between the coherent interac-tion rate and the decay rate can be estimated as

Rsn 4Najgcj

2

jcs 3

qnLa

k2s

do; ð9Þ

where qn and do denote the density and the on-resonance optical depth of the atomic ensemble,respectively, La is the length of the pencil-shape atomic ensemble, ks ¼ xs=c ¼ 2p=ks is the wave vec-tor of the cavity mode, and j is the cavity decay rate. j is relate to the quality factor Q of the cavityj ¼ xs=Q [35]. Thus, for the case of the cavity with a quality factor Q, we have the signal-to-noise ratio

Rsn 4Najgcj

2

jcs 3

qnLa

k2s

Q 3qnLak

2s

4p2 Q do; ð10Þ

which shows that the cavity quality factor Q and the atom number of the ensemble N play a similarrole in enhancing the atom–photon interaction. To estimate the magnitude of the signal-to-noise ratioRsn, we assume 3qnLak

2s =4p2 10�2 for the case of a single atom. Then we have Rsn 10 do for

Q ¼ 1000. According to the research [36], the maximum total efficiency for a single photon storagein an atomic ensemble followed by retrieval can be larger than 0.5 for do ¼ 10. Thus, the conversionefficiency ge1 larger than 0.01 is feasible if the atomic ensemble is placed in the cavity with a qualityfactor Q ¼ 1000.

Now we discuss imperfections in our architecture for quantum communication. In the basic entan-glement generation, the contamination of entanglement from processes containing two excitationscan be arbitrarily suppressed with unending advances in single-photon sources [26,34]. In the wholeprocess of basic entanglement generation, connection, and entanglement application, the photon lossincludes contributions from channel absorption, spontaneous emissions in atomic ensembles, conver-sion inefficiency of single-photon into and out of atomic ensembles, and inefficiency of single-photon

1024 F.-Y. Hong et al. / Annals of Physics 325 (2010) 1018–1025

detectors. This loss decreases the success probability but has no effect on the fidelity of the quantumcommunication performed. Decoherence from dark counts in the basic entanglement generation andthe entanglement connection can be excluded, for example, if a dark count occurs on the up side (D1

and D2) (Fig. 4), because in this case there are two clicks in the down side detectors (D2 and D4), thusthe protocol fails and the previous steps need to be repeated. Considering that the probability for adetector to give a dark count denoted by pd smaller than 5� 10�6 is within the reach of the currenttechniques [17], we can estimate the fidelity imperfection DF � 1� F for the generated long-distancePME states by

DF ¼ 2nþ2pd < 3:2� 10�4 ð11Þ

for n ¼ 4.The imperfection that the detectors cannot distinguish between one and two photons only reduces

the probability of successful entanglement generation and connection, but has no influence on both ofthe fidelity of the PME state generated and the quality of quantum communication. For instance, if twophotons have been miscounted as one click in detectors DI1 and DL1 in Fig. 5, then there is no click inthe detectors DI2 and DL2, thus the protocol says that the state transfer fails. Like DLCZ protocol, thephase shifts arising from the stationary quantum channels and the small asymmetry of the stationaryset-up can be eliminated spontaneously when we generate the PME state from the EME state, and thushave no effect on the communication fidelity. Because the basic entanglement between distance L0 isgenerated through two-photon interference, this scheme is robust against the phase fluctuation in thequantum channels [21].

In conclusion, we have proposed a robust scheme for long-distance quantum communication basedon ‘‘polarization” maximally entangled state. Through this scheme, the rate of long-distance quantumcommunication may increase compared with the SPS protocol. At the same time, higher fidelity oflong-distance quantum communication can be expected. Considering the simplicity of the physicalset-ups used, this scheme may opens up the probability of efficient long-distance quantum commu-nication.

Acknowledgments

This work was supported by the State Key Programs for Basic Research of China (2005CB623605and 2006CB921803), by National Foundation of Natural Science in China (Grant Nos. 10474033 and60676056), and the National Nature Science Foundation of China (Grant Nos. 50672088 and60571029).

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