articles Long-distance quantum communication with atomic ensembles and linear optics L.-M. Duan*², M. D. Lukin³, J. I. Cirac* & P. Zoller* * Institut fuÈr Theoretische Physik, UniversitaÈt Innsbruck, A-6020 Innsbruck, Austria ² Laboratory of Quantum Communication and Computation, USTC, Hefei 230026, China ³ Physics Department and ITAMP, Harvard University, Cambridge, Massachusetts 02138, USA ............................................................................................................................................................................................................................................................................ Quantum communication holds promise for absolutely secure transmission of secret messages and the faithful transfer of unknown quantum states. Photonic channels appear to be very attractive for the physical implementation of quantum communication. However, owing to losses and decoherence in the channel, the communication ®delity decreases exponentially with the channel length. Here we describe a scheme that allows the implementation of robust quantum communication over long lossy channels. The scheme involves laser manipulation of atomic ensembles, beam splitters, and single-photon detectors with moderate ef®ciencies, and is therefore compatible with current experimental technology. We show that the communication ef®ciency scales polynomially with the channel length, and hence the scheme should be operable over very long distances. The goal of quantum communication is to transmit quantum states segments only where the previous attempt fails. This is essential for between distant sites. Such transmission has potential application in ensuring polynomial scaling in the communication ef®ciency, the secret transfer of classical messages by means of quantum because if there were no available memory, the puri®cations for cryptography1, and is also an essential element in the construction all the segments would need to succeed at the same time; the of quantum networks. The basic problem of quantum communica- probability of such an event decreases exponentially with channel tion is to generate nearly perfect entangled states between distant length. The requirement of quantum memory implies that we need sites. Such states can be used, for example, to implement secure to store the local qubits in atomic internal states instead of photonic quantum cryptography using the Ekert protocol1, and to faithfully states, as it is dif®cult to store photons for a reasonably long time. transfer quantum states via quantum teleportation2. All realistic With atoms as the local information carriers, it seems to be very schemes for quantum communication are at present based on the hard to implement quantum repeaters: normally, one needs to use of photonic channels. However, the degree of entanglement achieve the strong coupling between atoms and photons by using generated between two distant sites normally decreases exponen- high-®nesse cavities for atomic entanglement generation, puri®ca- tially with the length of the connecting channel, because of optical tion, and swapping8,9, which, in spite of recent experimental absorption and other channel noise. To regain a high degree of advances10±12, remains a very challenging technology. entanglement, puri®cation schemes can be used3, but this does not Here we propose a different scheme, which realizes quantum fully solve the long-distance communication problem. Because of repeaters and long-distance quantum communication with simple the exponential decay of the entanglement in the channel, an physical set-ups. The scheme is a combination of three signi®cant exponentially large number of partially entangled states are advances in entanglement generation, connection, and applica- needed to obtain one highly entangled state, which means that for tions, with each of the steps having built-in entanglement puri®ca- a suf®ciently long distance the task becomes nearly impossible. tion and resilience to realistic noise. The scheme for fault-tolerant To overcome the dif®culty associated with the exponential ®delity entanglement generation originates from earlier proposals to decay, the concept of quantum repeaters can be used4. In principle, entangle single atoms through single-photon interference at this allows the overall communication ®delity to be made very photodetectors13,14. But the present approach involves collective close to unity, with the communication time growing only poly- excitations in atomic ensembles rather than in single particles, nomially with transmission distance. In analogy to fault-tolerant which allows simpler realization and greatly improved generation quantum computing5,6, the proposed quantum repeater is a ef®ciency. This is due to collectively enhanced coupling to light, cascaded entanglement-puri®cation protocol for communication which has been recently investigated both theoretically15±19 and systems. The basic idea is to divide the transmission channel into experimentally20±22. The entanglement connection is achieved many segments, with the length of each segment comparable to the through simple linear optical operations, and is inherently robust channel attenuation length. First, entanglement is generated and against realistic imperfections. Different schemes with linear optics puri®ed for each segment; the puri®ed entanglement is then have been proposed recently for quantum computation23 and extended to a greater length by connecting two adjacent segments puri®cation24. Finally, the resulting state of ensembles after the through entanglement swapping2,7. After this swapping, the overall entanglement connection ®nds direct applications in realizing entanglement is decreased, and has to be puri®ed again. The rounds entanglement-based quantum communication protocols, such as of entanglement swapping and puri®cation can be continued until quantum teleportation, cryptography, and Bell inequality detection. nearly perfect entangled states are created between two distant sites. In all of these applications, the mixed entanglement is puri®ed To implement the quantum repeater protocol, we need to gen- automatically to nearly perfect entanglement. As a combination of erate entanglement between distant quantum bits (qubits), store these three advances, our scheme circumvents the realistic noise and them for a suf®ciently long time and perform local collective imperfections, and provides a feasible method of long-distance operations on several of these qubits. Quantum memory is essential, high-®delity quantum communication. The required overhead in because all puri®cation protocols are probabilistic. When entangle- communication time increases with distance only polynomially. ment puri®cation is performed for each segment of the channel, quantum memory can be used to keep the segment state if Entanglement generation the puri®cation succeeds, and to repeat the puri®cation for the The basic element of our system is a cloud of Na identical atoms with NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 413 articles the relevant level structure shown in Fig. 1. A pair of metastable ensemble given by S²j0ai, where the ensemble ground state lower states jgi and jsi can correspond toÐfor exampleÐhyper®ne j0ai [ #i jgii). or Zeeman sublevels of the electronic ground state of alkali-metal We assume that the light±atom interaction time t¢ is short, so atoms. Long lifetimes for the relevant coherence have been observed that the mean photon number in the forward-scattered Stokes pulse both in a room-temperature dilute atomic gas (see, for example, is much smaller than 1. We can de®ne an effective single-mode ref. 21) and in a sample of cold trapped atoms (see, for example, bosonic operator a for this Stokes pulse with the corresponding refs 20, 22). To facilitate enhanced coupling to light, the atomic vacuum state denoted by j0pi. The whole state of the atomic medium is preferably optically thick along one direction. This can collective mode and the forward-scattering Stokes mode can now be achieved either by working with a pencil-shaped atomic be written in the following form (see Supplementary Information sample20±22 or by placing the sample in a low-®nesse ring for details) cavity17,25 (see Supplementary Information). p jfi j0 p S²a²j0 All the atoms are initially prepared in the ground state jgi. A ai j0pi c ai j0pi o pc 1 sample is illuminated by a short, off-resonant laser pulse that where pc is the small excitation probability, and o(pc) represents the induces Raman transitions into the states jsi. We are particularly terms with more excitations whose probabilities are equal to or interested in the forward-scattered Stokes light that is co-propagat- smaller than p2c. Before proceeding, we note that a fraction of light is ing with the laser. Such scattering events are uniquely correlated emitted in other directions owing to spontaneous emissions. But with the excitation of the symmetric collective atomic mode S p whenever Na is large, the contribution from the spontaneous (refs 15±22) given by S [ 1= Na Sijgii hsj, where the summation is emissions to the population in the symmetric collective mode is taken over all the atoms. In particular, an emission of the single small15±22. As a result, we have a large signal-to-noise ratio for the Stokes photon in a forward direction results in the state of atomic processes involving the collective mode, which greatly enhances the ef®ciency of the present scheme (see Box 1 and Supplementary Information). a e Box 1 Collective enhancement Long-lived excitations in atomic ensembles can be viewed as waves of excited spins. We are here particularly interested in the symmetric spin wave mode S. For a simple demonstration of collective enhancement, we assume that the atoms are placed in a low-®nesse ring cavity25, with a s relevant cavity mode corresponding to forward-scattered Stokes g radiation. The cavity-free case corresponds to the limit where the ®nesse tends to 1 (ref. 17). The interaction between the forward-scattered light mode and the atoms is described by the hamiltonian b L p H ~ Na­gc=¢ S²b² h:c: D1 Atoms where h.c. is the hermitian conjugation, b² is the creation operator for cavity photons, ­ is the laser Rabi frequency, and gc the atom±®eld coupling constant. In addition to coherent evolution, the photonic ®eld mode can leak out of the cavity at a rate k, whereas atomic coherence is Filter Channel dephased by spontaneous photon scattering into random directions that BS occurs at a rate g9s ­2=¢2gs for each atom, with gs being the natural R linewidth of the electronic excited state. We emphasize that in the D absence of superradiant effects, spontaneous emission events are 2 independent for each atom. Atoms In the bad-cavity limit, we can adiabatically eliminate the cavity mode, and the resulting dynamics for the collective atomic mode is described by the Heisenberg±Langevin equation (see Supplementary Information for Figure 1 Set-up for entanglement generation. a, The relevant level structure of the atoms details) in the ensemble, with jgi, the ground state, jsi, the metastable state for storing a qubit, k9 2 g9 p s and jei, the excited state. The transition jgi ! jei is coupled by the classical laser (the SÇ ² S² 2 k9b 2 in t noise pumping light) with the Rabi frequency ­, and the forward-scattered Stokes light comes where k9 4j­j2g2cNa= ¢2k , bin is a vacuum ®eld leading into the cavity, from the transition jei ! jsi, which has a different polarization and frequency to the and the last term represents the ¯uctuating noise ®eld corresponding to pumping light. For convenience, we assume off-resonant coupling with a large detuning spontaneous emission. We note that the nature of the dynamics is ¢. b, Schematic set-up for generating entanglement between the two atomic ensembles determined by the ratio between the build-up of coherence due to L and R. The two ensembles are pencil-shaped, and illuminated by the synchronized forward-scattered photons k9 and coherence decay due to spontaneous classical pumping pulses. The forward-scattered Stokes pulses are collected and coupled emission g9s. The signal-to-noise ratio is therefore given by to optical channels (such as ®bres) after the ®lters, which are polarization- and frequency- R k9=g9s [ 4Nag2c= kgs , which is large when a many-atom ensemble is selective to ®lter the pumping light. The pulses after the transmission channels interfere at used. In the cavity-free case, this expression corresponds to the optical a 50%-50% beam splitter BS, with the outputs detected respectively by two single-photon depth (density-length product) of the sample. The result should be detectors D compared with the signal-to-noise ratio in the single-atom case N 1 and D2. If there is a click in D1 or D2, the process is ®nished and we a 1, successfully generate entanglement between the ensembles L and R. Otherwise, we ®rst where to obtain R . 1 a high-Q microcavity is required10±12. The collective apply a repumping pulse (to the transition jsi ! jei) to the ensembles L and R, to set the enhancement takes place because the coherent forward scattering state of the ensembles back to the ground state j0 involves only one collective atomic mode S, whereas the spontaneous a iL # j0a iR, then the same classical laser pulses as the ®rst round are applied to the transition jgi ! jei and we detect again emissions distribute excitation over all atomic modes. Therefore only a the forward-scattered Stokes pulses after the beam splitter. This process is repeated until small fraction of spontaneous emission events in¯uences the symmetric ®nally we have a click in the D mode S, which results in a large signal-to-noise ratio. 1 or the D2 detector. 414 © 2001 Macmillan Magazines Ltd NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com articles We now show how to use this set-up to generate entanglement attenuation length, we want to extend the quantum communication between two distant ensembles L (left) and R (right) using the distance. This is done through entanglement swapping with the con®guration shown in Fig. 1. Here two laser pulses excite both con®guration shown in Fig. 2. Suppose that we start with two pairs ensembles simultaneously, and the whole system is described by the of entangled ensembles described by the state rLI # r 1 I2R, where rLI state jfi 1 L # jfiR, where jfiL and jfiR are given by equation (1) with and rI all the operators and states distinguished by the subscript L or R. 2R are given by equation (3). In the ideal case, the set-up shown in Fig. 2 measures the quan p tities corresponding to operators S²6S6 The forward-scattered Stokes light from both ensembles is com- with S6 SI 6 S = 2. If the measurement is successful (that is, 1 I2 bined at the beam splitter, and a photodetector click in either D1 or one of the detectors registers one photon), we will prepare the D2 measures the combined radiation p from two samples, a² a or ensembles L and R into another EME state. The new J-parameter is a²2a2 with a6 aL 6 eiJaR = 2. Here, J denotes an unknown given by J1 J2, where J1 and J2 denote the old J-parameters for difference of the phase shifts in the left and the right side channels. the two segment EME states. As will be seen below, even in the We can also assume that J has an imaginary part to account for the presence of realistic noise and imperfections, an EME state is still possible asymmetry of the set-up, which will also be corrected created after a detector click. The noise only in¯uences the success automatically in our scheme. But the set-up asymmetry can be probability of getting a click and the new vacuum coef®cient in the easily made very small, and for simplicity of expressions we assume EME state. In general, we can express the success probability p1 and that J is real in the following. Conditional on the detector click, we the new vacuum coef®cient c1 as p1 f 1 c0 and c1 f 2 c0 , where should apply a+ or a- to the whole state jfiL # jfiR, and the the functions f1 and f2 depend on the particular noise properties. projected state of the ensembles L and R is nearly maximally The above method for connecting entanglement can be cascaded entangled, with the form (neglecting the high-order terms o(pc)): to arbitrarily extend the communication distance. For the ith p jª (i 1; 2; ¼; n) entanglement connection, we ®rst prepare in par- Ji6 LR S²L 6 eiJS²R = 2j0aiL j0aiR 2 allel twopairs of ensembles in the EME states with the same vacuum The probability of getting a click is given by pc for each round, so we coef®cient ci-1 and the same communication length Li-1, and then need to repeat the process about 1/pc times for a successful perform entanglement swapping as shown in Fig. 2, which now entanglement preparation, and the average preparation time is succeeds with a probability pi f 1 ci21 . After a successful detector given by T0 < t¢=pc. The states jªJi LR and jªJi2LR can be transformed click, the communication length is extended to Li 2Li21, and the to each other by a simple local phase shift. Without loss of general- vacuum coef®cient in the connected EME state becomes ity, we assume in the following that we generate the entangled state ci f 2 ci21 . As the ith entanglement connection needs to be jªJi LR. repeated on average 1/pi times, the total time needed to establish As will be shown below, the presence of noise modi®es the an EME state over the distance Ln 2nL0 is given by projected state of the ensembles to Tn T0Pni 1 1=pi , where L0 denotes the distance of each segment 1 in the entanglement generation. rLR c0; J c c 0j0a0aiLRh0a0aj jªJi LRhªJj 3 0 1 Entanglement-based communication schemes where the `vacuum' coef®cient c0 is determined by the dark count After an EME state has been established between two distant sites, rates of the photon detectors. It will be seen below that any state in we would like to use it in communication protocols, such as the form of equation (3) will be puri®ed automatically to a quantum teleportation, cryptography, and Bell inequality detection. maximally entangled state in the entanglement-based communica- It is not obvious that the EME state of equation (3), which is tion schemes. We therefore call this state an effective maximally entangled in the Fock basis, is useful for these tasks, as in the Fock entangled (EME) state, with the vacuum coef®cient c0 determining basis it is experimentally hard to do certain single-bit operations. the puri®cation ef®ciency. We will now show how the EME states can be used to realize all these protocols with simple experimental con®gurations. Entanglement connection through swapping Quantum cryptography and Bell inequality detection are After the successful generation of entanglement within the achieved with the set-up shown by Fig. 3a. The state of the two a I I b e 1 2 BS Entangled Entangled sI s 1 I2 L D1 D2 R g Figure 2 Set-up for entanglement connection. a, Illustrative set-up for the entanglement 50%-50% beam splitter, and then detected by the single-photon detectors D1 and D2. If swapping. We have two pairs of ensemblesÐL and I1, and I2 and RÐdistributed at either D1 or D2 clicks, the protocol is successful and an EME state in the form of equation three sites L, I and R. Each of the ensemble-pairs L and I1, and I2 and R is prepared in an (3) is established between the ensembles L and R with a doubled communication EME state in the form of equation (3). The stored atomic excitations of two nearby distance. Otherwise, the process fails, and we need to repeat the previous entanglement ensembles I1 and I2 are converted simultaneously into light. This is achieved by applying a generation and swapping until ®nally we have a click in D1 or D2, that is, until the protocol retrieval pulses of suitable polarization that is near-resonant with the atomic transition ®nally succeeds. b, The two intermediate ensembles I1 and I2 can also be replaced by one jsi ! jei, which causes coherent conversion of atomic excitations into photons that have ensemble but with two metastable states I1 and I2 to store the two different collective a different polarization and frequency to the retrieval pulse18,21,22. The ef®ciency of this modes. The 50%-50% beam splitter operation can be simply realized by a p/2 pulse transfer can be very close to unity even at a single quantum level owing to collective applied to the two metastable states before the collective atomic excitations are enhancement18,21,22. After the transfer, the stimulated optical excitations interfere at a transferred to the optical excitations. NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 415 articles R perform the desired single-bit rotations in the corresponding basis. a L 1 1 For instance, to distribute a quantum key between the two remote BS DL BS DR1 sides, we simply choose JL randomly from the set {0; p=2} with an 1 equal probability, and keep the measurement results (to be 0 if DL1 R R L 2 clicks, and 1 if DL2 clicks) on both sides as the shared secret key if L 2 DL the two sides become aware that they have chosen the same 2 DR2 phase shift after the public declaration of JL. This is exactly the Ekert scheme1, and its absolute security follows directly from the proofs in refs 26 and 27. For the Bell inequality detection, b DI1 DL1 we infer the correlations E JL; JR [ PDL P 2 P 2 1 DR1 DL2DR2 DL1DR2 R1 L1 PDL cos J 2 DR1 L 2 JR from the measurement of the coincidences I PDL and so on. For the set-up shown in Fig. 3a, we would 1 p 1 DR1 have jE 0; p=4 E p=2; p=4 E p=2; 3p=4 2 E 0; 3p=4 j 2 2, BS whereas for any local hidden variable theories, the CHSH I R 2 2 L2 inequality28 implies that this value should be below 2. We can also use the established long-distance EME states for faithful transfer of unknown quantum states through quantum BS teleportation, with the set-up shown in Fig. 3b. As described in the DI2 DL2 ®gure legend, this set-up is used to teleport the polarization state of the collective atomic excitation in a probabilistic fashion. That is, Figure 3 Set-up for entanglement-based communication schemes. a, Schematic set-up even if the protocol succeedsÐthat is, two of the detectors register for the realization of quantum cryptography and Bell inequality detection. Two pairs of the counts on the left-hand sideÐan excitation is not necessarily ensembles L1, R1 and L2, R2 (or two pairs of metastable states as shown in b) have been present in the right (target) ensembles because the product of the prepared in the EME states. The collective atomic excitations on each side are transferred EME states rL # r contains vacuum components. However, if 1R1 L2R2 to the optical excitations, which, respectively after a relative phase shift JL or JR and a a collective excitation appears from the right-hand side, its `polari- 50%-50% beam splitter, are detected by the single-photon detectors DL1; DL2 and DR1; DR2. zation' state is exactly the same as the one input from the left side. We look at the four possible coincidences of DR1; DR2 with DL1; DL2, which are functions of the So, as in the experiment of ref. 29, such a probabilistic teleportation phase difference JL 2 JR. Depending on the choice of JL and JR, this set-up can realize needs posterior con®rmation of the presence of the excitation; but if both the quantum cryptography and the Bell inequality detection. b, Schematic set-up for the presence is con®rmed, the teleportation ®delity of its polariza- probabilistic quantum teleportation of the atomic `polarization' state. Similarly, two pairs tion state is nearly perfect. The success probability for the tele- of ensembles L1, R1 and L2, R2 are prepared in the EME states. We want to teleport an portation is also given by pa 1= 2 cn 1 2 , which determines atomic `polarization' state d0S²I d1S²I j0a0aiI with unknown coef®cients d0; d1 the average number of repetitions needed for a ®nal successful 1 2 1I2 from the left to the right side, where S²I ; S²I denote the collective atomic operators for the teleportation. 1 2 two ensembles I1 and I2 (or two metastable states in the same ensemble). The collective atomic excitations in the ensembles I1, L1 and I2, L2 are transferred to the optical Noise and built-in entanglement puri®cation excitations, which, after a 50%-50% beam splitter, are detected by the single-photon We now discuss noise and imperfections in our schemes for detectors DI1; DL1 and DI2; DL2. If, and only if, there is one click in DI1 DL1, and one click in DI2or entanglement generation, connection, and applications. In par- DI2, the protocol is successful. When the protocol succeeds, the collective excitation in the ticular, we show that each step contains built-in entanglement ensembles R1 and R2, if appearing, would be found in the same `polarization' state puri®cation, which makes the whole scheme resilient to realistic d0S²R d1S²R j0a0aiR up to a local p-phase rotation. noise and imperfections. 1 2 1R2 In the entanglement generation, the dominant noise is due to photon loss, which includes contributions from channel attenua- pairs of ensembles is expressed as rL # r , where r (i 1; 2) tion, spontaneous emissions in the atomic ensembles (which result 1R1 L2R2 LiRi denote the same EME state with the vacuum coef®cient cn if we in the population of the collective atomic mode with the accom- have done entanglement connection n times. The J-parameters panying photon going in other directions), coupling inef®ciency of in rL and r are the same, provided that the two states are the Stokes light into and out of the channel, and inef®ciency of the 1R1 L2R2 established over the same stationary channels. We register only the single-photon detectors. The loss probability is denoted by 1 2 hp coincidences of the two-side detectors, so the protocol is successful with the overall ef®ciency hp h9pe2L0=Latt, where we have separated only if there is a click on each side. Under this condition, the the channel attenuation e2L0=Latt (Latt is the channel attenuation vacuum components in the EME states, together with the state length) from other noise contributions h9p, with h9p independent of components S²L S² jvaci and S² S² jvaci, where jvaci denotes the the communication distance L 1 L2 R1 R2 0. The photon loss decreases the ensemble state j0a0a0a0aiL , have no contributions to the success probability for getting a detector click from p 1R1L2R2 c to hppc, but experimental results. So, for the measurement scheme shown by it has no in¯uence on the resulting EME state. Owing to this noise, Fig. 3, the ensemble state rL # r is effectively equivalent to the the entanglement preparation time should be replaced by 1R1 L2R2 following `polarization' maximally entangled (PME) state (the T0 < t¢= hppc . The second source of noise comes from the dark terminology of `polarization' comes from an analogy to the optical counts of the single-photon detectors. The dark count gives a case): detector click, but without population of the collective atomic p jªi mode, so it contributes to the vacuum coef®cient in the EME PME S²L S² S² S² = 2jvaci 4 1 R2 L2 R1 state. If the dark count comes up with a probability pdc for the time The success probability for the projection from rL # r to interval t 1R1 L2R2 ¢, the vacuum coef®cient is given by c0 pdc= hppc , which jªiPME (that is, the probability of getting a click on each side) is is typically much smaller than 1 as the Raman transition rate is given by pa 1= 2 cn 1 2 . We can also check that in Fig. 3, the much larger than the dark count rate. The ®nal source of noise, phase shift JL (L L or R) together with the corresponding beam- which in¯uences the ®delity of getting the EME state, is caused by splitter operation are equivalent to a single-bit rotation in the basis the event in which more than one atom is excited to the collective {j0iL [ S²L j0 ; j1i j0 } with the rotation angle mode S whereas there is only one click in D 1 a0aiL1L2 L [ S²L2 a0aiL1L2 1 or D2. The conditional v JL=2. Now it is clear how to do quantum cryptography and probability for that event is given by pc, so we can estimate Bell inequality detection, as we have the PME state and we can the ®delity imperfection DF0 [ 1 2 F0 for the entanglement 416 © 2001 Macmillan Magazines Ltd NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com articles generation by: Noise not correctable by our scheme includes the detector dark DF count in the entanglement connection, the non-stationary channel 0 < pc 5 noise and set-up asymmetries. The ®delity imperfection resulting Note that by decreasing the excitation probability pc, we can make from the dark count increases linearly with the number of segments the ®delity imperfection closer and closer to zeroÐwith the price of Ln/L0, and the imperfections from the non-stationary channel noise p a longer entanglement preparation time T0. This is the basic idea of and set-up asymmetries increase by the random-walk law Ln=L0. the entanglement puri®cation. So, in this scheme, the con®rmation For each time of entanglement connection, the dark count prob- of the click from the single-photon detector generates and puri®es ability is about 1025 if we make a typical choice that the collective entanglement at the same time. emission rate is about 10 MHz and the dark count rate is 102 Hz. So In the entanglement swapping, the dominant noise is still due to this noise is negligible, even if we have communicated over a long the losses, which include contributions from detector inef®ciency, distance (103 times the channel attenuation length Latt, for instance). the inef®ciency of the excitation transfer from the collective atomic The non-stationary channel noise and set-up asymmetries can also mode to the optical mode21,22, and the small decay of atomic be safely neglected for such a distance. For instance, it is relatively excitation during storage20±22. Note that by introducing the detector easy to control the non-stationary asymmetries in local laser inef®ciency, we have automatically taken into account the imper- operations to values below 10-4 with the use of accurate polarization fection that the detectors cannot distinguish between one and two techniques30 for Zeeman sublevels (as in Fig. 2b). photons. With all these losses, the overall ef®ciency in entanglement swapping is denoted by hs. The loss in entanglement swapping Scaling of the communication ef®ciency gives contributions to the vacuum coef®cient in the connected We have shown that each of our entanglement generation, connec- EME state, as in the presence of loss a single detector click might tion, and application schemes has built-in entanglement puri®ca- result from two collective excitations in the ensembles I1 and I2, tion, and as a result of this property, we can ®x the communication and in this case, the collective modes in the ensembles L and R have ®delity to be nearly perfect, and at the same time require the to be in a vacuum state. After taking into account the realistic noise, communication time to increase only polynomially with distance. we can specify the success probability and the new vacuum Assume that we want to communicate over a distance coef®cient for the ith entanglement connection by the recursion L Ln 2nL0. By ®xing the overall ®delity imperfection to be a relations pi [ f 1 ci21 hs 1 2 {hs= 2 ci21 1 } = ci21 1 and desired small value DFn, the entanglement preparation time ci [ f 2 ci21 2ci21 1 2 hs. The coef®cient c0 for the entangle- becomes T0 < tD= hpDF0 < Ln=L0 tD= hpDFn . For effective genera- ment preparation is typically much smaller than 1 2 hs, so we have tion of the PME state of equation (4), the total communication ci < 2i 2 1 1 2 hs Li=L0 2 1 1 2 hs , where Li denotes the time Ttot < Tn=pa with Tn < T0Pni 1 1=pi . So the total communica- communication distance after i times entanglement connection. tion time scales with distance by the law With the expression for the ci, we can evaluate the probability pi and the communication time T Ttot < 2 L=L0 2= hppaDFnPni 1pi 6 n for establishing an EME state over the distance Ln 2nL0. After the entanglement connection, the ®delity where the success probabilities pi; pa for the ith entanglement of the EME state also decreases, and after n times connection, the connection and for the entanglement application have been speci- overall ®delity imperfection DFn < 2nDF0 < Ln=L0 DF0. We need to ®ed above. make DFn small by decreasing the excitation probability pc in Equation (6) con®rms that the communication time Ttot equation (5). increases with distance L only polynomially. We show this We note that our entanglement connection scheme also has a explicitly by taking two limiting cases. In the ®rst case, the built-in entanglement-puri®cation function. This can be under- inef®ciency 1 2 hs for entanglement swapping is assumed to be stood as follows: each time we connect entanglement, the imperfec- negligibly small. We can deduce from equation (6) that in this tions of the set-up decrease the entanglement fraction 1= ci 1 in case the communication time Ttot < Tcon L=L0 2eL0=Latt, with the the EME state. However, this fraction decays only linearly with constant Tcon [ 2t¢= h9phaDFn being independent of the seg- distance (the number of segments), which is in contrast to the ment length and the total distance L0 and L. The exponential decay of entanglement for connection schemes without communication time Ttot increases with L quadratically. In entanglement puri®cation. The reason for the slow decay is that for the second case, we assume that the inef®ciency 1 hs is each time of entanglement connection, we need to repeat the fairly large. The communication time in this case is approxi- protocol until there is a detector click, and the con®rmation of a mated by Ttot < Tcon L=L0 log2 L=L0 1 =2 log2 1=hs21 2eL0=Latt, which click removes part of the added vacuum noise, as a larger vacuum increases with L still polynomially (or, more accurately, sub-expo- component in the EME state results in more repetitions. The built- nentially, but this makes no difference in practice as the factor in entanglement puri®cation in the connection scheme is essential log2(L/L0) is well bounded from above for any reasonably long for the polynomial scaling law of the communication ef®ciency. distance). If Ttot increases with L/L0 by the mth power law L=L0 m, As in the entanglement generation and connection schemes, our there is an optimal choice of segment length (L0 mLatt) to entanglement application schemes also have built-in entanglement minimize the time Ttot. As a simple estimation of the improvement puri®cation, which makes them resilient to realistic noise. First, we in communication ef®ciency, we assume that the total distance L is have seen that the vacuum components in the EME states are about 100Latt; for a choice of the parameter hs < 2=3, the commu- removed from the con®rmation of the detector clicks, and thus nication time Ttot=Tcon < 106 with the optimal segment length have no in¯uence on the ®delity of all the application schemes. L0 < 5:7Latt. This is a notable improvement over the direct com- Second, if the single-photon detectors and the atom-to-light excita- munication case, where the communication time Ttot for getting a tion transitions in the application schemes are imperfect, with the PME state increases with distance L by the exponential law overall ef®ciency denoted by ha, we can show that these imperfec- Ttot < TconeL=Latt. For the same distance L < 100Latt, we need tions only in¯uence the ef®ciency of getting detector clicksÐwith Ttot=Tcon < 1043 for direct communication, which means that for the success probability replaced by pa ha= 2 cn 1 2 Ðand have this example the present scheme is 1037 times more ef®cient. no effect on communication ®delity. Last, we have seen that the phase shifts in the stationary channels and the small asymmetry of Outlook the stationary set-up are removed automatically when we project We have presented a scheme for implementation of quantum the EME state to the PME state, and thus have no in¯uence on the repeaters and long-distance quantum communication. The pro- communication ®delity. posed technique allows the generation and connection of entangle- NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 417 articles ment, and its use in quantum teleportation, cryptography, and tests temporal coherence properties of stimulated Raman scattering. Phys. Rev. A 32, 332±344 of Bell inequalities. All of the elements of the present scheme are (1985). within reach of current experimental technology, and all have the 16. Kuzmich, A., MoÈlmer, K. & Polzik, E. S. Spin squeezing in an ensemble of atoms illuminated with squeezed light. Phys. Rev. Lett. 79, 481 (1998). important property of built-in entanglement puri®cationÐwhich 17. Kuzmich, A., Bigelow, N. P. & Mandel, L. Atomic quantum non-demolition measurements and makes them resilient to realistic noise. As a result, the overhead squeezing. Europhys. Lett. A 42, 481±486 (1998). required to implement the present scheme, such as the commu- 18. Lukin,M.D.,Yelin,S.F.&Fleischhauer,M.Entanglementofatomicensemblesbytrappingcorrelated photon states. Phys. Rev. lett. 84, 4232±4235 (2000). nication time, scales polynomially with the channel length. This is in 19. Duan,L.M.,Cirac,J.I.,Zoller,P.&Polzik,E.S.Quantumcommunicationbetweenatomicensembles marked contrast to direct communication, where an exponential using coherent light. Phys. Rev. Lett. 85, 5643±5646 (2000). overhead is required. Such ef®cient scaling, combined with the 20. Hald, J., Sorensen, J. L., Schori, C. & Polzik, E. S. Spin squeezed state: A macroscopic entangled relative simplicity of the experimental set-up, opens up realistic ensemble created by light. Phys. Rev. Lett. 83, 1319±1322 (1999). 21. Phillips, D. F. et al. Storage of light in atomic vapor. Phys. Rev. Lett. 86, 783±786 (2001). prospects for quantum communication over long distances. M 22. Liu, C., Dutton, Z., Behroozi, C. H. & Hau, L. V. Observation of coherent optical information storage in an atomic medium using halted light pulses. Nature 409, 490±493 (2001). Received 16 May; accepted 12 September 2001. 23. Knill, E., La¯amme, R. & Milburn, G. J. A scheme for ef®cient quantum computation with linear 1. Ekert, A. Quantum cryptography based on Bell's theorem. Phys. Rev. Lett. 67, 661±663 (1991). optics. Nature 409, 46±52 (2001). 2. Bennett, C. H. et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky- 24. Pan, J. W., Simon, C., Brukner, C. & Zeilinger, A. Feasible entanglement puri®cation for quantum Rosen channels. Phys. Rev. Lett. 73, 3081±3084 (1993). communication. Nature 410, 1067±1070 (2001). 3. Bennett, C. H. et al. Puri®cation of noisy entanglement and faithful teleportation via noisy channels. 25. Roch, J.-F. et al. Quantum nondemolition measurements using cold trapped atoms. Phys. Rev. Lett. 78, Phys. Rev. Lett. 76, 722±725 (1991). 634±637 (1997). 4. Briegel, H.-J., Duer, W., Cirac, J. I. & Zoller, P. Quantum repeaters: The role of imperfect local 26. Lo, H. K. & Chau, H. F. Unconditional security of quantum key distribution over arbitrarily long operations in quantum communication. Phys. Rev. Lett. 81, 5932±5935 (1998). distances. Science 283, 2050±2056 (1999). 5. Knill, E., La¯amme, R. & Zurek, W. H. Resilient quantum computation. Science 279, 342±345 (1998). 27. Shor, P. W. & Preskill, J. Simple proof of security of the BB84 quantum key distribution protocol. Phys. 6. Preskill, J. Reliable quantum computers. Proc. R. Soc. Lond. A 454, 385±410 (1998). Rev. Lett. 85, 441±444 (2000). 7. Zukowski, M., Zeilinger, A., Horne, M. A. & Ekert, A. ``Event-ready-detectors'' Bell experiment via 28. Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test local hidden- entanglement swapping. Phys. Rev. Lett. 71, 4287±4290 (1993). variable theories. Phys. Rev. Lett. 23, 880±884 (1969). 8. Cirac, J. I., Zoller, P., Kimble, H. J. & Mabuchi, H. Quantum state transfer and entanglement 29. Bouwmeester, D. et al. Experimental quantum teleportation. Nature 390, 575±579 (1997). distribution among distant nodes in a quantum network. Phys. Rev. Lett. 78, 3221±3224 (1997). 30. Budker, D., Yashuk, V. & Zolotorev, M. Nonlinear magneto-optic effects with ultranarrow width. Phys. 9. Enk, S. J., Cirac, J. I. & Zoller, P. Photonic channels for quantum communication. Science 279, 205± Rev. Lett. 81, 5788±5791 (1998). 207 (1998). Supplementary Information accompanies the paper on Nature's website 10. Ye, J., Vernooy, D. W. & Kimble, H. J. Trapping of single atoms in cavity QED. Phys. Rev. Lett. 83, (http://www.nature.com). 4987±4990 (1999). 11. Hood, C. J. et al. The atom-cavity microscope: Single atoms bound in orbit by single photons. Science 287, 1447±1453 (2000). Acknowledgements 12. Pinkse, P. W. H., Fischer, T., Maunz, T. P. & Rempe, G. Trapping an atom with single photons. Nature This work was supported by the Austrian Science Foundation, the Europe Union project 404, 365±368 (2000). EQUIP, the ESF, the European TMR network Quantum Information, and the NSF through 13. Cabrillo, C., Cirac, J. I., G-Fernandez, P. & Zoller, P. Creation of entangled states of distant atoms by a grant to the ITAMP and ITR program. L.-M.D. was also supported by the Chinese interference. Phys. Rev. A 59, 1025±1033 (1999). Science Foundation. 14. Bose, S., Knight, P. L., Plenio, M. B. & Vedral, V. Proposal for teleportation of an atomic state via cavity decay. Phys. Rev. Lett. 83, 5158±5161 (1999). Correspondence and requests for materials should be addressed to J.I.C. 15. Raymer, M. G., Walmsley, I. A., Mostowski, J. & Sobolewska, B. Quantum theory of spatial and (e-mail: ignacio.cirac@uibk.ac.at). 418 © 2001 Macmillan Magazines Ltd NATURE | VOL 414 | 22 NOVEMBER 2001 | www.nature.com