Quantum repeaters based on Rydberg-blockade coupled atomic ensembles
Yang Han, Bing He, Khabat Heshami, Cheng-Zu Li, Christoph Simon
aa r X i v : . [ qu a n t - ph ] M a r APS / Quantum repeaters based on Rydberg-blockade coupled atomic ensembles
Yang Han , , Bing He , Khabat Heshami , Cheng-Zu Li , and Christoph Simon Institute for Quantum Information Science and Department of Physics and Astronomy,University of Calgary, Calgary T2N 1N4, Alberta, Canada College of Science, National University of Defense Technology, Changsha 410073, China. (Dated: November 5, 2018)We propose a scheme for realizing quantum repeaters with Rydberg-blockade coupled atomic ensembles,based on a recently proposed collective encoding strategy. Rydberg-blockade mediated two-qubit gates ande ffi cient cooperative photon emission are employed to create ensemble-photon entanglement. Thanks to de-terministic entanglement swapping operations via Rydberg-based two-qubit gates, and to the suppression ofmulti-excitation errors by the blockade e ff ect, the entanglement distribution rate of the present scheme is higherby orders of magnitude than the rates achieved by other ensemble-based repeaters. We also show how to realizetemporal multiplexing with this system, which o ff ers an additional speedup in entanglement distribution. PACS numbers: 03. 67.Hk, 32.80.Ee, 42.50.Ex
I. INTRODUCTION
Quantum communication aims at secure message transmis-sion between remote locations by employing entanglement forquantum teleportation [1] or quantum cryptography [2]. Un-fortunately, since the inevitable photon loss scales exponen-tially with the length of channel, it is di ffi cult to establishhigh quality entanglement over long distances. This prob-lem may be overcome by quantum repeaters [3], which cre-ate and store shorter-distance entanglement in a heralded way,and then connect the elementary entangled states to establishlonger-distance entanglement via entanglement swapping.A highly influential protocol for realizing quantum re-peaters was proposed by Duan, Lukin, Cirac and Zoller(DLCZ)[4]. It is based on macroscopic atomic ensemblequantum memories, Raman scattering and linear optics. Thereis a significant body of theoretical [5–8] and experimental [9]work based on this general approach. In this paper, we re-fer to these schemes as DLCZ-type repeaters. The significantadvantage of DLCZ-type repeaters is that they use relativelysimple elements. However, there are two intrinsic limitationsin this approach. First, as Raman scattering is used to createentanglement between a single atomic excitation and a singlephoton, inevitable multi-excitation (and multi-photon) termscause errors in the final states. In order to suppress multipleexcitations, one has to work with very low excitation prob-ability [4]. Second, the Bell measurements in the swappingoperations are realized via linear optics, so the success prob-ability of the entanglement swapping is bounded by 1 / ffi ciencyof DLCZ-type repeaters.There are a number of proposals for realizing quantum re-peaters using ingredients other than atomic ensembles and lin-ear optics [11–13]. Most of them use individual quantum sys-tems as the quantum memory [12, 13]. An obvious advan-tage of using individual quantum systems is that the problemof multiple excitations is eliminated. If the two-qubit gatesfor Bell measurement in the swapping operations can alsobe realized e ffi ciently, repeaters based on individual quantumsystems have the potential to significantly outperform DLCZ-type repeaters [12, 13]. However, for the individual quantum systems one has to precisely address every single particle, andone may need cavities to achieve a high e ffi ciency of photoncollection [13].An attractive technique for quantum information processing(QIP) with atomic ensembles is based on the Rydberg block-ade mechanism, cf. below. There have been a number of pro-posals to use the Rydberg blockade for various QIP tasks(see[14] for an overview). In the present paper, we propose a con-crete scheme for realizing quantum repeaters in this way andanalyze its performance in detail. We show that the entangle-ment distribution rate o ff ered by repeaters based on Rydbergblockade coupled ensembles significantly surpasses the rate ofDLCZ-type repeaters. Compared to the schemes involving in-dividual quantum systems, repeaters based on Rydberg block-ade coupled ensembles achieve almost the same distributionrate and avoid addressing single particles and using cavities.Our proposed scheme also allows temporal multiplexing [15],which could further enhance the achievable distribution rate. II. RYDBERG BLOCKADE COUPLED ATOMICENSEMBLE
Rydberg states are states of alkali atoms characterized bya high principal quantum number. Atoms in such Rydbergstates have large size and can therefore have large dipolemoments, resulting in strong dipole-dipole interactions [16].Due to this strong long-range interaction, a single atom in anatomic ensemble excited to a Rydberg state shifts the Rydbergenergy level of its neighbors out of resonance and blocks fur-ther excitations, which is called the Rydberg blockade mech-anism [17], and this kind of ensembles is referred to as Ryd-berg blockade coupled ensembles [14]. Recently, experimentshave demonstrated an almost perfect blockade [18] as well asa blockade-based C-NOT gate [19] between a single pair oftrapped atoms at separation R ≤ µ m . Although no exper-iments have been done with an ensemble where the block-ade acts across the whole ensemble, a number of experimentsshow clear signs of the blockade e ff ect on larger samples [14].In this blockade regime, an e ff ective two-level system is re-alized between the state with all atoms in the ground level andthe single-excitation symmetric atomic state. This two-levelsystem has an e ff ective light-atom coupling that is a factor of √ N larger than the light-single-atom coupling. It is promisingfor a wide variety of quantum information processing applica-tions [20–26]. In the following two subsections, we briefly re-view the collective encoding strategy for a k -bit quantum reg-ister [23] and the cooperative photon emission e ff ect [25–28]in a Rydberg blockade coupled ensemble, which are directlyrelated to our repeater scheme. A. Collective encoding in a Rydberg blockade coupledensemble
A Rydberg coupled atomic ensemble consisting of N atomscan be used to build a k -bit quantum register ( N ≫ k ), wherethe qubits are collectively encoded in di ff erent single excita-tion symmetric atomic states [23]. As shown in Fig.1(a), k qubits are encoded in an ensemble of N atoms with 2 k + i -th qubit values zero and oneare identified with symmetric single-excitation atomic statespopulating | i i and | i i , respectively. The initialization of this k -bit register is as follows: Originally all the atoms are in thereservoir state | g i . Then due to the blockade mechanism, onecan transfer precisely one atom to each pair of levels ( | i i , | i i )via a Raman transition involving a Rydberg state | r i .In the following, for given atomic levels | x i i ( x = , , i = · · · k ), we will let the kets | ˜ x i i ( x = , , i = · · · k ) denotethe symmetric single-excitation collective atomic states, forinstance, | ˜1 i = √ N N X j = e − i ~ k · ~ r j | g i | g i · · · | i j · · · | g i N , (1)where ~ r j is the position of the j -th atom, | i j indicates thatthe j -th atom is in the state | i , and ~ k is the summation ofall the wave vectors of the light pulses used to transfer theensemble to the above state, i.e., ~ k = P m λ m ~ k m , and λ m = ± m -th pulse. Accordingly, the basis of the i -th qubit can bewritten as ( | ˜0 i i , | ˜1 i i ).It has been proposed in Ref. [23] that both single-bit rota-tions and two-qubit gates can be realized in this system. Thesingle-qubit rotations on the i -th qubit are straightforwardlyperformed using two-photon stimulated Raman beams cou-pling | i i and | i i . The two-qubit phase gate between the j -thand k -th qubits is implemented by a sequence of three laserpulses as shown in fig.1(b): ( i ) The excitation of the controlqubit internal state | j i into one Rydberg state | r i ; ( ii ) 2 π Rabirotation between the target qubit state | k i and another Ryd-berg state | r i ; ( iii ) The return of the population from | r i to | j i . If the control qubit is in state | i , the resulting unit occu-pancy of the | r i state blocks the Rabi cycle and nothing hap-pens to the target qubit, while a control qubit in state | i causesno blockade, and hence we obtain a controlled π phase shift onthe | k i state amplitude due to a full Rabi cycle. Hence, uni-versal quantum computing operations can be realized in thissystem [23]. r g k k j j r r g (a) (b) ( i ) ( iii ) ( ii )( i )( ii ) B k k FIG. 1: (a) Collective encoding of k qubits in an ensemble of N atoms(gray dots) with 2 k + | r i is a Rydberg state, and | x i i ( x = , i = · · · k ) and | g i are ground states. A magnetic field B is applied to the ensemble, which enables every ground state tobe selectively manipulated via appropriate choices of laser frequen-cies. The transitions ( i ) and ( ii ) show the initialization procedure ofthe second qubit. (b) Two-qubit phase gate on collectively encodedqubits j and k mediated by Rydberg blockade. See text for details. B. Cooperative photon emission from a Rydberg blockadecoupled ensemble
Now we discuss how to map a collectively encoded qubitin the ensemble into a flying photonic qubits. Assume theensemble is in the state | ˜1 f i . If one transfers state | ˜1 f i into anexcited state | ˜ e i via a π -pulse laser with wave vector ~ k e , thisstate will radiate into a variety of modes with all the atoms instate | g i and a single-photon propagating with the wave vector ~ k . The amplitude for emitting a photon with wave vector ~ k andpolarization ~ e , is proportional to h g · · · g N |h ~ k | ( ~ e · d )ˆ a † k | ˜ e i| vac i = − N X j = h g | ( ~ e · ~ d ) | e i e − i ( ~ k + ~ k e − ~ k ) · ~ r j √ N , (2)where ˆ a † ~ k is the creative operator for a photon in mode ~ k and ~ d is the dipole operator. Thus the transition probability P ( ~ k ) isproportional to P ( ~ k ) ∝ N | N X j = e − i ( ~ k + ~ k e − ~ k ) · r j | . (3)Note that if ~ k + ~ k e = ~ k all the phase terms are zero and P ( ~ k ) ∝ N ; otherwise the phase terms become random so that P ( ~ k ) ∝
1, which means the emission is highly directional.Although the typical size of a Rydberg blockade coupled en-semble is less than 10 µ m consisting of only several hundredsof atoms [14], the above cooperative emission e ff ect is stilllarge enough to ensure very good directed emissions of pho-tons [25–28], cf. below.There are two di ff erent ways to convert a collectively en-coded qubit α | ˜0 f i + β | ˜1 f i into a flying photonic polarizationqubit α | h i + β | v i , where h and v indicate the horizontal and h e g f f (a) (b) v eh v g e k ' k f f FIG. 2: Two di ff erent ways to convert a collectively encoded qubitinto a photonic qubit, cf.text. vertical polarization, respectively. On the one hand, using po-larization selection rules, one can transfer the state | ˜0 f i and | ˜1 f i to excited states | ˜ e h i and | ˜ e v i , respectively, and the sub-sequent atomic decay to the state | g · · · g N i leads to emissionof the photonic state (Fig.2(a)). However, the horizontal andvertical amplitudes may have di ff erent frequencies. While thisdoes not prevent the creation of entanglement between remoteensembles, it makes it more challenging to prove interme-diate ensemble-photon entanglement experimentally. On theother hand, one can use only one excited state | e i and appro-priately choose the wavevectors of the lasers for transitions( ~ k e : | ˜0 f i → | ˜ e i ) and ( ~ k ′ e : | ˜1 f i → | ˜ e i ). Thus the cooperativeemission from | ˜ e i to | g · · · g N i is directed into di ff erent direc-tions ~ k and ~ k ′ due to the phase matching condition, as shownin Fig.2(b). Then the photonic state α | ~ k i + β | ~ k ′ i can be easilychanged into photonic polarization state α | h i + β | v i by linearoptics. III. REPEATER BASED ON RYDBERG BLOCKADECOUPLED ENSEMBLESA. Main idea and e ffi ciency In our scheme, each repeater node contains a single ensem-ble collectively encoding three qubits. As shown in Fig.3,qubit s ( s = f is responsiblefor producing a flying photonic qubit. To establish entangle-ment between ensemble A and ensemble B , we first focus onqubit 1 and qubit f in these two ensembles. In each ensem-ble, we prepare the entangled state ( | ˜0 i| ˜0 f i + | ˜1 i| ˜1 f i ) / √ f in each ensemble is converted into aphotonic polarization state via the method shown in fig.2. Thejoint state of the two emitted photons and of ensembles A and B can be expressed as | ψ A i ⊗ | ψ B i = ( | ˜0 i A | h i A + | ˜1 i A | v i A ) ⊗ ( | ˜0 i B | h i B + | ˜1 i B | v i B ) / . (4)Combining the two emitted photons on a polarizing beam L +/-+/- A B qubit 1 qubit 2 C +/-+/- qubit fatomsphotons Swapping PBS FIG. 3: (Color online) Quantum repeater based on Rydberg block-ade coupled ensembles. Neighboring nodes are separated by a dis-tance L . Each node consists of one atomic ensemble encoding threequbits, qubit 1 (red circle), qubit 2 (blue circle) and qubit f (two-tonecircle). Each ensemble asynchronously emits two single photons (redand blue dots), which are entangled with qubits 1 and 2 (red and bluecircles), respectively. Then the photons will be measured in the mid-dle stations by polarizing beam splitters (PBSs) and photon detec-tors. When certain two-photon coincidences are observed, the ’red’(’blue’) qubits in neighboring ensembles are projected into entangledstates. Then entanglement is extended to long distances by entangle-ment swapping operations via two-qubit gates on qubits 1 and 2 inthe same ensemble. splitter (PBS) at a central station located half-way between en-sembles A and B , a probabilistic Bell state analysis can be per-formed by counting the photon number in each output mode d ± = √ ( | h i A ± | v i B ) and ˜ d ± = √ ( | h i B ± | v i A ). Such Bellanalysis projects non-destructively the two ensembles into anentangled state. In particular, the detection of two photons,one in each mode d + and ˜ d + , leads to the entangled state | ψ AB i = ( | ˜0 i A | ˜0 i B + | ˜1 i A | ˜1 i B ) / √ . (5)In the ideal case, the probability for such an event is 1 /
8. Tak-ing into account the coincidences between d + − ˜ d − , d − − ˜ d + and d − − ˜ d − combined with the appropriate one-qubit opera-tions, the probability to get the state (5) is 1 / A − B , C − D , etc. To entangle the remaininglinks, the procedure will be repeated with qubit 2 and qubit fbetween ensembles B − C , D − E , etc. Considering two links,say A − B and B − C , the resulting state after successful entan-glement creation is | ψ AB i ⊗ | ψ BC i = ( | ˜0 i A | ˜0 i B + | ˜1 i A | ˜1 i B )( | ˜0 i B | ˜0 i C + | ˜1 i B | ˜1 i C ) / . (6)We now calculate the time needed for entanglement cre-ation between two neighboring ensembles, which are sepa-rated by a distance L . Let us denote by p the success proba-bility for an ensemble to emit a photon, including the prob-ability to prepare the entangled state between qubit s andqubit f , the e ffi ciency of converting qubit f into a photon
400 500 600 700 800 900 1000 1100 120010 −3 −2 −1 T i m e ( s ) Distance(km)A B C D E
FIG. 4: (Color online) Performance of quantum repeaters based onRydberg blockade coupled ensembles. The quantity shown is the av-erage time needed to distribute a single entangled pair for the givendistance. We assume losses of 0.2 dB / km, corresponding to tele-com fibers at a wavelength of 1.5 µ m. Curve A: as a reference, thetime required using direct transmission of photons through opticalfibers with a single-photon generation rate of 10 GHz. Curve B:the most e ffi cient DLCZ-type repeater scheme known to us [8]. Inthis scheme high-fidelity entangled pairs are generated locally, andentanglement generation and swapping operations are based on two-photon detections. We have assumed memory and detector e ffi cien-cies of 0.9. Curves C and D: schemes based on Rydberg blockadecoupled ensembles with p = . p = .
9, respectively. CurveE: scheme based on trapped ions in high-finesse cavities, where thesuccess probability for the ion to emit a photon is 0.9. Notice thatwe imposed a maximum number of 16 links in the repeater chain forcurves B, C, D and E. and coupling it into the fiber. The probability to get the ex-pected twofold coincidence is thus given by P = p η d η t ,where η d is the photon detection e ffi ciency and η t = exp − L L att is the transmission e ffi ciency corresponding to a distance of L , where L att is the fiber attenuation length. Here we as-sume the losses in the fiber are 0.2dB / km, corresponding to L att = km . The entanglement creation attempts can be re-peated at time t p + t com , where t com = L / c is the communica-tion time and c = × m / s is the light velocity in the fiber[15]. We assume a typical preparation t p = µ s, cf. below.As a consequence, the average time required to entangle twoensembles separated by a distance L is given by T link = t p + t com P = t p + L / c ) p η d η t . (7)The entanglement can further be distributed over longer dis-tances by using successive entanglement swapping operationsbetween elementary links. Such swapping operations requirea local Bell state analysis, applied, for example, on the qubit1 and qubit 2 in ensemble B to entangle ensembles A and C . Thanks to the high-fidelity single-bit and two-qubit gatesavailable in our system, the success probability of entangle-ment swapping is only restrained by the read-out e ffi ciency. An e ff ective read-out mechanism in this context can be real-ized by state-selective ionization [29]. By coupling di ff erentground levels to di ff erent excited states and selectively ion-izing them, the resulting electron and ion can be detected bychannel electron multipliers. As it is su ffi cient to detect atleast one of the ionization fragments, the overall detection ef-ficiency can be as high as η ion =
95% [29]. Hence, the successprobability of entanglement swapping is P swap = /η ion , andthe total time for the distribution of an entangled pair over thedistance 2 L is given by T L = (cid:18) t p + L c (cid:19) P P swap = (cid:18) t p + L c (cid:19) p η t η d η ion . (8)The factor 3 / T , there will be a success for one of the two after T /
2; then one still has to wait a time T on average for the secondone, giving a total of 3 T /
2. This simple argument gives ex-actly the correct result in the limit of small P . In a repeaterwith n nesting levels, the precise values of analogous factorshave no analytic expression, but numerical results show thatthis remains a good approximation [30]. Hence, the total timerequired for a successful entanglement distribution over thedistance L = n L can be expressed as T tot ≈ P swap ! n (cid:18) t p + L c (cid:19) P = (cid:18) t p + L c (cid:19) n n − p η d η t η nion . (9)We calculate the performance of a quantum repeater basedon Rydberg blockade coupled ensembles with Eq.(9), asshown in Fig. 4. In the same figure we also show the per-formance of the most e ffi cient DLCZ-type repeater known tous [8], and that of a repeater based on trapped ions [13]. Onecan see that the achievable performance for repeaters basedon Rydberg blockade coupled ensembles greatly exceeds thebest DLCZ-type repeater, and is comparable with the repeaterbased on trapped ions with high-finesse cavities. Another fea-ture of repeaters based on Rydberg blockade coupled ensem-bles is that the average time for the distribution of an entan-gled pair scales only like 1 / p , in contrast to the DLCZ-typerepeaters which are much more sensitive to a reduction inmemory e ffi ciencies. As can be seen from Fig.4, even with p = .
2, the entanglement distribution time of our scheme isstill 10 times shorter than the time achievable with the bestknown DLCZ-type repeater protocol.
B. Implementation and noise analysis
In this paper, we focus on realizing the present scheme with Rb, whose nuclear spin I = / = B ∼ G is applied to the atoms, which gives splittingamong the above Zeeman states at least δ E = µ B B h ∼ B f=2f=1 m=1m=-1m=-2 m=2 S E (cid:71) P m=0 qubit 2qubit 1qubit f f=3
780 nm
FIG. 5: (Color online) Rb level scheme and identification of athree-qubit quantum register. See text for details. where µ B is the Bohr magneton. Initially, all the atoms are inthe reservoir state | S / , f = , m = i . Qubit s (s = | P / , f = , m = i is employed as | e i for mapping qubit f into photonic qubits sothat the wavelength of the emitted photons is λ = nm .In principle, any atomic ensembles that are suitable for col-lective encoding strategy can be used to implement the presentscheme [23]. To be specific, we first study a cubic lattice withseveral hundreds of atoms. It should be noted that our schemehas a significant flexibility of the shape and density of the en-semble, cf. below. For now let us suppose that an orderedthree-dimensional array of 7 × × = Rb atoms isloaded into an elongated optical lattice. With a lattice spacingof 0 . µ m , the maximum distance between any two atoms is R max ≈ µ m . We carefully choose two Rydberg states | r i and | r i , ensuring that the usual C / R or C / R van-der-Waals in-teraction is resonantly enhanced by F¨oster processes leadingto isotropic C / R long-range interaction [19, 22, 23]. As-suming the principal quantum number of the Rydberg statesis n =
80, a blockade shift as large as B / π ≥ µ m is achievable [19, 23].Based on the above specific physical system, we now an-alyze the error sources and determine the optimal Rabi fre-quencies for transitions in our scheme. The main errors inour scheme arise in the procedures involving Rydberg block-ade, including the initializations of qubit s (s = P and spontaneous emission probabil-ity P loss for the above four di ff erent procedures (as shown inTable I ).It should be noted that in our system the cooperative spon-taneous emission dominates the decay processes from Ryd- berg states. Thus the atoms in the Rydberg states are mostlikely to decay to the reservoir state (with a probability oforder d / ( d + d is the optical depth of the ensem-ble). If so, after spontaneous emission the state of the en-semble will be outside the subspace spanned by qubit s andqubit f , and thereby can be eliminated by the following post-selection measurement of our repeater scheme. Hence, all thespontaneous emission error terms are suppressed by a factor1 / ( d + − η ion ) η ion P ( is )2 ≈ . P ( is )2 . A double excitationin the procedure of initializing qubit f leads to a two-photonemission into the fiber, which gives an error as large as 2 P ( i f )2 .Double excitations which occur in the two-qubit gates for en-tanglement creation or entanglement swapping will cause thefinal state to be separable. They introduce errors with proba-bilities of P ( en )2 and P ( sw )2 , respectively. For clarity, we show allthe errors corresponding to the di ff erent procedures in Table I.The error in the entanglement creation E ( c ) can be written as E ( c ) = E ( is ) + E ( i f ) + E ( en ) = . Ω s B + π τ Ω s ( d + + Ω f B + π τ Ω f ( d + . (10)All the above errors result in a separable component ρ sep in thecreated state with the probability E ( c ) , where the specific formof ρ sep is not important to our discussion. Hence, the densitymatrix of each link after the entanglement creation reads ρ ≈ | ψ ih ψ | + E ( c ) ρ sep , (11)where | ψ i is the desired entangled state and ρ sep is the errorterm. The error term will be amplified by subsequent swap-ping operations. Taking into account the error in the entan-glement swapping E ( sw ) , after the n -th swapping operation thedensity matrix of final state is ρ n ≈ | ψ ih ψ | + [2 n E ( c ) + (2 n − E ( sw ) ] ρ sep . (12)Note that for ultra-cold atoms trapped in an optical lattice,the lifetime of a Rydberg state with n =
80 is about 500 µ s .The optical depth d = N λ / A in such a 7 × ×
15 opticallattice is around 10, where A is the cross section of the ensem-ble. Assuming a nesting level n = = Ω fopt / π = . Ω sopt / π = . F ≈ . p ,i.e., the success probability for an ensemble to emit a pho-ton. Suppose one can collect the photon emitted in a direc-tion within 0.3 rad o ff the axis of the ensemble as in Ref.[32]. Based on Eq.(3), we can predict that the photon is emit-ted into the collectable area with more than 93% probability.Taking spontaneous emission in the preparation of ensemble-photon entanglement state into account, we estimate p ≈ . p ,resulting in a significant flexibility of the requirements for theatomic ensemble. For example, instead of the optical lattice,we could use an atomic sample where 200 atoms are randomlypositioned within a 6 µ m diameter sphere, resulting in a mod-erate optical depth d ≈
1. Using the same method as above,we can derive that now p ≈ . F ≈ . C. Additional speed-up via temporal multiplexing
As seen in the previous subsections, the creation of entan-glement between neighboring nodes A and B is heralded onthe outcome of photon detections at a middle station. To ben-efit from a nested repeater, the entanglement swapping op-erations can only be performed once one knows the relevantmeasurement outcomes. This requires a communication timeof order L / c . If every node consists of a multiqubit register,and the entanglement creation in the register can be triggered m times in every communication time interval L / c , one candecrease the average time for entanglement creation T link bya factor of order m [15]. We here propose a realization ofthe same basic idea for quantum repeaters based on Rydbergblockade coupled ensembles.As said before, the collective encoding strategy providesa promising way to realize a multiqubit register. For in-stance, Ref. [33] proposed to use a single holmium ensem-ble to realize a 60-qubit register via collective encoding. Notethat in such a 60-qubit register every qubit can be separatelyaddressed and two-qubit gates between any two qubits areachievable [33]. Suppose we use such a 60-qubit register asthe quantum memory in each node. Take ensemble B as anexample, one of the 60 qubits in ensemble B is used to emitsingle photons, and other 58 qubits are equally divided intotwo groups (“red” and “blue”) as stationary qubits. Using thesame procedure shown in Fig.2, the qubits in the “red” and “blue” groups are alternately entangled with correspondingemitted single photons, which are sent toward ensembles Aor C, respectively. If there are two detections in the centralstation located between A and B for the k -th qubit for exam-ple, then we know that these qubits are entangled. Runningthe same scheme for the qubits in the other group, there maybe similar detections between B and C locations associated tothe l -th qubits. One then performs the Bell state analysis forentanglement swapping by applying a two-qubit gate on the k -th and l -th qubits, thus creating entanglement between en-sembles A and C. Using such a holmium ensemble registercould increase the entanglement distribution rate of the stud-ied scheme by up to a factor of 29. IV. CONCLUSION
We have shown that Rydberg blockade coupled ensemblesare very promising systems for the implementation of quan-tum repeaters. Compared with DLCZ-type repeaters, ourscheme improves the entanglement distribution rate by severalorders of magnitude. One reason is that for Rydberg blockadecoupled atomic ensembles the entanglement swapping oper-ations are performed almost deterministically, in contrast tosuccess probabilities below 0.5 per swapping for DLCZ-typerepeaters. Another reason is that the blockade mechanismsuppresses multiple emissions from individual ensembles sothat our scheme does not need to work with a very low emis-sion probability. Compared with repeaters based on trappedions, both the entanglement fidelity and the distribution rate ofour scheme are comparable. This is because a Rydberg block-ade coupled atomic ensemble behaves as one superatom witha two-level structure. However, by using an ensemble basedscheme we avoid the requirements of a high-finesse cavity,and of addressing and transporting single ions. Moreover, ourscheme is amenable to temporal multiplexing, which couldfurther improve the performance.
Acknowledgments.
We thank A. Lvovsky and A. MacRaefor useful discussions.
Note added.
After completion of this work we becameaware of a very recent similar proposal [34]. [1] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, andW. K. Wootters, Phys. Rev. Lett. , 1895 (1993).[2] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[3] H.-J. Briegel, W. D¨ur, J.I. Cirac, and P. Zoller, Phys. Rev. Lett. , 5932 (1998).[4] L.-M. Duan, M.D. Lukin, J.I. Cirac, and P. Zoller, Nature ,413 (2001).[5] L. Jiang, J.M. Taylor, and M.D. Lukin, Phys. Rev. A ,012301(2007).[6] Z.-B. Chen et al. , Phys. Rev. A , 022329 (2007);B. Zhao etal. , Phys. Rev. Lett. , 240502 (2007).[7] N.Sangouard et al. , Phys. Rev. A , 050301(R) (2007).[8] N. Sangouard et al. , Phys. Rev.A , 062301 (2008).[9] C.-W. Chou et al. , Science , 1316(2007); Z.-S. Yuan et al. ,Nature , 1098 (2008).[10] J. Calsamiglia and N. L¨utkenhaus, Appl. Phys. B , 67 (2001). [11] B. He et al. , Phys. Rev. A , 052323 (2009); M. S. Shahriar,G. S. Pati, and K. Salit, Phys. Rev. A , 022323 (2007).[12] L. Childress et al. , Phys. Rev. Lett. , 070504 (2006); C. Si-mon et al. , Phys. Rev. B , 081302(R) (2007); P. Van Loock,N. L¨utkenhaus, W.J. Munro, and K.Nemoto, Phys. Rev. A ,062319 (2008).[13] N. Sangouard, R. Dubessy, and C.Simon Phys. Rev. A ,042340 (2009)[14] M. Sa ff man, T. G. Walker, and K. Mølmer, arXiv:0909.4777.[15] C. Simon et al. , Phys. Rev. Lett. , 190503 (2007).[16] T. F. Gallagher, Rydberg Atoms (Cambridge University Press,Cambridge, 1994).[17] D. Jaksch et al. , Phys. Rev. Lett. , 2208 (2000).[18] A. Ga¨etan et al. , Nature Phys. , 115 (2009). E. Urban et al.,Nature Phys. , 110 (2009).[19] L. Isenhower et al. , Phys. Rev. Lett. , 010503 (2010); T. Wilk et al. , Phys. Rev. Lett. , 010502 (2010).[20] M. D.Lukin et al. , Phys. Rev. Lett. , 037901 (2001).[21] M. Sa ff man and T.G. Walker, Phys. Rev. A , 042302 (2005).[22] M. Sa ff man and T. G. Walker, Phys. Rev. A , 022347 (2005);T. G. Walker and M. Sa ff man, J. Phys. B , S309 (2005).[23] E. Brion, K. Mølmer, and M. Sa ff man, Phys. Rev. Lett. ,260501 (2007).[24] M. M¨uller et al. , Phys. Rev. Lett. 102, 170502 (2009); M.Sa ff man and K. Mølmer, Phys. Rev. Lett. , 240502 (2009).[25] M. Sa ff man and T. G. Walker, Phys. Rev. A , 065403 (2002).[26] L. H. Pedersen and K. Mølmer, Phys. Rev. A , 012320(2009). [27] A. E. B. Nielsen and K. Mølmer, arXiv:1001.1429.[28] D. Porras and J. I. Cirac, Phys. Rev. A , 053816 (2008).[29] W. Rosenfeld et al. , Adv. Sci. Lett. , 469 (2009).[30] O.A. Collins, S.D. Jenkins, A. Kuzmich, and T.A.B. Kennedy,Phys. Rev. Lett. , 060502 (2007); J.B. Brask and A.S.Sørensen, Phys. Rev. A , 012350 (2008).[31] M. Sa ff man and T. G. Walker,Phys. Rev. A , 022347 (2005).[32] S. Yoon et al. , J. Phys.: Conf. Ser. , 012046 (2007).[33] M. Sa ff man and K. Mølmer, Phys. Rev. A , 012336 (2008).[34] B. Zhao, M. M¨uller, K. Hammerer, and P. Zoller,arXiv:1003.1911. TABLE I: The double excitation probability P , spontaneous emission probability P loss and relevant errors in four di ff erent procedures involvingRydberg blockade in our scheme. We denote the Rabi frequencies corresponding to qubit s and qubit f by Ω s and Ω f , respectively; τ is thelifetime of Rydberg state and d is the optical depth of the ensemble. Since qubit s and qubit f play di ff erent roles in di ff erent procedures, theerrors caused by P and P loss need to be calculated separately for each. See text for details.Procedures involving Rydberg blockade P P loss Error caused by P and P loss Initializing qubit s P ( is )2 = Ω s / B P ( is ) loss = π/τ Ω s E ( is ) = − η ion ) η ion P ( is )2 + P ( is ) loss / ( d + P ( if )2 = Ω f / B P ( if ) loss = π/τ Ω f E ( if ) = P ( if )2 + P ( if ) loss / ( d + P ( en )2 = Ω f / B P ( en ) loss = π/ τ Ω f + π/ τ Ω s E ( en ) = P ( en )2 + P ( en ) loss / ( d + P ( sw )2 = Ω s / B P ( sw ) loss = π/ τ Ω s E ( sw ) = P ( sw )2 + P ( sw ) loss / ( d ++