Heralded high-efficiency quantum repeater with atomic ensembles assisted by faithful single-photon transmission
aa r X i v : . [ qu a n t - ph ] O c t Heralded high-efficiency quantum repeater with atomic ensembles assisted by faithfulsingle-photon transmission ∗ Tao Li and Fu-Guo Deng , † Department of Physics, Applied Optics Beijing Area Major Laboratory,Beijing normal University, Beijing 100875, China State Key Laboratory of Networking and Switching Technology,Beijing University of Posts and Telecommunications, Beijing 100876, China (Dated: July 25, 2018)Quantum repeater is one of the important building blocks for long distance quantum commu-nication network. The previous quantum repeaters based on atomic ensembles and linear opticalelements can only be performed with a maximal success probability of 1/2 during the entanglementcreation and entanglement swapping procedures. Meanwhile, the polarization noise during the en-tanglement distribution process is harmful to the entangled channel created. Here we introducea general interface between a polarized photon and an atomic ensemble trapped in a single-sidedoptical cavity, and with which we propose a high-efficiency quantum repeater protocol in which therobust entanglement distribution is accomplished by the stable spatial-temporal entanglement andit can in principle create the deterministic entanglement between neighboring atomic ensembles ina heralded way as a result of cavity quantum electrodynamics. Meanwhile, the simplified paritycheck gate makes the entanglement swapping be completed with unity efficiency, other than 1/2with linear optics. We detail the performance of our protocol with current experimental parametersand show its robustness to the imperfections, i.e., detuning and coupling variation, involved in thereflection process. These good features make it a useful building block in long distance quantumcommunication.
PACS numbers: 03.67.Pp, 03.65.Ud, 03.67.Hk
I. INTRODUCTION
Quantum mechanics provides some interesting waysfor communicating information securely between remoteparties [1–5]. However, in practice the quantum chan-nels such as optical fibers are noisy and lossy [6]. Thetransmission loss and the decoherence of photon systemsincrease exponentially with the distance, which makes itextremely hard to perform a long-distance quantum com-munication directly. To overcome this limitation, Briegel et al . [7] proposed a noise-tolerant quantum repeaterprotocol in 1998. The channel between the two remoteparties A and B is divided into smaller segments by sev-eral nodes, the neighboring nodes can be entangled effi-ciently by the indirect interaction through flying qubits,and the entanglement between non-neighboring nodes isimplemented by quantum entanglement swapping, whichcan be cascaded to create the entanglement between theterminate nodes A and B .The implementation of quantum repeaters is compati-ble with different physical setups assisted by cavity quan-tum electrodynamics, such as nitrogen vacancy centersin diamonds [8], spins in quantum dots [9–12], singletrapped ions or atoms [13, 14]. However, the mostwidely known approach for quantum repeaters is basedon atomic ensembles [15] due to the collective enhance-ment effect [16]. In a seminal paper by Duan et al. [17], ∗ Published in Sci. Rep. , 15610 (2015) † Corresponding author: [email protected] the atomic ensemble is utilized to act as a local memorynode. The heralded collective spin-wave entanglementbetween the neighboring nodes is established by the de-tection of a single Stokes photon, emitted indistinguish-ably from either of the two memory nodes via a Ramanscattering process. However, due to the low probabilityof Stokes photon emission required in the Duan-Lukin-Cirac-Zoller (DLCZ) proposal [17], the parties can hardlyestablish the entanglement efficiently for quantum entan-glement swapping. In order to improve the success proba-bility, photon-pair sources and multimode memories areused to construct a temporal multi-mode modification[18], and then the schemes based on the single-photonsources [19] and spatial multiple modes [20] are devel-oped. Besides these protocols based on Mach-Zehnder-type interference, Zhao et al. [21, 22] proposed a robustquantum repeater protocol based on two-photon Hong-Ou-Mandel-type interference, which relaxes the long-distance stability requirements and suppresses the vac-uum component to a constant item. Subsequently, thesingle-photon sources are embedded to improve the per-formance of robust quantum repeaters [23–25]. In ad-dition, Rydberg blockade effect [26] is used to performcontrolled-NOT gate between the two atomic ensemblesin the middle node [27, 28], which makes the quantum en-tanglement swapping operation be performed determin-istically.Since the two-photon interference is performed withthe polarization degree of freedom (DOF) of the pho-tons [21, 22], which is incident to be influenced by thethermal fluctuation, vibration, and the imperfection ofthe fiber [29], the fidelity of the entanglement createdbetween the neighboring nodes will be decreased whenthe photons are transmitted directly [6, 7]. In otherwords, the more the overlap of the initial photon stateused in the two-photon interference is, the higher thefidelity of the entanglement created is. Following theidea of Zhao’s protocol [21], quantum repeaters immuneto the rotational polarization noise are proposed withthe time-bin photonic state [30] and the antisymmetricBell state [31] | Ψ − i = ( | HV i − | V H i ) / √
2, respectively.When the noise on the two orthogonal polarized pho-ton states is independent, Zhang et al. [32] utilized thefaithful transmission of polarization photons [29] to sur-mount the collective noise. In the ideal case, the two-fold coincidence detection in the central node can suc-cessfully get the stationary qubits entangled maximallyin a heralded way. Apart from this type of entangle-ment distribution, Kalamidas [33] proposed an error-freeentanglement distribution protocol in the linear opticalrepeater. An entangled photon source is placed at thecenter node, and the entangled photons transmitted toneighboring nodes are encoded with their time-bin DOF.With two fast Pockels cells (PCs), the entanglement dis-tribution can be performed with a high efficiency whenthe polarization-flip-error noise is relatively small.In a recent work, Mei et al. [34] built a controlled-phase-flip (CPF) gate between a flying photon and anatomic ensemble embedded in an optical cavity, and con-structed a quantum repeater protocol, following someideas in the original DLCZ scheme [17]. In 2012, Brion et al. [35] constituted a quantum repeater protocol withRydberg blocked atomic ensembles in fiber-coupled cav-ities via collective laser manipulations of the ensemblesand photon transmission. Besides, Wang et al. [36] pro-posed a one-step hyperentanglement distillation and am-plification proposal, and Zhou and Sheng [37] designeda recyclable protocol for the single-photon entanglementamplification, which are quite useful to the high dimen-sional or multiple DOFs optical quantum repeater.In this paper, we give a general interface between apolarized photon and an atomic ensemble trapped in asingle-sided optical cavity. Besides, we show that a deter-ministic faithful entanglement distribution in a quantumrepeater can be implemented with the time-bin photonicstate when two identical fibers act as the channels of dif-ferent spatial DOFs of the photons. Interestingly, it doesnot require fast PCs and the time-slot discriminator [29–33] is not needed anymore. By using the input-outputprocess of a single photon based on our general interface,the entanglement between the neighboring atom ensem-bles can be created in a heralded way, without any clas-sical communication after the clicks of the photon de-tectors, and the quantum swapping can be implementedwith almost unitary success probability by a simplifiedparity-check gate (PCG) between two ensembles, otherthan 1 / II. RESULTSA. A general interface between a polarized photonand an atomic ensemble.
The elementary node in our quantum repeater pro-tocol includes an ensemble with N cold atoms trappedin a single-sided optical cavity [34, 35]. The atom hasa four-level internal structure and its relevant levels areshown in Fig. 1. The two hyperfine ground states aredenoted as | g i and | g h i . The excited state | e i and theRydberg state | r i are two auxiliary states. The | h i po-larized cavity mode a h couples to the transition between | g h i and | e i . Initially, all of the atoms are pumped tothe state | g i . With the help of the Rydberg state | r i ,one can efficiently perform an arbitrary operation be-tween the ground state | G i = | g , . . . , g j , . . . , g N i andthe single collective spin-wave excitation state [17] | S i = √ N P N j =1 | g , . . . , g h j , . . . , g N i via collective laser manip-ulations of the ensembles [34, 35, 38]. The single col-lective excited state | E i = √ N P N j =1 | g , . . . , e j , . . . , g N i .When the Rydberg blockade shift is of the scale 2 π × M Hz , the transition between | G i and | S i can be com-pleted with an effective coupling strength 2 π × M Hz andthe probability of nonexcited and doubly excited errors[39] is about 10 − -10 − . Recently, rotations along axes R x , R y , and R z of a spin-wave excitation with an aver-age fidelity of 99% are achieved in Rb atomic ensemblesand they are implemented by making use of stimulatedRaman transition and controlled Larmor procession [40].In other words, the high-efficiency single qubit rotationsof the atomic ensemble can be implemented faithfully. a in a out e r h g g E A (a) (b) FIG. 1: (Color online) (a) Schematic diagram for a single-side cavity coupled to an atomic ensemble system. (b) Atomiclevel structure.
Let us consider an | h i polarized input photon with thefrequency ω , which is nearly resonant to the cavity mode a h with the frequency ω c . The coupling rate between thecavity and the input photon can be taken to be a realconstant p κ π when the detuning | δ ′ | = | ω − ω c | is farless than the cavity decay rate κ ( | δ ′ | ≪ κ ) [41–43]. TheHamiltonian of the whole system, in the frame rotatingwith respect to the cavity frequency ω c , is (¯ h = 1) [41]ˆ H s = N X j =1 h(cid:16) ∆ − i γ e j (cid:17) ˆ σ e j e j + ig j (cid:0) ˆ a h ˆ σ e j s j − ˆ a † h ˆ σ s j e j (cid:1)i + i r κ π Z dδ ′ h ˆ b † ( δ ′ )ˆ a h − ˆ b ( δ ′ )ˆ a † h i + Z dδ ′ ˆ b † ( δ ′ )ˆ b ( δ ′ ) , (1)where ˆ a and ˆ b are the operators of the cavity mode andthe input photon with the properties [ˆ a, ˆ a † ] = 1 and[ˆ b ( δ ′ ) , ˆ b † ( δ ′′ )] = δ ( δ ′ − δ ′′ ), respectively. ∆ = ω − ω c isthe detuning between the cavity mode frequency ω c andthe dipole transition frequency ω , ˆ σ e j e j = | e j ih e j | , andˆ σ e j s j = | e j ih s j | . γ e j represents the spontaneous emissionrate of the excited state | e j i , while g j denotes the cou-pling strength between the j-th atom transition and thecavity mode ˆ a h . Here and after, we assume g j = g and γ e j = γ for simplicity .With the Hamiltonian ˆ H s shown in Eq.(1), theHeisenberg-Langevin equations of motion for cavity ˆ a h and the atomic operator ˆ σ − = | S ih E | taking into ac-count the atomic excited state decay γ can be detailedas [41] d ˆ a h dt = − (cid:16) iω c + κ (cid:17) ˆ a h − ig ˆ σ − − √ κ ˆ a in ,d ˆ σ − dt = − (cid:16) iω + γ (cid:17) ˆ σ − + ig ˆ σ z ˆ a h + √ γ ˆ σ z ˆ N . (2)Here the Pauli operator ˆ σ z = | E ih E | − | S ih S | , while ˆ N is corresponding to the vacuum noise field that helps topreserve the desired commutation relations for the atomicoperator. Along with the standard cavity input-outputrelation ˆ a out = ˆ a in + √ κ ˆ a h , one can obtain the reflectionand noise coefficients r ( δ ′ ) and n ( δ ′ ) in the weak excita-tion approximation where the ensemble is hardly in thestate | E i but predominantly in | S i , that is, r ( δ ′ ) = ( δ ′ − iκ/ ′ + iγ/ − g ( δ ′ + iκ/ ′ + iγ/ − g ,n ( δ ′ ) = ig √ κγ ( δ ′ + iκ/ ′ + iγ/ − g , (3)where ∆ ′ = ω − ω represents the frequency detun-ing between the input photon and the dipole transition. | r ( δ ′ ) | + | n ( δ ′ ) | = 1 means that when the noise fieldis considered, the energy is conserved during the input-output process of the single-sided cavity.If the atomic ensemble in the cavity is initialized tobe the state | G i , it does not interact with the cavitymode (i.e., g = 0). The input | h i polarized probe photonfeels an empty cavity and will be reflected by the cavitydirectly. Now, the reflection coefficient can be simplifiedto be [41] r ( δ ′ ) = δ ′ − iκ/ δ ′ + iκ/ . (4) -10 -5 0 5 100.00.20.41.0 | r | , | n | a nd | r | (cid:0) '/ ✁ |r| |n| |r | | r | g/ ✂ ✄ '/ ☎ ✆ ✝ ✄ '/ ☎ ✆ ✝✞✟ ✄ '/ ☎ ✆ ✠✞✝ (a) (b) FIG. 2: (Color online) (a) | r | , | n | and | r | vs the scaled de-tuning δ ′ /κ , with the scaled coupling rate g/κ = 4 . γ/κ = 0 . | r | vs the scaled coupling rate g/κ withdetuning δ ′ /γ = 0 , . , and 1. Note that the detuning is small | δ ′ | ≪ κ , the pulse band-width is much less than the cavity decay rate κ . If thestrong coupling condition γκ/ ≪ g is achieved, onecan get the input probe photon totally reflected with r ( δ ′ ) ≃ − r ( δ ′ ) ≃
1, shown in Fig. 2. The abso-lute phase shifts versus the scaled detuning are shown inFig. 3. -0.10 -0.05 0.00 0.05 0.100.00.20.40.60.81.0 (cid:0) ' ✁✂✄☎✆✝✞✄✟✄✆ ✠✝✞✄ a nd ✄✆✝✞✄ ✡☛☞✌✍✡✡☞✎✌✍✡✡☞✌✍✡ -10 -5 0 5 10-1.0-0.50.00.51.0 P h a s e s h i f t ✏ '/ ✑ ✒✓✔✕ ✒✔✕ FIG. 3: (Color online) The absolute phase shifts vs the scaleddetuning. The dashed and dashed-dot lines show the absolutephase shifts | θ /π | and | θ/π | that the reflected photon gets,with the ensemble in | G i and | S i , respectively. The solid linerepresents the absolute value of the phase shifts difference | ∆ θ/π | = | θ /π − θ/π | . The inset shows the phase shifts vsthe scaled detuning θ /π and θ/π that the reflected photongets, with the ensemble in | G i and | S i , respectively. B. Hybrid CPF gate on a photon-atomic-ensemblesystem and PCG on a two-atomic-ensemble system.
The principle of our CPF gate on a hybrid quantumsystem composed of a photon p and an atomic ensemble E A is shown in Fig. 4, following some ideas in previ-ous works [34, 42, 43]. Suppose that the photon p isin the state | ϕ p i = µ | h i + ν | v i ( | µ | + | ν | = 1) andthe ensemble E A is in the state | φ A i = µ ′ | G i + ν ′ | S i ( | µ ′ | + | ν ′ | = 1). The | h i polarized component of thephoton p transmits the polarization beam splitter (PBS)and then be reflected by the cavity, while the | v i polar-ized component is reflected by the mirror M . The opticalpathes of the | h i and | v i components are adjusted to beequal and they will be combined again at the PBS withan extra π phase shift on the | h i component if the en-semble is in the state | G i . This process can be describedas | φ A i ⊗ | ϕ p i → µ ′ | G i ⊗ ( − µ | h i + ν | v i )+ ν ′ | S i ⊗ ( µ | h i + ν | v i ) . (5)That is to say, the setup in Fig. 4(a) can be used toaccomplish a CPF gate on the atomic ensemble E A andthe photon p . E A PBS a in M a out D h HWP H D v a in (a) (b) E A HWP PBS PBS E B FIG. 4: (Color online) Schematic setup for implementing aCPF gate and a parity-check gate (PCG). M stands for amirror and the PBS transmits the | h i polarized photon andreflects the | v i component. HWP and HWP are half waveplates performing the bit-flip operation while H represent aHadamard rotation. The schematic diagram of our PCG on two atomicensembles E A and E B is shown in Fig. 4(b). Letus assume that E A and E B are initially in the states | φ i i = µ i | G i i + ν i | S i i ( | µ i | + | ν i | = 1 and i = A , B ). One can input a polarized photon p in the state | ϕ p i = √ ( | h i + | v i ) into the import of the setup.HWP (HWP ) is used to perform the bit-flip opera-tion | h i ↔ | v i on the photon p by using a half-wave plate(HWP) with its axis at π/ p are reflected bythe two cavities, they combine with each other at PBS .The state of the system composed of the two atom en-sembles and the photon evolves to be | Φ i p AB = 1 √ | h i⊗ ( − µ A | G i A + ν A | S i A )( µ B | G i B + ν B | S i B )+ | v i⊗ ( µ A | G i A + ν A | S i A )( − µ B | G i B + ν B | S i B )] . (6)And then, another HWP names H whose axis is placedat π/ | h i ↔ / √ | h i + | v i ) and | v i ↔ / √ | h i − | v i ) on the photon. The state of the system becomes | Φ i ′ p AB = | h i ⊗ ( ν A ν B | S i A | S i B − µ A µ B | G i A | G i B )+ | v i ⊗ ( ν A µ B | S i A | G i B − µ A ν B | G i A | S i B ) . (7)After the photon is measured with PBS and two single-photon detectors, the parity of E A and E B can be deter-mined. In detail, if the photon is in the state | h i , the twoensembles E A and E B have an even parity. If the pho-ton is in | v i , E A and E B have an odd parity. With aneffective input-output process of a single photon, one canefficiently complete the PCG on two atomic ensembles. Encoder l s PBS BS D ec od e r- A D ec od e r- B HWP HWP
Encoder
PBS BS l s (a) HWP b PBS BS D h M b P π H PBS E B D v (b) PBS
Decoder
HWP
HWP
FIG. 5: (Color online) Schematic setup for entanglement dis-tribution. p π is a π phase shifter. C. Entanglement distribution with faithfulsingle-photon transmission.
Suppose that there is an entanglement source whichis placed at a central station between two neighbor-ing nodes, say Alice and Bob. The source produces atwo-photon polarization-entangled Bell state | Ψ + i ab =1 / √ | h i a | v i b + | v i a | h i b ). Here the subscripts a and b denote the photons sent to Alice and Bob, respectively.As shown in Fig. 5 (a), the photons a and b will passthrough an encoder in each side before they enter thenoisy channels. The encoder is made up of a PBS, anHWP, and a beam splitter (BS). Here BS is used for aHadamard rotation on the spatial DOF of the photon,i.e., | u i ↔ √ ( | u i + | d i ) and | d i ↔ √ ( | u i − | d i ), where | u i and | d i represent the upper and the down ports ofthe BS, respectively.With our faithful single-photon transmission method(see Method), Alice and Bob can share photon pairs ina maximally entangled state, shown in Fig. 5. In detail,after a photon pair from the source passes through thetwo encoders, its state becomes | ϕ i ab = 12 √ | h i a | h i b ⊗ [( | u l i a + | d l i a ) ⊗ ( | u l i b + | d l i b )+( | u s i a −| d s i a ) ⊗ ( | u s i b −| d s i b )] . (8)As the two photons a and b suffer from independent col-lective noises from the two channels, the influence of thechannels on the two photons can be described with twounitary rotations U aC and U bC as follows: U aC | h i noise −−−→ δ a | h i + η a | v i , (9) U bC | h i noise −−−→ δ b | h i + η b | v i , (10)where | δ i | + | η i | = 1 ( i = a, b ). The influence on the po-larization of the photons arising from the channel noisescan be totally converted into that on the spatial DOF.The state of the photons a and b arriving at Alice andBob becomes | ϕ i ab = 1 √ | h i a | v i b + | v i a | h i b ) ⊗ ( δ a | a i + η a | a i ) ⊗ ( δ b | b i + η b | b i )= | ϕ i pab ⊗ | ϕ i sab . (11)This is a two-photon Bell state | ϕ i pab = √ ( | h i a | v i b + | v i a | h i b ) in the polarization DOF of the photon pair ab . Simultaneously, it is a separable superposition state | ϕ i sab = ( δ a | a i + η a | a i ) ⊗ ( δ b | b i + η b | b i ) in the spatialDOF.To entangle the stationary atomic ensembles E A and E B , which are initialized to be | φ A i = √ ( | G i A + | S i A )and | φ B i = √ ( | G i B + | S i B ), only two CPF gates arerequired if Alice and Bob have shared some photon pairsin the Bell state | ϕ i pab . Let us take the case that thephotons a and b come from the spatial modes a and b asan example to detail the entanglement creation process.As for the other cases, the same entanglement between E A and E B can be obtained by a similar procedure withor without some single-qubit operations.First, the photon a suffers a Hadamard operation bypassing through a half-wave plate H . Second, it is re-flected by the cavity or the mirror M , which is used tocomplete the CPF gate on the photon a and the ensemble E A . Third, Alice performs another Hadamard operationon the photon a . Now, the state of the composite systemcomposed of the photons a and b and the ensembles E A and E B evolves into | Φ i P E , | Φ i P E = 12 (cid:2) | h i a ( | v i b | S i A − | h i b | G i A )+ | v i a ( | h i b | S i A − | v i b | G i A ) (cid:3) ⊗ | ϕ i B . (12)Fourth, Alice measures the polarization state of the pho-ton a with a setup composed of PBS and single-photondetectors D h and D v . If an | h i polarized photon is de-tected, the hybrid system composed of b , E A , and E B will be projected into | Φ i P E = 1 √ | v i b | S i A − | h i b | G i A ) ⊗ | ϕ i B . (13)If a | v i polarized photon is detected, the remaining hybridsystem can also be transformed into the state | Φ i P E by a bit-flip operation ˆ σ Ax = | S i A h G | + | G i A h S | on theensemble E A .Up to now, the original entanglement of the photonpair ab is mapped to the hybrid entanglement betweenthe photon b and the ensemble E A . In order to createthe entanglement between E A and E B , Bob just performsthe same operations as Alice does. In brief, before andafter the CPF operation on the photon b and the ensem-ble E B , Bob performs two local Hadamard operationson the photon b with H . These operations result in theentanglement between the photon b and the two atomicensembles. The state | Φ i P E is changed into | Φ i P E | Φ i P E = 12 (cid:2) | v i b ⊗ ( | S i A | S i B + | G i A | G i B ) −| h i b ⊗ ( | G i A | S i B + | S i A | G i B ) (cid:3) . (14)If the detector D h at Bob’s node is clicked, the state ofthe system composed of E A and E B will be collapsed intothe desired entangled state | Ψ i AB = 1 √ | G i A | S i B + | S i A | G i B ) . (15)As for the case that the photon b is in the state | v i , theycan also obtain the desired entangled state | Ψ i AB withan additional bit-flip operation ˆ σ Bx on E B . D. Entanglement swapping on atomic ensembleswith a PCG.
After the parties produce successfully the entangle-ment between each two atomic ensembles in the neighbor-ing nodes, they can extend the entanglement to a furtherdistance by entanglement swapping. Let us use the casewith three nodes as an example to describe the principlefor connecting the two non-neighboring nodes.Suppose the atomic ensembles E A and E C belong tothe two non-neighboring nodes Alice and Charlie, respec-tively, and the two ensembles E B and E B belong to themiddle node Bob, shown in Fig. 6. The two ensem-bles E A E B are in the state | Ψ i AB = √ ( | G i A | S i B − (cid:1) (cid:1) D h HWP D v E A PBS a in E B1 (cid:1) (cid:1) HWP PBS E B2 E C (cid:1) (cid:1) H FIG. 6: (Color online) Schematic setup for entanglementswapping with the simplified PCG. | S i A | G i B ) and the two ensembles E B E C are in the state | Ψ i B C = √ ( | G i B | S i C + | S i B | G i C ). After a parity-check measurement performed on the two local ensembles E B and E B with a PCG shown in Fig. 3 (b), the stateof the system composed of the four ensembles E A , E C , E B , and E B evolves into an entangled one. If the out-come of the parity-check measurement on the ensembles B B is odd, the composite system composed of E B , E B , E A , and E C will be projected into the state | Ψ i E = 1 √ | G i B | S i B | S i A | G i C + | S i B | G i B | G i A | S i C ) , (16)which is a four-qubit Greenberger-Horne-Zeilinger state.The decoherence of both E B and E B has an awful in-fluence on the system composed of E A and E C as it de-creases the fidelity of the entanglement of the system.In order to disentangle the two ensembles E B and E B from the system, the party at the middle node could firstperform a Hadamard operation on the two ensembles andthen apply a parity-check measurement on them. If theoutcome of the second parity-check measurement is even,the composite system composed of the four ensembles E B , E B , E A , and E C is projected into the state | Ψ i E ′ = 12 ( | G i B | G i B + | S i B | S i B ) ⊗ ( | S i A | G i C + | G i A | S i C ) , (17)where the ensembles E B and E B are decoupled from thesystem composed of the two nonlocal ensembles E A and E C which are in the maximally entangled state | Ψ i AC = √ ( | G i A | S i C + | S i A | G i C ).In the discussion above, we use the outcomes (odd,even) of the two successive parity-check measurements asan example to describe the principle of the entanglementswapping between the four atomic ensembles. In fact,the other cases that the outcomes of each parity-checkmeasurement is either an odd one or an even one canalso be used for the entanglement swapping with onlya single-qubit operation on the ensemble E A , shown inTable. 1. TABLE I: The relation between the single-qbuit operationon the ensemble E A for entanglement swapping and the out-comes of the parity-check measurements on the two atomicensembles at the middle node. P and P denote the out-comes of the first and the second parity-check measurements.Here ˆ σ I = | G i A h G | + | S i A h S | , ˆ σ z = | G i A h G | − | S i A h S | ,ˆ σ y = | G i A h S | − | S i A h G | , and ˆ σ x = | G i A h S | + | S i A h G | .P P E A v h ˆ σ I v v ˆ σ z h v ˆ σ y h h ˆ σ x III. DISCUSSION
We would like to briefly discuss the imperfections ofour quantum repeater protocol. The photon loss is themain imperfection, which is also of crucial importancefor the previous quantum repeaters with photon inter-ference [8–15, 17–25]. The photon loss happens, due tothe fiber absorbtion, diffraction, the cavity imperfection,and the inefficiency of the single-photon detectors. Itwill decrease the success probability and prolong the timeneeded for establishing the quantum repeater. Since thememory node in this protocol is implemented with theatomic ensemble, the local operation between two col-lective quantum states | G i and | S i of the memory node,can be performed with collective laser manipulations [35],while excitations of higher-order collective states can besuppressed efficiently with the Rydberg blockade [38].During the entanglement swapping process, to detect thecollective state of two ensembles in the centering nodes,fluorescent detection [44] can be used, since the detectionefficiencies of 99 .
99% for trapped ions have been experi-mentally demonstrated [45]. Moreover, with the currentsignificant progress on the source of entangled photonpairs, the repetition rate as high as 10 / S − has beenachieved [46], so our entanglement distribution processcan be performed with a high efficiency.In summary, we have proposed a high-efficiency quan-tum repeater with atomic ensembles embedded in opticalcavities as the memory nodes, assisted by single-photonfaithful transmission. By encoding the polarization qubitinto the time-bin qubit, our faithful single-photon trans-mission can be completed with only linear-optical ele-ments, and neither time-slot discriminator nor fast PCsis required [29–33]. The heralded entanglement creationbetween the neighboring nodes is achieved with a CPFgate between the atomic ensemble and the photon inputin each node, which makes our scheme more convenientthan the one with post selection [35], although both effi-ciencies of our quantum repeaters are identical and max-imal among all the exciting quantum repeater schemeswhen multi-mode speed up is not considered [18, 20].Besides, no additional classical information is involved todeterminate the state of the entangled atomic ensembles,since the parties can create a deterministic entanglementup to a feedback upon the results of photon detection.The quantum swapping process is deterministically com-pleted with a simplified PCG involving only one input-output process, which makes our scheme far more effi-cient than the ones based on linear optical elements [15]. IV. METHODSA. Faithful single-photon transmission
Our protocol for deterministic polarization-error-freesingle-photon transmission can be details as follows. As-suming the initial state of the single photon to be trans-mitted is | ϕ i = µ | h i + ν | v i ( | µ | + | ν | = 1). After passingthrough the encoder, the photon launched into the noisychannel evolves into | ϕ ′ i = 1 √ | h i ⊗ ( ν | u l i + ν | d l i + µ | u s i − µ | d s i ) , (18)where the subscripts l and s represent the photons pass-ing through the long path and short path of the encoder,respectively. When the optical path difference between l and s is small, the two time bins are so close that theysuffer from the same fluctuation from the optical fiberchannels [3, 6, 29–33, 47–52]. The noise of the channelcan be expressed with a unitary transformation U C asfollows: U C | h i noise −−−−→ δ | h i + η | v i , (19)where | δ | + | η | = 1. After the photon passes throughthe channels, a π phase shifter P π on the d channel isapplied, and the state of the photon becomes | ϕ ′′ i = 1 √ δ | h i + η | v i ) ⊗ ( ν | u l i− ν | d l i + µ | u s i + µ | d s i ) . (20)With a decoder composed of a BS, an HWP, and a PBS,shown in Fig. 5 (b), the evolution of the photon can bedescribed as follows: | ϕ ′′ i BS −−−→ ( δ | h i + η | v i ) ⊗ ( ν | d ls i + µ | u sl i ) HWP −−−−→ ν | d ls i ⊗ ( δ | h i + η | v i )+ µ | u sl i ⊗ ( δ | v i + η | h i ) PBS −−−−→ δ | a i ( ν | h i + µ | v i )+ η | a i ( µ | h i + ν | v i ) HWP −−−−→ ( µ | h i + ν | v i ) ⊗ ( δ | a i + η | a i ) . (21)Here the subscripts ls ( sl ) represent the photon thatpasses through the long (short) path of the encoder andthe short (long) path of the decoder, respectively. Thedifference between the long path and the short one forthe encoder is designed to be the same as that for thedecoder. Without any time-slot discriminator, one canget the error-free photon in either the output a or a ata deterministic time slot. B. Performance of CPF and PCG with currentexperimental parameters.
Before we analyze the fidelity of the quantum entan-glement distribution and entanglement swapping in ourquantum repeater scheme, we first discuss the practi-cal performance of the CPF gate and the PCG basedon the recent experiment advances [53–55]. We definethe fidelity of a quantum process (or a quantum gate)as F = |h Ψ i | Ψ r i| , where | Ψ i i and | Ψ r i are the outputstates of the quantum system in the quantum process (orthe quantum gate) in the ideal condition and the realisticcondition, respectively [15].By combining a fibre-based cavity with the atom-chiptechnology, Colombe et al . [53] demonstrated the strongatom-field coupling in a recent experiment in which each Rb atom in Bose-Einstein condensates is identicallyand strongly coupled to the cavity mode. In this experi-ment, all the atoms are initialized to be the hyperfine zee-man state | S / , F = 2 , m f = 2 i . The dipole transitionof Rb | S / , F = 2 i 7→ | P / , F ′ = 3 i is resonantlycoupled to the cavity mode with the maximal single-atomcoupling strength g = 2 π ×
215 MHz. Meanwhile, thecavity photon decay rate is κ = 2 π ×
53 MHz and theatomic spontaneous emission rate of | P / , F ′ = 3 i is γ = 2 π × r ( δ ′ ) ≃ − r ( δ ′ ) ≃ | h i polarized photon a will get a π phase shift when the embedded atomic ensemble E A isin the state | G i ; otherwise, there is no phase shift on thephoton a . The fidelity of both the CPF gate (shown inFig.4 (a)) and the PCG (shown in Fig. 4 (b)) can reachunity. In a realistic atom-cavity system, the relationshipbetween the input and output field is outlined in Eqs.(3) and (4). In this time, after the party operates thephoton a and the ensemble E A with the CPF gate, theoutput state of the composite system becomes | Φ ′ i E p = 1 √ C [ µ ′ | G i ⊗ ( r µ | h i + ν | v i )+ ν ′ | S i ⊗ ( rµ | h i + ν | v i )] . (22)Here the normalized coefficient C = | r · µ ′ · µ | + | r · ν ′ · µ | + | µ ′ · ν | + | ν ′ · ν | . The fidelity of the CPF gate F cpf = | E p h Φ | Φ ′ i E p | depends on the input state of thesystem composed of the photon and the atomic ensemble.In the symmetric case with µ = µ = µ ′ = ν ′ = 1 / √
2, thefidelity F cpf can be simplified to be F cpf = 14 + 1 − Re ( r · r ∗ ) − Re ( r ) +2 Re ( r )2(2 + | r | + | r | ) . (23)Meanwhile, the efficiency η cpf of the CPF gate, which isdefined as the probability that the photon clicks eitherdetectors after being reflected by the CPF gate, can bedetailed as η cpf = 12 + | r | + | r | . (24)In a realistic condition, the output state of the com-posite system composed of a , E A , and E B in the PCGprocess before the single photon is detected becomes | Φ ′′ i p AB = 1 √ C {| v i ( r − r )( | G, S i − |
S, G i )+ | h i [ √ r | G, G i + √ r | S, S i +( r + r )( | G, S i + | S, G i )] } . (25)Compared with the ideal output state described inEq.(7), if an | h i polarized photon is detected, the fidelityof the PCG gate F pcg can be expressed as F pcg = | r | + | r | − Re ( r · r ∗ )3( | r | + | r | ) + 2 Re ( r · r ∗ ) . (26)When the photon in the state | v i is detected, the fi-delity of the PCG is F ′ pcg = 1. The success of the PCGis heralded when a single photon is detected after theparity-check process, no matter what the state the pho-ton evolves to be. The efficiency η pcg of the PCG processcan be defined as the probability that the probe photonis detected after it is reflected by the two cavities, thatis, η cpf = | r | + | r | . (27)Since the absolute value of the relative phase shift dur-ing the input-output process depends on the frequency ofthe input photon, it decreases smoothly with the detun-ing δ ′ between the input photon and the cavity mode,shown in Fig. 3.The fidelity of the CPF gate F cpf changes with thedetuning δ ′ , shown in Fig. 7(a). Here the parameters arechosen as g/κ = 2 . . γ/κ = 0 . δ = 2 | δ ′ | max with the maximal detuning | δ ′ | max = 0 . γ ( γ ), F cpf islarger than F cpf ( | δ ′ | max ) = 0 . . g/κ =4 . g/κ , as shown in Fig. 7(b) with the detuning | δ ′ | max =0 . γ or γ . When the maximal detuning of the inputphoton is | δ ′ | max = 0 . γ , the high-performance parity-check gate can be achieved with the fidelity F pcg higherthan F pcg ( | δ ′ | max ) = 0 . . g/κ = 2 . g/κ = 4 . F i d e lit y g/ k F' cpf F' pcg F cpf F pcg -0.10 -0.05 0.00 0.05 0.100.920.961.00 d ' /k F i d e lit y F' cpf F' pcg F cpf F pcg -0.02-0.01 0.00 0.01 0.020.9981.000 F i d e lit y d '/ k (a) (b) FIG. 7: (Color online) (a) Fidelities of CPF gate and PCGvs the scaled detuning. F ′ cpf and F ′ pcg is performed with thescaled coupling rate g/κ = 2 . γ/κ = 0 . F cpf and F pcg are performed with the scaled coupling rate g/κ = 4 . γ/κ = 0 . F ′ cpf and F ′ pcg are performedwith the scaled detuning δ ′ /κ = 0 . γ/κ = 0 . F cpf and F pcg is performed with δ ′ /κ = γ/κ = 0 . The efficiencies of the CPF gate and the PCG processversus the coupling rate g/κ are shown in Fig. 8. Whenthe bandwidth of the probe photon is on the scale of γ , both efficiencies η cpf and η pcg are robust to the vari-ation of g/κ with the parameters above [53]. In detail,when the maximal detuning | δ ′ | max of the input photon isless than 0 . γ , η cpf and η pcg are higher than 0 . . | δ ′ | max = γ , η cpf = 0 . η pcg = 0 . E ff i c i e n c y g/ k h ' cpf h ' pcg h cpf h pc FIG. 8: (Color online) Efficiencies of CPF gate and PCG gatevs the scaled coupling rate. η ′ cpf and η ′ pcg is performed withthe scaled detuning δ ′ /κ = 0 . γ/κ = 0 . η cpf and η pcg are performed with δ ′ /κ = γ/κ = 0 . -0.10 -0.05 0.00 0.05 0.100.920.940.960.981.00 d ' /k F i d e lit y F mh F mv F s -0.02 -0.01 0.00 0.01 0.020.9981.000 F i d e lit y d ' /k FIG. 9: (Color online) Fidelities of F mh , F mv and F s vs the detuing, with the scaled coupling rate g/κ = 2 . γ/κ = 0 . C. Performance of entanglement distribution andentanglement swapping.
Now, let us discuss the fidelities and the efficiencies ofthe entanglement distribution and entanglement swap-ping in our quantum repeater scheme. After Alice per-forms the local operations on the photon a and detectsan | h i polarized photon, the composite system composedof the photon b and the ensembles E A and E B will beprojected into the state | Φ ′ P E i , instead of | Φ i P E . Here | Φ ′ i P E = 1 √ C [( r − | h i ⊗ | G i A +( r + 1) | v i ⊗ | G i A + ( r − | h i ⊗ | S i A +( r + 1) | v i ⊗ | S i A ] ⊗ | ϕ i B , (28)where the normalized coefficient C = 2[ | r − | + | r +1 | + | r − | + | r +1 | ]. And then, the same operations, i.e., h ' m h ' s h m h s E ff i c i e n c y g /k FIG. 10: (Color online) Efficiencies of entanglement distri-bution and entanglement swapping processes vs the scaledcoupling rate. η ′ m and η ′ s is performed with the scaled detun-ing δ ′ /κ = 0 . γ/κ = 0 . η m and η s are performedwith δ ′ /κ = γ/κ = 0 . a CPF gate sandwiched by two Hadamard operations, areperformed by Bob on the photon b . After these opera-tions, the state of the composite system composed of thephoton b and the two ensembles E A and E B evolves into | ϕ i pE = 1 √ C ′ {| h i⊗ [( r − | G i A ⊗| G i B +( r · r − | G i A ⊗| S i B + | S i A ⊗| G i B )+( r − | S i A ⊗ | S i B ]+ | v i⊗ [( r + 1) | G i A ⊗| G i B +( r · r + 1)( | G i A ⊗| S i B + | S i A ⊗| G i B )+( r + 1) | S i A ⊗ | S i B ] } , (29)where the normalized coefficient C ′ = 2[ | r | + | r | +2 | r · r | + 4]. One can obtain the fidelity of the entanglementdistribution process F mh and F mv for the cases that D ′ h and D ′ v at the Bob’s node are clicked, respectively. F mh = 2 | r · r − | | r − | + | r − | + 2 | r · r − | ,F mv = | r + r + 2 | | r + 1 | + | r + 1 | + 2 | r · r + 1 | . (30)If one defines the efficiency η hm as the probability thatAlice detects an | h i polarized photon while Bob detectsa photon in either | h i or | v i polarization, one has η hm = C · C ′ C = | r | + | r | + 2 | r · r | + 416 . (31)In the above discussion, we detail the performance of ourentanglement distribution conditioned on the detectionof an | h i polarization photon at Alice’s node. Consider-ing the symmetric property of the system, one can easilyobtain the performance of the entanglement distribution0upon the detection of a | v i polarization photon at Alice’snode. Now, the fidelities F ′ mh and F ′ mv for the cases that D ′ h and D ′ v are clicked at Bob’s node, have the followingrelations to that for the cases that an | h i polarized pho-ton is detected by Alice, F ′ mh = F mv and F ′ mv = F mh ,see Eq. (30) for detail. Meanwhile, the efficiency η vm ofthe entanglement distribution process when Alice detectsa photon in | v i polarization is identical to η hm . The to-tal efficiency η m of the entanglement distribution can bewritten as η m = η hm + η vm = | r | + | r | + 2 | r · r | + 48 . (32)In our entanglement swapping process, two PCGs areapplied on the two ensembles E B and E B at the middlenode. In fact, only one PCG is enough if a single-atomic-ensemble measurement on each of the two ensembles E B and E B is utilized after the local Hadamard operations.After these measurements, the system composed of thetwo remote ensembles E A and E C is in the state | Ψ i AC with or without a local unitary operation. When thefluorescent measurement [44] or field-ionizing the atoms[28] with the help of Rydberg excitation are used, thestate detection on atomic ensembles could be performedwith a near-unity efficiency. In other words, the fidelityof the quantum entanglement swapping process can equalto that of the PCG operation. The fidelities of both the entanglement distributionand the entanglement swapping in our repeater schemeare shown in Fig. 9. One can see that all F mh , F mv , and F s = F pcg are larger than 0 . g , κ , γ ) = 2 π × (215, 53, 3) MHz achieved in experiment[53]. Meanwhile, all efficiencies involved in our quantumrepeater protocol, shown in Fig. 10, can be larger than0 . g/κ > . δ ′ /κ = γ/κ = 0 . - Perot cavity constituted by CO laser-machined mirrors [60], the maximal coupling strength ashigh as g = 2 π × . κ = 2 π × .
286 GHz ≃ γ .In this time, g/κ = 9 .
79 is achieved, and a better perfor-mance of our scheme is attainable.
ACKNOWLEDGEMENTS
TL was supported by the China Postdoctoral ScienceFoundation under Grant No. 2015M571011. FG wassupported by the National Natural Science Foundation ofChina under Grant Nos. 11174039 and 11474026, and theOpen Foundation of State key Laboratory of Networkingand Switching Technology (Beijing University of Postsand Telecommunications) under Grant No. SKLNST-2013-1-13. [1] Bennett, C. H. et al.
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