Parity-gate-based quantum information processing in decoherence-free subspace with nitrogen-vacancy centers

Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 Q2 63 64 65 66

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Parity-gate-based quantum information processing in decoherence-free subspace with nitrogen-vacancy centers Xiao-Ping Zhou a, Shi-Lei Su b, Qi Guo b, Hong-Fu Wang b, Ai-Dong Zhu b, Shou Zhang a,b,n a b

Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China Department of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

art ic l e i nf o

a b s t r a c t

Article history: Received 4 February 2015 Received in revised form 18 April 2015 Accepted 27 April 2015

We propose a parity-check gate (PCG) for logical qubits encoded in decoherence-free subspace (DFS) by the coherent optical pulse input–output process with nitrogen-vacancy (NV) centers fixed on the surface of microtoroidal resonators (MTRs) with quantized whispering-gallery modes (WGMs). Using the proposed PCG, we achieve controlled phase flip (CPF) gate, |χ 〉 state generation, and multipartite entanglement purification. The present schemes combine the advantages of the robustness of coherent pulse and the immunity to the dephasing noise of the DFS. The performances of these schemes are also analyzed, which shows that all the schemes works well in the low-Q cavity regime and are feasible under the current experimental technology. & 2015 Published by Elsevier B.V.

Keywords: Parity-check gate Decoherence-free subspace Nitrogen-vacancy centers

1. Introduction Over the past decades, due to controllable interaction between atoms and photons, cavity quantum electrodynamics (QED) that studies the coherent interactions of matter with quantized fields has been a paradigm for quantum information processing (QIP) [1–13]. However, the requirements of high-Q cavities and strong coupling to the confined atoms are stringent for current techniques. Because of the optical controllability and fine electron spin coherence even at room temperature [14], the NV center coupled to the microresonator cavity with a quantized WGM is a promising solid-state system to realize QIP effectively. In 2010, Yang et al. proposed an entanglement generation protocol by making use of a NV center coupled to a quantized WGM cavity [15]. The preparation of quantum entangled states between a single photon and the electronic spin of an NV center has been experimentally demonstrated by Togan et al. [16]. In 2011, Chen et al. proposed an efficient scheme to deterministically entangle two NV centers fixed on the exterior surface of two MTRs [17]. At the same time, Li et al. proposed an efficient scheme for realizing quantum information transfer and implementing entanglement with NV centers coupled to a high-Q WGM [18]. In 2013, Cheng et al. proposed efficient schemes of generating entangled state and transferring quantum state with NV centers in diamond confined in separated MTRs [19]. n Corresponding author at: Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China. E-mail address: [email protected] (S. Zhang).

In 2014, Wang et al. proposed an efficient entanglement concentration protocol on electron spin state in DFS by making use of the input–output process of low-Q confining NV centers. [20]. Most of the previous QIP schemes are focusing on the input–output process of single-photon pulse. But it is still not easy to accomplish QIP tasks with a single-photon resource because the perfect single-photon source and detection are demanding. An alternative way to deal with these problems is to use multiphoton fields such as coherent states, which are available from standard stabilized laser sources. The method to generate and use multimode entangled coherent states has been proposed [14]. The superiority of quantum information and computing comes from the quantum coherence, but the decoherence resulting from the unavoidable couplings between the system and its environment will destroy the quantum coherence and reduce the fidelity of entanglement. Generally, there are several methods to deal with the decoherence, namely, error correction [21–23], geometric phase [24–27], dissipative dynamics [28–35], entanglement purification [36–38] and DFS [39–42]. For the case of DFS, two physical qubits are used to encode one logic qubit when the system and environment have some certain symmetry. For instance, to avoid the influence originating from the collective-dephasing noise, which leads to the transformations |0〉 → eiψ0 |0〉 and |1〉 → eiψ1|1〉, the encoding |0˜ 〉 ≡ |0〉1|1〉2 and |1˜ 〉 ≡ |1〉1|0〉2 are introduced [43,44]. With this encoding method, Xue et al. [4] presented a universal quantum computation scheme via cavity-assisted interactions. Then, Deng et al. [8] proposed schemes too prepare multi-particle entanglement. After that, Wei et al. [10]

http://dx.doi.org/10.1016/j.optcom.2015.04.072 0030-4018/& 2015 Published by Elsevier B.V.

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101

X.-P. Zhou et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

considered two schemes to construct a many-logic- qubit conditional phase gate and teleportation of quantum states. In 2010, Chen and Feng proposed a scheme with the low-Q cavities to realize some QIP tasks in the same DFS [42]. Motivated by above schemes, we present a paradigm to construct a parity-check gate (PCG) with a higher probability and fidelity by utilizing the logical qubits encoded in the DFS. In our schemes, the coherent optical pulse picks up different phase shifts during the input–output process according to the states of the NV centers in the MTR due to the Faraday effect. The presented PCG is meaningful because it fits a current experimental condition and could be used widely in large-scale and nonlocal QIP protocols. Based on this PCG, we can realize the controlled phase flip (CPF) gate, the preparation of four-particle |χ 〉 state and multipartite entanglement purification. We analytically obtain the effect of relevant system parameters on the reflection characteristics. Several distinct advantages of the present schemes should be stressed. First, we employ a beam of coherent light pulse instead of singlephoton pulse to accomplish the input–output process, which is different from Ref. [45]. So, the low efficiencies of single-photon resource have been successfully avoided. Second, we should mention that the action of the photon detector in our protocol is to distinguish vacuum state from nonvacuum state, which is more feasible than a single-photon detector or homodyne measurement in other protocols. Third, the entangled qubits are encoded in DFS, which greatly weakened the decoherence, Moreover, the schemes are of scalability based on the MTR-NV system so that our schemes are of potential applications in large-scale quantum networks with optical channels. All these advantages will make this scheme beneficial for QIP and feasible with current technology. This paper is organized as follows. In Section 2, we discuss the input–output relation with the MTR-NV system. In Section 3, we describe the basic principle of the PCG in a DFS. In Section 4, we expound the process to achieve CPF gate based on the input– output process. In Section 5, we show how to generate the fourparticle |χ 〉 state. In Section 6, we expound the process of multipartite entanglement purification based on the PCG. Finally, discussions and the conclusion are given In Section 7.

2. The input–output relation in a MTR-NV system Consider two negatively charged NV centers positioned on the surface of a MTR with a single WGM, as shown in Fig. 1(a). Consider the ground state of the NV center is a spin triplet with the splitting at 2.87 GHz between levels |e〉(|A2 〉) and |0〉(ms = − 1), |1〉(ms =+ 1) [16,46]. The energy-level structure of the NV center is shown in Fig. 1(b). We encode the quantum information in the spin state |ms = − 1〉 of the A2 triplet such that |0〉 = |ms = − 1〉. The transition |0〉 → |e〉 is driven by the cavity mode with coupling

strength g. The WGM resonantly couples to the transition |0〉 → |e〉 for the j-th NV center with coupling constant gj, while the state |1〉 is decoupled from the cavity mode due to the large detuning. This system exhibits similar features with the Jaynes–Cummings (J–C) model so that the system is governed by the following Hamiltonian ( = 1):

⎤ σzj + igj (aσ+j − a†σ−j ) ⎥ + ωc a†a, ⎥⎦ 2 j=1 ⎣ N

H=

⎡ω

∑ ⎢⎢

0j

(1)

where a† (a) is the creation (annihilation) operator of the MTR field with frequency ωc; ω0 is the transition frequency of electronic energy levels; g represents the coupling strength between the MTR and the NV center; szj, s þ j, and s  j are inversion, raising, and lowering operators of the j-th NV center between the two corresponding transition levels, respectively. Then we introduce a beam of coherent light pulse with frequency ωp to input the MTR cavity with frequency ωc. In the rotating frame with respect to the frequency of the input pulse, the quantum Langevin equations of motion for cavity field operator and the NV center lowering operator are [47]

ȧ = [i (ωp − ωc ) −

κ ]a − 2

σ−̇ j = [i (ωp − ω 0 ) −

N

∑ gσ−j −

κ ain ,

j=1

γ ] σ−j − gj σzj a + 2

γ σzj bin ,

(2)

† where bin with the commutation relation [bin (t ) , bin (t′)] = δ (t − t′) is the vacuum input field felt by the NV centers, and κ is the cavity damping rate and γ is the dipolar decay of the NV center. Here we only consider the first order moment, in the Langevin equation [48,49] and we neglect the vacuum noise because the expectation value over the vacuum noise is zero under this situation. Providing the cavity decay rate κ is sufficiently large, the atom will have a weak excitation [50]. Now assuming 〈σz 〉 ≈ − 1 to make sure a weak excitation of the NV center initially prepared in the ground states. After adiabatically eliminating the cavity mode, the reflection coefficient of the system can be expressed as

r (ωp ) =

aout =1− ain

κ 2

gj κ i (ωc − ωp ) + + ∑ Nj=1 γ 2 ω 0 − ωp + 2

, (3)

here aout = ain + κ a , ain and aout are input and output field operators, respectively. For simplicity, we assume each of the coupling constants between the NV centers and the WGM are the same, i.e., gj = g . When there are n (0 < n < N ) identical NV centers confined in an MTR, Eqs. (3) reduces to

A2 NV1

NV2

g

α

1 0

D eg

m=0 Fig. 1. (a) The schematic of two NV centers positioned on the surface of the MTR to implement a phase change in a coherent beam induced by input–output process. (b) The energy level structure of a single NV center and the coupling induced by a cavity field.

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

X.-P. Zhou et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

⎡ κ ⎤⎡ γ⎤ i (ω − ωp ) − ⎥ ⎢i (ω 0 − ωp ) + ⎥ + ng 2 ⎣⎢ c 2 ⎦⎣ 2⎦ . rn (ωp ) = ⎡ κ ⎤⎡ γ⎤ i (ω − ωp ) + ⎥ ⎢i (ω 0 − ωp ) + ⎥ + ng 2 ⎣⎢ c ⎦ ⎣ ⎦ 2 2

3

|0〉1|0〉2 |α〉 → |0〉1|0〉2 |αeiϕ2 〉,

(4)

|0〉1|1〉2 |α〉 → |0〉1|1〉2 |αeiϕ1 〉,

In the case the input photons uncoupled to the WGM, i.e., g ¼0, we get the reflection coefficient for a cold cavity:

|1〉1|0〉2 |α〉 → |1〉1|0〉2 |αeiϕ1 〉,

κ 2 r0 (ωp ) = κ . i (ωc − ωp ) + 2 i (ωc − ωp ) −

|1〉1|1〉2 |α〉 → |1〉1|1〉2 |αeiϕ0 〉. (5)

Due to the high damping rate of the cavity, the coherent light pulse experiences a very tiny absorption, and thereby we may consider that the output coherent light pulse only experiences a pure phase shift. Based on the above discussions, we can adiabatically estimate the input–output relation. When the state of NV1 and NV2 is |0〉1|0〉2, the input–output relations could be expressed as

aout = r2 ain ,

(6)

⎡ κ ⎤⎡ γ⎤ 2 ⎢⎣i (ωc − ωp ) − 2 ⎥⎦ ⎢⎣i (ω 0 − ωp ) + 2 ⎥⎦ + 2g r2 (ωp ) = . ⎡ κ ⎤⎡ γ⎤ 2 ⎢⎣i (ωc − ωp ) + 2 ⎥⎦ ⎢⎣i (ω 0 − ωp ) + 2 ⎥⎦ + 2g

3. Basic model of PCG in a DFS Before explaining our schemes, we first introduce a basic element for QIP as shown in Fig. 2, i.e., a parity-check gate (PCG) in a DFS. Consider the low-Q case that the cavity parameters satisfy g = κ /2 and κ⪢γ , under the condition ωc = ω0 = ω + κ /2, according to Eqs. (4) and (5), we approximately obtain ϕ0 = π /2, ϕ1 = π , ϕ = − π /2. The logic qubits are encoded as |0˜ 〉 = |0〉1|1〉2 and 2

(7)

When the state of NV1 and NV2 is |0〉1|1〉2 or |1〉1|0〉2, the input– output relations could be expressed as

aout = r1ain ,

(8)

⎡ κ ⎤⎡ γ⎤ 2 ⎢⎣i (ωc − ωp ) − 2 ⎥⎦ ⎢⎣i (ω 0 − ωp ) + 2 ⎥⎦ + g . r1 (ωp ) = ⎡ κ ⎤⎡ γ⎤ 2 ( − ) + ( − ) + + ω ω ω ω i i g c p p 0 ⎣⎢ 2 ⎦⎥ ⎣⎢ 2 ⎦⎥

|1˜ 〉 = |1〉1|0〉2 [42]. To achieve the PCG, two sx1 operations are required to be performed on NV center 1, one before and the other after the coherent light pulse outputting the cavity. Suppose the two logic qubits of NV centers in the two separate MTRs are initially prepared in state α|0˜ 〉a + β|1˜ 〉a and δ|0˜ 〉b + γ |1˜ 〉b . The coherent beam, which is initially in the state | 2 α〉, successively gets injects into the two MTRs. Thus, initially, the state of the whole system is

|Ψ 〉0 = (α|0˜ 〉a + β|1˜ 〉a ) ⊗ (δ|0˜ 〉b + γ |1˜ 〉b ) ⊗ | 2 α〉0 ,

˜ ˜ 〉 + βδ|10 ˜ ˜ 〉 + βγ |11 ˜ ˜ 〉)ab | − α〉1]|α〉2 ˜ ˜ 〉)ab |α〉1 + (αδ|00 |Ψ 〉1 = [(αγ |01 (9)

aout = r0 ain ,

(10)

κ 2 r0 (ωp ) = κ . i (ωc − ωp ) + 2

(11)

i (ωc − ωp ) −

Suppose the initial input pulse is prepared in the coherent state |α〉, and the NV centers are prepared in state (|0〉 + |1〉)i / 2 , (i ¼1, 2). The initial state of the system in Fig. 1(a) and (b) is †

|ξ〉1 = 2 (|0〉1|0〉2 + |0〉1|1〉2 + |1〉1|0〉2 + |1〉1|1〉2 ) ⊗ e αain + α

BS 2

→ ˜ ˜ 〉 + βδ|10 ˜ ˜ 〉)ab )| − 2 α〉3 |0〉4 = (αγ |01 ˜ ˜ 〉 + βγ |11 ˜ ˜ 〉)ab |0〉3 | 2 α〉4 , + (αδ|00

⁎ain

|0〉in , (12)

|ξ 〉2 =

+ |0〉1|1〉2 |αe iϕ1〉 + |1〉1|0〉2 |αe iϕ1〉 + |1〉1|1〉2 |αe iϕ0 〉),

(16)

through the interactions of photon and NV centers. The measurement of photon number on path 3 can distinguish |0〉3 from | − 2 α〉3. And the combined state |Ψ 〉1 will evolve to |Ψ 〉2 according to the value of n¯ , i.e., for n¯ = 0, even parity

˜ ˜ 〉 + βγ |11 ˜ ˜ 〉)ab , |Ψ 〉2 = (αδ|00

(17)

for n¯ ≠ 0, odd parity

˜ ˜ 〉 + βδ|10 ˜ ˜ 〉)ab , |Ψ 〉2 = (αγ |01

(18)

in which n¯ denotes the measurement result of the PCG. Up to now, we have expounded the basic principle for the PCG in a DFS. Next, we will discuss the applications of the PCG in QIP.

If ω0 − ωp ≥ γ /2, after applying the cavity-assisted transformation, the system state would evolve to 1 (|0〉1|0〉2 |αe iϕ2 〉 2

(15)

which will be transformed to

When the state of NV1 and NV2 is |1〉1|1〉2, the input–output relations could be expressed as

1

(14)

PCG

in

DFS

(13)

i.e., when |α〉 is reflected by an MTR, it would achieve ϕ phase shift that determined by Eqs. (4) and (5). In detail, when both of the NV centers are in the state |1〉, it will experience a phase shift eiϕ0 ; When both of the NV centers are in the state |0〉, a phase shift eiϕ2 will be picked up by the coherent pulse; While if and only if one of the NV centers is in the state |0〉, the photon will obtain another phase shift eiϕ1. As a result, the output phase shifts of the coherent light pulse, corresponding to the previous states of the NV centers, can be expressed as

NV1

NV2

NV1

NV2

3

2α

0

BS1 1

D1

BS 2

2

D2

4

Fig. 2. A gate of logic qubit parity measurement; BSi (i¼ 1, 2) denotes the 50:50 beam splitter, which transforms |α〉|β〉 to |(α − β )/ 2 〉|(α + β )/ 2 〉. D is a photon detector.

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

X.-P. Zhou et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

4. The realization of the CPF gate based on the PCG in a DFS In this section, we will show how to realize a CPF gate based on the PCG proposed in Section 3. The setup sketch for the CPF gate is shown in Fig. 3. The initial state of the system could be expressed as

|Ψ 〉 = (c0 |0˜ 〉C + c1|1˜ 〉C ) ⊗

1 ˜ (|0〉A + |1˜ 〉A ) ⊗ (t0 |0˜ 〉T + t1|1˜ 〉T ), 2

(19)

where A, C, and T are auxiliary logic qubit, controlled logic qubit, and target logic qubit, respectively. After PCG1 running, the initial state will be transformed to

|Ψ 〉1 =

1 ˜ ˜ 〉CA + c1|11 ˜ ˜ 〉CA ) ⊗ (t0 |0˜ 〉T + t1|1˜ 〉T ). (c0 |00 2

(20)

Here, for simplicity, we only consider the case n¯ 1 = 0 to show how the setup works. Then, after performing Hadamard operations H˜ on logic qubit A and running PCG2. Here H˜ is the Hadamard operation on logic qubit, |0˜ 〉 → (|0˜ 〉 + |1˜ 〉) / 2 and |1˜ 〉 → (|0˜ 〉 − |1˜ 〉) / 2 , which has been implemented via different methods with high fidelity and success probability [51,52], the state will evolve to

˜ ˜ 〉 + |01 ˜ ˜ 〉)CA + c1 (|10 ˜ ˜ 〉)CA ] ⊗ (t0 |0˜ 〉T + t1|1˜ 〉T ), (21) ˜ ˜ 〉 − |11 |Ψ 〉2 = [c0 (|00 Similar to PCG1, in above analysis, we only consider the case n¯ 2 = 0 of PCG2. After performing bilateral Hadamard operations H˜ on logic qubit A again, the state of the system will be changed to

˜ ˜ ˜ 〉CAT + |010 ˜ ˜ ˜ 〉CAT ) + c0 t1 (|001 ˜ ˜ ˜ 〉CAT − |011 ˜ ˜ ˜ 〉CAT ) |Ψ 〉3 = c0 t0 (|000 ˜ ˜ ˜ 〉CAT − |111 ˜ ˜ ˜ 〉CAT ), ˜ ˜ ˜ 〉CAT ) − c1t1 (|101 ˜ ˜ ˜ 〉CAT + |110 + c1t0 (|100

(22)

Then, we perform single logic qubit measurement on A. If A is in state |0˜ 〉, the state in Eq. (22) will evolve to

˜ ˜ 〉CT + c0 t1|01 ˜ ˜ 〉CT + c1t0 |10 ˜ ˜ 〉CT . ˜ ˜ 〉CT − c1t1|11 |Ψ 〉4 = c0 t0 |00

1 |χ 〉 = 2 [(|0˜ 〉1|0˜ 〉2 + |1˜ 〉1|1˜ 〉2 )|0˜ 〉3 |0˜ 〉4

+ (|0˜ 〉1|1˜ 〉2 + |1˜ 〉1|0˜ 〉2 )|1˜ 〉3 |1˜ 〉4 ],

(24)

which is important in quantum computation and communication [50]. Here, we propose a simple but effective scheme to generate this state. The scheme sketch is shown in Fig. 4, to achieve our goals we complete this process in four low-Q cavities. Suppose the initial state of the logic qubit is

|ϕ〉0 = ⊗i4=1

1 ˜ (|0〉 + |1˜ 〉)i , 2

(25)

Then, with the help of PCG1, which will entangle logic qubits 1 and 2 together, if n¯ 1 = 0, the state will evolve to

|ϕ〉1 =

1 ˜ ˜ 1 ˜ 1 ˜ (|0〉1|0〉2 + |1˜ 〉1|1˜ 〉2 ) ⊗ (|0〉 + |1˜ 〉)3 ⊗ (|0〉 + |1˜ 〉)4 2 2 2 (26)

,

Conversely, if n¯ 1 ≠ 0, the state of the system could also be changed to |ϕ〉1 after performing σ˜x operation on logic qubit 2. Here σ˜xm is the bit flip operation on logic qubit m, |0˜ 〉m → |1˜ 〉m and |1˜ 〉m → |0˜ 〉m , 1 2 ⊗ σxm That is σ˜xm = σxm . Same to the role of PCG1, after running PCG2 and performing unitary operations, the state |ϕ〉1 will evolve to

|ϕ〉2 =

1 1 ˜ (|0˜ 〉1|0˜ 〉2 |0˜ 〉3 + |1˜ 〉1|1˜ 〉2 |1˜ 〉3 ) ⊗ (|0〉 + |1˜ 〉)4 , 2 2 2

(27)

Subsequently, we exploit unitary operation U on logic qubits 1, 2, and 3, and keep PCG3 working. n¯ 3 = 0 will evolve |ϕ〉2 to the |χ 〉 state as shown in Eq. (24). While for the case n¯ 3 ≠ 0, the corresponding output state will be transformed to the |χ 〉 state by performing σ˜x operation on logic qubit 4.

(23)

That is, the CPF gate in the DFS is obtained. Note that only one case is considered in this section to simplify the contents. While, for the other cases, the CPF gate could also be realized through the different unitary operations, which is corresponding to the values of n¯ 1, n¯ 2, and single logic qubit measurement result. The detailed process is listed in Table 1.

5. Four-logic-qubit |χ 〉 state preparation based on the PCG in a DFS In this section, we will turn our attention to generate a specific multipartite entangled state, four-logic-qubit |χ 〉 state in the DFS with the form

Fig. 3. Setup sketch for deterministic CPF gate. A, C and T are auxiliary qubit, controlled qubit and target qubit, respectively. σ˜i , σ˜ j are unitary operations. H˜ denotes Hadamard operation. M means single logic qubit measurement.

6. Multipartite entanglement purification based on the PCG in a DFS Now, let us detail how this entanglement purification scheme works for multi-logic-qubit system. We first take three-logic-qubit Greenberger–Horne–Zeilinger (GHZ) state as an example to show the principle of purification scheme and then extend it to n-logicqubit system. A three-logic-qubit entangled GHZ state can be described as

|ψ +〉ABC =

1 ˜˜˜ ˜ ˜ ˜ 〉) ABC , (|000〉 + |111 2

(28)

Here, the subscripts A, B, and C represent the three logic qubits belonging to the three parties Alice, Bob, and Charlie, respectively. The bit-flip channel noise and phase-flip channel noise will degrade the state and change the pure state into the other forms which can be written as

|ψ −〉ABC =

1 ˜˜˜ ˜ ˜ ˜ 〉) ABC , (|000〉 − |111 2

|ψ1±〉ABC =

1 ˜˜˜ ˜ ˜ ˜ 〉) ABC , (|100〉 ± |011 2

|ψ2±〉ABC =

1 ˜˜ ˜ ˜ ˜ ˜ 〉) ABC , (|010〉 ± |101 2

1 ˜˜˜ ˜ ˜ ˜ 〉) ABC . (|001〉 ± |110 (29) 2 Here we only discuss the purification of bit-flip errors, because |ψ3±〉ABC =

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

X.-P. Zhou et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

5

Table 1

˜ ˜ 〉CT + c0 t1|01 ˜ ˜ 〉CT + c1t0 |10 ˜ ˜ 〉CT + c1t1|11 ˜ ˜ 〉CT , if the Results of PCG, output states and corresponding operations to achieve CPF gate. The initial state of logic qubit C and T is c0 t0 |00 ˜ ˜ 〉CT + c0 t1|01 ˜ ˜ 〉CT + c1t0 |10 ˜ ˜ 〉CT − c1t1|11 ˜ ˜ 〉CT , the CPF gate is obtained. output state is c0 t0 |00 Measurement results

Output states

Operations

|0˜ 〉A |1˜ 〉A

˜ ˜ 〉CT + c0 t1|01 ˜ ˜ 〉CT + c1t0 |10 ˜ ˜ 〉CT − c1t1|11 ˜ ˜ 〉CT c0 t0 |00 ˜ ˜ 〉CT − c0 t1|01 ˜ ˜ 〉CT + c1t0 |10 ˜ ˜ 〉CT + c1t1|11 ˜ ˜ 〉CT c0 t0 |00

σ˜ zT

|0˜ 〉A |1˜ 〉A

˜ ˜ 〉CT + c0 t1|00 ˜ ˜ 〉CT + c1t0 |11 ˜ ˜ 〉CT − c1t1|10 ˜ ˜ 〉CT c0 t0 |01

σ˜ xT

˜ ˜ 〉CT − c0 t1|00 ˜ ˜ 〉CT + c1t0 |11 ˜ ˜ 〉CT + c1t1|10 ˜ ˜ 〉CT c0 t0 |01

σ˜ xT ⊗ σ˜ zT

˜ ˜ 〉CT − c0 t1|01 ˜ ˜ 〉CT + c1t0 |10 ˜ ˜ 〉CT + c1t1|11 ˜ ˜ 〉CT c0 t0 |00

σ˜ zT I

n¯ 1 = 0

n¯ 2 = 0

n¯ 1 = 0

n¯ 2 = 0

n¯ 1 = 0

n¯ 2 ≠ 0

n¯ 1 = 0

n¯ 2 ≠ 0

n¯ 1 ≠ 0

n¯ 2 = 0

n¯ 1 ≠ 0

n¯ 2 = 0

|0˜ 〉A |1˜ 〉A

n¯ 1 ≠ 0

n¯ 2 ≠ 0

|0˜ 〉A

˜ ˜ 〉CT + c0 t1|01 ˜ ˜ 〉CT + c1t0 |10 ˜ ˜ 〉CT − c1t1|11 ˜ ˜ 〉CT c0 t0 |00 ˜ ˜ 〉CT − c0 t1|00 ˜ ˜ 〉CT + c1t0 |11 ˜ ˜ 〉CT + c1t1|10 ˜ ˜ 〉CT c0 t0 |01

n¯ 1 ≠ 0

n¯ 2 ≠ 0

|1˜ 〉A

˜ ˜ 〉CT + c0 t1|00 ˜ ˜ 〉CT + c1t0 |11 ˜ ˜ 〉CT − c1t1|10 ˜ ˜ 〉CT c0 t0 |01

DFS of Hilbert space is used to protect quantum information against the detrimental effects from dephasing noise. In detail, logic qubits |0˜ 〉 = |0〉1|1〉2 and |1˜ 〉 = |1〉1|0〉2 are immune from the dominant source of collective dephasing caused by ambient magnetic field fluctuation [42], so the errors of phase-flip can be avoided. Suppose that the initial pure state ensemble ρ = |ψ +〉〈ψ +| is suffering by the bit flip on the i-th logic qubit in the noisy transmission channel, then the final state ensemble can be written as

ρ′ = F0 |ψ +〉〈ψ +| + F1|ψ1+〉〈ψ1+| + F2 |ψ2+〉〈ψ2+| + F3 |ψ3+〉〈ψ3+| ,

1

σ~1

(30)

here F0 is the probability of the state |ψ +〉 after the transmitting in the noisy channel. Fi (i = 1, 2, 3) denotes the probability that the ith logic qubit in GHZ state takes place a bit-flip error. They satisfy the relation F0 þ F1 þ F2 þ F3 ¼1. The schematic diagram of this three-logic-qubit entanglement purification for bit-flip errors is described in Fig. 5. During the GHZ state purification process, Alice, Bob and Charlie should share two three-logic-qubit |GHZ〉A1B1C1 and |GHZ〉A2 B2 C2 , Alice possesses the logic qubits A1 and A 2, Bob possesses the logic qubits B1 and B2, and Charlie possesses the logic qubits C1 and C2. After the parity-check measurements, Alice, Bob, and Charlie communicate their outcomes. According to the outcomes, the odd parity states will be discarded, the even parity will be preserved. The three parties Alice, Bob, and Charlie pick up the states |ψ +〉|ψ +〉, |ψ1+〉|ψ1+〉, |ψ2+〉|ψ2+〉, |ψ3+〉|ψ3+〉 with the probabilities

1

(31)

σ˜ xT

PCG2

U

σ~ 2

1

U 4

of F02, F12 , F22 , F32. The state of the complicated system composed of the six logic qubits A1 B1C1A2 B2 C2 becomes a mixed one ρeven (without normalization),

ρeven = 2 (F02 |ϕ +〉〈ϕ +| + F12 |ϕ1+〉〈ϕ1+| + F22 |ϕ2+〉〈ϕ2+| + F32 |ϕ3+〉〈ϕ3+|).

σ˜ xT ⊗ σ˜ zT

3

PCG1

2

I

PCG3

U

2

σ~3

4

3 Fig. 4. The setup sketch for four-logic-qubit |χ 〉 state generation. si is dependent on the result of the PCGi, σi = I for n¯i = 0 and σi = σ˜x for n¯i ≠ 0 . U = σ˜x ⊗ H˜ is the unitary operation (|0˜ 〉 → (|0˜ 〉 − |1˜ 〉)/ 2 ; |1˜ 〉 → (|0˜ 〉 + |1˜ 〉)/ 2 ).

where

|ϕ0+〉 =

1 ˜˜˜˜˜˜ ˜ ˜ ˜ ˜ ˜ ˜ 〉) A1B1C1A 2B2C2 . (|000000〉 + |111111 2

(32)

qubits are in the GHZ state |ψ +〉, |ψ1+〉, |ψ2+〉, |ψ3+〉 with the prob-

1 ˜ ˜˜˜ ˜˜ ˜ ˜ ˜ ˜ ˜ ˜ 〉) A B C A B C . = (|100100〉 + |011011 1 1 1 2 2 2 2

(33)

|ϕ2+〉 =

1 ˜˜ ˜˜˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ 〉) A1B1C1A 2B2C2 . (|010010〉 + |101101 2

(34)

|ϕ3+〉 =

1 ˜˜˜ ˜˜˜ ˜ ˜ ˜ ˜ ˜ ˜ 〉) A1B1C1A 2B2C2 . (|001001〉 + |110110 2

(35)

|ϕ1+〉

˜ ˜ ˜ 〉A B C , |011 ˜ ˜ ˜ 〉A B C , |101 ˜ ˜ ˜ 〉A B C or If they obtain the results |000 2 2 2 2 2 2 2 2 2 ˜ ˜ ˜ |110〉A2B2C2, they would confirm that the remaining three logic abilities of F02/2, F12/2, F22/2, F32/2, respectively. On the contrary, if ˜ ˜ ˜ 〉A B C , |010 ˜ ˜ ˜ 〉A B C , |100 ˜ ˜ ˜ 〉A B C or they obtain the results |001 2 2 2 2 2 2 2 2 2 ˜ ˜ ˜ |111〉A B C , the three participants need to flip the relative phase of 2 2 2

the logic qubit system A1, B1, C1 to obtain the pure state. After the above purification process, Alice, Bob, and Charlie will obtain a new ensemble ρ″:

We need only discuss the case that the system is in the state |ϕi 〉 with the probabilities Fi below. In the following, the three parties perform the Hadamard (H˜ ) operations on the logic qubits A 2B2 C2 by an external classical field. After the H˜ operations, Alice, Bob, and Charlie measure their logic qubits A 2, B2 , C2 with the basis {|0˜ 〉, |1˜ 〉}.

ρ″ = F0′ |ψ +〉〈ψ +| + F1′ |ψ1+〉〈ψ1+| + F2′ |ψ2+〉〈ψ2+| + F3′ |ψ3+〉〈ψ3+| ,

(36)

where

Fi′ =

Fi2 F02 + F12 + F22 + F32

(i = 0, 1, 2, 3)

(37)

The fidelity of the new ensemble F0′ > F0 when the initial

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

X.-P. Zhou et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

P

Alice

D

~ H

A1

C

B1 A2

B2

P C G

G

C1

~ H

0.9

Bob

0.8

D

C2

0.8

P C G

0.6

0.6 0.5

0.4 0.2

~ H

Charlie

0.7

1

Psuc

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

0.4

0 0

1

D

1 2

Fig. 5. Setup sketch for the principle of multipartite entanglement purification. PCG denotes the parity check gate. H˜ indicates a Hadamard operation on the logic qubit. D represents the single-logic-qubit measurements with the basis {|0˜ 〉, |1˜ 〉} done by Alice, Bob, Charlie.

1 [3 − 2F1 − 2F2 − 4

1 + 4 (F1 + F2 ) − 12 (F12 + F22 ) − 8F1F2 ].

(38)

If we take the case F1 ¼F2 ¼ F3 (a symmetric noise mode), the fidelity of the state |ψ +〉 will be improved by our EPP just when F0 > 1/4 . By far, we have discussed the principle of the entanglement purification for three-logic-qubit entangled system. This method can also be generalized to multipartite entangled system. We assume that the initial state of a n-logic-qubit system can be described as

|ψ +〉N =

1 ˜˜ ˜ ˜ ˜ ⋯1˜ 〉). (|00⋯0〉 + |11 2

α

0.6 3

0.4 4

0.2 5

ξ

0.2 0.1

0

Fig. 6. Success probability as function of α and the efficiency ξ of photon detector. here, we have assumed that the photon loss rate η = 1/3 for the sake of discussion.

fidelity F0 satisfies the relation

F0 >

0.3

0.8

(39)

And the entanglement purification process of bit-flip errors can also use the same method mentioned above. After performing all operations, the N parties will obtain a new ensemble in which the ′ which is larger than FN0 when fidelity of the initial state |ψ +〉N is F N0 FN 0 > 1/2N − 1. Based on our PCG, through repetition the purification process the remaining entangled logic qubits can be expected to obtain a higher fidelity.

7. Discussion and conclusion Up to now, we have described our proposed relevant quantum communication and computation schemes in detail in an ideal case. In the following, we will give a brief discussion about the success probability, fidelity under ideal condition, and experimental feasibility of our protocol. The basic element of this paper is PCG composed by two NV centers fixed on the surface of an MTR with a quantized WGM. Here, we use a photon detector that only distinguish the vacuum and the non-vacuum state instead of the homodyne measurement to distinguish outgoing coherent states |0〉 and | 2 α〉. That is to say, the measurement itself and the efficiency ξ of detector will reduce the success probability of our schemes. Because we just need the photon detector to determine whether there is photon, the efficiency ξ have a little effect on our scheme. Moreover, the photon loss rate ηwhich is mainly caused by the fiber absorption can also reduce the success probability. After considering these factors, error rate can be expressed as Pe = exp [ − 2ξ (1 − η) α 2] [53]. The success probability is show in Fig. 6, which shows the PCG could work in a higher probability after considering the above factors. In fact, the PCG is imperfect due to real noise in controlling the reflectivity of the pulse, though it has a high success probability. The state before measurement should be

|Ψ 〉ideal = while

1 ˜ ˜〉 [(|01 2

in

˜ ˜ 〉)ab |α〉1 + |00 ˜ ˜ 〉ab |αe−iπ 〉1 + |11 ˜ ˜ 〉ab |αeiπ 〉1]|α〉2, + |10 1 ˜ ˜ 〉 + |10 ˜ ˜ 〉)ab |r0 r2 α〉1+ practice, the state |Ψ 〉real = [(|01 2

˜ ˜ 〉ab |r 2 α〉1 + |11 ˜ ˜ 〉ab |r 2 α〉1]|α〉2 . Hence we can calculate the fidelity |00 2 0 F = |real 〈Ψ |Ψ 〉ideal |2 to characterize the performance of the PCG. The numerical simulation of the fidelity is shown in Fig. 7(a) and (b). Here are to be highlighted when the coherent light increases, the need to cavity damping rate κ is large enough to satisfy the weak excitation condition 〈σz 〉 ≈ − 1. For the implementation of our schemes, there will be some requirements for NV center and MTR. First of all, the detunings of coherent light pulse with respect to NV center and cavity mode are required to be same, which could be satisfied by changing the input coherent light pulse frequency ωp if the NV center and cavity mode have a resonant interaction (ω0 = ωc ). On the other hand, the coupling of the NV center and MTR is also a key ingredient in our scheme. For more general cases, we plot the phase shifts ϕ2 and ϕ0 versus the values of (ωc − ω)/κ and g/κ in Fig. 7(c) and (d) respectively, which illustrates that in the case of moderate coupling with g = κ /2, the expected phase shifts can be obtained when ω0 = ωc = ω + κ /2 is satisfied. As discussed in Refs. [54,55], in realistic experiments, some detrimental effects will weaken the coupling rate within the narrow band zero photon line (ZPL), so stronger field-matter coupling strength between the WGM and NV centers is required. Consider the quality factor Q determined by Q = c/ λκ with the speed of light c and the transition wavelength λ ≃ 637 nm , the quality factor Q of the MTR approaches to 105 (corresponding to κ ∼ 1 GHz ) or 104 (corresponding to κ ∼ 10 GHz ). It means that the reflection coefficient r (ωp ) approaches to unit when the coupling strength can be on the order of hundreds of megahertz [56]. Since low-Q is sensitive to the number of the NV centers confined in the cavity, we must seriously control the number of the coupled NV centers, and the NV centers should be close to the surface of diamond in order to facilitate manipulating and extending the MTR-NV system [57]. As discussed in Ref. [45], even if there exits deviation between the coupling rates of two NV centers, it is a small impact for our schemes. Similar to the schemes proposed in Refs. [8,10,39–42], the current one requires the two physical qubits in one cavity influenced by the same noise. However, the encoding method is also feasible when more than one cavity is concerned even if the noises in different cavities are different. We now take two cavities as an example, suppose the physical qubits in cavity 1 are subject to the phase noise |0〉 → eiψ0 |0〉 and |1〉 → eiψ1|1〉, while the physical qubits

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

X.-P. Zhou et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 1

1

0.95

0.9 1 0.8 F

0.8

1

0.7

0.9

0.9 0.85 0.8

0.8

0.6

0.6

0.4

0.5

0.7

0.75

0.4

0.6

0.7

0.2 0 0

0.3 1

15 α

2

10 3

5 4 0

F

0.5 0

0.65 1

0.2 0.1

2

10 3

5 4 0

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −6

0.6

15 α

κ/γ

Phase/π

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

7

−4

−2

0

0.55

κ/γ

2

4

6

(ωc−ω)/κ

1

Fig. 7. (a) Fidelity of the PCG as function of α and κ /γ with η = when n¯ ≠ 0 . (b) Fidelity of the PCG as function of α and κ /γ with η = 3 versus (ωc − ω)/κ and g/κ with γ = 0.01κ . (d) The phase shift ϕ0 as function of (ωc − ω)/κ with γ = 0.01κ and g being arbitrary values.

in cavity 2 are subjected to the phase noise |0〉 → eiψ0′ |0〉 and

|1〉 → eiψ1′|1〉. Thus, under the influence of the phase noise, the transformations of a random states of cavities 1 and 2 could be written as

a|0˜ 〉1|0˜ 〉2 + b|0˜ 〉1|1˜ 〉2 + c|1˜ 〉1|0˜ 〉2 + d|1˜ 〉1|1˜ 〉2 = a|01〉1|01〉2 + b|01〉1|10〉2 + c|10〉1|01〉2 + d|10〉1|10〉2 noise

→ a (eiψ 0 + iψ1 |01〉1 ⊗ eiψ 0′+ iψ1′ |01〉2 ) + b (eiψ 0 + iψ1 |01〉1 ⊗ eiψ1′+ iψ 0′ |10〉2 ) + c (eiψ1+ iψ 0 |10〉1 ⊗ eiψ 0′+ iψ1′ |01〉2 ) + d (eiψ1+ iψ 0 |10〉1 ⊗ eiψ1′+ iψ 0′ |10〉2 )

= ei (ψ 0 + ψ1+ ψ 0′+ ψ1′) (a|01〉1|01〉2 + b|01〉1|10〉2 + c|10〉1|01〉2 + d |10〉1|10〉2 ) = ei (ψ 0 + ψ1+ ψ 0′+ ψ1′) (a|0˜ 〉1|0˜ 〉2 + b|0˜ 〉1|1˜ 〉2 + c|1˜ 〉1|0˜ 〉2 + d|1˜ 〉1|1˜ 〉2 ).

(40)

where (|a|2 + |b|2 + |c|2 + |d|2 = 1). One could see from Eq. (40) that the scheme is also feasible when two cavities are concerned even if the noises in different cavities are different since the state is unchanged if we ignore the global phase. Another thing should be noted here is the range of values of α. The success probability of the scheme relies on the distinction between a coherent state and vacuum state. Thus, to achieve a high success probability, the strength of the coherent state should not be too low. On the other hand, the input–output process relies on the condition 〈σz 〉 ≈ − 1, which means that the strength of the coherent pulse should not be too large or the cavity decay rate κ should not be too small. As pointed out in the scheme in Ref. [50] for coherent pulse with larger strength, the cavity decay rate κ should be also sufficiently large to make the atom have a weak excitation. In Ref. [50], to have a better distribution for the final output coherent states, α ¼3 and α ¼5 are discussed, respectively. For our scheme, Fig. 5 indicates that α should be higher than 3 to have a better success probability (better distribution between coherent state and vacuum state). On

1 3

when n¯ = 0 . (c) The value of ϕ2/π

the other hand, Fig. 7(a) and (b) shows that κ/γ should be higher than 10 to have a better fidelity when 3 < α < 4 . Thus, based on above analysis, we estimate that 3 < α < 4 and κ /γ ≥ 10 should be satisfied for our scheme. Before ending this paper, we would like to discuss the differences of the present work from the relevant works. Compared with Refs. [58,59] working in strong atom-cavity coupling, our scheme focuses on the intermediate coupling of the low-Q WGM cavities and moderate coupling between the NV centers and the cavities. Different from Ref. [45], the demanding single-photon resource and single-photon detector have been successfully avoided in our scheme, and our scheme is more efficient because the homodyne measurement has been replaced by vacuum or nonvacuum photon state measurement. Besides, we encode the quantum information in DFS, which makes our schemes robust to collective dephasing errors, so one of the most important advantage in the process of entanglement purification is that we only need to consider bit-flip errors. And we have achieved the quantum logic CPF gate and |χ 〉 state generation. In principle, our schemes are scalable and could be accomplished in cryogenic environment with current experiment techniques. In summary, with the input–output relation in a composite system consisting of two NV centers fixed on the surface of a MTR, and a coherent optical pulse as a quantum channel, we first proposed a protocol to construct a PCG with a higher probability and fidelity. Quantum CPF gate, |χ 〉 state generation and entanglement purification were realized by using the proposed PCG. We analysis the feasible of the scheme in detail. With significant advancement techniques in the fabrication of qualified WGM cavities, our protocol is feasible for large-scale quantum information processing and fits the experimental condition very well.

Acknowledgments This work is supported by the National Natural Science

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

X.-P. Zhou et al. / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Foundation of China under Grant nos. 61465013, 11465020, and 11264042; the Program for Chun Miao Excellent Talents of Jilin Provincial Department of Education under Grant no. 201316; and the Talent Program of Yanbian University of China under Grant no. 950010001.

References [1] J.I. Cirac, P. Zoller, H.J. Kimble, H. Mabuchi, Phys. Rev. Lett. 78 (1997) 3221; J.I. Cirac, A.K. Ekert, S.F. Huelga, C. Macchiavello, Phys. Rev. A 59 (1999) 4249. [2] Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, H.J. Kimble, Phys. Rev. Lett. 75 (1995) 4710. [3] M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 77 (1996) 4887; K. Mattle, H. Weinfurter, P.G. Kwiat, A. Zeilinger, Phys. Rev. Lett. 76 (1996) 4656. [4] P. Xue, Y.F. Xiao, Phys. Rev. Lett. 97 (2006) 140501. [5] L.-M. Duan, H.J. Kimble, Phys. Rev. Lett. 92 (2004) 127902; Y.F. Xiao, X.M. Lin, J. Gao, Y. Yang, Z.F. Han, G.C. Guo, Phys. Rev. A 70 (2004) 042314; J. Cho, H.W. Lee, Phys. Rev. Lett. 95 (2005) 160501; L.-M. Duan, B. Wang, H.J. Kimble, Phys. Rev. A 72 (2005) 032333. [6] X.-M. Lin, Z.-W. Zhou, M.-Y. Ye, Y.-F. Xiao, G.-C. Guo, Phys. Rev. A 73 (2006) 012323. [7] H. Goto, K. Ichimura, Phys. Rev. A 72 (2005) 054301. [8] Z.J. Deng, M. Feng, K.L. Gao, Phys. Rev. A 75 (2007) 024302. [9] Z.J. Deng, X.L. Zhang, H. Wei, K.L. Gao, M. Feng, Phys. Rev. A 76 (2007) 044305. [10] H. Wei, Z.J. Deng, X.L. Zhang, M. Feng, Phys. Rev. A 76 (2007) 054304; H. Wei, W.L. Yang, Z.J. Deng, M. Feng, Phys. Rev. A 78 (2008) 014304. [11] A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 83 (1999) 5166; S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 87 (2001) 037902. [12] J. Beugnon, M.P.A. Jones, J. Dingjan, B. Darqui, G. Messin, A. Browaeys, P. Grangier, Nature (London) 440 (2006) 779. [13] Y.-A. Chen, S. Chen, Z.-S. Yuan, B. Zhao, C.-S. Chuu, J. Schmiedmayer, J.-W. Pan, Nat. Phys. 4 (2008) 103. [14] F. Jelezko, T. Gaebel, I. Popa, A. Gruber, J. Wrachtrup, Phys. Rev. Lett. 92 (2004) 076401. [15] W.L. Yang, Z.Y. Xu, M. Feng, J.F. Du, New J. Phys. 12 (2010) 113039. [16] E. Togan, Y. Chu, A.S. Trifonov, L. Jiang, J. Maze, L. Childress, M.VG. Dutt, A. S. Sørensen, P.R. Hemmer, A.S. Zibrov, M.D. Lukin, Nature (London) 466 (2010) 730. [17] Q. Chen, W.L. Yang, M. Fang, J.F. Du, Phys. Rev. A 83 (2011) 054305. [18] P.B. Li, S.Y. Gao, F.L. Li, Phys. Rev. A 83 (2011) 054306. [19] L.Y. Cheng, H.F. Wang, S. Zhang, Y.H. Yeon, Opt. Express 21 (2013) 5988–5997. [20] C. Wang, Y. Zhang, G.S. Jin, Phys. Rev. A 84 (2011) 032307. [21] A. Steane, Proc. R. Soc. A 452 (1996) 2551.

[22] A.R. Calderbank, P.W. Shor, Phys. Rev. A 54 (1996) 1098. [23] J. Chiaverini, D. Leibfried, T. Schaetz, M.D. Barrett, R.B. Blakestad, J. Britton, W. M. Itano, J.D. Jost, E. Knill, C. Langer, R. Ozeri, D.J. Wineland, Nature (London) 432 (2004) 602. [24] L.M. Duan, J.I. Cirac, P. Zoller, Science 10 (2001) 292 1695. [25] J.A. Jones, V. Vedral, A. Ekert, G. Castagnoli, Nature (London) 403 (2000) 869. [26] S.L. Zhu, Z.D. Wang, Phys. Rev. Lett. 91 (2003) 187902. [27] X.L. Feng, Z. Wang, C. Wu, L.C. Kwek, C.H. Lai, C.H. Oh, Phys. Rev. A 75 (2007) 052312. [28] M.J. Kastoryano, F. Reiter, A.S. Sørensen, Phys. Rev. Lett. 106 (2011) 090502. [29] D.D. Bhaktavatsala Rao, K. Mølmer, Phys. Rev. Lett. 111 (2013) 033606. [30] A.W. Carr, M. Saffman, Phys. Rev. Lett. 111 (2013) 033607. [31] L.T. Shen, X.Y. Chen, Z.B. Yang, H.Z. Wu, S.B. Zheng, Europhys. Lett. 99 (2012) 20003. [32] J. Song, Y. Xia, X.D. Sun, H.S. Song, Phys. Rev. A 86 (2012) 034303. [33] X.Q. Shao, J.B. You, T.Y. Zheng, C.H. Oh, S. Zhang, Phys. Rev. A 89 (2014) 052313. [34] S.L. Su, X.Q. Shao, H.F. Wang, S. Zhang, Phys. Rev. A 90 (2014) 054302. [35] S.L. Su, X.Q. Shao, H.F. Wang, S. Zhang, Sci. Rep. 4 (2014) 7566. [36] C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W. K. Wootters, Phys. Rev. Lett. 76 (1996) 722. [37] J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, A. Zeilinger, Nature (London) 423 (2003) 417. [38] Y.-B. Sheng, F.-G. Deng, H.-Y. Zhou, Phys. Rev. A 77 (2008) 042308. [39] L.-M. Duan, G.C. Guo, Phys. Rev. Lett. 79 (1997) 1953. [40] P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79 (1997) 3306. [41] D.A. Lidar, I.L. Chuang, K.B. Whaley, Phys. Rev. Lett. 81 (1998) 2594. [42] M. Feng, Phys. Rev. A 63 (2001) 052308. [43] D. Kielpinski, C. Monroe, D.J. Wineland, Nature (London) 417 (2002) 709. [44] Q. Chen, M. Feng, Phys. Rev. A 82 (2010) 052329. [45] A.P. Liu, L.Y. Cheng, L. Chen, S.L. Su, H.F. Wang, S. Zhang, Opt. Commun. 313 (2014) 180–185. [46] N.B. Manson, J.P. Harrison, M.J. Sellars, Phys. Rev. B 74 (2006) 104303. [47] D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin, 1994. [48] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, WK. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [49] F. Mei, Y.F. Yu, X.L. Feng, S.L. Zhu, Z.M. Zhang, Eur. Phys. Lett. 91 (2010) 10001. [50] F. Mei, Y.F. Yu, X.L. Feng, Z.M. Zhang, C.H. Oh, Phys. Rev. A 82 (2010) 052315. [51] C.W. Gardiner, Quantum Noise, Springer-Verlag, Berlin, 1991. [52] H. Wei, Z.J. Deng, X.L. Zhang, M. Feng, Phys. Rev. A 76 (2007) 054304. [53] D. Gottesman, I.L. Chuang, Nature (London) 402 (1999) 390–393. [54] X.W. Wang, D.Y. Zhang, S.Q. Tang, L.J. Xie, Z.Y. Wang, LM. Kuang, Phys. Rev. A 85 (2012) 052326. [55] P.E. Barclay, C. Santori, K.M. Fu, R.G. Beausoleil, O. Painter, Opt. Express 17 (2009) 8081. [56] J. Wolters, A.W. Schell, G. Kewes, N. Nüsse, M. Schoengen, H. Döscher, T. Hannappel, B. Löchel, M. Barth, O. Benson, Appl. Phys. Lett. 97 (2010) 141108. [57] C. Santori, P.E. Barclay, K.M.C. Fu, RG. Beausoleil, Phys. Rev. B 79 (2009) 125313. [58] L.M. Duan, H.J. Kimble, Phys. Rev. Lett. 92 (2004) 127902. [59] J. Cho, H.W. Lee, Phys. Rev. Lett. 95 (2005) 160501.

Please cite this article as: X.-P. Zhou, et al., Optics Communications (2015), http://dx.doi.org/10.1016/j.optcom.2015.04.072i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132