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Efﬁcient single-photon-assisted entanglement concentration for an arbitrary entangled photon pair with the diamond nitrogen-vacancy center insides cavity Lin-Lin Fan, Yan Xia n Department of Physics, Fuzhou University, Fuzhou 350002, China

art ic l e i nf o

a b s t r a c t

Article history: Received 12 March 2014 Received in revised form 24 September 2014 Accepted 25 September 2014 Available online 30 October 2014

In this paper, a protocol for single-photon-assisted entanglement concentration is proposed. Resorting to the nonlinear optics of a nitrogen-vacancy (NV) center in a diamond embedded in a photonic crystal cavity coupled to a waveguide, remote parties can share the maximally entangled photon pair with a certain probability. Compared with other entanglement concentration protocols (ECPs), the current one does not need to know the accurate coefﬁcients of the initial state and can be repeated to get a higher success probability. Meanwhile, this protocol is more suitable for multiphoton system concentration. All these advantages make the protocol useful in long-distance quantum communication. & Elsevier B.V. All rights reserved.

Keywords: Entanglement concentration Nitrogen-vacancy center Bell state

1. Introduction Entanglement plays a crucial role in quantum information processing [1–10]. In order to optimally complete the quantum information processing, maximally entangled states are usually required. The entanglement, to our knowledge, is ﬁrst produced locally and distributed to different distant locations. Therefore, the particles will inevitably contact with the environment during the transmission, and the channel noise will degrade the maximally entanglement into a less-entangled pure state or even make the entanglement into a mixed state, which will decrease the ﬁdelity and the security of long-distance quantum communication. For example, the noise will make the ﬁdelity of quantum teleportation degrade, quantum dense coding fail, and the quantum cryptography protocol insecure, etc. To overcome this ﬂaw, one can exploit entanglement puriﬁcation [11–17] or entanglement concentration [18–33,40] to improve the entanglement of the quantum system ﬁrst, and then achieve the goals of the above applications with the maximally entangled states. The methods of converting an ensemble of less-entangled mixed state into a maximally entangled state is called entanglement puriﬁcation [11–17]. Since Bennett et al. proposed the ﬁrst entanglement puriﬁcation protocol in 1996 [11], the entanglement puriﬁcation has been studied widely [12–17]. Another way to convert the partially entangled state into the maximally entangled state is called entanglement concentration [18–33,40], which n

Corresponding author. E-mail address: [email protected] (Y. Xia).

http://dx.doi.org/10.1016/j.optcom.2014.09.085 0030-4018/& Elsevier B.V. All rights reserved.

operates in an ensemble of less-entangled pure state. Bennett et al. ﬁrstly proposed an entanglement concentration protocol (ECP) in 1996 [18], which is the so-called Schmidt projection protocol. In their protocol, they used collective measurements which are difﬁcult to manipulate experimentally. Bose et al. also proposed an ECP based on entanglement swapping [19], which needs to know the coefﬁcients of the entangled states. Ren et al. investigated the possibility of concentrating the two-photon fourqubit systems in partially hyperentangled states in both the spatial mode and the polarization DOFs with linear optics [29]. In this paper, they ﬁrst introduced the parameter-splitting method to concentrate the systems in the partially hyperentangled states with known parameters, including partially hyperentangled Bell states and cluster states. Then, they presented another two nonlocal hyperentanglement concentration protocols (hyper-ECPs) for the systems in partially hyperentangled unknown states, resorting to the Schmidt projection method. The results show that the parameter-splitting method is very efﬁcient for the concentration of the quantum systems in partially entangled states with known parameters, resorting to linear-optical elements only. Till now, many interesting ECPs have been proposed [18–33,40]. And among these protocols, the methods of introducing ancillary photons to complete the entanglement concentration have been proposed in Refs. [26–28,31–33]. For instance, Sheng et al. presented an ECP to concentrate partially entangled photon pair to a maximally entangled pair with only one ancillary single photon [26]. They also presented a two-step practical entanglement concentration protocol for concentrating an arbitrary three-particle less-entangled W state into a maximally entangled W state

L.-L. Fan, Y. Xia / Optics Communications 338 (2015) 174–180

assisted with two single photons [28]. Almost at the same time, Deng proposed an optimally nonlocal ECP for multiphoton systems in a partially entangled pure state by resorting to the projection measurement on an additional photon [27]. Du et al. [31] and Gu et al. [32] proposed two different ECPs for the special W state resorted to ancillary photon. Zhou et al. [33] proposed an efﬁcient ECP for arbitrary less-entangled NOON state assisted with a single photon. Most of the above ECPs are accomplished with the help of the cross-Kerr nonlinearity. Although a lot of works have been studied on cross-Kerr nonlinearity, a clean cross-Kerr nonlinearity in the optical single-photon regime is still quite a controversial assumption, especially the strong cross-Kerr nonlinearity, which is still a big challenge in the experiment at present. The nitrogen vacancy (NV) center is an attractive spin qubit since it exhibits a unique combination of robust room-temperature spin coherence and efﬁcient optical addressability, controllability, and readout [34,35]. Therefore, the NV magnetometry is very potential for future practical applications. Till now, many theoretical and experimental protocols have been proposed on quantum information processing based on the NV center. For example, Togan et al. [36] proposed a protocol for the quantum entangled state generation between a photon and an NV center. Yang et al. [37] proposed a protocol to generate quantum entangled state between electrons associated with the NV centers. Ren et al. [38] proposed a protocol to implement the hyperentanglement puriﬁcation of two-photon systems in nonlocal hyperentangled Bell states with the help of the diamond NV centers inside photonic crystal cavities. Wei et al. [39] proposed some compact quantum circuits for a deterministic solid-state quantum computing, including the CNOT, Toffoli, and Fredkin gates on the diamond NV centers conﬁned inside cavities, achieved by some input–output processes of a single photon. In Ref. [39], the quantum circuits for these universal quantum gates are simple and economic. Additional electron qubits are not employed, but only a single-photon medium. The authors have discussed the feasibility of these universal solid-state quantum gates, concluding that they are feasible with current technology. In this paper, we propose a single-photon-assisted ECP with the assistance of the NV center. In the protocol, only one lessentangled state and one single photon are required. Due to long electronic spin lifetime, fast initialization, good qubit readout, and coherent manipulation at room temperature, the diamond NV center embedded in a photonic crystal cavity coupled to a waveguide is considered as a promising candidate for constructing the parit-check quantum nondemolition detectors (QNDs). Comparing with conventional ECPs, the single-photon-assisted ECPs are more economical, and with the help of the diamond NV center insides photonic crystal cavity, it can be repeated to get a higher success probability. In addition, we can extend the present ECP to concentrate the arbitrary N-photon Greenberger–Horne–Zeilinger (GHZ) state |Φ〉a1b1⋯ z1 = α|RR⋯R〉a1b1⋯ z1 + β|LL⋯L〉a1b1⋯ z1, the arbitrary spacial W state |Ψ 〉a1b1⋯ z1 = γ |HHV 〉 + δ (|HVH 〉 + |VHH 〉) and the arbitrarily less-entangled standard W state |Ψ 〉a1b1c1 = ζ |RRL〉a1b1c1 + ς|RLR〉a1b1c1 + ξ|LRR〉a1b1c1. These advantages make the protocol more meaningful in practical applications. The paper is organized as follows: In Section 2, we will explain the principle of the single-photon-assisted ECP with the diamond NV center insides cavity, and calculate the success probability. In Section 3, we will extend the protocol to complete the entanglement concentration for the GHZ state and the W-class state. And in Section 4, we will present the discussion and summary.

175

2. Single-photon-assisted entanglement concentration of lessentangled bell-state with the diamond NV center insides cavity To discuss the principle of concentration in detail, we ﬁrst brieﬂy consider the diamond NV center insides cavity. The schematic diagram for an NV center in a diamond embedded in photonic crystal cavity coupled to a waveguide is shown in Fig. 1. The negatively charged NV center consisted of a substitutional nitrogen atom and an adjacent vacancy with six electrons from the nitrogen and three carbons surrounding the vacancy. As described in Refs. [41,42], the |A2 〉 state is robust with the stable symmetries, and decays to the ground state sublevels | − 1〉 and | + 1〉 with radiation of left (L) and right (R) polarizations, respectively. Therefore the zero phonon line (ZPL) is observed after the optical resonant excitation at 637 nm. In our protocol, the Λ-type three-level system is realized by employing one of the speciﬁc excited states |A2 〉 as an ancillary state. Moreover, when a Λ-type three-level diamond NV center is conﬁned into a cavity, the Heisenberg equations of the motion for the annihilation operator of the cavity mode a^ and the lowing operator of the NV center operation σ − and the input–output relation for the cavity can be given by [43]

⎡ η κ⎤ da^ = − ⎢i(ωc − ω) + + ⎥a^ − gσ^− − dt 2 2⎦ ⎣

η a^in,

⎡ γ⎤ dσ^− = − ⎢i(ωe − ω) + ⎥σ^− − gσ^za^, dt 2⎦ ⎣ ^ ^ ^ a = a + η a, out

in

(1)

where ωc, ω, and ωe are the frequencies of the cavity, the single photon, and the NV center, respectively. Ω is the coupling strength between the cavity and the NV center. γ, η, and κ are the decay rates of the NV center, the cavity ﬁeld, and the cavity side leakage mode, respectively. If the NV center is dominantly in the ground state as well as in a weak excitation, i.e., taking 〈σz〉 = − 1, the form of the reﬂection coefﬁcient r(ω) can be described as

⎡ κ ⎤⎡ γ⎤ 2 ⎢⎣i(ωc − ωp) − 2 ⎥⎦⎢⎣i(ω0 − ωp) + 2 ⎥⎦ + Ω , r (ωp) = ⎡ κ ⎤⎡ γ⎤ 2 ⎢⎣i(ωc − ωp) + 2 ⎥⎦⎢⎣i(ω0 − ωp) + 2 ⎥⎦ + Ω

(2)

where ω0 is the transition frequency between energy levels | − 1〉 and |A2 〉. As shown in Ref. [44], Chen et al. showed that when Ω ≥ 5 γκ with ωc = ω0 + ωp , r(ω) ≃ 1 and r0(ω) = − 1. Therefore, the change of the input photon is summarized as [44,45]

| R , − 1〉 → | R , − 1〉 , | R , + 1〉 → − | R , + 1〉 , | L , − 1〉 → − | L , − 1〉 , | L , + 1〉 → | L , + 1〉 .

(3)

And as described in Ref. [38], a polarizing beam splitter in the circular basis (CPBS) and the NV center can be used to construct a

Fig. 1. Schematic diagram of a diamond NV center embedded in a photonic crystal cavity with the circularly polarized photon, and the possible Λ-type optical transition in an NV center.

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L.-L. Fan, Y. Xia / Optics Communications 338 (2015) 174–180

Fig. 2. Schematic diagram of the single-photon-assisted entanglement concentration for an arbitrary entangled photon pair with diamond NV center insides cavity. CPBSi (i = 1, 2, 3, 4 ) represents a polarizing beam splitter in the circular basis, which transmits the photon in the right-circular polarization |R〉 and reﬂects the photon in the leftcircular polarization |L〉. R45 represents a half-wave plate which is used to perform a Hadamard operation on the polarization DOF of the photon. Dj (j¼1,2) is the photon detector.

polarization parity-check quantum nondemolition detector (QND) which is exploited below. Now, we show the basic principle of our concentration protocol, by exploiting a previously known protocol in Ref. [26]. We suppose that the initial two-particle less-entangled state is in the form

|Φ〉a1b1 = α|RR〉a1b1 + β|LL〉a1b1,

(4)

where α and β are two real parameters and satisfy the relation |α|2 + |β|2 = 1. As shown in Fig. 2, the photons 1 and 2 (emitted from source S1) of the less-entangled pair are sent to Alice and Bob, respectively. That is, the photon in the spatial mode a1 is sent to Alice, and that in the spatial mode b1 is sent to Bob. In the meantime, Alice prepares an ancillary photon 3 (emitted from source S2) in the spatial mode a2. The form of the ancillary photon 3 is [26,28]

|Φ〉a2 = α|R〉a2 + β|L〉a2 .

(5)

The initial state of the three photons 1, 2 and 3 can be written as

|Ψ 〉a1b1a2 = |Φ〉a1b1 ⊗ |Φ〉a2 = α 2|RRR〉a1b1a2 + β 2|LLL〉a1b1a2 + αβ(|RRL〉a1b1a2 + |LLR〉a1b1a2 ).

(6)

Alice ﬁrst lets the photons in spatial a1 and a2 to pass through CPBS (CPBS1 and CPBS2), which transmits the photon in the rightcircular polarization R and reﬂects the photon in the left-circular polarization L, respectively. Then the two photons meet NV, CPBS3, and CPBS4, shown in Fig. 2. Suppose that the initial state of NV is

|Φ〉e =

1 2

(| − 1〉 + | + 1〉)e . After interaction, the state of the system

composed of NV and three photons with polarization becomes |Ψ 〉a1b1a2e = |Ψ 〉a1b1a2 ⊗ |Ψ 〉e = (α2|RRR〉a1b1a2 + β 2|LLL〉a1b1a2 ) + αβ (|RRL〉a1b1a2 + |LLR〉a1b1a2 )

1 2

1 2

(| − 1〉 − | + 1〉)e .

(| − 1〉 + | + 1〉)e (7)

By measuring the state of NV in the orthogonal basis {| + 〉e , | − 〉e } (| ± 〉e =

1 2

(| − 1〉 ± | + 1〉)e ), it is evident that the items |RRR〉a1b1a2

and |LLL〉a1b1a2 will lead to the result of measuring the state of NV in | + 〉e . However, the items |RRL〉a1b1a2 and |LLR〉a1b1a2 will lead to the result of measuring the state of NV in | − 〉e . Firstly, we only consider that the initial state of the three photons is projected into

|Ψ 〉′ =

1 (|RRL〉a1b1a2 + |LLR〉a1b1a2 ). 2

(8)

The probability that Alice and Bob get the state in Eq. (8) is 2|αβ|2. In order to generate a maximally entangled Bell state between Alice and Bob, as shown in Fig. 2, Alice lets the photon in spatial mode a2 to pass through a λ /4 − wave plate (HWP45) whose optical axis is set at 22.5° to complete the Hadamard gate on the

polarization photon, it can yield the following transformation:

|R〉a2 →

1 (|R〉a2 + |L〉a2 ), 2

|L〉a2 →

1 (|R〉a2 − |L〉a2 ). 2

(9)

After the Hadamard gate operation, Eq. (8) evolves to

|Ψ 〉″ =

1 [(|RR〉 + |LL〉)a1b1|R〉a2 + (|RR〉 − |LL〉)a1b1|L〉a2 ]. 2

(10)

We easily ﬁnd that if detector D1 ﬁres, we will get

|Ψ +〉m =

1 (|RR〉 + |LL〉)a1b1. 2

(11)

If detector D2 ﬁres, we will get

|Ψ −〉m =

1 (|RR〉 − |LL〉)a1b1. 2

(12)

Both Eqs. (11) and (12) are the maximally entangled states. In order to get the same state |Ψ +〉m , if D2 ﬁres, one of the two parties, Alice or Bob should perform a phase rotation on her or his photon to covert Eq. (12) to Eq. (11). And then we have accomplished the concentration process and distilled the maximally entangled Bell state from the less one with the success probability of P1 = 2|αβ|2. In above description, we only pick up the items which lead to the result of measuring the state of NV in | − 〉e , and it is necessary to discuss the items which are discarded. The neglectful items in Eq. (7) are in the state

|Φ〉′ = α 2|RRR〉a1b1a2 + β 2|LLL〉a1b1a2 ,

(13)

with probability |α|4 + |β|4 . After Alice performing HWP45 in spatial mode a2, just like the Eqs. (9) and (10), if D1 ﬁres, Eq. (13) becomes

|ϕ+〉a1b1 = α 2|RR〉a1b1 + β 2|LL〉a1b1,

(14)

and if D2 ﬁres, Eq. (13) becomes

|ϕ−〉a1b1 = α 2|RR〉a1b1 − β 2|LL〉a1b1.

(15)

One of the two parties, Alice or Bob performs a phase shift which can covert Eq. (15) to Eq. (14). Then we need to only replace α2 and

β2 with α′ = α 2/( α 4 + β 4 ) and β′ = β 2/( α 4 + β 4 ), which make the

state in Eq. (14) have the same form as the state |Φ〉a1b1 shown in Eq. (4). And we can see that this two-photon system is not in a maximally entangled state, but it can be used to get a maximally entangled state with entanglement concentration. That is, Alice only needs to exploit another single-photon state with the form α′|R〉 + β′|L〉 and follows the same method as described in above Eqs. (6)–(12). Alice and Bob can distill the maximally entangled state |Ψ +〉m from Eq. (14) with the success probability of

L.-L. Fan, Y. Xia / Optics Communications 338 (2015) 174–180

177

Fig. 3. The relation between success probability P of getting a maximally entangled Bell state and the coefﬁcient α of the initial less-entangled state. Here we chose n¼ 6 as a approximation.

P2 = 2(|α|4 + |β|4 )|α′β′|2 = 2|αβ|4 /(|α|4 + |β|4 ). Moreover, they also can distill the photon pairs in the maximally entangled state |Ψ +〉m from the less-entangled systems in the next round yet, and the probability of the success is P3 = 2|αβ|8 /(|α|4 + |β|4 )(|α|8 + |β|8). In this way, one can easily obtain the probability of the nth iteration: n

Pn =

2|αβ|2

n

n

(|α|4 + |β|4 )(|α|8 + |β|8)⋯(|α|2 + |β|2 )

, (16)

where n is the iterative times of the entanglement concentration processes. The total success probability of this ECP is P = P1 + P2 + ⋯ + Pn = 2|αβ |2 +

2|αβ |4 |α|4 + |β |4

+

2|αβ |8

+⋯

(|α|4 + |β |4 )(|α|8 + |β |8)

n

+

2|αβ |2

n

n

(|α|4 + |β |4 )(|α|8 + |β |8)⋯(|α|2 + |β |2 )

.

1 2

α when 0 < α < 2 , and decreases with the increase of the coefﬁcients α when 1

increases with the increase of the coefﬁcient

2

|Φ〉a1b1⋯ z1 = α|RR⋯R〉a1b1⋯ z1 + β|LL⋯L〉a1b1⋯ z1,

< α < 1. Moreover, we can easily ﬁnd that when α =

1 2

, the

success probability P reaches up to 1. In other words, the success probability is increasing with the entanglement of the initial lessentangled state. In addition, we also discuss the relationship between success probability and the times for the iteration of the entanglement concentration process in Fig. 4. As shown in Fig. 4, we can ﬁnd that if we choose a stationary initial state and iterate 6 times, the success probability P can reach up to the maximum P → 1. In another word, 6 times for the iteration is enough to obtain an optimal success probability.

3. Entanglement concentration of N-photon GHZ state, specially W state and arbitrary W state The ECP in Section 2 is still workable even if we extend the present protocol to distill the maximally N-photon entangled Greenberger–Horne–Zeilinger (GHZ) state from partially N-photon

(18)

where |α|2 + |β|2 = 1. Now, let us show the basic principle of the concentration protocol. The photons in spatial mode a1, b1, …, and z1 are sent to Alice, Bob, …, and Zach, respectively. Alice exploits an auxiliary photon |Φ〉a2 = α|R〉a2 + β|L〉a2 which has the same form as the state shown in Eq. (5). So the N parties can also obtain the maximally entangled N-photon system with the total success probability

2|αβ|2 +

, the success

probability P ¼1. While if α ≠ β , the success probability P < 1, but P approximately reaches up to 1. And the relation between the success probability P and the coefﬁcient α is shown in Fig. 3. From Fig. 3, we can see that the success probability P of the ECP

1

entangled GHZ state:

(17)

It can be found that if the initial entangled state is the maximally entangled Bell state, where α = β =

Fig. 4. The success probability P vs. the coefﬁcient of the initial partially entangled state α and the iteration times of entanglement concentration n.

2|αβ|4 4

| α| + | β|

4

+

2|αβ|8 4

(|α| + |β|4 )(|α|8 + |β|8)

+⋯

n

+

2|αβ|2

n

n

(|α|4 + |β|4 )(|α|8 + |β|8)⋯(|α|2 + |β|2 )

,

if Alice deals with the photon in spatial mode a1 and another additional photon in spatial mode a2 in the same way as the case of the less-entanglement Bell state, shown in Section 2, where n is the iterative times of the entanglement concentration processes. In addition, it is interesting that the present ECP can also be used to concentrate a N-photon system in a specially entangled W state:

|Ψ 〉a1b1⋯ z1 = γ |RR⋯RL〉a1b1⋯ z1 + δ(|RR⋯LR〉a1b1⋯ z1 + ⋯ + |RL⋯RR〉a1b1⋯ z1 + |LR⋯RR〉a1b1⋯ z1),

(19)

where the parameters γ and δ satisfy the relation |γ |2 + (N − 1)|δ|2 = 1. To obtain a standard N-photon maximally W state from the system |Ψ 〉a1b1⋯ z1 described in Eq. (19), Zach needs to prepare an ancillary photon in the state

|Ψ 〉z2 =

γ |γ |2 + |δ|2

|L〉z2 +

δ |γ |2 + |δ|2

|R〉z2 ,

(20)

and deals with the photon in spatial mode z1 and another additional photon in spatial mode z2 in the same way as the case of the less-entanglement Bell state ECP discussed in Section 2. After interacting, the state of the system composed of NV and

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L.-L. Fan, Y. Xia / Optics Communications 338 (2015) 174–180

N + 1 photons becomes 2

2

|γ |

|Ψ 〉a1b1⋯ z1z2e = [

2

2

| γ | + | δ|

| δ|

|RR⋯RLL〉a1b1⋯ z1z2 +

2

|γ | + |δ|2

(|RR⋯LRR〉

Firstly, Alice performs the same operations as shown in Section 2 on the photons in spatial modes a1, a2 and measures the state of NV in the orthogonal basis {| + 〉e , | − 〉e } (|± 〉e = 1 (| − 1〉 ± | + 1〉)e ). When 2

the result of the measurement is in | + 〉, the whole state collapses to

|Ψ 〉′w = (ζς / |ζ |2 + |ς|2 )|LRRL〉a1b1c1a2 + (ζς / |ζ |2 + |ς|2 )|RLRR〉a1b1c1a2+

+⋯ + |RL⋯RRR〉 + |LR⋯RRR〉)a1b1⋯ z1z2] 1 (| − 1〉 + | + 1〉)e + 2

After the photon in the spatial (ζξ/ |ζ |2 + |ς|2 )|RRLR〉a1b1c1a2. mode a2 passes through the half-wave plate R45 , one of the ° parties, Alice, Bob, or Charlie, should perform a local operation of phase rotation on her or his photon, the state |Ψ 〉′w be-

γδ |γ |2 + |δ|2

(|RR⋯RLR〉a1b1⋯ z1z2 + |RR⋯LRL〉a1b1⋯ z1z2

.

comes |Ψ 〉″w = (ζς / |ζ |2 + |ς|2 )|LRR〉a1b1c1 + (ζς / |ζ |2 + |ς|2 )|RLR〉a1b1c1+

+ ⋯ + |RL⋯RRL〉a1b1⋯ z1z2

(ζξ/ |ζ |2 + |ς|2 )|RRL〉a1b1c1

1 (| − 1〉 − | + 1〉) + |LR⋯RRL〉a1b1⋯ z1z2 ) 2 e

2

(21)

By measuring the state of NV in the orthogonal basis {| + 〉e , | − 〉e } 1

(| ± 〉e =

2

(| − 1〉 ± | + 1〉)e ),

we

can

get

the

maximally

(N + 1)-photon entangled state:

γδ

|Ψ 〉N + 1 =

2

2

| γ | + | δ|

(22)

+ ⋯ + |RL⋯RR〉a1b1⋯ z1z2 + |LR⋯RR〉a1b1⋯ z1z2 ) with the success probability P = N |γδ|2 /(|γ |2 + |δ|2). And by iterating the entanglement concentration process n times, the total success probability of the maximally entangled W state shared by N parties is ⎡ |γδ|2 |γδ|4 |γδ|8 + Ptotal = N ⎢⎢ 2 + 2 2 2 4 4 2 2 4 | γ | + | δ | γ δ γ δ γ δ γ ( | | + | | )( | | + | | ) ( )( | | + | | | | + |δ|4 )(|γ|8 + |δ|8) ⎣ n

|γδ|2

n

(|γ|2 + |δ|2)(|γ|4 + |δ|4 )(|γ|8 + |δ|8)⋯(|γ|2

⎤ ⎥. 2 ⎥ + |δ|2 ) ⎦

(23)

Based on the description above, if two of the three parties, for example Alice and Charlie choose ancillary photon |Ψ 〉a2 =

(ζ / |ζ |2 + |ς|2 )|R〉a2 + (ς / |ζ |2 + |ς|2 )|L〉a2 , and |Ψ 〉c2 = (ξ/ |ς|2 + |ξ|2 )

|R〉c2 + (ς / |ς|2 + |ξ|2 )|L〉c2 , respectively, we can concentrate an arbitrary

less-entangled 2

W 2

state 2

1 with a success probability of Pw =

2

3

(|LRR〉 + |RLR〉 + |RRL〉)a1b1c1

with the success probability of

1 2 Pw Pw .

Then, Zach performs the operations on the photon in the spatial mode z2 as Alice does in Section 2, and one of the N parties executes an operation of phase ﬂip on her or his photon. After the above operations the maximally (N + 1)-photon entangled state in Eq. (19) becomes the maximally entangled N-photon W state |Ψ 〉Nm = (1/ N )(|RR⋯RL〉a1b1⋯ z1z2 + |RR⋯LR〉a1b1⋯ z1z2 + |RL⋯RR〉a1b1⋯ z1z2

+⋯ +

2

the success probability of Pw2 = 3ζςξ/(ζ 2 + ς 2)(ς 2 + ξ 2) . Finally, Charlie rotates his photon in the mode c2 by 45° with a half-wave plate, the state |Ψ 〉‴w becomes the maximally entangled W state |Ψ 〉wm = 3

+ |RL⋯RRL〉a1b1⋯ z1z2 + ⋯ + |RL⋯RRL〉a1b1⋯ z1z2 + |LR⋯RRL〉a1b1⋯ z1z2 .

2

ζ (2ς + ξ )/(ζ + ς ) . Secondly, Charlie performs the same operations as what Alice has done above. We can get the maximally four-photon state |Ψ 〉w ‴= 1 (|LRRR〉 + |RLRR〉 + |RRLL〉)a1b1c1c2 with

1

(|RR⋯RLR〉a1b1⋯ z1z2 ) + |RR⋯LRL〉a1b1⋯ z1z2

2

|Ψ 〉a1b1c1 = ζ |LRR〉a1b1c1

+ς|RLR〉a1b1c1 + ξ|RRL〉a1b1c1 (|ζ | + |ς| + ξ| = 1) to the maximally one with two steps, shown in Fig. 5.

Moreover, the items which are ignored also can be Pw = regarded as a less-entangled state, and similar to operations before, we can concentrate this less-entangled state in next round. We can also get a high success probability by repeating the concentration process n times as shown in Section 2.

4. Discussion and summary Till now, we have accomplished the ECP for photons lessentangled states by exploiting the diamond NV center insides photonic crystal cavity. The NV center in a diamond provides us a powerful tool to make a quantum-nondemolition measurement which can check the number of photons in the Fock state without destroying them. Now we need to discuss the feasibility of the nonlinear optics of an NV center in a diamond embedded in a photonic crystal cavity coupled to a waveguide. In recent years, several attempts have been pursued to couple the NV centers to cavities such as microsphere resonators [46,47], microtoroids [48,49] or photonic crystal cavities (PCCs) [50,51]. And as shown in Ref. [52], PCCs have the particular advantages that they can be fabricated with high quality factors as well as small mode volumes, both being ﬁgures of merit for obtaining a large Purcell enhancement. Moreover, PCCs can be easily integrated into more complex systems of coupled cavities and waveguides, which have been shown in Ref. [53]. On the other hand, the interaction between an NV center and a circularly polarized photon may be greatly enhanced by coupling with the photonic crystal cavity [54]. Therefore, we can estimate the ﬁdelity of our protocol discussed above by deﬁning the ﬁdelity as F = |〈ψreal|ψideal〉|2. Here, |ψideal〉 is the target state of the NV-cavity system encoded for the QND in the

Fig. 5. Schematic diagram of the single-photon-assisted entanglement concentration for arbitrary less-entangled W state.

L.-L. Fan, Y. Xia / Optics Communications 338 (2015) 174–180

ideal case Ω ≥ 5 κγ , and |ψreal〉 is the ﬁnal state that depends on experimental factors. In addition, the ﬁdelity is mainly inﬂuenced by the coupling strength and cavity side leakage. One can see that the ﬁdelity of the present protocol can be calculated as

F=

[(r + 1)2 + (r0 − 1)2]2 8[(r 2 + 1)2 + (r02 + 1)2]

.

(24)

Here, we set the decay rate γtotal = 2π × 15 MHz [41], the Q factor is determined by Q = c/λκ where c is the speed of light and the transition wavelength ¼637 nm between the states | − 1〉 and |A2 〉. We can see from Ref. [41] that when Ω / κγ ≥ 5 with ωc = ωp = ω0 , Q∼105 or Q∼104, r(ωp) ∼ 1. Fig. 6 shows the ﬁdelity of the protocol as a function of Ω / κγ in the resonant condition ωc = ω0 = ωp . The result shows that the ﬁdelity of our protocol is increasing with Ω / κγ . When Ω / κγ = 5, the ﬁdelity F = 99.98% . That is, the present protocol is feasible with current technology. Now, it is necessary to give a brief discussion on the robust to low strain regime and magnetic ﬁeld. The electron-spin triple ground states of an NV center used in the present scheme are split into |0〉 and | ± 〉 by 2.88 GHz with zero ﬁeld, due to the spin–spin interactions [55]. Because of the NV center's C3ν symmetry, spin– orbit, and spin–spin interaction in the absence of an external magnetic ﬁeld or crystal strain, the structure of the excited states is relatively complex. However, the level conﬁguration described in Fig. 1 is simple one. Ref. [36] has showed that employing one of the speciﬁc excited state |A2 〉, the simple Λ-type three-level system is realized. Using a particular magnetic ﬁeld to mix the ground states, we can obtain the Λ-type system in which optical control is required [56]. Alternatively, it is possible to ﬁnd a Λ-type system at zero magnetic ﬁeld as the inevitable strain in diamond reduces the symmetry and primarily modiﬁes the excited-state structure according to their orbital wave functions. As illustrated in [57,58], the excited state is separated into |A1〉, |A2 〉, |Ex〉, and |Ey〉, |E1〉, |E2〉 at moderate and high strain. An energy gap protects |A2 〉 against low strain and magnetic ﬁelds and the zero phonon line (ZPL) was observed after the optical resonant excitation at 637 nm (| − 〉 → |A2 〉 driven by a L-polarized photon and | + 〉 → |A2 〉 driven by a R-polarized photon) has been demonstrated by Togan et al. [36]. And the mutually orthogonal circular polarization will be destroyed by high strain. Therefore, the proposals are robust against low strain and magnetic ﬁelds because of the special auxiliary energy level we employed. On the other hand, the read-out techniques rely on resonant excitation of spin-selective optical transitions of the NV, which can be spectrally resolved at low temperatures [59]. In Ref. [60], the authors have found that

Fig. 6. The ﬁdelity F of the present protocol vs. g/ κγ . Here μ = g/ κγ .

179

the ﬁdelity of readout and addressed at low temperature (T ¼8.6 K) is 93.2 ± 0.5% using the Exand A1transitions in their experiments. The state |A1〉 connects the ground states with spin projection mS = ± 1 to an excited state with a primarily ms = ± 1 character, whereas Exand connects states with ms ¼0. With only linear optics elements [29,61–63], the entanglement concentration processes cannot be repeated, and we cannot get a higher success probability. For example, the scheme in Ref. [29] has completed hyperentanglement concentration for two-photon four-qubit systems resorting to linear-optical elements only. They ﬁrst introduced the parameter-splitting method to concentrate the systems in the partially hyperentangled states with known parameters. And the scheme in Ref. [29] does not need two copies of partially entangled states or the ancillary single photon. However, because the entanglement concentration processes cannot be repeated, the success probability of the scheme in Ref. [29] is not high. Comparing with Ref. [29], the present scheme has several advantages. Firstly, the parties do not need to know the accurate coefﬁcients of the initial state. Secondly, the present scheme can be iterated to get a higher success probability. Moreover, the present scheme is also suitable for multiphoton system concentration. Therefore, nonlinear optics material is required to implement the entanglement concentration. For example, the diamond NV center and the cross-Kerr nonlinearity are two perfect platforms for quantum information processing. By utilizing such platforms, we can concentrate the photon less-entangled state to a maximally entangled one with the high success probability in principle by the iteration of the ECP process with the states that have been discarded in the ECP with linear optics, because the photon states are not destroyed. In fact, we should acknowledge that, although a lot of works have been studied on the cross-Kerr nonlinearity [64,65], a clean cross-Kerr nonlinearity in the optical single-photon regime is still quite a controversial assumption, especially the strong cross-Kerr nonlinearity, which is still a big challenge in the experiment at present. In addition, in the optical singe-photon regime, Gea–Banacloche pointed out that the large phase shifts via the giant Kerr effect with single-photon wave packets is impossible [66], and as pointed out by Kok et al. the Kerr phase shift is only τ E10 18, even with electromagnetically induced transparent materials, one can obtain the cross-Kerr nonlinearities of τ E10 5 [67]. By using homodyne detector, it is difﬁcult to determine the phase shift due to the impossible discrimination of two overlapping coherent states. However the quantum system combining a cavity and an artiﬁcial atom has long coherence time and good scalability, and it has nanosecond manipulation time for an individual NV center. Therefore, the diamond NV center is also the competitive candidate for constructing quantum nondemolition detectors to complete some quantum operations. By now, solid-state systems are much more attractive as they have a good scalability. The solid-state systems include superconductor, quantum-dot, and NV-center. By utilizing such systems, many schemes have been proposed for realizing quantum information processing. For example, based on superconductor, Romero et al. [68] and Stojanović et al. [69] proposed some interesting schemes for realizing controlled-phase and Toffoli gates in nanosecond time scale, respectively. Based on single electron spin conﬁned in a charged quantum dot inside a microcavity, Hu et al. [70] present a scheme to generate photon polarization entanglement. Ren et al. [71] proposed a scheme for completing hyperentangled-Bell-state analysis for photon systems. Based on diamond NV centers inside photonic crystal cavities, Ren et al. [38] proposed a protocol to implement the hyperentanglement puriﬁcation of two-photon systems in nonlocal hyperentangled Bell states. Wei et al. [39] present some scheme for realizing CNOT gate, Tofolli gate and Fredkin gate.

180

L.-L. Fan, Y. Xia / Optics Communications 338 (2015) 174–180

In summary, we have presented an ECP to concentrate the partially entangled Bell state with a higher success probability. In this protocol, We exploit an NV center in a diamond embedded in a photonic crystal cavity coupled to a waveguide to construct the quantum nondemolition detectors and achieve the task. Compared with other concentration protocols, the ECP does not require two same copies of less entangled photon states, which makes it more economical. Moreover, the remote parties do not need to know the accurate coefﬁcients of the initial state. With the NV center in a diamond, the ECP can be repeated to get a higher success probability, and does not require sophisticated single-photon detectors. In addition, we also have demonstrated that the ECP is more suitable to concentrate the less-entangled multiphoton GHZ state, the N-photon spatially less-entangled W state, and the arbitrary less-entangled W state. All these advantages make our protocol more useful and meaningful in practical applications.

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Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant nos. 11105030 and 11374054, the Major State Basic Research Development Program of China under Grant no. 2012CB921601, and the Foundation of Ministry of Education of China under Grant no. 212085.

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