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Entanglement concentration for multi-particle partially entangled W state using nitrogen vacancy center and microtoroidal resonator system Ling-yan He a, Cong Cao a, Chuan Wang b,n a b

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 December 2012 Received in revised form 14 February 2013 Accepted 14 February 2013 Available online 6 March 2013

In this paper, we present an efﬁcient entanglement concentration protocol (ECP) for N spatially separated nitrogen-vacancy (N-V) centers coupled to microtoroidal resonators in a partially entangled W state. The robustness of W-type entanglement contrasts strongly with the Greenberger-HorneZeilinger (GHZ) state which is fully separable after loss of one qubit. We construct the parity-check gate by introducing an auxiliary N-V center in microcavity and a single photon. Based on the parity-check measurement and local operations, the remote parities can distill a maximally entangled W state from a partially entangled state with a certain success probability. By iterating the concentration process several times, the present ECP can obtain an optimal yield. With its robustness and scalability, the present ECP for entangled N-V center ensembles can be widely applied to quantum repeaters and longdistance quantum communications. Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.

Keywords: Entanglement concentration N-V center W state Parity-check measurement

1. Introduction During the past decades, quantum entanglement has become an indispensable quantum resource in quantum information processing (QIP). It can speed up quantum computation powerfully, and be widely used in many branches of quantum communication, such as quantum key distribution (QKD) [1,2], quantum teleportation [3,4], quantum secure direct communication (QSDC) [5–9], and so on. As Kimble [10] noted in 2008 that the realization of quantum networks composed of many nodes and channels should require new scientiﬁc capabilities for generating and characterizing quantum coherence and entanglement, quantum networks and repeaters should resort to quantum entanglement. However, in the practical entanglement distribution through a quantum channel, entangled quantum systems will irresistibly and inevitably interact with noise, which gives rise to decoherence. The intrinsic decoherence will generate the nonmaximally entangled state [11,12], cause the irreversible loss of entanglement shared by remote parties, and reduce the success possibility of QIP, leaving the quantum communication insecure. Aiming to improve the ﬁdelity of entanglement after entanglement distribution through a noisy channel, the parties in quantum communication can recur to entanglement puriﬁcation and entanglement concentration.

n

Corresponding author. Tel.: þ86 10 62282050. E-mail addresses: [email protected], [email protected] (C. Wang).

Entanglement puriﬁcation deals with mixed entangled states, distilling a subset system with high ﬁdelity entanglement from a mixed entangled ensemble [13–24], while entanglement concentration is used to extract the maximally entangled state from a partially entangled pure state. Though entanglement puriﬁcation can be used to purify mixed states, entanglement concentration is of interest in its own rights. As entanglement concentration protocols (ECPs) provide us a way to generate maximally entangled states, while entanglement puriﬁcation protocols (EPPs) only generate highly entangled states by consuming more than half the quantum resource in each round. The ﬁrst ECP was proposed by Bennett et al. [25] which is based on the Schmidt projection method. They utilized the collective and nondestructive measurement to obtain information about the coefﬁcients. Since then, there have been many interesting and typical ECPs. In 1999, Bose et al. [26] proposed an ECP based on entanglement swapping. Subsequently, Shi et al. [27] designed an ECP by exploiting a two-particle collective unitary evaluation. In 2001, Yamamoto et al. [28] and Zhao et al. [29] presented two similar ECPs based on polarization beam splitters (PBSs), independently. They also experimentally demonstrated their ECP using linear optics in their related works [30]. Later in 2008, Sheng et al. [31] proposed an efﬁcient ECP based on cross-Kerr nonlinearity. They also present an efﬁcient polarization entanglement concentration for electrons with charge detection [32]. Later, they designed an ECP for single-photon entangled systems [33] in 2010. In 2011, Wang et al. [22,34] put forward an ECP using the quantum dot (QD) and microcavity coupling system in which spin entangled

0030-4018/$ - see front matter Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.02.031

L. He et al. / Optics Communications 298–299 (2013) 260–266

state can be distilled efﬁciently. In 2012, Sheng et al. [35] proposed an ECP for entangled photon pairs in a less-entangled state, assisted by an additional single-photon. Deng [36] presented an optimal nonlocal multipartite ECP based on projection measurements. Recently, Peng et al. [37] proposed an ECP for entangled atomic and photonic systems via photonic Faraday rotation. Then, Zhou [38] presents two ECPs for arbitrary threeelectron W state based on their charges and spins. Soon afterwards, Zhou et al. put forward an efﬁcient entanglement concentration protocol (ECP) for arbitrary less-entangled NOON state [39]. Among so many interesting ECPs, most of them are used to distill some maximally entangled Bell states or multi-partite Greenberger-Horne-Zeilinger (GHZ) states, while there are few schemes for concentrating the less-entangled pure W-class states. The W state has an essential difference from the GHZ state, as it cannot be transformed into each other under stochastic local operations and classical communication (SLOCC). Nowadays, W state has begun to gain widespread attention both in theory and experiment, as the robustness of W-type entanglement contrasts strongly with the GHZ state which is fully separable after loss of one qubit. The W state’s robustness against particle loss is a very beneﬁcial property ensuring good storage properties of these ensemble based quantum memories [40]. W state can be used to realize the teleportation of an unknown state probabilistically [41,42] and superdense coding [43,44]. Consequently, it is of practical signiﬁcance to discuss the entanglement concentration on the less-entangled pure W state. In 2010, Wang et al. [45] proposed an ECP for two partially entangled three-photon W states based on linear optics. Subsequently, he designed another linear-optics-based ECP of unknown partially entangled three-photon W states [46]. In 2011, Xiong and Ye [47] proposed an ECP using cross-Kerr nonlinearity for a partially entangled W state. All these interesting ECPs are used to deal with an unknown multi-photon W-class state. In 2012, Sheng et al. proposed an efﬁcient two-step ECP for arbitrary three-photon W states [48] and an ECP with parity check measurement, which do not need two copies of less-entangled states but only need one pair of lessentangled state and two single photons. Recently, solid state systems such as nitrogen-vacancy (N-V) centers consisting of a substitutional nitrogen atom and an adjacent vacancy in diamond are drawing more attention in solid-state based QIP, for their coherence time over 180 s [49]. Especially, N-V defect in diamond that combines with the whispering-gallery mode (WGM) system has emerged as a promising solid-state candidate to meet the requirements in the QIP schemes. On one hand, the N-V centers possess a good electron spin coherence even at room temperature. In a recent article [50], Maurer et al. demonstrate that the qubit consisting of a single 13C nuclear spin in the vicinity of a nitrogen-vacancy center within an isotopically puriﬁed diamond crystal can preserve its polarization for several minutes and feature coherence lifetimes exceeding 1 s at room temperature, which provides us enough time for coherent manipulation. Furthermore, its optical controllability is another attractive advantage. On the other hand, WGM can obtain a high Q factor. The study of WGMs was started almost a century ago with the work of Lord Rayleigh, who studied propagation of sound over a curved gallery surface [51,52]. The whispering-gallery modes originally were introduced for sound waves propagating close to the cylindrical wall in St. Pauls Cathedral, London, where the body of the modes was partially conﬁned due to the suppression of the wave diffraction by the sound reﬂection from the curved dome walls. Nowadays, Whispering-gallery-mode microresonators are a promising cavity to study, due to the ability to obtain quality factors exceeding 100 million in micron-scale volumes. The

261

microtoroidal resonator is a potential system to realize WGM. Due to its capability to obtain a high Q factor, ranging from 13M [53] to 50G [54], meanwhile keeping a low mode volume, we utilize the microtoroidal resonator with WGM to accomplish our ECP. Many theoretical and experimental works have been devoted to this kind of hybrid system composed of solid-state qubit and cavity, and some progress has been achieved. For example, in 2009, Jiang et al. [55] argued that N-V centers could be rapidly readout with high ﬁdelity. Also the entanglement generation process on N-V centers coupled to micro-cavity systems has been experimentally studied [56]. In 2010, Yang et al. [57] presented a W state and Bell state entanglement generation protocol based on separate nitrogen-vacancy centers and a WGM cavity coupling system. Later, Li et al. [58] proposed a quantum information transfer protocol with N-V centers coupled to a WGM resonator. In 2012, Wang et al. [59] proposed an efﬁcient ECP for separate N-V centers via coupling to microtoroidal resonators. The preservation and concentration of entanglement between N-V centers is a potentially practical approach for solid state based QIP. In this paper, we present an efﬁcient scheme of ECP for three spatially separated N-V centers coupled to microtoroidal resonators in a partially entangled W state, by introducing an auxiliary N-V center coupled to a microtoroidal resonator. The key ingredient of this scheme is the parity-check gate (PCG) based on the N-V centers and the microtoroidal resonators coupling system and then exploit it to accomplish our ECP. In the present ECP, we use an ancillary photon which is prepared locally as the paritycheck index, then we can obtain the parity information of two N-V centers by detecting the spatial modes of the output photon. Finally, we apply the PCG to extract the N-particle maximally entangled W state from a less-entangled W state of N spatially separated N-V centers coupled to microtoroidal resonators with high efﬁciency, by extending the three qubits ECP to N qubits ECP. Based on the recent experimental progress, the ECP may be useful in long-distance communication, especially in the condition where N-V centers interacting with microtoroidal resonators as quantum nodes in quantum repeaters. This paper is organized as follows. In Section 2, we demonstrate the entanglement concentration process based on three N-V centers coupled to microtoroidal resonators, assisted by an auxiliary N-V center and a single photon. In Section 3, we extend the three-particle ECP to N-particle ECP for a less-entangled W state of N spatially separated N-V centers coupled to microtoroidal resonators. Finally, summary and discussion are given in Section 4.

2. Single N-V center assisted entanglement concentration based on N-V center and microtoroidal resonator system 2.1. The model We consider the model that a single N-V center is coupled to the microtoroidal resonator. The microtoroidal resonator can be described as a double-sided optical cavity. The N-V center can be viewed as an additional electron with a negatively charge which consists of a substitutional nitrogen atom and an adjacent vacancy. The coupling system exhibits similar features with the Jaynes–Cummings model and the energy level structure is illustrated in Fig. 1. The states 9g 1 S and 9g þ 1 S correspond to the Zeeman sublevels of an alkali atom in the degenerate ground state, and 9A2 S represents the excited state. We label the states as 9A2 S ¼ 9E0 S 9ms ¼ 0, 71S here 9E0 S is the orbital state with zero angular momentum projection along the N-V axis. As the N-V center is in resonance with the cavity modes, the possible cavity-mode-induced transitions are 9g 1 S-9A2 S and

262

L. He et al. / Optics Communications 298–299 (2013) 260–266

described as follows: 0

91S91S

0

9LS þ 9RS ðeiðf þ f Þ 9HS þeiðf0 þ f0 Þ 9VSÞ pﬃﬃﬃ pﬃﬃﬃ -91S91S 2 2 ð2Þ

9þ 1S9 þ 1S

iðf0 þ f00 Þ

9LSþ 9RS ðe pﬃﬃﬃ -9 þ 1S9 þ1S 2

0

9HSþ eiðf þ f Þ 9VSÞ pﬃﬃﬃ 2 ð3Þ

Fig. 1. The energy level structure of the N-V center coupled to microtoriodal cavity, where the lower levels are Zeeman sublevels of the ground state and the upper level is the excited one. The dotted line represents the R circularly polarized mode and the solid line represents the L circularly polarized mode.

9g þ 1 S-9A2 S by absorbing left L or right R circularly polarized photon, respectively. The transitions 9g i S-9A2 S (i ¼ 71,0) can be realized by the classical laser interaction. The quantum information is encoded in the spin state 91S ¼ 9g 1 S and 9 þ1S ¼ 9g þ 1 S. The reﬂection coefﬁcient of the system can be investigated by solving the Heisenberg equations as discussed in Ref. [60] h g ih k ks i 2 iðoc op Þ þ þg iðo0 op Þ þ 2 2 2 rðoÞ ¼ h ð1Þ gih k ks i 2 iðo0 op Þ þ iðoc op Þ þ þ þg 2 2 2 Here g=2 denotes the decay rate of the N-V center. k and ks are the cavity decay rate and the cavity leaky rate, respectively. The frequencies of the input photon, cavity mode and the atomic level transition are represented by op , oc , o0 , respectively. The dynamics of the composite system can be described as follows: if the N-V center is initially prepared in 9g 1 S, the only possible transition is 9g 1 S-9A2 S, which indicates that only the L circularly polarized single-photon pulse 9LS with angular momentum S ¼ þ 1 will interact with the N-V center-cavity system. The output pulse will be 9Cout SL ¼ rðoÞ9LS eif 9LS where the corresponding phase shift is determined by the parameter values. In addition, it also implies that an input R circularly polarized single-photon pulse 9RS with angular momentum S ¼ 1 would experience the empty cavity. The corresponding output pulse is 9Cout SR ¼ r 0 ðoÞ9RS eif0 9RS with f0 a phase shift different from f. Therefore, for an input linearly polarized photon pulse 9Cin S ¼ p1ﬃﬃ2 ð9LSþ 9RSÞ, the output pulse is 9Cout S ¼ p1ﬃﬃ ðeif 9LS þ eif0 9RSÞ. Similarly, if the N-V center is initially pre2 pared in Jg þ 1 S, then only the R circularly polarized photon could interact with the N-V center, whereas the L circularly polarized photon only experiences the empty cavity. So we have 9Cout S þ ¼ p1ﬃﬃ2 ðeif0 9LS þ eif 9RSÞ. The proposed system can be generalized as the parity check gate (PCG) for N-V center spins by distinguishing the phases of the ancillary photon. We input a single-photon pulse in superposition of horizontal and vertical polarizations, i.e, 9Cin S ¼ p1ﬃﬃ ð9HS þ9VSÞ which changes to 9Cin S ¼ p1ﬃﬃ ð9LS þ9RSÞ after 2 2 going through the quarter-wave plate (QWP) into the ﬁrst N-V center coupled to the microtoroidal resonator, and then we direct the output pulse by a ﬁber to the second cavity. The detection of the output photon from the second cavity, assisted by a QWP and a half-wave plate (HWP), would dedicate the parity of the states of two N-V centers by distinguishing the phase of the ancillary photon. The principle of our PCG is similar to Ref. [59]. The purpose of PCG is to distinguish the parity of the states of two N-V centers. Following the evolution of the setup proposed above, the state of the photon and N-V centers composite system can be

if the two N-V centers are in the same spin state, where the affect of the QWPs has been included. Apparently, the state of the ancillary photon depends on the phase shift f and f0 . Here we 0 have f ¼ f ¼ p and f0 ¼ f00 ¼ p=2. Otherwise, if the two N-V centers are in the state 91S9 þ 1S or 9 þ 1S91S, the output photon will trigger the 9HS þ9VS mode detector which indicates that the parity of the two particles is odd. If we adjust the frequency of the input pulse to op ¼ oc k=2, g ¼ k=2 and o0 ¼ oc , the corresponding phase shift can be realized. And this device can be used to accomplish a nondemolition measurement of spin parity on two N-V centers spin entangled system. 2.2. Three-particle entanglement concentration We take three-particle entanglement as an example to describe the principle of our ECP. The principle of the ECP is illustrated in Fig. 2. Assume that three separate N-V centers coupled to the microtoroidal resonators are in the following partially entangled W states: 9fSCBA ¼ a91SC 91SB 9 þ 1SA þ bð91SC 9 þ 1SB 91SA þ9 þ 1SC 91SB 91SA Þ

ð4Þ

here the subscripts A, B and C denote the three spatially separate N-V centers owned by Alice, Bob and Charlie, respectively. a and b 2 satisfy the following relation: a2 þ 2b ¼ 1. To facilitate the implementation of our protocol, we prepare an ancillary N-V center a, which is in the superposition state: 1 9fSa ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ða9 þ 1Sþ b91SÞa a2 þ b2

ð5Þ

Then the composite system made up of the four N-V centers is in the state 9FSCBAa ¼ 9fSCBA 9fSa 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fab½91SC 91SB 9 þ 1SA 91Sa 2 a þ b2 þ ð91SC 9 þ 1SB þ 9 þ1SC 91SB Þ91SA 9 þ1Sa þ a2 91SC 91SB 9 þ 1SA 9 þ 1Sa 2

þ b ð91SC 9 þ 1SB þ 9 þ1SC 91SB Þ91SA 91Sa g

ð6Þ

With the parity-check measurement on N-V centers A and a at Alice’s side, 9FSCBAa can be divided into two classes based on the

Fig. 2. Schematic diagram shows the principle of ECP process for entangled N-V centers. PCG represents the parity-check gate operations. Here the dashed lines represent the entanglement between the N-V centers. The X in the box indicates the local single-particle measurements after PCG.

L. He et al. / Optics Communications 298–299 (2013) 260–266

1 9C1 SCBAa ¼ pﬃﬃﬃ ½91SC 91SB 9 þ 1SA 91Sa 3 þð91SC 9 þ 1SB þ 9þ 1SC 91SB Þ91SA 9 þ 1Sa

þ

ð7Þ

which corresponds to the odd parity with a success probability of 2 2 P 1 ¼ 3a2 b =a2 þ b . The other class is in a less-entangled state 1 9C01 SCBAa ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½a2 91SC 91SB 9 þ 1SA 9 þ 1Sa 2 a þ 2b2 2

þ b ð91SC 9 þ 1SB þ 9 þ1SC 91SB Þ91SA 91Sa

ð8Þ

#

n

n

post selection principle, one is in the state

263

a2 b2 2

4

n

ð13Þ

2n

ða2 þ b Þða4 þ b Þ ða2 þ b Þ

The relation between the total success probability P(n) and the coefﬁcient a is illustrated in Fig. 3. After repeating entanglement concentration process ﬁve times, the total success probability P reaches a high value. So for a partially entangled W-state class of N-V center, we need only iterate the process in the repeateduntil-success fashion to obtain a high total success probability.

3. Entanglement concentration of N particles in a partially entangled W state

which corresponds to the even parity. Then we measure the ancillary N-V center a with the basis X, that is 9 7 xS ¼ p1ﬃﬃ2 ð91S7 9 þ1SÞ. If we get 9þ xSa , the three N-V centers are in the maximally entangled W state 9W 3þ S:

Theoretically, we can extend the three-particle ECP to N N-V centers coupled to microtoroidal resonators in a partially entangled W state. Suppose that the N-particle state is initially in the partially entangled W-class state

1 9W 3þ SCBA ¼ pﬃﬃﬃ ½91SC 91SB 9 þ 1SA 3 þð91SC 9 þ 1SB þ 9 þ1SC 91SB Þ91SA

9fSZCBA ¼ a1 91SZ 91SC 91SB 9 þ1SA þ b1 ð91SZ 91SC 9 þ 1SB 91SA ð9Þ

Otherwise, if we get 9xSa , the three N-V centers will collapse to another maximally entangled W state 1 ¼ pﬃﬃﬃ ½91SC 91SB 9 þ1SA 3 ð91SC 9 þ 1SB þ9 þ 1SC 91SB Þ91SA

9W 3 SCBA

ð10Þ

9W 3þ S

which can be transformed to the state by performing a phase-ﬂip operation sz ¼ 9 þ 1S/ þ 1991S/19 on N-V center A. Likewise, if we get the state 9C01 SCBAa after the parity-check measurement, we can also measure the ancillary N-V center a with the basis X to transform it into a less-entangled state, which can be described as

a2

9C02 SCBA ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 91SC 91SB 9 þ1SA a2 þ2b2

þ 91SZ 9 þ 1SC 91SB 91SA þ þ 9 þ 1SZ 91SC 91SB 91SA Þ

ð14Þ

the subscripts A, B, C,y, and Z represent the N N-V centers owned by Alice, Bob, Charlie,y, and Zash, respectively. Here, the coefﬁ2 cients a1 and b1 satisfy the following relation: a21 þ ðn1Þb1 ¼ 1. Similar to the case in three-particle situation, we prepare an qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ancillary N-V center a1 in the state 9fSa1 ¼ 1= a21 þ b1 ða1 9 þ1Sþ b1 91SÞa1 to obtain a standard maximally entangled W state for the N-V centers in the state 9fSZCBA . Then the state of the composite system can be written as 9FSZCBAa ¼ 9fSZCBA 9fSa1 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fa1 b1 ½91SZ 91SC 91SB 9 þ 1SA 91Sa1 a21 þ b21 þð91SZ 91SC 9 þ1SB þ 91SZ 9 þ1SC 91SB

b2

þ þ9 þ 1SZ 91SC 91SB Þ91SA 9 þ1Sa1

þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð91SC 9 þ1SB þ9 þ 1SC 91SB Þ91SA a2 þ 2b2

þ a21 91SZ 91SC 91SB 9þ 1SA 9 þ1Sa1 ð11Þ 1

Apparently, 9C02 SCBA has the same form as the state 9fSCBA if qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 we replace the coefﬁcients a and b with a0 ¼ a2 = a2 þ2b and qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b0 ¼ b2 = a2 þ 2b2 , respectively. In this way, 9C02 SCBA can be seen

3

P2 ¼

b4

a4

4 3a4 b a4 þ 2b4 a4 þ2b4 ¼ 4 4 4 4 a b ða4 þ b Þða4 þ 2b Þ þ 4 4 a4 þ 2b a4 þ2b

ð12Þ

After iterating the entanglement concentration process N times, the total success probability that we obtain the standard maximally entangled W state 9W 3þ S for the composite system is PðnÞ ¼ P 1 þP 01 P 2 þ P01 P02 P3 þ "

¼3

þ

þ P 01 P 02

P0n1 P n

a2 b2 a4 b4 þ 2 2 4 a2 þ b ða2 þ b Þða4 þ b Þ

a8 b8 þ 4 8 ða2 þ b Þða4 þ b Þða8 þ b Þ 2

n=1 n=2 n=3 n=4 n=5

0.8 Success probability

as another initial pure state and be concentrated in the next round similar to the above steps. The success probability that we obtain the standard maximally entangled W state for the N-V centers in the state 9C02 SCBA is

0.9

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

α

0.6

0.8

1

Fig. 3. The relation between the total success probability P(n) and the parameter a. The dotted line, the dot-dashed line, the thin solid line, the dashed line represent the success probability of the entanglement after performing the concentration process one time, twice, three times, and four times. Finally, the thick solid line represents the success possibility after the concentration process is iterated ﬁve times.

264

L. He et al. / Optics Communications 298–299 (2013) 260–266 2

þ b1 ð91SZ 91SC 9 þ 1SB þ 91SZ 9 þ 1SC 91SB þ þ 9 þ1SZ 91SC 91SB Þ91SA 91Sa1 g

ð15Þ

We can perform a parity-check measurement on N-V center A and a1 on Alice’s side, if the parity of the two N-V centers is odd, then the Nþ1 N-V centers composite system is in the state 1 9C1 SZCBAa1 ¼ pﬃﬃﬃﬃ ½91SZ 91SC 91SB 9 þ 1SA 91Sa1 N þð91SZ 91SC 9þ 1SB þ91SZ 9 þ1SC 91SB þ þ9 þ 1SZ 91SC 91SB Þ91SA 9 þ1Sa1

ð16Þ 2 2 1 b1 =

which takes place with the success probability P1 ¼ N a

a21 þ b21 .

If the parity of the two N-V centers A and a1 is even, then the N þ 1 N-V centers composite system is in the state 1 9C01 SZCBAa1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½a21 91SZ 91SC 91SB 9 þ1SA 9þ 1Sa1 4 a1 þðN1Þb41 2

þ b1 ð91SZ 91SC 9 þ 1SB þ 91SZ 9 þ 1SC 91SB þ þ 9 þ1SZ 91SC 91SB Þ91SA 91Sa1

ð17Þ

4 2 4 2 1 þðN1Þb1 = 1 þ b1 .

a which takes place with the probability P 01 ¼ a If the composite system is in the state 9C1 SZCBAa1 , we can obtain the standard N-particle maximally entangled W state 1 9W Nþ SZCBA ¼ pﬃﬃﬃﬃ ½91SZ 91SC 91SB 9 þ 1SA N þ ð91SZ 91SC 9 þ 1SB þ 91SZ 9 þ 1SC 91SB þ þ 9 þ 1SZ 91SC 91SB Þ91SA

ð18Þ

with or without a phase-ﬂip operation on the N-V center A, after measuring the ancillary N-V center a1 with the basis X. If the composite system is in the state 9C01 SZCBAa1 , the state collapses to 1 9C01 SZCBA ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½a21 91SZ 91SC 91SB 9 þ 1SA 4 a1 þðN1Þb41 2

þ b1 ð91SZ 91SC 9 þ 1SB þ 91SZ 9 þ 1SC 91SB þ þ 9 þ1SZ 91SC 91SB Þ91SA :

At ﬁrst, the equal superposition state of the NV center can be prepared by excited using a p=2 microwave pulse with current technology [61]. Also, the PCG is a basic element for us to complete the ECP task which depends on the practical implementation of the system. It is necessary to verify the model and study the performance of this protocol under realistic circumstances through numerical simulations. Here we take the three-particle ECP as an example. In the simulation, two main decoherence processes ought to be taken into consideration, i.e., cavity photon loss(decay rate k) and decay of the N-V centers. For the N-V centers, spontaneous emission from the excited state should be included. In Fig. 4, we numerically simulate the relation between the coupling strength g=k and the success probability in our protocol. Here we assume the decay rate of the N-V center is g=2k ¼ 0:05, and the decoherence effect denoted by ks ¼ 0:1k. From Fig. 4, we can see the success probability of the entanglement after performing the concentration process one time is lower than 0.5, which can only be obtained in theory. Given this constraint, the decoherence of the photon in the WGM cavity may reduce the success probability of the proposed ECP. We assumed that the decoherence effect by the coefﬁcient ks ¼ 0:1k and simulated the success probability of the ECP in current experiment parameters. The reduce of success probability is less than 5% under current decoherence effect. On the other hand,we ﬁnd that a higher success probability can be obtained after several repetitions of the ECP process as shown in Fig. 3. By ﬁve times iteration, the success probability may increase to 95%. We can effectively gain a high success probability by using this method. As shown in Fig. 4, we can conclude that the success probability reaches the maximal value when the coupling strength g=k is about 3, while if the coupling strength g=k keeps increasing, the success probability remains unchanged. It’s just like a paradox. We need a high-Q factor to sustain a sufﬁciently high single-photon-NV coupling rate, while the scheme also demands the contrasting requirement of a fast decay rate into the waveguide so that the photon can couple into the second cavity. We require an optimal to both efﬁciently excite the modes of the cavity and to efﬁciently extract optical energy from the cavity. To balance these two mechanisms, here, we choose g=k ¼ 3 as our optimal coupling strength to accomplish our ECP. In the realistic implementation, practical achievements in experiment with N-V centers and microtoroidal resonators are used. The electronic spins are easy for initialization, manipulation and measurement. As discussed in Ref. [62], the spin and orbital

ð19Þ

By iterating the entanglement concentration process N times, the total success probability that we obtain a system in a maximally entangled W state of N N-V centers is " 2 4

a21 b1 a41 b1 þ 2 2 2 a1 þ b1 ða1 þ b21 Þða41 þ b41 Þ

a81 b81 þ þ 2 4 8 2 ða1 þ b1 Þða41 þ b1 Þða81 þ b1 Þ þ

n

2 2 1 b1 2 þ b2 Þð 4 þ b4 Þ ð 2n 1 1 1 1 1

a

ða

a

#

n

n

a þ b21 Þ

ð20Þ

0.4 Success Probability

P 0 ðnÞ ¼ N

0.3 0.2 0.1 0 1

3

4. Discussion and summary The key ingredient of our proposed protocol is the N-V center and microtoroidal resonator coupling system. In the proposed ECP, the ancillary photon is sent through the microtoroidal cavity and the phase information of the output photon will reveal the parity information of the two local N-V centers.

0.5 α

2 1 0

0

g/κ

Fig. 4. The relation between the total success probability P(n), the parameters a and g=k in practical conditions. ks ¼ 0:1k represents the photon loss and the cavity leakage in the process, and the decay rate of the N-V center is g=2k ¼ 0:05.

L. He et al. / Optics Communications 298–299 (2013) 260–266

of single N-V centers in diamond are studied and the spin state of N-V centers can be revealed by electron spin resonance measurement. Moreover, the coupling strength of the N-V center with microcavity resonator is on the order of hundreds of megahertz, for instance, strong coupling between a single N-V center and silica microsphere [63,64]. The coupling strength between N-V centers and the WGM can reach g=2p 0:321 GHz [63–66]. The Q factor of the whispering gallery mode microresonator has a value exceeding 109, which can lead to a photon loss rate of k ¼ o=Q 2p 0:5 MHz. Experimentally, the N-V centers are coupled with chip-based microcavity with Q 425,000 and the coupling strength between a single microdisk photon and the N-V center zero photon line is g=2p ¼ 0:3 GHz [64]. The total spontaneous emission rate of the N-V center is g=2p ¼ 0:013 GHz at 637 nm. In summary, we propose an efﬁcient scheme of ECP for less entangled W state of N N-V centers coupled to microtoroidal resonators by the assistance of an ancillary N-V center prepared locally. In the present ECP, the single photon is sent through the two microtoroidal resonators and the phase information of the output photon will indicate the parity information of the two locally N-V centers. The PCG is a fundamental element for us to complete the ECP task. Compared with the current ECPs based on practical systems, our proposal has several merits as given below. Let’s take the earlier paper [22] as an example. The earlier protocol is based on two pairs of electron-spin entangled states system. The users need two copies of the entangled states in each round concentration. Moreover, the proposed entanglement concentration protocol is based on the Schmidt projection method [25], no matter whether the information about the initial state is known or not. Additionally, the authors discussed the nonlocal entanglement concentration for an unknown partially electron-spin entangled pure state. There are some distinct differences between the present entanglement concentration protocol and the earlier paper. In the present work, we exploit parity-check gate as a platform to concentrate a partially entangled W state. The present ECP does not depend on a pair of systems in a partially entangled state in each round of concentration, just each system itself and one additional single N-V center prepared locally, which makes it far different from the earlier paper. It works for a known partially entangled W state. Second, the N-V centers possess long coherence time at room temperature and optical controllability, which make it a promising candidate for solid state QIP. Third, the robustness of W-type entanglement contrasts strongly with the GHZ state which is fully separable after loss of one qubit. The W state’s robustness of this system may ensure good storage properties of these ensemble based quantum memories. Furthermore, it has a higher total success probability by iterating the entanglement concentration process several times. In a word, our ECP has some differences in principle compared with all the typical existing ECPs. And the ECP based on N-V centers may be useful in long-distance quantum communication, especially in the case where N-V centers coupled to microtoroidal resonators are used to store quantum information as a quantum repeater.

Acknowledgments This work is supported by the National Fundamental Research Program Grant No. 2010CB923202, Specialized Research Fund for the Doctoral Program of Education Ministry of China No. 20090005120008, the Fundamental Research Funds for the Central Universities, China National Natural Science Foundation Grant Nos. 61205117.

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