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Discussion

Generation and entanglement concentration for electron-spin entangled cluster states using charged quantum dots in optical microcavities Jie Zhao a, Chun-Hong Zheng a,b, Peng Shi a, Chun-Nian Ren a, Yong-Jian Gu a,n a b

Department of Physics, Ocean University of China, Qingdao 266100, PR China School of Science, Qingdao Technological University, Qingdao 266033, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 9 November 2013 Received in revised form 2 February 2014 Accepted 4 February 2014 Available online 15 February 2014

We present schemes for deterministically generating multi-qubit electron-spin entangled cluster states by the giant circular birefringence, induced by the interface between the spin of a photon and the spin of an electron conﬁned in a quantum dot embedded in a double-sided microcavity. Based on this interface, we construct the controlled phase ﬂip (CPF) gate deterministically which is performed on electron-spin qubits and is the essential component of the cluster-state generation. As one of the universal gates, the CPF gate constructed can also be utilized in achieving scalable quantum computing. Besides, we propose the entanglement concentration protocol to reconstruct a partially entangled cluster state into a maximally entangled one, resorting to the projection measurement on an ancillary photon. By iterating the concentration scheme several times, the maximum success probability can be achieved. The ﬁdelities and experimental feasibilities are analyzed with respect to currently available techniques, indicating that our schemes can work well in both the strong and weak (Purcell) coupling regimes. & 2014 Elsevier B.V. All rights reserved.

Keywords: Cluster state Quantum dot inside microcavities Entanglement concentration

1. Introduction Under the motivation to develop large-scale quantum algorithms, a revolutionary computing paradigm, i.e. measurement-based quantum computation (MBQC), was introduced by Raussendorf and Briegel [1], where quantum computation can be performed by implementing local measurements on highly entangled quantum states. Apart from the alternative Afﬂeck–Kennedy–Lieb–Tasaki (AKLT) states [2], cluster states have been considered as the most promising candidate to be the universal resource due to its high connectedness and large persistency of entanglement. Additionally, they can be utilized as a resource for generating other multi-qubit entangled states. Thus, a wide range of quantum information processing (QIP) proposals using cluster states have been proposed [3]. Meanwhile, of particular interest is the generation of cluster states in a variety of systems, including linear optics [4], where the scalability remains a stumbling block due to the need to generate the initial multi-qubit photonic entanglement through, for example, concatenating parametric down conversion process, and cavity QED [5], which often requires the system to operate in the strong-coupling regime, which

n

Corresponding author. Tel.: þ 86 13869826103. E-mail address: [email protected] (Y.-J. Gu).

http://dx.doi.org/10.1016/j.optcom.2014.02.019 0030-4018 & 2014 Elsevier B.V. All rights reserved.

means the vacuum Rabi frequency of the dipole exceeds both the cavity and the dipole decay rates. The scheme that generates cluster states using neutral atoms in optical lattices [6] has also been proposed, which remains to be difﬁcult concerning the lack of individual addressing. Recently, Hu et al. also mulled over the possibility to perform QIP based on endohedral fullerene systems, proposing the direct and indirect methods, respectively, to generate cluster states with arrays of endohedral fullerenes residing in single-walled carbon nanotubes [7]. On the other hand, there have recently been theoretical and experimental breakthroughs into the hybrid quantum system, consisting of a singly charged electron conﬁned in a quantum dot (QD) inside an optical resonant microcavity and the ancillary photons to be the ﬂying qubits. This system has attracted extensively exploration and is considered as a viable candidate for QIP tasks. Within this system, incorporating QDs into solid-state cavities is comparatively easy, which largely enhances its scalability and stability. Besides, the electron-spin coherence time is rather long that has been prolonged to μs range using spin-echo techniques [8–11], which greatly promotes the feasibility of realizing large numbers of unitary operations in QIP. Additionally, the initialization of electron-spin states can be realized by means of optical pumping or optical cooling [12], where, in the case of superposition states, ultrafast optical rotation of the single spin can be employed [13]. The spin-state preparation with a ﬁdelity

J. Zhao et al. / Optics Communications 322 (2014) 32–39

exceeding 99.8% was reported by Press et al. [14]. Ultrafast coherent control over the electron-spin states using picosecond optical pulses were reported, so that a large number of spin rotations could be carried out while the coherence is maintained [13,15–18]. Efﬁcient optical methods for readout of spins in QDs have been developed in recent years [19–21]. In 2008 and 2009, Hu et al. proposed three schemes to entangle remote spins [22], photons [23] and to realize entanglement beam splitter [24], respectively, using the coupled system of QD spins and optical resonant cavities. Subsequently, a variety of technology have been employed resorting to this spinQD-cavity system, including the constructions of universal hybrid quantum gates, namely CNOT gate, Fredkin and Toffoli gates [25,26]. In 2013, Deng et al. proposed the deterministic two-qubit and three-qubit universal quantum gates on photonic circularpolarization qubits [27]. Schemes for efﬁcient entanglement teleportation, swapping [28], concentration and puriﬁcation [29,30] of electron-spin entangled states and quantum repeaters [31] were presented in 2011 and 2012, respectively. Recently, two proposals concerning the generation and complete analysis of hyper-entangled Bell states for photons were addressed assisted by the electron-spin interface in this system [32]. In addition, for utilizing entanglement in real QIP, it is mandatory to take into account the decoherence resulting from the coupling of quantum systems to the environment, which destroys the fragile quantum information rapidly. Two comprehensive methods that depress the effect of noise are quantum errorcorrection code (QECC) [33] and quantum fault tolerant operations [34–36]. However, the entire framework of quantum fault tolerance and the encoding, decoding processes of QECC promise to be comparatively difﬁcult and expensive in practice in terms of the number of ancillary qubits required and the number of physical gates implemented, especially the controlled gates. There exist yet two other more appealing methods when we are concerned with recovering a subset of shared entanglement from a larger number of shared but less entangled states between distant parties. Here, when the initial shared but less entangled states are mixed, this recovery of entanglement is termed as entanglement puriﬁcation or distillation [37]. Alternatively, entanglement concentration copes with the cases that distilling maximally entangled states from less entangled pure states. The importance of such a scheme is obvious as maximally entangled states are essential for various QIP with perfect ﬁdelity. In 1996, C. H. Bennett proposed the original entanglement concentration protocol (ECP) [38] (called Schmidt projection method). The achievement of this proposal requires the collective measurement for the joint state of n pairs of particles, which is difﬁcult to perform in practice, especially for long-distance quantum communication using photons. Subsequently, similar schemes, namely ECPs based on quantum swapping [39] and merely linear optical elements [40], were investigated independently where local measurements on two photons associated with classical communication realize the required projection. The corresponding experiments were reported later in 2008 [41]. To date, assorted breakthroughs have been made tackling entanglement concentration for Bell-class states, GHZ-class states as well as W states, such as the ones based on Nielsen's theorem [42] which realize concentration in a deterministic manner by performing positive-operator-valued measurement locally combined with classical communication, and the ones resorting to cavity QED [43], among which an ECP using photonic Faraday rotation that requires low-Q cavity was presented recently [44]. Various ECPs have been proposed with the help of cross-Kerr nonlinearity, where the nonlinear effect can be used to realize parity-check quantum nondemolition detection (QND) [45]. Moreover, three recent studies showed that ECP(EPP) can be performed more efﬁciently with a higher success probability(deterministically) when hyperentanglement is available [46–48]. For concentration, Ren et al. [48]

33

proposed the parameter-splitting method to extract the maximally entangled photons in both the polarization and spatial degrees of freedom when the coefﬁcients of the initial partially hyperentangled state are known. This fascinating method is demonstrated to be very efﬁcient and simple in terms of concentrating partially entangled state in one degree of freedom, especially a lessentangled polarization state. It can be achieved with the maximum success probability by performing the protocol only once, resorting to linear optical elements only. ECPs for electron-spin entangled Bell states [29,49] and W states [50] were presented, respectively, using quantum-dot spins inside optical microcavities. Currently, most of the protocols aim at concentration for entangled Bell-class, GHZclass states or W states on photonic systems. However, concentration for cluster states on solid-state systems deserves to be developed in the regime of measurement-based quantum computation. Arguably, Choudhury et al. [51] reported the ECP for cluster states based mainly on the property of polarization beam splitter (PBS), where the success probability is relatively low and its postselection nature leads to the fact that once the required state is chosen, the composite state is detected and then destroyed. The ECP for cluster states with the help of cross-Kerr nonlinearity was also proposed [52]. However, it remains a challenge to initially achieve a high order of phase shift at the single photon level, even with electromagnetically induced transparency [53]. Besides, it is currently controversial whether these nonlinearities are sufﬁcient for single-photon quantum applications [54]. The difﬁculties associated with these schemes prevent the realization and utilization of ECP for cluster states in large-scale quantum computation. In this paper, we explore ﬁrstly the deterministic generation of electron-spin entangled cluster states using the hybrid system (spin-QD-cavity system). The double-sided cavity is preferred rather than the single-sided one due to its improved robust and ﬂexible feature, i.e. the reﬂectance for the uncoupled and coupled cavity is not strictly required to be balanceable to obtain high ﬁdelity [28]. The core of the scheme, namely the CPF gate, is constructed deterministically that is the essential building block in a host of QIP. Then, the concentration scheme for four-qubit cluster states is introduced, which is then be generalized to the multiqubit cases. Within the protocol, merely one copy of the initial partially entangled state and one ancillary photon are required. The generation and concentration schemes are proved to work well in both the strong and weak(Purcell) coupling regimes. The paper is organized as follows. In Section 2, we propose a scheme for implementing CPF gate deterministically on electronspin qubits. Then, the generation for cluster states is discussed. In Section 3, we explain our concentration scheme in an ideal situation. The ﬁdelities and experimental feasibilities from a practical point of view are discussed in Section 4, while some discussions and conclusions are shown in Section 5.

2. Schemes for generating cluster states in QD-cavity system Consider a singly charged QD, e.g. a self-assembled GaAs/InAs or GaAs interface QD, placed in a double-sided optical resonant microcavity. As shown in Fig. 1, there are four relevant electronic levels and two optically allowed transitions of the trion X (also called a negatively charged exciton), which are spin dependent due to the Pauli's exclusion principle. This optical property introduces large differences in the phase or the amplitude of the reﬂection and transmission coefﬁcients between the two circularly polarized photons, one involving a sz ¼ þ 1 photon and the other involving a sz ¼ 1 photon. This giant circular birefringence works in both the strong and weak coupling regimes [25,28]. In detail, as Fig. 1 illustrates, the quantization axis for angular momentum is set along the normal direction of the cavity axis, that is, the z-axis.

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J. Zhao et al. / Optics Communications 322 (2014) 32–39

tðwÞ ¼ h

iðwX

h γi κ iðwX wÞ þ 2 ; γ ih κs i wÞ þ iðwc wÞ þκ þ þ g2 2 2

ð3Þ

Thus, when consider the resonant interaction case that wX ¼ wc ¼ w0 , the reﬂection and transmission coefﬁcients for the uncoupled cavity(cold cavity that means g ¼ 0) can be simpliﬁed as κs iðw0 wÞ þ κ 2 ; t ðwÞ ¼ r 0 ðwÞ ¼ ð4Þ 0 κs κs : iðw0 wÞ þ κ þ iðw0 wÞ þ κ þ 2 2 The speciﬁc reﬂection and transmission operator can be deﬁned as [29] t^ ðwÞ ¼ t 0 ðwÞðjR〉〈Rj j↑〉〈↑j þ jL〉〈Lj j↓〉〈↓jÞ þ tðwÞðjR〉〈Rj j↓〉〈↓j þjL〉〈Lj j↑〉〈↑jÞ;

Fig. 1. (a) Schematic diagram for the spin-QD-cavity system which is double-sided and (b) energy levels and spin selection rule for optical transitions of the negatively charged exciton X , consisting of two electrons in the antiparallel spin states bound to one hole. Here j*〉 and j + 〉 represent the spin states of the hole with J z ¼ 7 32 , respectively. The superscript arrow ↑ð↓Þ of the circularly polarized photon states jR↑ 〉 and jL↓ 〉 indicates their propagation directions along(against) the z-axis.

Given that the photon is in the state jR↑ 〉 and jL↓ 〉ðsz ¼ þ 1Þ, the dipole interaction only occurs when the excess electron is in j↑〉ðJ z ¼ þ 12Þ and, in this case, the photon is reﬂected. Since the handedness of the circularly polarized photon changes upon reﬂection while the rotation direction of its electromagnetic ﬁeld remains unchanged, the propagation direction and, thus, the polarization of the input photon are ﬂipped. Otherwise, if the electron spin is in the state j↓〉ðJ z ¼ 12Þ, the photon does not couple to the dipole and is transmitted through the cavity feeling a π mod 2π phase shift relative to the reﬂected photonic state. Similarly, the photonic state jR↓ 〉 and jL↑ 〉ðsz ¼ 1Þ only couple the electron in the spin state j↓〉 to the trion state j↓↑ + 〉 and transmit through the cavity when the spin is j↑〉. The speciﬁc transmission and reﬂection coefﬁcients of this spin-QD-cavity system can be calculated by solving the Heisenberg equation of motion for the cavity ﬁeld operator a^ and X dipole operator s- [55]: h pﬃﬃﬃ pﬃﬃﬃ 0 da^ κs i ^ ¼ iðwc wÞ þ κ þ a^ g s κ a^ in κa^ in þ H; dt 2 h ds γi ^ ¼ iðwX wÞ þ s g sz a^ þ G: 2 dt

ð1Þ

combining with the input–output relations: pﬃﬃﬃ ^ a^ r ¼ a^ in þ κ a; pﬃﬃﬃ 0 ^ a^ t ¼ a^ in þ κa:

ð2Þ

where w, wX and wc are the frequencies of the photon, the cavity and the X transition, respectively; g denotes the coupling strength between the exciton and the cavity mode; γ=2 is the exciton dipole decay rate; κ and κs =2 are the cavity ﬁeld decay rate and the leaky rate; H^ and G^ are the noise operators related to the 0 reservoirs. a^ in ða^ in Þ and a^ r ða^ t Þ are the input and output ﬁeld operators, respectively. Hu et al. proposed the explicit expression of the reﬂection and transmission coefﬁcients of this spin-QD-cavity system, concerning the weak excitation approximation where the charged QD is predominantly in the ground state [29]: rðwÞ ¼ 1 þ tðwÞ;

r^ ðwÞ ¼ r 0 ðwÞðjR〉〈Rj j↑〉〈↑jþ jL〉〈Lj j↓〉〈↓jÞ þ rðwÞðjR〉〈Rj j↓〉〈↓j þ jL〉〈Lj j↑〉〈↑jÞ:

ð5Þ

To explicitly clarify the difference of transmittance and reﬂectance between the uncoupled(cold) cavity with g ¼ 0 and the coupled (hot) cavity with g a 0. Hu et al. studied the reﬂection and transmission spectra versus the frequency detuning w w0 for different coupling strength g, under the resonant interaction and the condition γ ¼ 0:1κ [29]. It is indicated that, in the strongcoupling regime g 4 ðκ; γÞ, when the main cavity decay rate signiﬁcantly outweighs the side leakage rate(κ⪢κ s ), the coefﬁcients jrðwÞj-1 and jtðwÞj-0 for the hot cavity while jr 0 ðwÞj-0 and jt 0 ðwÞj-1 for the cold cavity at w C w0 . So the rules governing the spin-dependent transition in this case can be described as follows [22]. jR↑ ; ↑〉-jL↓ ; ↑〉 ↓

jL↓ ; ↑〉-jR↑ ; ↑〉

↓

jL ; ↓〉- jL ; ↓〉; jR↓ ; ↓〉-jL↑ ; ↓〉 ↑

↑

ðsz ¼ þ 1Þ;

jL↑ ; ↓〉-jR↓ ; ↓〉

jL ; ↑〉- jL ; ↑〉;

jR↑ ; ↓〉- jR↑ ; ↓〉 jR↓ ; ↑〉- jR↓ ; ↑〉

ðsz ¼ 1Þ;

ð6Þ

In the following, we present how the electron-spin entangled cluster states can be deterministically generated using this system, where we neglect the effect of side leakage and coupling strength ﬁrst but come back to it later. In general, cluster states can be ða þ 1Þ written in the form jϕN 〉 ¼ 1=2N=2 N þ j1〉a Þ, where a ¼ 1 ðj0〉a sz þ 1Þ siz ¼ ðj0〉i 〈0j j1〉i 〈1jÞ with the convention sðN 1 [1]. It can be z generated by ﬁst, initializing all the pelectron-spin qubits in the ﬃﬃﬃ superposition state: j þ 〉e ¼ ðj0〉 þ j1〉Þ= 2 and then, by performing the controlled phase ﬂip gate (CPF i;i þ 1 ) on each pair of neighboring qubits with the control qubit the ith one and the target qubit the ði þ 1Þ th one. Within this process, the transformation: jx〉jy〉-ð 1Þxy jx〉jy〉ðx; y ¼ 0; 1Þ is realized. In our case, we deﬁne j↑〉 ¼ j0〉 and j↓〉 ¼ j1〉 for the electron-spin qubit and the initial state j þ〉e can be prepared by optical pumping or cooling followed by single spin rotations using nanosecond microwave pulses or picosecond-scale optical pulses [12,13]. The realization of the CPF gate assisted by a ﬂying photon is depicted in Fig. 2 and this scheme can be used directly to prepare the two-qubit cluster states:jϕ2 〉 ¼ 12ðj0〉1 sð2Þ z þ j1〉1 Þðj0〉2 þ j1〉2 Þ. The p inputting single photon prepared in the state ﬃﬃﬃ jþ 〉p ¼ 1= 2ðjR〉þ jL〉Þ is injected into the ﬁrst cavity to interact with the electron(as shown in Eq. (6) whose state represents the control qubit. After the reﬂection and transmission, the combined state of the electron and the photon evolves as 1 1 jΨ 〉1 ¼ ½j↑〉 ðjL〉2 jL〉1 Þ þ j↓〉 ðjR〉2 jR〉1 Þ pﬃﬃﬃðj↑〉 þ j↓〉Þ; 2 2

ð7Þ

where the subscripts of the photonic states represent certain spatial modes. Then, the emitting photon is splitting into two

J. Zhao et al. / Optics Communications 322 (2014) 32–39

35

The whole process can be formatively illustrated as N 1

1

i¼2

2N=2

jφ〉N ¼ ∏ CPF i;i þ 1

Fig. 2. Schematic setup for constructing the CPF gate on stationary electron-spin qubits. S1 and S2 are two optical switches leading the photon to speciﬁc spatial mode. The polarizing beam splitter in the circular basis(c-PBS) transmits the right-circularly polarized photon and reﬂects the left-circularly one. HWP1 is the half-wave plate which is set to 22.51 to realize the Hadamard gate on photonic states while HWP2 is set to implement operation jR〉-jL〉 and the inverse process. π phase shifters are included in spatial modes 2 and 5 to make the passing photon pick up a π phase.

spatial modes, where the photon in spatial mode 2 picks up a π phase until it reaches c-PBS1. Next, the photon in state jL↑ 〉3 and jL↓ 〉3 is lead back to the ﬁrst cavity by using S1 and S2 (dashed line with a half-wave plate to implement the Hadamard operation). The whole state of the system at this time can be expressed as 1 1 j↑〉 jL〉8 j↓〉 ðjR〉4 jL〉4 Þ ðj↑〉 þj↓〉Þ: jΨ 〉2 ¼ ð8Þ 2 4 In the following, the photon in spatial mode 4 passes through c-PBS2 and is separated into two trajectories: jL〉5 , propagating in spatial mode 5 and feeling a phase shift π and jR〉6 , propagating in mode 6. It is then interacted with the electron conﬁned in the second cavity. Subsequently, c-PBS3 combines the emitting photon in two spatial modes and two Hadamard operations are performed before and after c-PBS4 to accomplish the transformation: 1 jR〉2pﬃﬃﬃðjR〉 þjL〉Þ 2

;

1 jL〉2pﬃﬃﬃðjR〉 jL〉Þ: 2

ð9Þ

Then, the whole state turns to be 1 jR〉 ½j↑〉 ðj↑〉 þ j↓〉Þ þ j↓〉 ðj↑〉 j↓〉Þ 2 1 þ jL〉 ½j↑〉 ðj↑〉 þ j↓〉Þ j↓〉 ðj↑〉 j↓〉Þ; 2

N i ¼ 1 ðj↑〉 þj↓〉Þi :

ð12Þ

Given the more universal MBQC, the two-dimensional(2D) cluster state is intriguingly more preferable than the one-dimensional(1D) one. It is noteworthy that by applying similar transitions from 1D to 2D cluster states as sketched in [7], our scheme can be directly generalized to generate 2D cluster states assisted by merely one auxiliary photon. Taking a concrete example, an equivalent 2 4 cluster state can be prepared in two steps. First, all the electronic spins are entangled in a 1D cluster state as presented above. Second, implement CPF i;i þ 3 ði ¼ 1; 3; 5Þ sequentially on the non-nearest neighboring qubits assisted by a ﬂying photon which can be reused in every round as Fig. 2. Therefore, by cascading the 1D entangling process and appropriate CPF gates, we can deterministically obtain the 2D M M cluster states.

3. Entanglement concentration protocol for cluster states based on the photon-spin interface Apart from the various entanglement concentration protocols in photonic systems, the ECP for matter-qubit systems is rather signiﬁcant in depressing the effect of decoherence in quantum computation. In this section, we propose an ECP for four-qubit electron-spin cluster states at ﬁrst, which requires only a single copy of the partially entangled state initially. In the whole concentration process, only local operations are performed by merely one of the distant parties. Next, we generalize this scheme to the n-qubit ðn 4 4Þ case. Consider that the four electrons inside QDs are in the less-entangled cluster state: jφ〉i ¼ αðj↑〉j↑〉j↑〉j↑〉 þ j↑〉j↑〉j↓〉j↓〉ÞBCDA þ βðj↓〉j↓〉j↑〉j↑〉 j↓〉j↓〉j↓〉j↓〉ÞBCDA ; ð13Þ 2

2

where 2jαj þ 2jβj ¼ 1 and ,without loss of generality, we suppose jαjo jβj. The subscripts B, C, D, A denote the four remotely located owners of spin states, respectively. The schematic diagram of concentration is shown in Fig. 3. One of the remote parties (e.g. Alice) prepares a single photon in the state j þ〉p and injects it into her cavity from the input port. Now the composite state of the system consisting of four excess electrons and a single photon is transformed from the initial one

jΨ 〉3 ¼

ð10Þ

and the photon is, then, detected by detectors D1 and D2. If D2 ﬁres, the composite system collapses into the cluster state, which means the CPF operation is completed. Otherwise, if D1 ﬁres, a phase shifter Δφ ¼ π is implemented on the electron spin to accomplish the CPF gate. Thus, no matter which detector ﬁres, the electronspin state will collapse into a two-qubit cluster state deterministically jΨ 〉f ¼

1 ðj↑↑〉 þ j↑↓〉 þ j↓↑〉 j↓↓〉Þ 2

ð11Þ

and the CPF gate between two adjacent excess electrons is realized. As for the generation for multi-qubit cluster states, we ﬁrst implement Hadamard operation on the photon being detectedto re-prepare it in the state jþ 〉p and then, inject it into the ith and (iþ1)th cavity in sequence according to the scheme in Fig. 2. Then we repeat the same process on the (iþ1)th and (iþ2)th electron spins, and so on, until a multi-qubit cluster state is obtained ﬁnally.

Fig. 3. Schematic setup for concentrating the four-qubit cluster states. HWP is the half-wave plate used to realize the unitary rotation of the single ancillary photon, while two c-PBSs are placed before the four single photon detectors to spilt the photon trajectories.

36

J. Zhao et al. / Optics Communications 322 (2014) 32–39

jφ〉i to jφ〉1 . Here pﬃﬃﬃ pﬃﬃﬃ jφ〉1 ¼ α= 2ðj↑↑〉 þ j↓↓〉Þ j↑↑〉jR↓ 〉 þ α= 2ðj↑↑〉 þj↓↓〉Þ j↑↑〉jL↓ 〉 pﬃﬃﬃ pﬃﬃﬃ þ β= 2ðj↑↑〉 j↓↓〉Þ j↓↓〉jL↓ 〉 þ β= 2ðj↑↑〉 j↓↓〉Þ j↓↓〉jR↓ 〉; ð14Þ After the photon–electron interface, it evolves as pﬃﬃﬃ pﬃﬃﬃ jφ〉1 ¼ α= 2ðj↑↑〉 þ j↓↓〉Þ j↑↑〉jR↓ 〉 þ α= 2ðj↑↑〉 þ j↓↓〉Þ j↑↑〉jR↑ 〉 pﬃﬃﬃ p ﬃﬃﬃ β= 2ðj↑↑〉 j↓↓〉Þ j↓↓〉jL↓ 〉 þ β= 2ðj↑↑〉 j↓↓〉Þ j↓↓〉jL↑ 〉: ð15Þ Then, both the reﬂected and transmitted photon trajectories feel a unitary rotation 1 1 jR〉-qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðβjR〉 þ αjL〉Þ; jL〉-qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðαjR〉 βjL〉Þ α2 þ β 2 α2 þ β 2

ð16Þ

and pass through two c-PBSs which split the left- and rightcircularly polarized photon components into different spatial modes before they are detected. Either the detector D2 or D4 ﬁres, they will get jΨ 〉 ¼ 12 ðj↑↑↑↑〉 þ j↓↓↑↑〉 þj↑↑↓↓〉 j↓↓↓↓〉Þ; 2

ð17Þ

1 jΨ 0 〉 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðα2 j↑↑↑↑〉 þ α2 j↓↓↑↑〉 β2 j↑↑↓↓〉 þβ2 j↓↓↓↓〉Þ: 2ðα4 þ β4 Þ

ð18Þ

In this case, the concentration fails with a probability P 1f ¼ 4jαj4 þ 4jβj4 . Here, the subscript s(f) means the success (failure) probability, while the superscript 1 represents the ﬁrst round. It is interesting to ﬁnd that, in this case, the concentration process does not truly fail, since the state in Eq. (18) is actually a lessentangled cluster state similar to that in Eq. (13). Thus, it is still possible to obtain a maximally entangled cluster state by repeating the concentration process for a second round. In detail, Alice performs a Hadamard operation on the photon being detected to rotate it to the state j þ 〉p and leads it enter the microcavity to couple with the electron spin again. Under the same procedure as discussed above, if the transmitted or reﬂected photon is detected by detector D2 or D4, the concentration step succeeds with a probability 8jαj4 jβj4 ; ðjαj4 þ jβj4 Þ

ð19Þ

where P 20 s means the probability that the ﬁrst round fails but the second round succeeds. Similarly, if D1 or D3 is clicked, the failure probability can be calculated as P 2f ¼ P 1f P 20 f ¼

4ðjαj8 þjβj8 Þ ; jαj4 þ jβj4

ð20Þ

where P 20 f represents the probability that both the ﬁrst and second round fail. At this time, Alice can restart the aforementioned steps and distill the maximally entangled state in the next round yet. Thus, by iterating the concentration process n times, the total success probability of our ECP is P s ¼ ∑P ns ¼ 8jαj2 jβj2 þ ∑ n

2n

2n

8jαj jβj

2m n n ¼ 2∏m ¼ 2 ðjαj

jφ〉2N ¼ αðj↑↑↑↑〉⋯j↑↑↑↑〉þ j↑↑↓↓〉⋯j↑↑↑↑〉 þ j↑↑↓↓〉⋯j↓↓↑↑〉 þ ⋯ þ j↓↓↓↓〉⋯j↓↓↑↑〉Þ123⋯N þ βðj↑↑↑↑〉⋯j↑↑↓↓〉 þ j↑↑↓↓〉⋯j↑↑↓↓〉 þ j↑↑↓↓〉⋯j↓↓↓↓〉 þ ⋯ þ j↓↓↓↓〉⋯j↓↓↓↓〉Þ123⋯N :

ð22Þ

Here, the subscripts represent the N QD spins inside in neighboring optical microcavities, respectively. Similar to the process of four-qubit cluster state concentration, one of the distant parties (e.g. Alice) should, ﬁrstly, prepare an ancillary photon initialized in state j þ 〉p and inject it into her cavity; subsequently, she performs the unitary rotation on photon in different spatial modes and, ﬁnally, detects the photon. Compared with other ECPs [30–38], our protocol has many attractive virtues: ﬁrst, we need only one single copy of a less-entangled cluster state as the initial resource, which is more efﬁcient than the ECPs based on Schmidt projection method [38–42]; second, the success probability can reach near unity when iterating the protocol for four times; third, only one photonic qubit is demanded to be the ancilla which can be reused in each round of iterations.

2

with a success probability P s ¼ 8jαj jβj and the concentration process is completed. Otherwise, if D1 or D3 ﬁres, the whole state will collapse into

P 2s ¼ P 1f P 20 s ¼

It is clear that this scheme can be directly extended to extract maximally multi-qubit entangled cluster state from a lessentangled one with even number of qubits. For example, suppose the initial partially entangled cluster state has the following form:

m

þ jβj2 Þ

:

ð21Þ

Clearly, the success probability of our protocol (shown in Fig. 6(a)) alters with the initial coefﬁcient of smaller magnitude. When the deviation between α and 1/2 is not signiﬁcantly dramatic, the total success probability is near unit, which means the present protocol is nearly deterministic.

4. Fidelities and experimental feasibility In this section, we should brieﬂy mention the facts relevant from the experimental viewpoint. Here, a promising system with GaAs- or InAs-based QDs in a double-sided optical resonant microcavity is utilized, which supports the circularly polarized light and has negligible mode mismatching between the ﬂying and cavity photons [56]. In the practical cases, where the cavity side leakage, the coupling strength and other factors are taken into consideration, we should use complete operator formalism shown in Eq. (5), which implies that the dynamic of the electron-photon interface in Eq. (6) should be modiﬁed as jR↓ ↓〉-jrjjL↑ ↓〉 þ jtjjR↓ ↓〉; jL↑ ↓〉-jrjjR↓ ↓〉 þjtjjL↑ ↓〉; jR↑ ↑〉-jrjjL↓ ↑〉 þ jtjjR↑ ↑〉; jL↓ ↑〉-jrjjR↑ ↑〉 þjtjjL↓ ↑〉; jR↓ ↑〉- jt 0 jjR↓ ↑〉 jr 0 jjL↑ ↑〉; jL↑ ↑〉- jt 0 jjL↑ ↑〉 jr 0 jjR↓ ↑〉; jR↑ ↓〉- jt 0 jjR↑ ↓〉 jr 0 jjL↓ ↓〉; jL↓ ↓〉- jt 0 jjL↓ ↓〉 jr 0 jjR↑ ↓〉:

ð23Þ

Thus, by substituting the changes of input photon presented in Eq. (23) for that in Eq. (6), the generated cluster state in our scheme described in Eq. (11) turns to be 2η þ 2μ þ χ ðjrj2 jt 0 j þ jt 0 j3 Þ ν þ ω pﬃﬃﬃ þ ðjtj þ jrjÞ jΨ 〉p ¼ j↑↑〉 8 8 4 2 0 2η þ2μ0 þχ 0 ðjrj2 jt 0 j þ jt 0 j3 Þ ν þ ω pﬃﬃﬃ þ ðjt 0 j þ jr 0 jÞ j↑↓〉 þ 8 8 4 2 0 2 3 2η þ 2μ þχ ðjr 0 j jtj þjtj Þ ν ω0 pﬃﬃﬃ þ ðjtj þ jrjÞ þ j↓↑〉 8 8 4 2 0 2η þ 2μ þχ ðjr 0 j2 jtj þjtj3 Þ ν ω0 pﬃﬃﬃ ðjt 0 jþ jr 0 jÞ ; j↓↓〉 8 8 4 2 ð24Þ where χ ¼ ðjtj þ jtjÞðjrr 0 j jtt 0 jÞ; ν ¼ ðr 2 jtj t 20 jtj 2jrtt 0 jÞ; η ¼ ðr 2 jtj þ jr 0 t 0 tjÞ; μ ¼ ðjrjt 2 þ jrr 0 t 0 jÞ; ω ¼ ðr 2 þt 20 Þjr 0 j; and χ 0 ¼ ðjt 0 jþ jr 0 jÞðjrr 0 j jtt 0 jÞ; ν0 ¼ ðr 20 jt 0 j t 2 jt 0 j 2jr 0 tt 0 jÞ; η0 ¼ ðjrr 0 tj þ jr 0 jt 20 Þ; μ0 ¼ ðjrtt 0 j þ r 20 jt 0 jÞ; ω0 ¼ ðr 20 þ t 2 Þjrj; We denote F ¼ jΨ 〉0p as the normalized state of jΨ 〉p . Thus, the ﬁdelity can be deﬁned as F ¼ jf 〈Ψ jΨ 〉0p j2 , where F ¼ jΨ 〉f and F ¼ jΨ 〉0p represent the ﬁnal state of generation protocol in the

J. Zhao et al. / Optics Communications 322 (2014) 32–39

Fig. 4. Calculated the ﬁdelity of the CPF gate and the two-qubit cluster state generation in our scheme versus the coupling constant with respect to different side leakage rates (κ s =κ ¼ 0 dash, κ s =κ ¼ 0:05 solid, κ s =κ ¼ 0:1 dot, κ s =κ ¼ 0:2 dashdot). The resonant interaction is considered here other than the dispersive interaction, that is, wc ¼ wX ¼ w0 and γ=κ ¼ 0:1 is taken by considering the practical QD-cavity parameters (i.e. experimentally γ is about several μeV) [32,48].

ideal and practical cases, respectively. Similarly, the ﬁdelity of the concentration protocol is calculated as F con ¼

jtjðα2 þ β2 Þ 2jt 0 jαβ 4½ðjtjα2 jt

0 jαβÞ

2

þðjtjβ2 jt 0 jαβÞ2

:

ð25Þ

The relation between the ﬁdelity of our generation protocol and the cavity coupling strength is plotted in Fig. 4, where the side leakage is taken into account. It is clear that a low side leakage rate is highly required in our protocol, whereas the protocol can achieve high ﬁdelity in both the strong and weak coupling regimes. As shown in [28], the cavity side leakage can be decreased by thinning down the top mirrors of the high-Q micropillars, which increases κ while κ s remains approximately unchanged. κ s ¼ 0:05κ was also demonstrated possible in a pillar microcavity with the quality factor reaching 1:65 105 [42,56]. Besides, κ s =κ ¼ 0:2 is also achievable with quality factor and coupling strength of the micropillar cavity arriving at Q 4:3 104 and g=ðκ þ κs Þ C 0:58, respectively [32]. It is worth pointing out that g and κ can be controlled independently, for the coupling strength is determined by the trion oscillator strength and the cavity modal volume, while the side leakage and cavity loss rate by the quality factor only [28]. Thus, it is possible to reduce κ, whereas remaining the system in the strong coupling regime. We notice that recent experiments are quite promising, where the strong coupling was obtained with g ¼ 16 μeV in a 7:3 μm diameter micropillar with a relatively small side leakage [57]. For our protocol, in the regime g=κ ¼ 0:5 and κs =κ ¼ 0:05, the cluster state generation ﬁdelity can reach F gen ¼ 67:4%, while in the case that g=κ ¼ 2:4, the ﬁdelity can reach F gen ¼ 87:56% and F gen ¼ 75:79% for κ s =κ ¼ 0:05 and κ s =κ ¼ 0:1, respectively. If the cavity leakage can be neglected, the ﬁdelity can reach near unity (F gen ¼ 98:79%) in the strong coupling regime and high values (F gen ¼ 80:26%) in the Purcell regime(g=κ ¼ 0:5). Through calculation, we found that the performance of our ECP for cluster states varies relative to the cavity coupling strength, whereas the effect of the cavity side leakage on the ﬁdelity is so slight that can be neglected. Thus, we employ the coupling constant g=κ to test the performance of concentration using the coupled system in realistic cases. Fig. 5 illustrates the

37

Fig. 5. The ﬁdelity of entanglement concentration vs 2jαj2 for different coupling strength, where α is the smaller coefﬁcient of the initial less-entangled cluster state. (g=κ ¼ 0:5 solid, g=κ ¼ 0:75 dash, g=κ ¼ 1:0 dot, g=κ ¼ 2:0 dash-dot). Similarly, wc ¼ wX ¼ w0 and γ=κ ¼ 0:1 are taken here.

concentration ﬁdelity vs twice the modulus square of the smaller initial coefﬁcient of the less-entangled cluster state when different g=κ are chosen. Interestingly, the scheme demonstrates excellent ﬁdelity in both the weak and strong coupling regimes. More speciﬁcally, the ﬁdelity stably remains more than 90% when the coefﬁcient 2jαj2 ranges from 0.02 to 0.5 and 0.25 to 0.5 in the strong and weak coupling regimes, respectively. The success probability of our ECP in realistic cases is illustrated in Fig. 6(b) and (c). It is shown that the side leaky rate and the coupling strength only exert slight inﬂuence on the probability, which can, thus, be neglected. In result, the concentration scheme can work ideally in real experiments regardless of the conditions of coupling strength and cavity leakage rate. Nevertheless, it is still necessary to analyze the feasibility of strong-coupling regime, especially under the consideration that larger g=κ could enable near unity ﬁdelity to be achieved. In real experiments, it is easy to observe weak coupling in spin-QD-cavity systems, while strong coupling, though more challenging, has been achieved in various microcavities and nanocavities [57–61]. g=ðκ þ κ s Þ C0:5 was achieved for micropillar microcavities with d ¼ 1:5 μm and a quality factor Q ¼ 8:8 103 [58]. The strong coupling regime, 4g=κ ¼ 3, was reported recently within a single InGaAs annealed quantum dot with d ¼ 7:3 μm and the cavity spectral width κ ¼ 20:5 μeVðQ ¼ 6:5 104 Þ [57]. Besides, g=ðκ þ κs Þ C 2:4 was demonstrated to be observable for spin-QD-cavity systems with micropillars around 1:5 μm, coupling strength g ¼ 80 μeV and quality factor more than 4 104 (corresponding to κ ¼ 33 μeV) [59]. The effects of other factors that decrease the ﬁdelity have been brieﬂy discussed by Hu et al. in [24], including the electron-spin decoherence, the trion dephasing and the imperfect optical selection rule due to the heavy-light hole mixing, which are demonstrated to reduce the ﬁdelity by only a few percent, respectively. In self-assembled In(Ga)As QDs, the optical coherence time of excitons can be as long as several hundred picoseconds [62], which is ten times longer than the cavity photon lifetime(usually around tens of picoseconds), while the trion coherence time of the excitons can reach T h2 4 100 ns [63], which is at least three orders of magnitude longer than the cavity photon lifetime. These facts result in that either the optical dephasing or the spin dephasing can slightly reduce the ﬁdelity by only a few percent. As for the heavy-light hole mixing [64], it could be reduced by engineering the shape and size of QDs or choosing different types of QDs. We thus prove that the current schemes can achieve high ﬁdelity and

38

J. Zhao et al. / Optics Communications 322 (2014) 32–39

Fig. 6. (a) The relationship between the success probability of the present ECP and twice the modulus square of the initial coefﬁcient of smaller magnitude in ideal case with no leakage. The solid, dash-dot-dot, dashed, dotted and dash-dot lines represent, respectively, the success probability of our protocol when iterating our concentration protocol for six times, once, twice, three and four times. (b) and (c) denote the success probability in the ideal (solid line) and practical (dotted line) cases when the detectors D3 and D4 ﬁre or D1 and D2 ﬁre in Fig. 3. Here, we assume wc ¼ wX ¼ w0 and γ=κ ¼ 0:1. By considering the experimental feasibility, different QD-cavity parameters g=κ ¼ 2:4 and κ s =κ ¼ 0:5(g=κ ¼ 1:0 & κ s =κ ¼ 0:7; g=κ ¼ 0:5 & κ s =κ ¼ 0:25) are taken.

success probability within the current experimental technology in both the strong- and weak-coupling regimes. Moreover, it is worth investigating the feasibility of manipulations on electron-spin state using currently available experimental techniques that are required in the generation and entanglement concentration schemes. The technology, including electron-spin initialization, manipulation and detection on timescales much shorter than the spin coherence time, has been improved dramatically currently [8–21]. Fast initialization of the electron-spin state in QD by optical cooling with the initialization rate of order 5 108 s 1 was demonstrated [12]. The spin-state preparation with a ﬁdelity exceeding 99.8% was reported in [14] using resonant excitation of the charged QD transitions and the heavy-light hole mixing. Within the spin coherence time ( μs), about 105 operations on spin states can be performed [15]. The arbitrary coherent rotations of the spin states in QD were completed on the picoseconds timescale using ultrafast optical pulses [13] or periodic optical pumping [16]. A coherent beam splitter, the initialization, readout and unitary manipulation of a single spin, based on GCB, were reported, where the spin rotation can be achieved by tuning the assisted photon frequency [29]. The quantum nondemolition (QND) measurement of a single spin in QD was also observed [19–21].

5. Discussions and conclusions In summary, we have proposed a scheme for implementing the CPF gate deterministically based on the giant optical circular birefringence induced by quantum dot spins inside double-sided optical microcavities, resulting from the spin selective photon reﬂection from the cavity. Consequently, the 1D and 2D multiqubit electron-spin entangled cluster states are generated by concatenating the CPF gate properly. This generation schemes can be achieved assisted by only one single ancillary photon with high ﬁdelity in both ideal and realistic cases. Then, the entanglement concentration protocols for cluster states (4 qubits as well as n qubits where n 4 4) were proposed. Different from most of the existing ECPs which are, in essence, based on Schmidt projection method [38–42], the current scheme only need a single-copy of the less-entangled state initially. Besides, its success probability can reach near unity by iterating the protocol for four times. In the process of iteration, the less-entangled cluster state is not destroyed and the entanglement is conserved. Besides, only one single photon is required to be the ancilla. Numerous calculation denotes that the present schemes can be realized with high

J. Zhao et al. / Optics Communications 322 (2014) 32–39

ﬁdelity and are feasible from the experimental viewpoint in both strong- and weak-coupling regimes. The present generation and concentration schemes for cluster states take advantage of the hybrid system which are intriguingly promising for scalable quantum communication and computation, for example, to build a quantum network. In this regime, photons are regarded to be excellent candidates for long-distance transmission, while the solid states are preferred for local storage and processing. Compared with the currently proposed CNOT gate [25] and universal quantum gates [26] between ﬂying photonic qubits and stationary electron-spin qubits based on the spin-QD-cavity system, the CPF gate presented here can realize the controlled operation between electron spins, which may be of signiﬁcant usage in solid-state QIP. Thus, we argue that the aforementioned two schemes, resorting to the spin-QD-cavity system, have potentially practical applications for scalable QIP. Note Added: After the completion of our work, we learned of a similar result by H. R. Wei and F. G. Deng, who construct the deterministic controlled-not gate on stationary qubits using quantum-dot spins inside double-sided optical microcavities. This scheme removes the use of two optical switches required in our CPF gate and can be used to achieve scalable quantum computing [65].

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