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A hybrid-system approach for W state and cluster state generation
Optics Communications 310 (2014) 166–172
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A hybrid-system approach for W state and cluster state generation Xin Tong a,b, Chuan Wang a,b,n, Cong Cao a,b, Ling-yan He a,b, Ru Zhang a,b 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
art ic l e i nf o
a b s t r a c t
Article history: Received 29 May 2013 Received in revised form 28 July 2013 Accepted 29 July 2013 Available online 12 August 2013
Here we demonstrate an efficient scheme for W state and cluster state generation based on the hybridsystem where nitrogen-vacancy centers are coupled to microcavities. In our proposed scheme, entanglement between solid-state qubits can be generated with the assistance of an ancillary photon. We also discuss the efficiency of the entanglement generation protocols and generalize them to multiqubit cases. With current and near-future technology, the entanglement of nitrogen-vacancy centers can be achieved and our scheme can further be used for quantum information processing and long-distance quantum communication. & 2013 Elsevier B.V. All rights reserved.
Keywords: W state generation Cluster state generation Nitrogen-vacancy centers Microtoroidal resonator
1. Introduction Quantum entanglement is an essential resource for the implementation of quantum communication [1,2] and quantum information processing (QIP) [3–5], such as quantum teleportation [6–8] and quantum computation [9–11]. Over the past few decades, much effort has been devoted to the generation of entanglement, especially for multi-particle entangled states in different quantum systems [12,13]. Among these works, generation of W state [14–16] and cluster state [17] are current topics in experiments. Recently, there have been several theoretical protocols for W state and cluster state generation by different systems, such as optical systems [18], atomic ensembles [19], trapped ions [20], cavity quantum electrodynamics (QED) [21,22] and so on [23]. One of the promising candidates for QIP is the cavity QED system [24–27]. Here we consider a nitrogen vacancy (N-V) center and a microtoroidal resonator (MTR) coupled system. The N-V center [28–32] consisting of a nearest neighbor pair of a nitrogen atom substituted for a carbon atom and a lattice vacancy has been considered as an outstanding candidate for QIP due to its long electronic spin decoherence time at room temperature [33]. The energy level of the N-V center can be controlled by electromagnetic field or optical pulse [34]. It seems that the resonant zero phonon line (ZPL) relevant to the emitted photons from the N-V center could be improved by confining N-V centers in cavities [35]. These N-V centers could be read out rapidly with high fidelity in n Corresponding author at: School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China. Tel.: +86 10 62282050. E-mail addresses: [email protected], [email protected] (C. Wang).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.083
experiment, which makes entanglement generation based on N-V centers achievable in practice [36]. Besides that, MTR exhibits high Q factor, excellent scalability and small mode volume [37]. The novel solid-state cavity system combines the advantages of the N-V center having exceptional spin properties with the microtoroidal resonator possessing ultra high quality factor and small mode volume [38,39]. Owing to the outstanding feature of solid-state cavity QED system, much theoretical and experimental effort has been devoted to the QIP process related to this field [40,41]. In 2008, Dayan et al. experimentally realized the single photon input– output process using the model consisting of a microtoroidal resonator (MTR) coupled to an optical fiber [42]. In 2009, An et al. proposed some schemes for quantum information processing with Faraday rotation using single photon input–output process, which is much different from the high-Q cavity and strongcoupling cases [43]. In 2011, Chen et al. proposed a potentially practical scheme to entangle negatively charged N-V centers in distant diamonds [44]. In 2012, Zheng et al. proposed a scheme to generate N-qubit GHz state with distant N-V centers confined in spatially separated photonic crystal nanocavities via input–output process of photons [45]. In 2013, Cheng et al. demonstrated efficient schemes of deterministic entanglement generation and quantum state transfer with the N-V centers in diamond confined in separated microtoroidal resonators via single photon input– output process [46]. Compared with previous proposals, our new method has the similar superiority and some improvements. Firstly, the quantum information is encoded in the electronic spin ground states of the N-V center, which is convenient to store the generated entanglement. Secondly, owing to the characteristic of N-V centers, the spin state has long decoherence time, which is
X. Tong et al. / Optics Communications 310 (2014) 166–172
easy to be manipulated even at room temperature. Thirdly, our scheme can work well on the condition that the N-V center is coupled to low-Q MTR. The successful entanglement generations are still available under weak coupling conditions and large cavity damping rate. In recent years, this system consisting of N-V centers coupled to MTR has been widely used in many aspects, such as entanglement purification of separate N-V centers [47], entanglement concentration for multi-particle entanglement [48], universal quantum gates with N-V centers in diamond [49]. This paper is organized as follows: in Section 2, we present the physical principle of our scheme by explaining the photon input– output process of the N-V center and MTR coupled system. In Section 3, we put forward a scheme based on the solid-state system to generate W state among three N-V centers and then generalize it to an arbitrary multi-qubit case. We analyze the fidelity of the generated W state at the end of this part. In Section 4, we propose a scheme to generate cluster state between two remote N-V centers and then extend this protocol to N-particle cases. At the end of this part, we analyze the fidelity of the cluster states. In Section 5, we analyze experimental feasibility of our scheme, then give our conclusion at last.
2. The input–output process of the N-V center and MTR coupling system The energy level structure of the N-V center is illustrated in Fig. 1, which can be represented by a three-level Λ-type atom [50,51]. The N-V center is made up of a substitutive nitrogen atom and an adjacent vacancy in diamond. Actually, the electronic spin state of the N-V center is composed of spin triplet 3A ground state and spin triplet 3E excited state [52]. The ground state 3A is split into the upper levels j3 A; ms ¼ 7 1〉 and the lower level j3 A; ms ¼ 0〉 by 2.88 GHz owing to spin–spin interaction. The N-V center has six excited states because of the effect of electronic spin–orbit interaction and spin–spin interaction [53,54]. Among them, jA2 〉 state is quite special, which can decay to the ground state j3 A; ms ¼ þ 1〉 and j3 A; ms ¼ 1〉 with the radiation of left (L) and right (R) circularly polarized photon, respectively. As shown in Fig. 1, we encode the quantum information on the ground states j1〉 ¼ j3 A; ms ¼ 1〉 and j0〉 ¼ j3 A; ms ¼ þ 1〉. We can obtain a Λ-type transition by choosing the excited state jA2 〉 as auxiliary level je〉 because of its stable symmetric properties [55]. Here one optical transition is allowed between j1〉 ¼ j3 A; ms ¼ 1〉 and je〉, which is resonantly coupled to the left polarized photon jL〉. The other optical transition is between j0〉 ¼ j3 A; ms ¼ þ 1〉 and je〉, which is resonantly coupled to the right polarized photon jR〉. Under the Jaynes–Cummings model, the Hamiltonian of the system can be
A2
|e>
|L>
|R>
167
expressed as H ¼ H sys þ H bath þ H int
ð1Þ
here Hsys represents the interaction between the MTR cavity and the N-V center hω i j0 ð2Þ H sys ¼ ∑ sjz þ ωjc a†j aj þ ig j aj sjþ aþ j sj j ¼ R;L 2 and Hbath represents the Hamiltonian of the input field Z 1 † ωbj ðωÞbj ðωÞ dω H bath ¼ 1
ð3Þ
The interaction Hamiltonian between the external photon and the MTR field can be described as Z 1 † H int ¼ ∑ iGðωÞ½ibj ðωÞaj ibj ðωÞa†j dω ð4Þ j ¼ R;L
1
a†j
and aj are the creation and annihilation operators of the here MTR cavity which satisfy the relation ½aj ; a†j ¼ 1. ω0 is the transition frequency of the N-V center electronic energy levels and ωc is the frequency of the MTR cavity field. sz , sþ and s are the inversion, raising and lowering operators of the N-V center between the ground levels and the excited level, respectively. GðωÞ is the coupling constant between the external photon and the †
N-V center. bj ðωÞ and bj ðωÞ are the creation and annihilation operators of the external photon. We introduce a single photon pulse with the frequency of ωp to enter the N-V center and MTR cavity coupling system, which is described in Fig. 2. Considering the rotating frame with respect to the frequency of the input pulse, we can get the quantum Langevin equations for operator of MTR cavity field and the lowing operator of N-V center [56]: κi pffiffiffi daj h ¼ i ωp ωc aj ðt Þgsj ðt Þ κ aj;in ðt Þ; ð5Þ 2 dt γi dsj h ¼ i ωp ω0 sj ðt Þgsj;z ðt Þaj ðt Þ; 2 dt
ð6Þ
here g represents the coupling strength between the N-V center and the microtoroidal resonator, κ is the damping rate of the cavity and γ denotes the spontaneous emission of the N-V center. The input and output fields of the cavity are related by the intracavity field as pffiffiffi aj;out ðtÞ ¼ aj;in ðtÞ þ κ aj ðtÞ ð7Þ Considering the relations above, we can adiabatically eliminate the cavity mode and solve the reflection coefficient as follows: h κ ih γi i ω0 ωp þ þ g2 i ωc ωp aout ðtÞ 2 ih 2i ¼h : ð8Þ r ωp ¼ κ γ ain ðtÞ i ω0 ωp þ þ g2 i ωc ωp þ 2 2 The coefficient for the uncoupled system can be described as κ i ωc ωp 2 ð9Þ r 0 ωp ¼ κ; i ωc ωp þ 2 by choosing g¼ 0. Here we assume that ωp ¼ ωc ¼ ω0 , then the
3
A |1>=|ms=-1>
N-V
|0>=|ms=+1>
|ms=0> Fig. 1. Diagram indicates the energy level structure of the N-V center coupled to MTR cavity. Quantum information is encoded on the spin-triplet ground states of the N-V center. The transition j0〉 3 je〉 is driven by the right (R) circularly polarized photon and the transition j1〉 3 je〉 is driven by the left (L) circularly polarized photon.
Input pulse
MTR Detector
Fig. 2. Diagram illustrates the basic model of single photon input–output process. A single photon pulse is imported to interact with the N-V center coupled to microtoroidal cavity.
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reflection coefficient changes to r 0 ðωp Þ ¼ 1. From Eq. (8), it can be seen that under the resonance condition, the reflection coefficient becomes:
r ωp ¼
κγ þ 4g κγ þ 4g 2
2
ð10Þ
pffiffiffiffiffi In this case, if we set the coupling strength g Z5 κγ , the reflection coefficient becomes rðωp Þ 1, which illustrates that strong coupling and high-Q cavities are not necessary conditions for the photon to achieve nearly unchanged reflection during the input– output process [46]. In our scheme, if the N-V center is initially prepared in the state j1〉, the only optical transition is j1〉 3 je〉 which is driven by a left (L) circular polarized photon. If a single photon in the state jL〉 is sent into the system, we will get the output jψ out 〉L ¼ eiϕ jL〉 with the corresponding phase ϕ when the N-V center is prepared in j1〉 and the other output jψ out 〉L ¼ eiϕ0 jL〉 with a different phase ϕ0 when the N-V center is prepared in j0〉. On the other hand, if the N-V center is initially prepared in the state j0〉, the only optical transition is j0〉 3 je〉 which is driven by a right (R) circular polarized photon. So if another single polarized photon jR〉 is sent into the system, we will get the opposite result as jψ out 〉R ¼ eiϕ jR〉 with the phase ϕ when the N-V center is in j0〉 and the other output jψ out 〉R ¼ eiϕ0 jR〉 with the phase ϕ0 when the N-V center is in j1〉 [44]. By choosing g ¼0 and ωc ¼ ω0 ¼ ωp , we will have r 0 ðωp Þ ¼ 1 ¼ eiϕ0 with the corresponding phase ϕ0 ¼ π. Under pffiffiffiffiffi the condition of g Z 5 κγ and ωc ¼ ω0 ¼ ωp , we approximately iϕ have rðωp Þ 1 ¼ e with the other phase ϕ 0. According to the relations discussed above, we will get the results: jR〉j1〉-jR〉j1〉; jL〉j1〉-jL〉j1〉;
jR〉j0〉-jR〉j0〉; jL〉j0〉-jL〉j0〉:
ð11Þ
3. W state generation based on N-V center and MTR coupling system Here in this section, we proposed a W state generation protocol based on N-V centers coupled to MTR cavities. The basic setup of W state generation is shown in Fig. 3. Initially p the ffiffiffi three N-V centers are prepared in the state jpþffiffiffi〉 ¼ ðj1〉 þ j0〉Þ= 2. We sent a pffiffiffi photon pulse in the state ðjH〉 þ 2jV〉Þ= 3 to the polarization D2
N-V1
D1 Input
H
L
V
V V
N-V3 H H
H
L
M
M : Polarization beam splitter
: Bit-flip using half wave plate
:
:
Quarter wave plate
ð12Þ
Obviously, the jH〉 polarization component of the photon pulse is transmitted by the first PBS, while the jV〉 polarization component is reflected. Then the jV〉 polarization component of the photon pulse arrives at the second PBS after passing through a HWP with 22.51 along its axis which performs a Hadamard operation on the photon. Then the state of the system can be described as p1ffiffi H〉a þ 〉1 j þ 〉2 j þ 〉3 p1ffiffijV〉b j þ 〉1 j þ 〉2 j þ 〉3 3 3 þp1ffiffi3H〉c þ〉1 þ〉2 þ〉3 ð13Þ here the subscripts a, b and c of the photon state represent different path information of the photon pulse. Let us consider the photon pulse along path a at first. jH〉 polarization component of the photon pulse passes through the QWP, the state changes to jL〉 and then interacts with the first N-V center and MTR coupled system. The photon pulse along path b passes through HWP and QWP sequentially, which change the state jV 〉b -jH〉b and jH〉b -jL〉b , respectively. Then jL〉b enters and interacts with the second N-V center and MTR coupled system. The photon pulse along path c passes through a QWP which changes the state jH〉c -jL〉c , then jL〉c enters and interacts with the third N-V center and MTR coupled system. After that, the state of the composite system evolves as p1ffiffi L〉a 〉1 j þ 〉2 j þ 〉3 p1ffiffijL〉b j þ 〉1 j〉2 j þ 〉3 3 3 þp1ffiffi3L〉c þ〉1 þ〉2 〉3 ð14Þ By using the linear optical devices, the circular polarization state of the single photons can be changed back to the linear polarization state. The photon pulse along path b is combined with the photon pulse along path c at the PBS, then the state of the composite system can be expressed as p1ffiffi H〉a 〉1 j þ 〉2 j þ 〉3 p1ffiffijV〉b j þ 〉1 j〉2 j þ 〉3 3 3 þp1ffiffiH〉c þ〉1 þ〉2 〉3 ð15Þ 3
The three components of the photon pulse will pass through a HWP with 22.51 along its axis separately, which performs a Hadamard operation on each of the photon pulse, then the state of the system evolves to 1 jH〉a þ jV〉a 1 jH〉b jV〉b pffiffiffi pffiffiffi pffiffiffi 〉1 þ 〉2 j þ 〉3 pffiffiffi 3 3 2 2 þ jV〉c þ〉1 〉2 þ 〉3 þ p1ffiffiffi jH〉cp ffiffiffi j þ 〉1 j þ 〉2 j〉3 3 2
ð16Þ
The jH〉 polarization component of path a, jV 〉 polarization component of path b and path c will be detected by D1 after passing through PBS. However, the jV〉 polarization component of path a, jH〉 polarization component of path b and path c will be detected by D2 after PBS. If D1 clicks, the state of the three N-V centers is projected to p1ffiffi 〉1 þ〉2 þ〉3 þ þ〉1 〉2 þ〉3 þ þ〉1 þ〉2 〉3 ð17Þ 3
N-V2
L
beam-splitter (PBS). The whole system can be described as pffiffiffi ! 2 1 pffiffiffiH〉 þ pffiffiffiV〉 þ〉1 þ〉2 þ〉3 : 3 3
Hadamard using half wave plate
Fig. 3. Schematic of the setup for W entanglement among three remote N-V centers. PBS represents the polarization beam splitter. HWP stands for the half wave plate. QWP denotes the quarter wave plate. M is the mirror.
Obviously, this is a standard W state among the three remote N-V centers. On the contrary, if D2 clicks, the state of the three N-V centers is projected to p1ffiffi 〉1 þ〉2 þ〉3 þ〉1 〉2 þ〉3 þ þ〉1 þ〉2 〉3 ð18Þ 3 The systemic state can easily evolve to the standard W state by applying a unitary operation on the second NV center. This proposal can be easily extended to generate W state among N remote N-V centers, which is shown in Fig. 4. Firstly, the state of input photon pulse is expressed as jH〉. Secondly, there
X. Tong et al. / Optics Communications 310 (2014) 166–172
169
1 0.9 0.8
Fidelity
0.7 0.6 0.5 0.4 0.3 0.2 0
1
2
3
4
5
pffiffiffiffiffi Fig. 5. The fidelity of the generated W state versus the coupling strength g= κγ .
Fig. 4. Schematic of the setup for W entanglement among N remote N-V centers. PBS represents the polarization beam splitter. HWP stands for the half wave plate. QWP denotes the quarter wave plate. BS represents beam splitter and M is the mirror.
are (N 1) BS in the scheme with the distribution proportion of 1=ðNiÞ ði ¼ 1; 2; …; N1Þ separately. According to the same principle with three-qubit W state, all the output ports are combined at the PBS as shown in [57,58]. When D1 gets a click, the state of the N N-V centers is projected to a standard N-qubit W state. On the other hand, if D2 clicks, the state of the N N-V centers can easily evolve to the standard N-qubit W state by unitary operations. In the realistic implementations, the fidelity of the generated W state lies on the accuracy of phase shift during the input–output process of the photon, which relies on the key parameters of the system, including the damping rate of MTR cavity κ, the spontaneous emission of the N-V center γ and the coupling strength g. Here we numerically simulate the fidelity of the generated W state, which is defined as F ¼ j〈ψ ideal jψ real 〉j2 . jψ ideal 〉 refers to the state in ideal case and jψ real 〉 refers to the state of the N-V center in practical implementation. We label the fidelity of the generated pffiffiffiffiffi W state versus the coefficient g= κγ in Fig. 5. In the process of W state generation, the components of the input photon pass through different optical paths interacting with the N-V and MTR coupled system, respectively. In theory, the fidelity of multi-qubit W state should be equal to the three-qubit W state, assuming that all other linear optical operations can be implemented accurately. So the fidelity of four-qubit and five-qubit W states should overlap with the curve in Fig. 5. Actually, the fidelity of the generated W state decreases when the scheme is extended to multi-qubit case considering the inevitable errors brought by the linear optical devices. Theoretically, the fidelity of our scheme relies on the large coupling strength between the N-V center and the MTR cavity, low spontaneous emission of the N-V center and small damping rate of the MTR cavity. We can notice that the fidelity is equal to unity when the coupling strength approaches to 0 because reflection coefficient approaches to 1 under the uncoupled condition due to the special Faraday rotation in Eq. (11). After that, the fidelity drops rapidly because the special Faraday rotation in Eq. (11) cannot be achieved in this extremely small coupling strength regime. The output photon achieves a phase flip between 0 and
π leading to the undesirable reflection coefficient rðωÞ, which influences the fidelity of our generated W state. Then the fidelity increases gradually with the increase of the coupling strength g. The fidelity of entangled W state is close to unity when the value pffiffiffiffiffi of g= κγ is increased to 5 which satisfies the weak coupling condition. Actually, we can see that the fidelity of entangled W pffiffiffiffiffi states obtain F 0:95 when the value of g= κγ is increased to 3, which is not a strict requirement of the values to achieve high fidelity in our system. Effective reflection coefficient of the input photon and the phase shift of the output photon can be achieved in our scheme, so the W state can be generated with a high fidelity.
4. Cluster state generation based on N-V centers and MTR coupled system In 2001, Raussendorf and Briegel proposed a quantum computation protocol consisting of one-qubit measurements and the cluster states, which is called one-way quantum computing (QC) [59]. The key ingredient of one-way QC is the preparation of cluster state. In 2005, Walther et al. [60] implemented the oneway QC by experimentally realizing four-qubit cluster states encoded in the polarization four photons. In 2007, Louis et al. [61] proposed an approach for generating cluster states with weak nonlinearities. The cluster state usually has a large persistency of entanglement, which can be regarded as a resource for other entangled states. Here in this section, we describe the procedure of preparing the cluster state among two remote N-V centers in detail. The setup of cluster state generation is proposed in Fig. 6. Initially, pffiffiffithe two N-V centers are prepared in the state j〉 ¼ ðj1〉j0〉Þ= 2. A photon pffiffiffi pulse with the state ðjH〉 þ jV〉Þ= 2 enters and interacts with the solid system. The initial state of the composite system can be expressed as p1ffiffi H〉 þ V〉 〉1 〉2 ð19Þ 2 Obviously, the jH〉 polarization component of the photon pulse is transmitted by the first PBS, while the jV〉 polarization component is reflected. After passing through the first QWP, the jL〉 polarization component interacts with the first N-V and MTR coupling system changing the j〉 state of the first N-V center to j þ 〉. After passing through the second QWP, the jH〉 polarization component of the photon is combined with the jV〉 polarization component at
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of two adjacent N-V centers. The N-qubit cluster state
the second PBS. The state of the system evolves as p1ffiffi H〉 þ〉1 〉2 þ V 〉 〉1 〉2 2
ð20Þ
jψ〉n ¼
A Hadamard operation is operatedpon ffiffiffi the photon which changes the jH〉 polarization to ðjH〉 þ jV〉Þ= 2 and the jV 〉 polarization to pffiffiffi ðjH〉jV〉Þ= 2. Then the photon interacts with the second N-V center and MTR system, then the state of the system evolves as H〉 þ〉1 þ〉2 þ 〉1 þ〉2 þ V〉 þ〉1 〉2 〉1 〉2 :
ð21Þ
By performing the single photon measurement on the photon in the bases j þ 〉 ¼ p1ffiffi2ðjH〉 þ jV〉Þ and j〉 ¼ p1ffiffi2ðjH〉jV〉Þ, if the detector records a j þ 〉 result, we will get the final state 1 2
þ〉1 þ〉2 þ 〉1 þ〉2 þ þ〉1 〉2 〉1 〉2
ð22Þ
Then the state of the system can be expressed as 1 1 þ〉1 þ 〉1 2 þ〉2 þ 〉2 sz
ð23Þ
Obviously, this is a standard cluster state between two remote N-V centers. The proposed protocol can be easily generalized to N-particle cluster state generation between remote N-V centers. Suppose that the first two N-V centers have been entangled in a standard cluster state and another pffiffiffiN-V center is prepared in the superposition state j〉 ¼ ðj1〉j0〉Þ= 2. Then we iterate the above interaction between the second and the third N-V centers again. The state of the three N-V centers system will become jψ〉3 ¼
1 2
3=2
j þ 〉3 þ j〉3 s2z j þ 〉2 þ j〉2 s1z j þ 〉1 þ j〉1
ð24Þ
⋯ ðj þ 〉2 þ j〉2 s1z Þ ðj þ 〉1 þ j〉1 Þ
1 0.9 0.8 0.7
Detector
Input H
H
V
V
0.6 0.5 0.4 2−qubit cluster state 3−qubit cluster state 4−qubit cluster state
0.3 0.2 0.1 0
0
1
2
3
4
5
N-V2
N-V1
L
ð25Þ
can be generated by the above simple method. The fidelity of the generated N-qubit cluster state relies on the high fidelity operation of nearest two N-V centers interaction. The interaction is influenced by the damping of the MTR cavity κ and the spontaneous emission of the N-V center γ. The coupling strength g between N-V center and MTR cavity also needs to be considered. Here we describe the fidelity of cluster state F as a pffiffiffiffiffi function of g= κγ with different qubits. In Fig. 8, the solid line represents the fidelity of two-qubit cluster state, the dashed line and the dot line denote the fidelity of three-qubit and four-qubit cluster state separately. From Fig. 8, it is noticed that the fidelity of pffiffiffiffiffi generated cluster states are close to unity when the value of g= κγ
which represents the three-qubit cluster state among three remote N-V centers. By iterating the steps between the Nth particle and the (N1)th particle, the N-particle cluster state between N-V centers can be generated. As shown in Fig. 7, the black circles represent N-V centers and the line between them denotes a previous interaction
H
ðj þ 〉n þ j〉n sn1 Þ z
Þ ðj þ 〉n1 þ j〉n1 sn2 z
Fidelity
1 2
1 2n=2
H
L
V
Fig. 8. The fidelity F of the generated cluster state among remote N-V centers as a pffiffiffiffiffi function of g= κγ . Here the solid line, dashed line and dot line represent the fidelity of the final cluster state with 2-qubit, 3-qubit and 4-qubit, respectively.
V
1 0.8 : Polarization beam splitter
Fig. 6. Schematic diagram shows the setup of the cluster state generation between two remote N-V centers coupled to MTR cavities. In the diagram, PBS represents the polarization beam splitter. HWP represents the half wave plate. D is the singlephoton detector.
N-1
0.6 0.4 0.2
r(ω)
: Half wave plate
0 −0.2
N
N-1
−0.4 N
N+1
N
−0.6 −0.8 −1
N+1
0
0.02
0.04
0.06
0.08
0.1
g/κ Fig. 7. Schematic diagram shows the setup of generating N-qubit cluster state among N remote N-V centers by a simple add-on strategy based on our new scheme. The black circle represents N-V center and the line between them denotes a previous interaction.
Fig. 9. The reflection coefficient rðωÞ as a function of g=κ under the resonant condition ωc ¼ ω0 ¼ ωp . Here the solid line, dashed line and dot line represent γ ¼ 6 105 κ, γ ¼ 6 104 κ and γ ¼ 3 104 κ, respectively.
X. Tong et al. / Optics Communications 310 (2014) 166–172
1 0.998 0.996 0.994
r(ω)
0.992 0.99 0.988 0.986 0.984 0.982 0.98 200
300
400
500
600
700
800
900
1000
g /γ Fig. 10. The reflection coefficient rðωÞ as a function of g=γ under the resonant condition ωc ¼ ω0 ¼ ωp . Here the solid line, dashed line and dot line represent κ ¼ 500γ, κ ¼ 1000γ and κ ¼ 1500γ, respectively.
171
schemes. Moreover, current techniques can decrease the photon leakage during the input–output process, making it negligible compared with the cavity decay. Secondly, although the performance of our scheme can be decreased significantly by the imperfect single-photon preparation, it is fortunate that 300 000 high quality single photons can be generated within 30 s by recent experimental technology [66]. The highly efficient single-photon source ensures that our scheme can be accomplished fast and near deterministically even if we implement the scheme with low success rate caused by photon loss during the input–output process. In addition, the weak excitation condition during the input–output process can be satisfied by such high quality singlephoton sources [67]. In summary, we have put forward a new scheme to generate the W state and cluster state among remote N-V centers via the photon input–output process in the solid-state system. The experimental requirement has been greatly reduced due to the long electronic spin decoherence time of N-V center at room temperature and the ultra high quality factor of microtoroidal resonators. The analysis shows that our scheme is feasible in current experimental conditions, which is expected to be meaningful for the realization of scalable quantum computation network.
is increased to 5, which means strong coupling and high-Q cavities are not required to ensure high efficiency of our schemes. Acknowledgments 5. Conclusion We investigate an efficient protocol for W state and cluster state generation using the N-V center and MTR cavity coupled system. The main challenge of the realization is the coupling between the input photons and the transition of the N-V center. Another characteristic of our system is the long decoherence time due to the dephasing time and the electron spin relaxation time of N-V centers [62]. Recently, much effort has been made in experiment where N-V centers are coupled to different optical resonators, such as SiN crystals [63], GaP microdisks [64] and silica microspheres [65]. Among them, one experiment with a low-Q cavity gives the values of parameters as ½g; κ; γ=2π ¼ ½0:3; 26; pffiffiffiffiffi pffiffiffiffiffi 0:0004 GHz [64]. Since κγ 2π 0:1 GHz, we get g 3 κγ and rðωÞ 0:95 under the condition of ωc ¼ ω0 ¼ ωp , which practically meet the requirement of our scheme. In our scheme, the reflection coefficient of the output photon plays an important role during the process of successfully generating W state and cluster state. The efficiency of our entanglement generation may be reduced if the specific Faraday rotation in Eq. (11) cannot be realized perfectly. To consider the effect of spontaneous emission of the N-V centers γ and the damping of the MTR cavity κ in our schemes, we have plotted the reflection coefficient as functions of g=κ and g=γ in Figs. 9 and 10. We can see that the reflection coefficient rðωÞ increases with the increase of the coupling strength g between the N-V center and MTR cavity. Actually, as shown in Fig. 9, even when g o 0:1κ reflection coefficient can obtain rðωÞ 0:95, which means our schemes do not essentially need strong coupling and small damping. It is shown in Fig. 10 that even when κ ¼ 1500γ, reflection coefficient is nearly unity on the condition of g=γ ¼ 600, which means that the requirement on g and κ is relaxed. From the above, we can conclude that our proposed schemes can still work well even when the cavity is bad. At last, we would like to consider the experimental feasibility of our scheme. Firstly, the photon loss due to photon leakage during the input–output process, the fiber absorption, the cavity absorption and scattering and inefficient photon detection will bring ineffectiveness to our schemes. Actually, the photon loss just affects the success efficiency rather than the fidelity of our
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
A hybrid-system approach for W state and cluster state generation Xin Tong a,b, Chuan Wang a,b,n, Cong Cao a,b, Ling-yan He a,b, Ru Zhang a,b 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
art ic l e i nf o
a b s t r a c t
Article history: Received 29 May 2013 Received in revised form 28 July 2013 Accepted 29 July 2013 Available online 12 August 2013
Here we demonstrate an efficient scheme for W state and cluster state generation based on the hybridsystem where nitrogen-vacancy centers are coupled to microcavities. In our proposed scheme, entanglement between solid-state qubits can be generated with the assistance of an ancillary photon. We also discuss the efficiency of the entanglement generation protocols and generalize them to multiqubit cases. With current and near-future technology, the entanglement of nitrogen-vacancy centers can be achieved and our scheme can further be used for quantum information processing and long-distance quantum communication. & 2013 Elsevier B.V. All rights reserved.
Keywords: W state generation Cluster state generation Nitrogen-vacancy centers Microtoroidal resonator
1. Introduction Quantum entanglement is an essential resource for the implementation of quantum communication [1,2] and quantum information processing (QIP) [3–5], such as quantum teleportation [6–8] and quantum computation [9–11]. Over the past few decades, much effort has been devoted to the generation of entanglement, especially for multi-particle entangled states in different quantum systems [12,13]. Among these works, generation of W state [14–16] and cluster state [17] are current topics in experiments. Recently, there have been several theoretical protocols for W state and cluster state generation by different systems, such as optical systems [18], atomic ensembles [19], trapped ions [20], cavity quantum electrodynamics (QED) [21,22] and so on [23]. One of the promising candidates for QIP is the cavity QED system [24–27]. Here we consider a nitrogen vacancy (N-V) center and a microtoroidal resonator (MTR) coupled system. The N-V center [28–32] consisting of a nearest neighbor pair of a nitrogen atom substituted for a carbon atom and a lattice vacancy has been considered as an outstanding candidate for QIP due to its long electronic spin decoherence time at room temperature [33]. The energy level of the N-V center can be controlled by electromagnetic field or optical pulse [34]. It seems that the resonant zero phonon line (ZPL) relevant to the emitted photons from the N-V center could be improved by confining N-V centers in cavities [35]. These N-V centers could be read out rapidly with high fidelity in n Corresponding author at: School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China. Tel.: +86 10 62282050. E-mail addresses: [email protected], [email protected] (C. Wang).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.083
experiment, which makes entanglement generation based on N-V centers achievable in practice [36]. Besides that, MTR exhibits high Q factor, excellent scalability and small mode volume [37]. The novel solid-state cavity system combines the advantages of the N-V center having exceptional spin properties with the microtoroidal resonator possessing ultra high quality factor and small mode volume [38,39]. Owing to the outstanding feature of solid-state cavity QED system, much theoretical and experimental effort has been devoted to the QIP process related to this field [40,41]. In 2008, Dayan et al. experimentally realized the single photon input– output process using the model consisting of a microtoroidal resonator (MTR) coupled to an optical fiber [42]. In 2009, An et al. proposed some schemes for quantum information processing with Faraday rotation using single photon input–output process, which is much different from the high-Q cavity and strongcoupling cases [43]. In 2011, Chen et al. proposed a potentially practical scheme to entangle negatively charged N-V centers in distant diamonds [44]. In 2012, Zheng et al. proposed a scheme to generate N-qubit GHz state with distant N-V centers confined in spatially separated photonic crystal nanocavities via input–output process of photons [45]. In 2013, Cheng et al. demonstrated efficient schemes of deterministic entanglement generation and quantum state transfer with the N-V centers in diamond confined in separated microtoroidal resonators via single photon input– output process [46]. Compared with previous proposals, our new method has the similar superiority and some improvements. Firstly, the quantum information is encoded in the electronic spin ground states of the N-V center, which is convenient to store the generated entanglement. Secondly, owing to the characteristic of N-V centers, the spin state has long decoherence time, which is
X. Tong et al. / Optics Communications 310 (2014) 166–172
easy to be manipulated even at room temperature. Thirdly, our scheme can work well on the condition that the N-V center is coupled to low-Q MTR. The successful entanglement generations are still available under weak coupling conditions and large cavity damping rate. In recent years, this system consisting of N-V centers coupled to MTR has been widely used in many aspects, such as entanglement purification of separate N-V centers [47], entanglement concentration for multi-particle entanglement [48], universal quantum gates with N-V centers in diamond [49]. This paper is organized as follows: in Section 2, we present the physical principle of our scheme by explaining the photon input– output process of the N-V center and MTR coupled system. In Section 3, we put forward a scheme based on the solid-state system to generate W state among three N-V centers and then generalize it to an arbitrary multi-qubit case. We analyze the fidelity of the generated W state at the end of this part. In Section 4, we propose a scheme to generate cluster state between two remote N-V centers and then extend this protocol to N-particle cases. At the end of this part, we analyze the fidelity of the cluster states. In Section 5, we analyze experimental feasibility of our scheme, then give our conclusion at last.
2. The input–output process of the N-V center and MTR coupling system The energy level structure of the N-V center is illustrated in Fig. 1, which can be represented by a three-level Λ-type atom [50,51]. The N-V center is made up of a substitutive nitrogen atom and an adjacent vacancy in diamond. Actually, the electronic spin state of the N-V center is composed of spin triplet 3A ground state and spin triplet 3E excited state [52]. The ground state 3A is split into the upper levels j3 A; ms ¼ 7 1〉 and the lower level j3 A; ms ¼ 0〉 by 2.88 GHz owing to spin–spin interaction. The N-V center has six excited states because of the effect of electronic spin–orbit interaction and spin–spin interaction [53,54]. Among them, jA2 〉 state is quite special, which can decay to the ground state j3 A; ms ¼ þ 1〉 and j3 A; ms ¼ 1〉 with the radiation of left (L) and right (R) circularly polarized photon, respectively. As shown in Fig. 1, we encode the quantum information on the ground states j1〉 ¼ j3 A; ms ¼ 1〉 and j0〉 ¼ j3 A; ms ¼ þ 1〉. We can obtain a Λ-type transition by choosing the excited state jA2 〉 as auxiliary level je〉 because of its stable symmetric properties [55]. Here one optical transition is allowed between j1〉 ¼ j3 A; ms ¼ 1〉 and je〉, which is resonantly coupled to the left polarized photon jL〉. The other optical transition is between j0〉 ¼ j3 A; ms ¼ þ 1〉 and je〉, which is resonantly coupled to the right polarized photon jR〉. Under the Jaynes–Cummings model, the Hamiltonian of the system can be
A2
|e>
|L>
|R>
167
expressed as H ¼ H sys þ H bath þ H int
ð1Þ
here Hsys represents the interaction between the MTR cavity and the N-V center hω i j0 ð2Þ H sys ¼ ∑ sjz þ ωjc a†j aj þ ig j aj sjþ aþ j sj j ¼ R;L 2 and Hbath represents the Hamiltonian of the input field Z 1 † ωbj ðωÞbj ðωÞ dω H bath ¼ 1
ð3Þ
The interaction Hamiltonian between the external photon and the MTR field can be described as Z 1 † H int ¼ ∑ iGðωÞ½ibj ðωÞaj ibj ðωÞa†j dω ð4Þ j ¼ R;L
1
a†j
and aj are the creation and annihilation operators of the here MTR cavity which satisfy the relation ½aj ; a†j ¼ 1. ω0 is the transition frequency of the N-V center electronic energy levels and ωc is the frequency of the MTR cavity field. sz , sþ and s are the inversion, raising and lowering operators of the N-V center between the ground levels and the excited level, respectively. GðωÞ is the coupling constant between the external photon and the †
N-V center. bj ðωÞ and bj ðωÞ are the creation and annihilation operators of the external photon. We introduce a single photon pulse with the frequency of ωp to enter the N-V center and MTR cavity coupling system, which is described in Fig. 2. Considering the rotating frame with respect to the frequency of the input pulse, we can get the quantum Langevin equations for operator of MTR cavity field and the lowing operator of N-V center [56]: κi pffiffiffi daj h ¼ i ωp ωc aj ðt Þgsj ðt Þ κ aj;in ðt Þ; ð5Þ 2 dt γi dsj h ¼ i ωp ω0 sj ðt Þgsj;z ðt Þaj ðt Þ; 2 dt
ð6Þ
here g represents the coupling strength between the N-V center and the microtoroidal resonator, κ is the damping rate of the cavity and γ denotes the spontaneous emission of the N-V center. The input and output fields of the cavity are related by the intracavity field as pffiffiffi aj;out ðtÞ ¼ aj;in ðtÞ þ κ aj ðtÞ ð7Þ Considering the relations above, we can adiabatically eliminate the cavity mode and solve the reflection coefficient as follows: h κ ih γi i ω0 ωp þ þ g2 i ωc ωp aout ðtÞ 2 ih 2i ¼h : ð8Þ r ωp ¼ κ γ ain ðtÞ i ω0 ωp þ þ g2 i ωc ωp þ 2 2 The coefficient for the uncoupled system can be described as κ i ωc ωp 2 ð9Þ r 0 ωp ¼ κ; i ωc ωp þ 2 by choosing g¼ 0. Here we assume that ωp ¼ ωc ¼ ω0 , then the
3
A |1>=|ms=-1>
N-V
|0>=|ms=+1>
|ms=0> Fig. 1. Diagram indicates the energy level structure of the N-V center coupled to MTR cavity. Quantum information is encoded on the spin-triplet ground states of the N-V center. The transition j0〉 3 je〉 is driven by the right (R) circularly polarized photon and the transition j1〉 3 je〉 is driven by the left (L) circularly polarized photon.
Input pulse
MTR Detector
Fig. 2. Diagram illustrates the basic model of single photon input–output process. A single photon pulse is imported to interact with the N-V center coupled to microtoroidal cavity.
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reflection coefficient changes to r 0 ðωp Þ ¼ 1. From Eq. (8), it can be seen that under the resonance condition, the reflection coefficient becomes:
r ωp ¼
κγ þ 4g κγ þ 4g 2
2
ð10Þ
pffiffiffiffiffi In this case, if we set the coupling strength g Z5 κγ , the reflection coefficient becomes rðωp Þ 1, which illustrates that strong coupling and high-Q cavities are not necessary conditions for the photon to achieve nearly unchanged reflection during the input– output process [46]. In our scheme, if the N-V center is initially prepared in the state j1〉, the only optical transition is j1〉 3 je〉 which is driven by a left (L) circular polarized photon. If a single photon in the state jL〉 is sent into the system, we will get the output jψ out 〉L ¼ eiϕ jL〉 with the corresponding phase ϕ when the N-V center is prepared in j1〉 and the other output jψ out 〉L ¼ eiϕ0 jL〉 with a different phase ϕ0 when the N-V center is prepared in j0〉. On the other hand, if the N-V center is initially prepared in the state j0〉, the only optical transition is j0〉 3 je〉 which is driven by a right (R) circular polarized photon. So if another single polarized photon jR〉 is sent into the system, we will get the opposite result as jψ out 〉R ¼ eiϕ jR〉 with the phase ϕ when the N-V center is in j0〉 and the other output jψ out 〉R ¼ eiϕ0 jR〉 with the phase ϕ0 when the N-V center is in j1〉 [44]. By choosing g ¼0 and ωc ¼ ω0 ¼ ωp , we will have r 0 ðωp Þ ¼ 1 ¼ eiϕ0 with the corresponding phase ϕ0 ¼ π. Under pffiffiffiffiffi the condition of g Z 5 κγ and ωc ¼ ω0 ¼ ωp , we approximately iϕ have rðωp Þ 1 ¼ e with the other phase ϕ 0. According to the relations discussed above, we will get the results: jR〉j1〉-jR〉j1〉; jL〉j1〉-jL〉j1〉;
jR〉j0〉-jR〉j0〉; jL〉j0〉-jL〉j0〉:
ð11Þ
3. W state generation based on N-V center and MTR coupling system Here in this section, we proposed a W state generation protocol based on N-V centers coupled to MTR cavities. The basic setup of W state generation is shown in Fig. 3. Initially p the ffiffiffi three N-V centers are prepared in the state jpþffiffiffi〉 ¼ ðj1〉 þ j0〉Þ= 2. We sent a pffiffiffi photon pulse in the state ðjH〉 þ 2jV〉Þ= 3 to the polarization D2
N-V1
D1 Input
H
L
V
V V
N-V3 H H
H
L
M
M : Polarization beam splitter
: Bit-flip using half wave plate
:
:
Quarter wave plate
ð12Þ
Obviously, the jH〉 polarization component of the photon pulse is transmitted by the first PBS, while the jV〉 polarization component is reflected. Then the jV〉 polarization component of the photon pulse arrives at the second PBS after passing through a HWP with 22.51 along its axis which performs a Hadamard operation on the photon. Then the state of the system can be described as p1ffiffi H〉a þ 〉1 j þ 〉2 j þ 〉3 p1ffiffijV〉b j þ 〉1 j þ 〉2 j þ 〉3 3 3 þp1ffiffi3H〉c þ〉1 þ〉2 þ〉3 ð13Þ here the subscripts a, b and c of the photon state represent different path information of the photon pulse. Let us consider the photon pulse along path a at first. jH〉 polarization component of the photon pulse passes through the QWP, the state changes to jL〉 and then interacts with the first N-V center and MTR coupled system. The photon pulse along path b passes through HWP and QWP sequentially, which change the state jV 〉b -jH〉b and jH〉b -jL〉b , respectively. Then jL〉b enters and interacts with the second N-V center and MTR coupled system. The photon pulse along path c passes through a QWP which changes the state jH〉c -jL〉c , then jL〉c enters and interacts with the third N-V center and MTR coupled system. After that, the state of the composite system evolves as p1ffiffi L〉a 〉1 j þ 〉2 j þ 〉3 p1ffiffijL〉b j þ 〉1 j〉2 j þ 〉3 3 3 þp1ffiffi3L〉c þ〉1 þ〉2 〉3 ð14Þ By using the linear optical devices, the circular polarization state of the single photons can be changed back to the linear polarization state. The photon pulse along path b is combined with the photon pulse along path c at the PBS, then the state of the composite system can be expressed as p1ffiffi H〉a 〉1 j þ 〉2 j þ 〉3 p1ffiffijV〉b j þ 〉1 j〉2 j þ 〉3 3 3 þp1ffiffiH〉c þ〉1 þ〉2 〉3 ð15Þ 3
The three components of the photon pulse will pass through a HWP with 22.51 along its axis separately, which performs a Hadamard operation on each of the photon pulse, then the state of the system evolves to 1 jH〉a þ jV〉a 1 jH〉b jV〉b pffiffiffi pffiffiffi pffiffiffi 〉1 þ 〉2 j þ 〉3 pffiffiffi 3 3 2 2 þ jV〉c þ〉1 〉2 þ 〉3 þ p1ffiffiffi jH〉cp ffiffiffi j þ 〉1 j þ 〉2 j〉3 3 2
ð16Þ
The jH〉 polarization component of path a, jV 〉 polarization component of path b and path c will be detected by D1 after passing through PBS. However, the jV〉 polarization component of path a, jH〉 polarization component of path b and path c will be detected by D2 after PBS. If D1 clicks, the state of the three N-V centers is projected to p1ffiffi 〉1 þ〉2 þ〉3 þ þ〉1 〉2 þ〉3 þ þ〉1 þ〉2 〉3 ð17Þ 3
N-V2
L
beam-splitter (PBS). The whole system can be described as pffiffiffi ! 2 1 pffiffiffiH〉 þ pffiffiffiV〉 þ〉1 þ〉2 þ〉3 : 3 3
Hadamard using half wave plate
Fig. 3. Schematic of the setup for W entanglement among three remote N-V centers. PBS represents the polarization beam splitter. HWP stands for the half wave plate. QWP denotes the quarter wave plate. M is the mirror.
Obviously, this is a standard W state among the three remote N-V centers. On the contrary, if D2 clicks, the state of the three N-V centers is projected to p1ffiffi 〉1 þ〉2 þ〉3 þ〉1 〉2 þ〉3 þ þ〉1 þ〉2 〉3 ð18Þ 3 The systemic state can easily evolve to the standard W state by applying a unitary operation on the second NV center. This proposal can be easily extended to generate W state among N remote N-V centers, which is shown in Fig. 4. Firstly, the state of input photon pulse is expressed as jH〉. Secondly, there
X. Tong et al. / Optics Communications 310 (2014) 166–172
169
1 0.9 0.8
Fidelity
0.7 0.6 0.5 0.4 0.3 0.2 0
1
2
3
4
5
pffiffiffiffiffi Fig. 5. The fidelity of the generated W state versus the coupling strength g= κγ .
Fig. 4. Schematic of the setup for W entanglement among N remote N-V centers. PBS represents the polarization beam splitter. HWP stands for the half wave plate. QWP denotes the quarter wave plate. BS represents beam splitter and M is the mirror.
are (N 1) BS in the scheme with the distribution proportion of 1=ðNiÞ ði ¼ 1; 2; …; N1Þ separately. According to the same principle with three-qubit W state, all the output ports are combined at the PBS as shown in [57,58]. When D1 gets a click, the state of the N N-V centers is projected to a standard N-qubit W state. On the other hand, if D2 clicks, the state of the N N-V centers can easily evolve to the standard N-qubit W state by unitary operations. In the realistic implementations, the fidelity of the generated W state lies on the accuracy of phase shift during the input–output process of the photon, which relies on the key parameters of the system, including the damping rate of MTR cavity κ, the spontaneous emission of the N-V center γ and the coupling strength g. Here we numerically simulate the fidelity of the generated W state, which is defined as F ¼ j〈ψ ideal jψ real 〉j2 . jψ ideal 〉 refers to the state in ideal case and jψ real 〉 refers to the state of the N-V center in practical implementation. We label the fidelity of the generated pffiffiffiffiffi W state versus the coefficient g= κγ in Fig. 5. In the process of W state generation, the components of the input photon pass through different optical paths interacting with the N-V and MTR coupled system, respectively. In theory, the fidelity of multi-qubit W state should be equal to the three-qubit W state, assuming that all other linear optical operations can be implemented accurately. So the fidelity of four-qubit and five-qubit W states should overlap with the curve in Fig. 5. Actually, the fidelity of the generated W state decreases when the scheme is extended to multi-qubit case considering the inevitable errors brought by the linear optical devices. Theoretically, the fidelity of our scheme relies on the large coupling strength between the N-V center and the MTR cavity, low spontaneous emission of the N-V center and small damping rate of the MTR cavity. We can notice that the fidelity is equal to unity when the coupling strength approaches to 0 because reflection coefficient approaches to 1 under the uncoupled condition due to the special Faraday rotation in Eq. (11). After that, the fidelity drops rapidly because the special Faraday rotation in Eq. (11) cannot be achieved in this extremely small coupling strength regime. The output photon achieves a phase flip between 0 and
π leading to the undesirable reflection coefficient rðωÞ, which influences the fidelity of our generated W state. Then the fidelity increases gradually with the increase of the coupling strength g. The fidelity of entangled W state is close to unity when the value pffiffiffiffiffi of g= κγ is increased to 5 which satisfies the weak coupling condition. Actually, we can see that the fidelity of entangled W pffiffiffiffiffi states obtain F 0:95 when the value of g= κγ is increased to 3, which is not a strict requirement of the values to achieve high fidelity in our system. Effective reflection coefficient of the input photon and the phase shift of the output photon can be achieved in our scheme, so the W state can be generated with a high fidelity.
4. Cluster state generation based on N-V centers and MTR coupled system In 2001, Raussendorf and Briegel proposed a quantum computation protocol consisting of one-qubit measurements and the cluster states, which is called one-way quantum computing (QC) [59]. The key ingredient of one-way QC is the preparation of cluster state. In 2005, Walther et al. [60] implemented the oneway QC by experimentally realizing four-qubit cluster states encoded in the polarization four photons. In 2007, Louis et al. [61] proposed an approach for generating cluster states with weak nonlinearities. The cluster state usually has a large persistency of entanglement, which can be regarded as a resource for other entangled states. Here in this section, we describe the procedure of preparing the cluster state among two remote N-V centers in detail. The setup of cluster state generation is proposed in Fig. 6. Initially, pffiffiffithe two N-V centers are prepared in the state j〉 ¼ ðj1〉j0〉Þ= 2. A photon pffiffiffi pulse with the state ðjH〉 þ jV〉Þ= 2 enters and interacts with the solid system. The initial state of the composite system can be expressed as p1ffiffi H〉 þ V〉 〉1 〉2 ð19Þ 2 Obviously, the jH〉 polarization component of the photon pulse is transmitted by the first PBS, while the jV〉 polarization component is reflected. After passing through the first QWP, the jL〉 polarization component interacts with the first N-V and MTR coupling system changing the j〉 state of the first N-V center to j þ 〉. After passing through the second QWP, the jH〉 polarization component of the photon is combined with the jV〉 polarization component at
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X. Tong et al. / Optics Communications 310 (2014) 166–172
of two adjacent N-V centers. The N-qubit cluster state
the second PBS. The state of the system evolves as p1ffiffi H〉 þ〉1 〉2 þ V 〉 〉1 〉2 2
ð20Þ
jψ〉n ¼
A Hadamard operation is operatedpon ffiffiffi the photon which changes the jH〉 polarization to ðjH〉 þ jV〉Þ= 2 and the jV 〉 polarization to pffiffiffi ðjH〉jV〉Þ= 2. Then the photon interacts with the second N-V center and MTR system, then the state of the system evolves as H〉 þ〉1 þ〉2 þ 〉1 þ〉2 þ V〉 þ〉1 〉2 〉1 〉2 :
ð21Þ
By performing the single photon measurement on the photon in the bases j þ 〉 ¼ p1ffiffi2ðjH〉 þ jV〉Þ and j〉 ¼ p1ffiffi2ðjH〉jV〉Þ, if the detector records a j þ 〉 result, we will get the final state 1 2
þ〉1 þ〉2 þ 〉1 þ〉2 þ þ〉1 〉2 〉1 〉2
ð22Þ
Then the state of the system can be expressed as 1 1 þ〉1 þ 〉1 2 þ〉2 þ 〉2 sz
ð23Þ
Obviously, this is a standard cluster state between two remote N-V centers. The proposed protocol can be easily generalized to N-particle cluster state generation between remote N-V centers. Suppose that the first two N-V centers have been entangled in a standard cluster state and another pffiffiffiN-V center is prepared in the superposition state j〉 ¼ ðj1〉j0〉Þ= 2. Then we iterate the above interaction between the second and the third N-V centers again. The state of the three N-V centers system will become jψ〉3 ¼
1 2
3=2
j þ 〉3 þ j〉3 s2z j þ 〉2 þ j〉2 s1z j þ 〉1 þ j〉1
ð24Þ
⋯ ðj þ 〉2 þ j〉2 s1z Þ ðj þ 〉1 þ j〉1 Þ
1 0.9 0.8 0.7
Detector
Input H
H
V
V
0.6 0.5 0.4 2−qubit cluster state 3−qubit cluster state 4−qubit cluster state
0.3 0.2 0.1 0
0
1
2
3
4
5
N-V2
N-V1
L
ð25Þ
can be generated by the above simple method. The fidelity of the generated N-qubit cluster state relies on the high fidelity operation of nearest two N-V centers interaction. The interaction is influenced by the damping of the MTR cavity κ and the spontaneous emission of the N-V center γ. The coupling strength g between N-V center and MTR cavity also needs to be considered. Here we describe the fidelity of cluster state F as a pffiffiffiffiffi function of g= κγ with different qubits. In Fig. 8, the solid line represents the fidelity of two-qubit cluster state, the dashed line and the dot line denote the fidelity of three-qubit and four-qubit cluster state separately. From Fig. 8, it is noticed that the fidelity of pffiffiffiffiffi generated cluster states are close to unity when the value of g= κγ
which represents the three-qubit cluster state among three remote N-V centers. By iterating the steps between the Nth particle and the (N1)th particle, the N-particle cluster state between N-V centers can be generated. As shown in Fig. 7, the black circles represent N-V centers and the line between them denotes a previous interaction
H
ðj þ 〉n þ j〉n sn1 Þ z
Þ ðj þ 〉n1 þ j〉n1 sn2 z
Fidelity
1 2
1 2n=2
H
L
V
Fig. 8. The fidelity F of the generated cluster state among remote N-V centers as a pffiffiffiffiffi function of g= κγ . Here the solid line, dashed line and dot line represent the fidelity of the final cluster state with 2-qubit, 3-qubit and 4-qubit, respectively.
V
1 0.8 : Polarization beam splitter
Fig. 6. Schematic diagram shows the setup of the cluster state generation between two remote N-V centers coupled to MTR cavities. In the diagram, PBS represents the polarization beam splitter. HWP represents the half wave plate. D is the singlephoton detector.
N-1
0.6 0.4 0.2
r(ω)
: Half wave plate
0 −0.2
N
N-1
−0.4 N
N+1
N
−0.6 −0.8 −1
N+1
0
0.02
0.04
0.06
0.08
0.1
g/κ Fig. 7. Schematic diagram shows the setup of generating N-qubit cluster state among N remote N-V centers by a simple add-on strategy based on our new scheme. The black circle represents N-V center and the line between them denotes a previous interaction.
Fig. 9. The reflection coefficient rðωÞ as a function of g=κ under the resonant condition ωc ¼ ω0 ¼ ωp . Here the solid line, dashed line and dot line represent γ ¼ 6 105 κ, γ ¼ 6 104 κ and γ ¼ 3 104 κ, respectively.
X. Tong et al. / Optics Communications 310 (2014) 166–172
1 0.998 0.996 0.994
r(ω)
0.992 0.99 0.988 0.986 0.984 0.982 0.98 200
300
400
500
600
700
800
900
1000
g /γ Fig. 10. The reflection coefficient rðωÞ as a function of g=γ under the resonant condition ωc ¼ ω0 ¼ ωp . Here the solid line, dashed line and dot line represent κ ¼ 500γ, κ ¼ 1000γ and κ ¼ 1500γ, respectively.
171
schemes. Moreover, current techniques can decrease the photon leakage during the input–output process, making it negligible compared with the cavity decay. Secondly, although the performance of our scheme can be decreased significantly by the imperfect single-photon preparation, it is fortunate that 300 000 high quality single photons can be generated within 30 s by recent experimental technology [66]. The highly efficient single-photon source ensures that our scheme can be accomplished fast and near deterministically even if we implement the scheme with low success rate caused by photon loss during the input–output process. In addition, the weak excitation condition during the input–output process can be satisfied by such high quality singlephoton sources [67]. In summary, we have put forward a new scheme to generate the W state and cluster state among remote N-V centers via the photon input–output process in the solid-state system. The experimental requirement has been greatly reduced due to the long electronic spin decoherence time of N-V center at room temperature and the ultra high quality factor of microtoroidal resonators. The analysis shows that our scheme is feasible in current experimental conditions, which is expected to be meaningful for the realization of scalable quantum computation network.
is increased to 5, which means strong coupling and high-Q cavities are not required to ensure high efficiency of our schemes. Acknowledgments 5. Conclusion We investigate an efficient protocol for W state and cluster state generation using the N-V center and MTR cavity coupled system. The main challenge of the realization is the coupling between the input photons and the transition of the N-V center. Another characteristic of our system is the long decoherence time due to the dephasing time and the electron spin relaxation time of N-V centers [62]. Recently, much effort has been made in experiment where N-V centers are coupled to different optical resonators, such as SiN crystals [63], GaP microdisks [64] and silica microspheres [65]. Among them, one experiment with a low-Q cavity gives the values of parameters as ½g; κ; γ=2π ¼ ½0:3; 26; pffiffiffiffiffi pffiffiffiffiffi 0:0004 GHz [64]. Since κγ 2π 0:1 GHz, we get g 3 κγ and rðωÞ 0:95 under the condition of ωc ¼ ω0 ¼ ωp , which practically meet the requirement of our scheme. In our scheme, the reflection coefficient of the output photon plays an important role during the process of successfully generating W state and cluster state. The efficiency of our entanglement generation may be reduced if the specific Faraday rotation in Eq. (11) cannot be realized perfectly. To consider the effect of spontaneous emission of the N-V centers γ and the damping of the MTR cavity κ in our schemes, we have plotted the reflection coefficient as functions of g=κ and g=γ in Figs. 9 and 10. We can see that the reflection coefficient rðωÞ increases with the increase of the coupling strength g between the N-V center and MTR cavity. Actually, as shown in Fig. 9, even when g o 0:1κ reflection coefficient can obtain rðωÞ 0:95, which means our schemes do not essentially need strong coupling and small damping. It is shown in Fig. 10 that even when κ ¼ 1500γ, reflection coefficient is nearly unity on the condition of g=γ ¼ 600, which means that the requirement on g and κ is relaxed. From the above, we can conclude that our proposed schemes can still work well even when the cavity is bad. At last, we would like to consider the experimental feasibility of our scheme. Firstly, the photon loss due to photon leakage during the input–output process, the fiber absorption, the cavity absorption and scattering and inefficient photon detection will bring ineffectiveness to our schemes. Actually, the photon loss just affects the success efficiency rather than the fidelity of our
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