Quantum cloning based on iSWAP gate with nitrogen-vacancy centers in photonic crystal cavities

Optics Communications 333 (2014) 187–192

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

Quantum cloning based on iSWAP gate with nitrogen-vacancy centers in photonic crystal cavities A-Peng Liu a, Jing-Ji Wen b, Liu-Yong Cheng c, Shi-Lei Su c, Li Chen d, Hong-Fu Wang a, Shou Zhang a,c,n a

Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China College of Foundation Science, Harbin University of Commerce, Harbin, Heilongjiang 150028, China c Department of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China d School of Science, Changchun University, Changchun 130022, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 1 July 2014 Received in revised form 25 July 2014 Accepted 29 July 2014 Available online 10 August 2014

we propose a scheme for physical implementation for the optimal asymmetric (symmetric) 1-2 universal quantum cloning, the optimal symmetric economical 1-3 phase-covariant cloning and the optimal asymmetric (symmetric) real state cloning based on an iSWAP gate between separated nitrogenvacancy (NV) centers embedded inphotonic crystal cavities. This two-qubit iSWAP gate is produced by the long-range interaction between two distributed NV centers mediated by the vacuum fields of the cavities. The analysis results show that our scheme is efficient and may be useful for scalable quantum information processing. & 2014 Elsevier B.V. All rights reserved.

Keywords: Nitrogen-vacancy center Photonic crystal cavity Cavity quantum electrodynamics Quantum cloning

1. Introduction Nowadays quantum information science, for its fascinating applications in information processing, has attracted much attention and undergone a rapid development. For quantum information processing (QIP), flying qubits, such as photon and motion electrons, are suitable for transferring information; while solidstate systems, e.g., quantum dots (QDs), nitrogen-vacancy (NV) centers and superconducting qubits (SQs), are of natural advantages for information storage. Among various solid-state-based emitters, the diamond NV center, consisting of a substitutional nitrogen atom and an adjacent vacancy, has emerged as a promising building block for solid-state QIP, owing to its extremely long electronic and nuclear spin lifetimes at room temperature [1]. Furthermore, the capability for optical coherent manipulation, such as fast initialization, well qubit readout, and information storage makes NV center suitable for QIP [2–4]. Diamond NV center embedded in various “cavities”, such as microtoroidal resonator [5,6], superconducting flux qubit [7] and nanoscale photonic crystal (PC) cavities [8–10] has been widely studied. In particular, for the strong interactions between solid state-based emitters and radiation-field modes of high-Q, PC cavities have attracted much attention and many theoretical and experimental n

Corresponding author. E-mail address: [email protected] (S. Zhang).

http://dx.doi.org/10.1016/j.optcom.2014.07.087 0030-4018/& 2014 Elsevier B.V. All rights reserved.

efforts have been devoted to quantum communication and quantum computation based on this composite PC-NV system recently [11–18]. Especially, planar PC nanocavities have gained widespread interest because of their high-Q factor and small mode volumes [19]. Strong coupling between NV centers and the modes of silicon nitride PC or gallium phosphide PC have also been experimentally demonstrated [8–10]. All these features permit the PC-NV system an ideal platform for QIP. In the field of quantum information science, the quantum nocloning theorem [20,21] forbids the creation of identical copies of an arbitrary unknown quantum state and ensures the absolute security of quantum communication. Although it is impossible to perfectly copy an unknown quantum state, one can construct approximate copies in a deterministic way [22–28]. In 1996, Bužek and Hillery [22] first designed a symmetric universal quantum cloning (UQC) machine that acts on any unknown quantum state and produces identical approximate copies equally well. Later, real state cloning (RSC) [29], which has better quality than UQC when the amplitude of the input pure state is unknown while the phase of the input state is known, was also reported. Otherwise, when the amplitude of the input pure state is known while the phase is unknown, the so-called phase-covariant cloning (PCC) [30,31] might be a better way. Recently, many protocols for realizing quantum cloning machines (QCMs) have been experimentally proposed [32–36]. In this work, we propose a scheme for optimal asymmetric (symmetric) 1-2 UQC, the optimal symmetric economical 1-3

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splitting between j3 A; ms ¼ 0〉 and j3 A; ms ¼ 7 1〉. Meanwhile, the degenerate levels j3 A; ms ¼ 7 1〉 can be split by employing an external magnetic field B0 along the quantized symmetry axis of the NV center, which induces a level splitting Deg. The NV center has a relatively complicated structure of excited states, which includes six excited states determined by the NV center's C3v symmetry. Here we choosepone ffiffiffi of the excited states jA2 〉 ¼ ðjE  ; ms ¼ þ 1〉 þ jE þ ; ms ¼  1〉Þ= 2 as an ancillary state, where jE 7 〉 are orbital states with angular momentum projection 71 along the NV axis. We encode level jA2 〉 as jr〉 and choose the ground-state sublevels jg〉 ¼ j3 A; ms ¼ þ 1〉 and je〉 ¼ j3 A; ms ¼ 1〉 [43]. As shown in Fig. 1, in our scheme, the transition between jg〉 and jr〉 of each NV center is dispersively coupled to the cavity mode with coupling constant gj and detuning ΔC;j , while the transition between je〉 and jr〉 is driven by a largely detuned laser field with Rabi frequency Ωj and detuning ΔL;j . In this case the total Hamiltonian of the system is written as (ℏ¼ 1)

PCC and the optimal asymmetric (symmetric) RSC. The quantum cloning process is based on an efficient two-qubit gate, i.e., an iSWAP gate, produced by the long range interaction between two NV centers embedded in two spatially separated singlemode nanocavities in a planar PC. In the scheme, two ground states of each NV center are coupled via the respective cavity mode and a classical field in the Raman manner. The bosonic modes are never populated and the NV centers undergo no transitions, therefore the gate operation is insensitive to the decoherence effect.

2. iSWAP gate between two NV centers embedded in two spatially separated PC cavities The PC nanocavities are point defects embedded within periodic dielectric structures, where photons are completely localized in the vicinity of the defects [37]. Furthermore, this kind of cavities, which can be easily tuned by bringing a nano-object into the near field of the cavity [38,39], can strongly confine photons in a tiny space of optical-wavelength dimension within a full band gap [40]. As illustrated in Fig. 1(a), two distant NV centers located in two directly coupled single-mode nanocavities with the cavity frequency tunable by changing the geometrical parameters of the defects [41,42]. As shown in Fig. 1, each NV center consists of a substitutional nitrogen atom and an adjacent vacancy having trapped an additional electron, whose electronic ground state has a spin S ¼1. The spin–spin interaction leads to the energy

H tot ¼ H C þH NV þ H CC þ H int ;

ð1Þ

with H C ¼ ∑ ωjC a†j aj ; j ¼ 1;2

H NV ¼ ∑ ðεjr jr〉j 〈rjþ εjg jg〉j 〈gj þ εje je〉j 〈ejÞ; j ¼ 1;2

H CC ¼ Jða†1 a2 þ a1 a†2 Þ; H int ¼ ∑ ½g j aj jr〉j 〈gje  iΔC;j t þ Ωj jr〉j 〈eje  iΔL;j t þ H:c:;

ð2Þ

j ¼ 1;2

A2

ΔL

ΔC

Ω

g Deg

ms = +1

m s = -1

3

A

ms = 0

ΔC,1 Δ L,1

r

1

Ω1

g1 e g

1

1

Laser

Δ C,2

J

ΔL,2

r

2

Ω2

g2 e g

2

2

Laser

Fig. 1. (a) The system under consideration consists of a planar PC cavity. The bubble shows the level structure of an NV center, where the electronic ground state j3 A〉 is a spin triplet state, and Deg ¼ γ e B0 is the level splitting induced by an external magnetic field B0 with γe p being the electron gyromagnetic ratio. We encode logical states ffiffiffi jg〉 ¼ j3 A; ms ¼  1〉 and je〉 ¼ j3 A; ms ¼ þ 1〉, while state jr〉 denotes jA2 〉 ¼ ðjE  ; ms ¼ þ 1〉 þ jE þ ; ms ¼  1〉Þ= 2. (b) The schematic representation of the coupled-PC-NV system, where two identical NV centers in diamond nanocrystals are located in two spatially separated nanocavities.

A.-P. Liu et al. / Optics Communications 333 (2014) 187–192

where aj ða†j Þ is the annihilation (creation) operator for the jth cavity mode. εjk (k¼ r,e,g) denotes the energy of level jk〉 of the jth NV center, J is the cavity–cavity coupling strength which can be tuned by the distance between the two nanocavities in PC [18]. This photon tunneling can be realized by the evanescent fields of the two cavities overlapped through the cavity dissipation, or by quantum channels, e.g., conventional optical fibers [44–46] or PC waveguides [47]. For simplicity, we assume g j ¼ g, Ωj ¼ Ω, ΔC;j ¼ ΔC and ΔL;j ¼ ΔL . Under the large detuning conditions, ΔC {g, ΔL {Ω, the level jr〉 can be adiabatically eliminated. The interaction Hamiltonian describing the NV-cavity and cavity– cavity interactions can be written as H I ¼ H 0I þ H CC ; H 0I ¼ ∑ ½ξje〉j 〈ej þ ηa†j aj jg〉j 〈gj þ λðeiδt aj je〉j 〈gjþ e  iδt a†j jg〉j 〈ejÞ;

ð3Þ

j ¼ 1;2

189

# ! c1 c†1 c2 c†2 þ þ ðje〉j 〈ej jg〉j 〈gjÞ 2 δJ δþJ )   λ2 1 1 þ þ jg〉j 〈gj 2 δ  J δ þJ

λ2

"

η2

ðc† c1  c†2 c2 Þðjg〉1 〈gj jg〉2 〈gjÞ2 8J 1 þ χ ðjg〉1 〈ej  je〉2 〈gjþ je〉1 〈gj  jg〉2 〈ejÞ; þ

ð7Þ

where χ ¼ λ =2ð1=ðδ  JÞ þ 1=ðδ þ JÞÞ. If we assume both the cavities are initially in the vacuum state, then the bosonic modes will remain in the vacuum state and the Hamiltonian H eff will reduce to 2

H 0eff ¼ ∑ μje〉j 〈ej þ χ ðjg〉1 〈ej  je〉2 〈gjþ je〉1 〈gj  jg〉2 〈ejÞ;

where μ ¼ ξ þ λ =2ð1=ðδ JÞ þ1=ðδ þ JÞÞ. The quantum information in the present scheme is stored in the states jg〉 and je〉, i.e.: j0〉  jg〉 and j1〉  je〉. This Hamiltonian in Eq. (8) is very useful for QIP, such as entanglement generation and transferring of quantum state [46]. Here we focus on the two-qubit operation produced by Hamiltonian in Eq. (8) acting on qubits 1 and 2, i.e., an iSWAP gate. This operation for the coupled-PC–NV system shown in Fig. 1 can be expressed as 0 1 0 0 0 e  i2μt B C B 0  ie  iμt sin χ t 0 C e  iμt cos χ t 0 C U iSWAP ¼ e  iHeff t ¼ B B C e  iμt cos χ t 0A  ie  iμt sin χ t @ 0 0 0 0 1 2

where ξ ¼ Ω =ΔL , η ¼ g =ΔC , δ ¼ ΔC  ΔL , and λ ¼ ðg Ω=2Þð1=ΔC þ 1=ΔL Þ. As done in Ref. [48], we introduce new bosonic modes which are given by the antisymmetric and symmetric superpositions of the states at the localized sites: 2

c1 ¼ p1ffiffi2 ða1 þ a2 Þ;

2

c2 ¼ p1ffiffi2 ða1  a2 Þ:

ð4Þ

In terms of these operators, the above Hamiltonians H0I and HCC become H CC ¼ Jðc†1 c1 c†2 c2 Þ; h i η H 0I ¼ ∑ ξje〉j 〈ej þ ðc1 c†1 þc2 c†2 Þjg〉j 〈gj 2 j ¼ 1;2

η

ð9Þ

þ ½ðc†1 c2 þc†2 c1 Þjg〉1 〈gj ðc†1 c2 þ c†2 c1 Þjg〉2 〈gj 2

λ

þ pffiffiffi½eiδt ðc1 þ c2 Þje〉1 〈gjþ eiδt ðc1  c2 Þje〉2 〈gj þ H:c:: 2

ð5Þ

Taking HCC as the “free Hamiltonian” and performing the unitary transformation eiHCC t , we obtain H 0I

iHCC t

¼e

¼ ∑

on the basis fje〉1 je〉2 ; je〉1 jg〉2 ; jg〉1 je〉2 ; jg〉1 jg〉2 g. With the choice of χ t ¼ π =2 and performing the single-qubit phase shifts: je〉j -eiμt je〉j (j¼1,2), an iSWAP gate is obtained such that je〉1 je〉2 -je〉1 je〉2 ; je〉1 jg〉2 -  ijg〉1 je〉2 ; jg〉1 je〉2 -  ije〉1 jg〉2 ; jg〉1 jg〉2 -jg〉1 jg〉2 :

i nη η ξje〉j 〈ejþ ðc†1 c1 þ c†2 c2 Þjg〉j 〈gj þ ei2Jt c†1 c2 ðjg〉1 〈gj jg〉2 〈gjÞ 2

2

λ

þ pffiffiffi½eiðδ  JÞt c1 je〉1 〈gjþ eiðδ þ JÞt c2 je〉1 〈gj þ eiðδ  JÞt c1 je〉2 〈gj 2 o þ eiðδ þ JÞt c2 je〉2 〈gj þ H:c: :

ð6Þ

pffiffiffi Under the conditions 2J{η=2, δ 7 J{λ= 2, the bosonic modes not only do not exchange quantum numbers with the NV system, but also do not exchange quantum numbers with each other. The Stark shifts and Heisenberg XY coupling between the NV centers are induced by the off-resonant Raman coupling. We thus have the effective Hamiltonian [46,49,50] nh i η H eff ¼ ∑ ξje〉j 〈ej þ ðc†1 c1 þ c†2 c2 Þjg〉j 〈gj 2 j ¼ 1;2

3. Optimal quantum circuit and physical implementation of quantum cloning In this section, we would like to show physical implementation of quantum cloning. An alternative way to realize optimal quantum cloning has been proposed [23,24]. As we know, the conventional two-qubit controlled-NOT (CNOT) gate can be constructed by applying iSWAP gate twice (together with six single-qubit rotation gates), namely  π  π   π U CNOT ¼ eiπ =4 Rz1  Rz2 U iSWAP Rx1  U iSWAP 4 π   π  2 π  4 Rx2  Rz1  ; ð11Þ Rz2 4 4 4 where Rαj ¼ eiϕσ α ðj ¼ 1; 2; α ¼ x; zÞ represent rotation operators acting on the jth qubit with σ α being the Pauli operators. Thus if we

p rep aratio n ψ 0 0

clo ning

1

2

3

ð10Þ

H 0I e  iHCC t

h

j ¼ 1;2

ð8Þ

j ¼ 1;2

R (θ1)

U1

U2

U3

R (θ 2 )

iSWAP gate

U1

U2

U3

R (θ 3 )

U1

U3

U1

U2

U2

U2

U1

U3

U3

Fig. 2. Three-bit quantum circuit representation for cloning a quantum state. Here the gate    corresponds to a two-qubit iSWAP gate and Uj (j¼ 1,2,3) are single-qubit rotation operations.

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A-P. Liu et al. / Optics Communications 333 (2014) 187–192

adopt the way in Ref. [24], we need nearly 10 two-qubit iSWAP operations and 40 single-qubit operations, this would be very complicate. Here using the present iSWAP gate, we propose a quantum circuit for physical realization quantum cloning which should be easier. The quantum circuit for the implementation of optimal quantum cloning is shown in Fig. 2, which can be divided into two processes: the preparation and the pffiffifficloning. we assume the state of NV 1 is jψ 〉1 ¼ ðαje〉1 þ βeiφ jg〉1 Þ= 2, and NV 2 is in the state jg〉2 . The NV center in cavity 3 is selected as the assist particle, it is also initially in the state jg〉3 . To realize the optimal quantum cloning, qubits 2 and 3 are entangled at first by applying single- and twoqubit gate operations, as shown in Fig. 2, where U1 ¼

e  iπ =4 0

U3 ¼

cos ðπ =4Þ sin ðπ =4Þ

!

0

;

eiπ =4

ei3π=4 cos ðπ =4Þ

U2 ¼

ei3π =4 sin ðπ =4Þ

 e  i3π =4 sin ðπ =4Þ

!  sin ðπ =4Þ ; cos ðπ =4Þ

Rðθi Þ ¼

cos sin

θi θi

þ β eiφ ½ðp þ qÞjeee〉321 þ pjgge〉321 þ qjgeg〉321 

cos 2θ1 þ sin 2θ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 þ sin 4θ1 sin 2θ1 C 3 ¼ q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: 2 þ sin 4θ1

;

F1 ¼

3 þ 2 sin 4θ1  cos 4θ1 ; 2ð2 þ sin 4θ1 Þ

If we choose the angles

π θ1 ¼ θ3 ¼ ;

C 1 ¼ cos θ1 cos θ2 cos θ3 þ sin θ1 sin θ2 sin θ3 ;

C 2 ¼ cos θ1 cos θ2 sin θ3  sin θ1 sin θ2 cos θ3 ; C 3 ¼ sin θ1 cos θ2 cos θ3  cos θ1 sin θ2 sin θ3 ; ð14Þ

Then the input state jψ 〉1 is copied to qubits 1 and 2 by the cloning work as shown in Fig. 2. After the cloning process, the state of the whole system is

π θ1 þ θ 3 ¼ ; 4

θ 1 þ θ3 ¼ ;

ð23Þ

1 þ cos 2θ1 þ sin 2θ1 pffiffiffi ; 2 2 1 þ cos 2θ1  sin 2θ1 pffiffiffi ; C2 ¼ 2 2 1  cos 2θ1 þ sin 2θ1 pffiffiffi C3 ¼ ; 2 2  1 þ cos 2θ1 þ sin 2θ1 pffiffiffi C4 ¼ : 2 2 C1 ¼

4 pffiffiffi 2 cos θ1 ð cos θ1 þ sin θ1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; cos θ2 ¼ 2 þ sin 4θ1 pffiffiffi 2 sin θ1 ð sin θ1  cos θ1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin θ2 ¼ ; 2 þ sin 4θ1 cos θ1 þ sin θ1 pffiffiffi ; 2

1

Δ C,2

J

Δ L,2

r

2

Ω2

g2 e

1

g

Laser

θ2 ¼ 0;

where θ1 ; θ3 A ½0; π =4, we can easily obtain that

π

e

ð22Þ

which corresponds to the optimal symmetric economical 1-3 PCC, with the corresponding fidelities being F 1 ¼ F 2 ¼ F 3 ¼ 5=6. pffiffiffi When the input information state jψ 〉1 ¼ ðαje〉1 þ β jg〉1 Þ= 2, if we set the angles of single-qubit rotations Rðθi Þ as

In the following, we show different types of optimal cloning can be realized by choosing different angles of single-qubit rotations. For the optimal asymmetric UQC, we choose the parameters as

1

ð20Þ

6

þ p1ffiffi3 eiφ ½jeee〉321 þ jgge〉321 þ jgeg〉321 ;

ð15Þ

g



1

jΨ 〉123 ¼ p1ffiffi3 ½jggg〉321 þ jeeg〉321 þ jege〉321 

þ βeiφ ½C 1 jeee〉321 þC 2 jgge〉321 þ C 3 jgeg〉321 þ C 4 jegg〉321 :

Ω1

3

ð19Þ

we can obtain

jΨ 〉123 ¼ α½C 1 jggg〉321 þC 2 jeeg〉321 þ C 3 jege〉321 þC 4 jgee〉321 

g1

1

3 þ 2 sin 4θ1 þ cos 4θ1 : 2ð2 þ sin 4θ1 Þ

the optimal asymmetric 1-2 UQC can be reduced p toffiffiffi the symmetric one with the cloning coefficients p ¼ q ¼ 1= 6 and the corresponding fidelities are F 1 ¼ F 2 ¼ 5=6. If the pffiffiffiinput information state is encoded in jψ 〉1 ¼ ðje〉1 þ eiφ jg〉1 Þ= 2, with the choice of sffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffi 1 3 þ 15 5 ; ð21Þ θ1 ¼ θ3 ¼ arccos þ ; θ2 ¼  arccos 2 10 6

ð13Þ

where

r



F2 ¼

θ2 ¼  arccos pffiffiffi þ pffiffiffi ;

8

jΨ 〉prep 23 ¼ α½C 1 jgg〉23 þ C 2 jge〉23 þ C 3 jge〉23 þ C 4 jee〉23 

Δ L,1

ð18Þ

The fidelities of the optimal asymmetric 1-2 UQC are

After the preparation process, the state of qubits 2 and 3 is

Δ C,1

cos 2θ1 C 2 ¼ p ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 2 þ sin 4θ1

C 1 ¼ p þq ¼

ði ¼ 1; 2; 3Þ:

C 4 ¼ cos θ1 sin θ2 cos θ3 þ sin θ1 cos θ2 sin θ3 :

ð17Þ

where

!

þ βeiφ ½C 1 jgg〉23 þ C 2 jge〉23 þ C 3 jge〉23 þC 4 jee〉23 ;

ð16Þ

jΨ 〉123 ¼ α½ðp þ qÞjggg〉321 þ pjeeg〉321 þ qjege〉321 

ð12Þ

cos θ3 ¼

cos θ1  sin θ1 pffiffiffi ; 2

where θ1 ; θ3 A ½0; π =4. Under our choice of the angles of the single-qubit rotations Rðθi Þ, the cloning coefficients of the optimal asymmetric 1-2 UQC can be expressed as

!

e  i3π =4 cos ðπ =4Þ

 sin θi cos θi

sin θ3 ¼

ð24Þ

Δ C,3

J

3

Ω3

e g

Laser

r

g3

2

2

Δ L,3

3

3

Laser

Fig. 3. Schematic setup to implement the quantum cloning shown in Fig. 2. Here the cavity–cavity coupling strength J which control two adjacent cavities whether have interaction or not can be modulated by the distance between the two nanocavities in PC or optical switches [18,49,50].

A.-P. Liu et al. / Optics Communications 333 (2014) 187–192

191

This situation corresponding to the optimal asymmetric RSC. Choosing the angles θ1 ¼ θ3 ¼ π =8, the optimal asymmetric RSC can be reduced to the symmetric one with the output state

probability that the two modes c1 and c2 are excited due to nonresonant coupling with the classical modes is

jΨ 〉123 ¼ α½ð12 þ p1ffiffi8 Þjggg〉321 þ p1ffiffi8 jeeg〉321 þ p1ffiffi8 jege〉321 þ ð12  p1ffiffi8 Þjgee〉321 

P1 ¼

þ ð12  p1ffiffi8 Þjegg〉321 :

ð25Þ

The schematic setup for physical implementation of the aforementioned several types of quantum cloning is shown in Fig. 3. Each NV center is embedded in a spatially separated cavity and each two cavities are coupled to each other. At the beginning, all the couplings between the cavities are turned off and the cavities are in a vacuum state. According to the orders of the gate operations shown in Fig. 2, the single-qubit operations required for implementing quantum cloning in the quantum circuits are realized by classical laser fields. The interaction between two NV centers is controlled by adjusting the cavity–cavity coupling strength J by modulating the distance between the two nanocavities in PC or optical switches [18,49,50]. Each time only the nearest-neighbor two NV centers interact with each other, while the others do not, by turning on or off the couplings between the cavities. The interaction time between two NV centers in the cavities is chosen as τ ¼ π =2χ such that the two-qubit iSWAP gate operating on the nearest-neighbor two NV centers is achieved. Therefore, the quantum cloning based on the iSWAP gate could be implemented in a deterministic way by using the PC–NV system.

4. Analysis and discussion We now briefly analyze and discuss some practical issues in relation to the experimental feasibility of the present scheme. In this composite PC–NV system, we exploit the dipole transition between j3 A; ms ¼ 7 1〉2jA2 〉 with a zero phonon line at Λ ¼ 637 nm of the NV centers in a diamond nanocrystal. Experimentally, the strong coupling case g 2 =κ 0 γ 0 ¼ 325 in a silicon nitride PC nanocavity with a Q factor of 1.4  106 and a mode volume Vm  2:5  10  20 m3 has been reported [10]. In another experiment, they show strong coupling between the mode of PC nanocavity and the NV center has been achieved with the Q factor 1.5  106, where the typical experimental parameters are ðg; κ 0 ; γ 0 Þ=2π ¼ ð2:25; 0:16; 0:013Þ GHz [8]. For the presently available technology on the coupled-PC–NV system, we set g=2π ¼ 1 GHz and Ω=2π ¼ 1 GHz, and the detuning ΔC =2π ¼ 12 GHz, ΔL =2π ¼ 10 GHz, the effective PC-NV coupling rates could be η=2π C 8:33  10  2 GHz, ξ=2π ¼ 0:01 GHz, λ=2π C 9:17 10  2 GHz. Considering pffiffiffi δ=2π ¼ 2 GHz, J=2π ¼ 1 GHz, the conditions 2J{η=2, δ 7 J{λ= 2 can be satisfied. The probabilities that NV centers undergo transitions from jg〉 and je〉 states to jr〉 2 2 2 state are P gr ¼ g 2 =ΔC ¼ 6:94  10  3 and P er ¼ Ω =ΔL ¼ 10  2 . The

0.83

F2

0.80 0.79 0.00

F

F

0.82 F1

0.05

0.10 k

0.15

0.20

1

2 ðδ  JÞ

þ 2

#

1 ðδ þ JÞ2

C 4:67  10  3 :

ð26Þ

Therefore, the effective Hamiltonian in Eq. (8) is valid. The time required for realizing the iSWAP gate is τ ¼ π =2χ C 4:46  10  2 μs. Our operating time for realization of quantum cloning is T ¼ 5τ þ 23t C 0:52 μs with t being the time for performing a single-qubit rotation operation which is nearly 1:3  10  2 μs. Besides, the effective decoherence rates due to the decay of the bosonic modes are κ ¼ κ 0 P 1 ¼ 2π  2:34  10  2 MHz with κ 0 ¼ c=ΛQ ¼ 2π  5 MHz, where c is the speed of light and Q¼108 is the cavity quality factor. The effective spontaneous decay rate from the state jr〉 to je〉 and jg〉 could be estimated as γ e ¼ γ 0 P er C 2π  0:13 MHz and γ g ¼ γ 0 P gr C 2π  0:09 MHz with γ 0 ¼ 2π  13 MHz [51,52], here we have assumed the equal spontaneous decay rates from jr〉 to je〉 and to jg〉, respectively. Thus we can calculate the average infidelity induced by the decoherence is F inf ¼ 3:64  10  2 . Besides, the imprecise interaction time between two NV centers and a slight deviation of the single-qubit rotation angles could have an effect on the fidelities of quantum cloning. Therefore, we assume that the interaction time and the angles of the single qubit rotations may have a tiny deviation from the ideal 0 angles, i.e., t 0 ¼ t 7kt, θi ¼ θi 7 εθi ði ¼ 1; 2; 3Þ ε being a small constant. We plot the effects of k and ε on the fidelity of realizing the optimal symmetric 1-2 QUC in Fig. 4. We note that the effects of the errors on the fidelities of the resulting state are very little. When ε ¼0.2 the fidelity F ¼ F 1 ¼ F 2 still can be larger than 0.78. Besides, even when k ¼0.2, the fidelities F1 and F2 still can be larger than 0.81 and 0.80, respectively. Also we can get the same results by using the same methods in the implementation of other types of quantum cloning. Furthermore, since the interaction between two distributed NV centers is mediated by the vacuum fields of the cavities, the scheme is insensitive to the cavity decay and the spontaneous emission of NV center. These features make the present scheme feasible for experimental realization and for scalable quantum communications and quantum computation. Along with the progress in manipulation of the coupling between coupled-PC–NV nanocavities, we believe our scheme could be useful for optimal quantum cloning. In summary, we have proposed a scheme of an iSWAP gate between two NV centers located in two spatially separated nanocavities in a planar PC. The optimal quantum circuit and physical implementation of optimal asymmetric (symmetric) 1-2 universal quantum cloning, the optimal symmetric economical 1-3 phase-covariant cloning and the optimal asymmetric (symmetric) real state cloning have been proposed. During the process of implementing quantum cloning, no one subsystem is excited

0.84

0.81

"

0.84 0.83 0.82 0.81 0.80 0.79 0.78 0.77 0.00

0.05

0.10

∋

þ β½ð12 þ p1ffiffi8 Þjeee〉321 þ p1ffiffi8 jgge〉321 þ p1ffiffi8 jgeg〉321

λ2

0.15

0.20

Fig. 4. Fidelities of realizing the optimal 1-2 UQC versus (a) k and (b) ε. The parameters used in the numerical calculation are θ1 ¼ θ3 ¼ π=8, θ2 ¼  arccosðp1ffiffi3 þ p1ffiffi6 Þ, t ¼ π=2χ.

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and the scheme is insensitive to the cavity decay or the spontaneous emission of the NV centers. This scheme is simple and feasible and we hope it could be useful for future quantum information processing based on solid-state systems. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 61068001 and 11264042; the Program for Chun Miao Excellent Talents of Jilin Provincial Department of Education under Grant no. 201316; and the Talent Program of Yanbian University of China under Grant no. 950010001. References [1] M. Stoneham, Physics 2 (2009) 34. [2] F. Jelezko, T. Gaebel, I. Popa, A. Gruber, J. Wrachtrup, Phys. Rev. Lett. 92 (2004) 076401. [3] R. Hanson, F.M. Mendoza, R.J. Epstein, D.D. Awschalom, Phys. Rev. Lett. 97 (2006) 087601. [4] V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, J. Wrachtrup, Phys. Rev. Lett. 102 (2009) 057403. [5] Q. Chen, W.L. Yang, M. Feng, J.F. Du, Phys. Rev. A 83 (2011) 054305. [6] L.Y. Cheng, H.F. Wang, S. Zhang, K.H. Yeon, Opt. Express 21 (2013) 5988. [7] X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W.J. Munro, Y. Tokura, M.S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, K. Semba, Nature (Lond.) 478 (2011) 221. [8] P.E. Barclay, K.M. Fu, C. Santori, R.G. Beausoleil, Opt. Express 17 (2009) 9588. [9] D. Englund, B. Shields, K. Rivoire, F. Hatami, J. Vǔ cković, H. Park, M.D. Lukin, Nano Lett. 10 (2010) 3922. [10] M.W. McCutcheon, M. Lončar, Opt. Express 16 (2008) 19136. [11] I. Aharonovich, S. Castelletto, D.A. Simpson, C.H. Su, A.D. Greentree, S. Prawer, Rep. Prog. Phys. 74 (2011) 076501. [12] J. Wolters, A.W. Schell, G. Kewes, N. Nüsse, M. Schoengen, H. Döscher, T. Hannappel, B. Löchel, M. Barth, O. Benson, Appl. Phys. Lett. 97 (2010) 141108. [13] T. van der Sar, J. Hagemeier, W. Pfaff, E.C. Heeres, S.M. Thon, H. Kim, P. M. Petroff, T.H. Oosterkamp, D. Bouwmeester, R. Hanson, Appl. Phys. Lett. 98 (2011) 193103. [14] M. Barth, N. Nüsse, B. Löchel, O. Benson, Opt. Lett. 34 (2009) 1108. [15] S. Tomljenovic-Hanic, A.D. Greentree, C. Martijn de Sterke, S. Prawer, Opt. Express 17 (2009) 6465. [16] C.H. Su, A.D. Greentree, L.C.L. Hollenberg, Phys. Rev. A 80 (2009) 052308.

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