Deterministic implementation of optimal symmetric ... - OSA Publishing

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J. Opt. Soc. Am. B / Vol. 31, No. 10 / October 2014

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Deterministic implementation of optimal symmetric quantum cloning with nitrogen-vacancy centers coupled to a whispering-gallery microresonator Zhao Jin, Yan-Qiang Ji, Ai-Dong Zhu,* Hong-Fu Wang, and Shou Zhang Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China *Corresponding author: [email protected] Received June 26, 2014; revised August 27, 2014; accepted August 27, 2014; posted August 28, 2014 (Doc. ID 214834); published October 1, 2014 Schemes for implementing quantum cloning are proposed with nitrogen-vacancy (N-V) centers in diamond confined in separated microtoroidal resonators via a single-photon input–output process and linear optics. By properly arranging and adjusting the linear optical elements to control the interaction between photons and N-V centers, we realize quantum cloning transformations, including the optimal symmetric 1 ! 2 universal, phase-covariant, and 1 ! M economical phase-covariant ones. In these schemes, photons are used as data buses and information is copied from one N-V center to the other. We also discuss the cloning fidelities and dissipative effects. The results show that the present schemes are robust under the current experimental conditions. © 2014 Optical Society of America OCIS codes: (270.5565) Quantum communications; (270.5585) Quantum information and processing; (270.5570) Quantum detectors; (030.5260) Photon counting. http://dx.doi.org/10.1364/JOSAB.31.002516

1. INTRODUCTION Since perfect quantum copies for arbitrary quantum states are forbidden [1], approximate cloning [2] and probabilistic cloning [3] of quantum states are attracting great interest. Approximate quantum cloning plays an important role in quantum cryptography, and has useful applications in quantum information processing (QIP) [4,5]. There are usually several types of quantum cloning machines in terms of different input states, e.g., the universal quantum cloning machine (UQCM), which clones an arbitrary state with the same fidelity, and the phase-covariant cloning machine (PCCM), which clones a partially known state with a higher fidelity than the UQCM [2,6–11]. The 1 ! 2 economical PCCM (EPCCM) without ancilla was first proposed by Niu and Griffiths in 1999 [12], and then developed by Fiurášek in 2003 [13]. It includes two different optimal cloning transformations, one for the input qubit on the northern hemisphere of the Bloch sphere and another for the input qubit on the southern hemisphere [14–17]. It has been studied extensively because of its usage in quantum cryptography and has been verified by the experiment using nuclear magnetic resonance techniques [18]. The 1 ! M EPCCM transformation by preparing multipartite W states has also been derived explicitly [19,20]. Recently, solid-state systems, including quantum dot, nitrogen-vacancy (N-V) centers, and so on, have been considered as promising candidates for QIP [21–26]. Among the various proposals, the N-V center in diamond coupled to a microtoroidal resonator (MTR) with a quantized whisperinggallery mode (WGM) has attracted great interest. It consists of a nearest-neighbor pair of a nitrogen atom substituted for 0740-3224/14/102516-08$15.00/0

a carbon atom and a lattice vacancy in diamond. The N-V center has a long-lived spin triplet even at room temperature and can be manipulated by an electromagnetic field or optical pulse [27,28]. It seems that the resonant zero phonon line (ZPL) relevant to the photons emitted from the N-V center could be enhanced significantly by embedding the N-V centers in cavities [29]. These unique characteristics make the N-V center one of the most optimal candidates for QIP. Since this MTR-NV system can serve as an efficient node for distributed quantum networks, it may be useful for constructing a quantum network on a large scale. In 2008, an experimental apparatus for a singlephoton input–output process from a MTR with a coupled optical fiber was proposed by Dayan et al. [30]. In 2009, An et al. proposed some schemes for QIP based on Faraday rotation and single-photon input–output processes [31], which are much different from the situation of high-Q cavity and strong coupling cases. In 2011, Chen et al. proposed an efficient scheme to deterministically entangle two N-V centers fixed on the exterior surface of two MTRs [24]. In 2012, Zheng et al. proposed a scheme to generate an N-qubit Greenberger–Horne–Zeilinger state with distant N-V centers confined in spatially separated photonic crystal nanocavities via a photon input–output process [32]. Most recently, investigations on multipartite entanglement with N-V centers and a MTR-coupled system have also been reported [25,33]. Experimentally, the quantum nondemolition measurement for a single spin of an N-V center by Faraday rotation has also been accomplished [34]. Enlightened by these works, we present theoretical schemes for implementing quantum cloning machines with N-V centers in diamond confined in separated MTRs via a © 2014 Optical Society of America

Jin et al.

Vol. 31, No. 10 / October 2014 / J. Opt. Soc. Am. B

single-photon input–output process. The schemes only need low-Q cavities and moderate coupling; thus, they are less challenging experimentally than those in previous works, which required high-Q cavities and strong coupling. The remainder of the paper is organized as follows: in Section 2, the interaction between an N-V center and a polarized photon is introduced. With such systems, in Section 3 we implement optimal symmetric 1 ! 2 UQCM and PCCM. In Section 4, we implement an optimal symmetric 1 ! M EPCCM for the input state on the northern and southern hemispheres. In Section 5, we discuss the dissipative effects caused by cavity decay, dipole decay, and spectral diffusion on the fidelity of quantum cloning. The paper concludes with a summary in Section 6.

2. INPUT–OUTPUT INTERACTION BETWEEN A SINGLE-PHOTON AND AN N-V CENTER CONFINED IN A MTR We consider that an N-V center, consisting of a substitutional nitrogen atom and an adjacent vacancy, is fixed on the surface of a MTR with a WGM and is coupled to the cavity mode. The energy-level configuration of the N-V center is illustrated in Fig. 1 where j3 A2 i and jA2 i are the ground state and excited state of an electronic spin triplet, respectively. The ground state j3 A2 i is split into the upper levels j3 A2 ; ms  1i and the lower level j3 A2 ; ms  0i by 2.88 GHz owing to spin–spin interaction. Here j3 A2 i  jE0 i ⊗ jms  0; 1i, and jE 0 i is the orbital state with zero angular momentum projection along thepN-V axis. If the excited state jei is chosen as jA2 i 


1∕ 2jE− ij  1i  jE  ij − 1i, then a Λ-type three-level system can be constructed [35] where jE  i are orbital states with 1 angular momentum projection along the N-V axis. The state jA2 i is induced by spin–orbit and spin–spin interactions and C 3v symmetry [36], which is robust with stable symmetry. It decays to the ground state j − 1i and j  1i with radiation of left (L) and right (R) polarizations, respectively. This system exhibits similar features to the Jaynes–Cummings model, so the Hamiltonian of the system can be expressed as  X ω0j H (1) σ zj  ωcj a†j aj  igj aj σ j − a†j σ −j  ; 2 jR;L where ω0 and ωc are the transition frequencies of the electronic energy levels and the WGM, respectively; a† and a are the creation and annihilation operators of the MTR field, respectively; and σ z , σ  , and σ − are, respectively, the inversion,

A2 = e

σ+

σ−

L− light

3

A2

{

raising, and lowering operators of the N-V center between the two corresponding transition levels. A single-photon pulse with frequency ωp is injected into the MTR as shown in Fig. 2. Considering the low-temperature reservoir and neglecting the vacuum input field, along with the input–output relation [31,37,38], we get the quantum Langevin equations   p


daj κ κ  iωp − ωc  − − s aj t − gσ j− t − κaj;in t  Γ; 2 2 dt   dσ j− γ (2)  iωp − ω0  − σ j− t − gσ j;z taj t  Ω; 2 dt where κ, κ s , and γ are, respectively, the cavity decay rate, side leakage rate, and dipole decay rate of the N-V center; Γ and Ω the noise operators related to the reservoirs; and g is the coupling strength between the N-V center and the MTR. Under the condition of a weak incoming field, i.e.,hσ z i ≃ −1, we can adiabatically eliminate the cavity mode and find the reflection coefficient for the photon pulse as [31] h ih i iωc −ωp − 2κ  κ2s iω0 −ωp  2γ g2 aout t ih i rωp   ; (3) h ain t iω −ω  κ  κs iω −ω  γ g2 c

p

2

r 0 ωp  

Fig. 1. Energy-level configuration of an N-V center and relevant transition coupling with the input polarized photon jRi and jLi.

0

p

2

iωc − ωp  − 2κ  κ2s : iωc − ωp   2κ  κ2s

(4)

By taking ω0  ωc  ωp in Eqs. (3) and (4), we have r 0 ωp  

κs − κ ; κs  κ

rωp  

κs − κγ  4g2 : κs  κγ  4g2

(5)

If the N-V center is initially prepared in the state j − 1i or j  1i, for an input L-polarized photon, the output will be in the state jΨout i  rωjLi =eiφ jLi or jΨout i  r 0 ωjLi  eiφ0 jLi, where the parameters φ and φ0 represent the phase shifts determined by the input–output relation. On the other hand, if an R-polarized photon is sent into the system, it will just experience a phase shift of φ0 regardless of the state of the N-V centers because j  1i is decoupled to the input pulse with frequency ωp due to the large detuning. So we have φ  0 p




and φ0  π by taking κs ≪ κ, g ≥ 5 κγ , and ω0  ωc  ωp . Hence, a controlled phase flip gate is obtained conditioned on the incident photon pulse and the N-V center after a π-phase shifter on the photon reflection path [25,26]:

N-V center

ms = +1

ms = 0

2

where the relation between the output and input fields is p


aout t  ain t  κat. When the input photon is uncoupled to the WGM, i.e., g  0, we get the reflection coefficient for a cold cavity:

R− light

ms = −1

2517

π

Input pulse

Detector MTR

Fig. 2. N-V center is confined to a MTR with a quantized WGM and a single-photon pulse is introduced to interact with the N-V center.

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jRij  1i ! jRij  1i;

jRij − 1i ! jRij − 1i;

jLij  1i ! jLij  1i;

jLij − 1i ! −jLij − 1i:

Jin et al.

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3. IMPLEMENTATION OF OPTIMAL SYMMETRIC 1 ! 2 UQCM AND PCCM The systematic setup for implementing a UQCM and a PCCM is shown in Fig. 3, where C-PBS j (j  1, 2, 3, 4) are polarization beam splitters that transmit the jRi polarization component and reflect the jLi polarization component. The dotted diagonals in C-PBS2 and C-PBS4 represent the status after a 90° rotation. TRk (k  1, 2, 3, 4) are optical devices exactly controllable for transmitting or reflecting photons with very fast switch. The half-wave plates HWPϑi (i  1, 2) with their axes at an angle of ϑi rotate the polarization state of the photons by the transformation fjRi ! cos 2ϑi jRi  sin 2ϑi jLi, jLi ! − cos 2ϑi jLi  sin 2ϑi jRig. D1 and D2 are single-photon detectors. Mirrors M 1 and M 2 reflect the polarization component without change of shape and phase, while DLs are delay lines. N-V1 is initially in an arbitrary state αji1  βeiϕ j−i1 , which is to be cloned; the other two N-V centers are prepared in state ji, p


and the logical qubits are encoded p


in the states ji  1∕ 2j  1i  j − 1i and j−i  1∕ 2j  1i − j − 1i. The TRk (k  1, 2, 3, 4) are initially switched to transmission status. The half-wave plates HWPϑi i  1; 2 are adjustable and ϑi i  1; 2 are initially arranged as ϑ1  90° and ϑ2  67.5°. First, a single-photon pulse A in the state cos ψjRi  sin ψjLi is injected into input port 1 and reaches C-PBS1. The jRi polarization component is transmitted, whereas the jLi polarization component is reflected to path a and then passes through TR1–TR2–N-V1 to reach DL1. The state of the system composed of the three N-V centers and the single photon can be described as cos ψjRib αji1  βeiϕ j−i1   sin ψjLia αj−i1  βeiϕ ji1 ji2 ji3 :

(7)

HWP p


22.5° rotates the jRib polarization component to the state 1∕ 2jRib  jLib ; then the state of the system becomes D2 Input 1 HWP 22 .5 o

a

N− V 1

DL 3

HWP 22 .5 o

DL 1

Input 2

r


1 cos ψjRib  jLib αji1  βeiϕ j−i1 ji2 ji3 2  sin ψjLia αj−i1  βeiϕ ji1 ji2 ji3 ;

then the photon passes through C-PBS2, which reflects the jLib polarization component to path b and transmits the jRic polarization component to path c, so the state of the system can be represented as r


1 cos ψjLib αji1 j−i2 ji3  βeiϕ j−i1 j−i2 ji3  2  jLic αji1 ji2 j−i3  βeiϕ j−i1 ji2 j−i3  × sin ψjLia αj−i1 ji2 ji3  βeiϕ ji1 ji2 ji3 ;

TR2

HWP 22 .5 oT R 3

N− V 2

C PBS 1 HWP 22 .5 o

DL 2

C PBS 4

b HWPϑ 2

N− V3

C PBS 2

TR 4

C PBS 3

c

HWP 45 o

HWP 45 o

M2

M1 HWPϑ 1

Fig. 3. Setup of the UQCM and PCCM by which quantum information from N-V1 is distributed to N-V2 and N-V3. HWPθ stands for half-wave plate with its axis at θ. TRs are optical devices exactly controllable for transmitting or reflecting a photon with very fast switch. C-PBSs are circular polarization beam splitters, M12 is a mirror, DLs are delay lines, and D1 (D2) is a single-photon detectors (similarly hereinafter).

(9)

here the subscripts a, b, and c represent different paths of the single photon. The photon pulse in path c is then reflected by mirror M 2 . Finally, the photons of the three paths a, b, and c are collected by C-PBS4. The jRi polarization component of path a, and the jLi polarization component of path b and path c will be detected by D1 after passing through C-PBS4. However, the jLi polarization component of path a, and the jRi polarization component of path b and path c will be detected by D2. If D2 clicks, then the state of the three N-V centers is projected to r


 1 α sin ψj−i1 ji2 ji3  cos ψji1 j−i2 ji3 2    ji1 ji2 j−i3   βeiϕ sin ψji1 ji2 ji3 

r


 1 cos ψj−i1 j−i2 ji3  j−i1 ji2 j−i3  : 2

(10)

On the other hand, if D1 clicks, then the systemic state evolves to Eq. (10) after a unitary operation j − 1i↔j  1i is applied on the second N-V center. Before the second photon B is injected into the setup, we switch TR1, TR2, and TR4 to reflection status, and perform a 90° rotation on C-PBS2 and C-PBS4. By applying a single-qubit operation fji1 ↔j∓1i1 g on the first N-V1 and adjusting to ϑ1  45° and ϑ2  90°, the state of Eq. (10) evolves to r


1 cos ψj − 1i1 j−i2 ji3 α sin ψj  1i1 ji2 ji3  2    j − 1i1 ji2 j−i3   βeiϕ sin ψj − 1i1 ji2 ji3 

D1 TR1

(8)



r


 1 cos ψj  1i1 j−i2 ji3  j  1i1 ji2 j−i3  : 2

(11)

At this time, let the photon pulse B in state jRi enter input port 2. It goes forward along the path TR1–HWP22.5° –TR2–N-V1– DL1–HWP22.5° –TR3–C-PBS4. The system state becomes r


 1 cos ψj − 1i1 j−i2 ji3 α sin ψj  1i1 ji2 ji3 jRia  2    j − 1i1 ji2 j−i3 jLia  βeiϕ sin ψj − 1i1 ji2 ji3 jLia r


 1 cos ψj  1i1 j−i2 ji3  j  1i1 ji2 j−i3 jRia :  2

(12)

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Vol. 31, No. 10 / October 2014 / J. Opt. Soc. Am. B

Obviously, the jRia polarization component transmitted by C-PBS4 passes through TR4 to reach DL3, and the jLia polarization component passes through HWPϑ2 to reach C-PBS3. At this time, a 90° rotation on the C-PBS4 should be performed again to make sure it returns to the initial position, and we need to adjust ϑ2  22.5°, and switch TR3 to reflection status and TR4 to transmission status. Then the jLi polarization photon travels along the path C-PBS3–DL2–N-V2–C-PBS2– HWP45° –M1 –HWPϑ1 –N-V3–M2 –HWP45° –C-PBS3. The above state can be written as

r


 1 cos ψj  1i1 j−i2 ji3  j  1i1 ji2 j−i3 jRia : (13)  2 Next, the jRib polarization component passes through HWPϑ2 to reach C-PBS4. At the same time, the polarization photon jRia passes through HWP22.5° –TR3 to reach C-PBS4 again. jRia and jLib will be detected by D1, and jLia and jRib will be detected by D2. Whichever detector clicks, the state of the three N-V center is projected to r


 1 cos ψj − 1i1 ji2 j−i3 α sin ψj  1i1 ji2 ji3  2    j − 1i1 j−i2 ji3   βeiϕ sin ψj − 1i1 j−i2 j−i3 r


 1 cos ψj  1i1 j−i2 ji3  j  1i1 ji2 j−i3  : 2

transmit

TR1 (TR2 )

r =1

transmit

t =1

t0

t =1

t 0 + t1

t TR3

t TR4

r =1

reflect

r =1

reflect

transmit t0

t1

t

reflect

Fig. 4. Status of TRk (k  1, 2, 3, 4) can be switched accurately with computers. t0 is the interval for photon A from input port 1 to D1 or D2; t1 is time for the second photon to travel along the path input port 2–C-PBS1–TR1–HWP22.5° –TR2–N-V1–DL1–HWP22.5° –TR3– C-PBS4–TR4–DL3.

4. IMPLEMENTATION OF AN OPTIMAL SYMMETRIC 1 ! M EPCCM

r


1 α sin ψj  1i1 ji2 ji3 jRia  cos ψj − 1i1 ji2 j−i3 2    j − 1i1 j−i2 ji3 jRib  βeiϕ sin ψj − 1i1 j−i2 j−i3 jRib 



t =1

2519

(14)

Obviously, by choosing the input state of the single photon as ψ  arcsin 2∕31∕2 , we can obtain an optimal symmetric 1 ! 2 UQCM as r


r


 2 1 j  1i1 ji2 ji3  j − 1i1 ji2 j−i3  j−i2 ji3  3 6 r


r


 2 1 j − 1i1 j−i2 j−i3  j  1i1 j−i2 ji3  ji2 j−i3  :  βeiϕ 3 6 (15) p


In case N-V1 is initially in an equatorial state 1∕ 2ji1  eiϕ j−i1  with an unknown ϕ, by choosing ψ  45°, we can obtain an optimal symmetric 1 ! 2 PCCM as

Based on similar coupling systems of N-V centers and MTRs, we can implement an optimal symmetric 1 ! M EPCCM effectively. The setup, shown in Fig. 5, is composed of nn  M  1 N-V centers and a series of linear optical elements. We take the optimal symmetric 1 ! 3 EPCCM as an example in the following demonstration. Initially, ϑ1 , ϑ2 , and ϑ3 are arranged as ϑ1  12 arcsin 1∕31∕2 , ϑ2  22.5°, and ϑ3  45°; TR1 is switched to transmission status and TR2 to reflection status. N-V1 is initially in a state to be cloned on the northern hemisphere fcosθ∕2ji1  eiϕ sinθ∕2j−i1 0 ≤ θ ≤ π∕2g, and the other N-V centers, 2, 3, and 4, are initially prepared in state ji. First, a Hadamard transformation (e.g., using a π∕2 microp


wave pulse) fji ! 1∕ 2ji  j−ig should be performed on N-V1, p


and then a single-photon pulse in the state jRi  jLi∕ 2 is injected into the input port. After the interaction between the photon and N-V1, a Hadamard transformation is performed on N-V1 again; then the state of the system can be written as 1 θ p


jRi1  jLi1  cos ji1 ji2 ji3 ji4 2 2 1 θ  p


jRi1 − jLi1  expiϕ sin j−i1 ji2 ji3 ji4 ; 2 2 N− V 1

Laser

α

r


r


 1 1 1 j  1i1 ji2 ji3  j − 1i1 ji2 j−i3  j−i2 ji3  2 2 2 r


r


 1 1 1 j − 1i1 j−i2 j−i3  j  1i1 j−i2 ji3  ji2 j−i3  :  eiϕ 2 2 2 (16) In the above process, we need to switch the TRk (k  1, 2, 3, 4) and rotate C-PBS2 and C-PBS4. The switching moments can be accurately controlled with computers [39–41]. The time switching sequences for the 1 ! 2 UQCM and PCCM are shown in Fig. 4.

Input

TR2

TR1

(17)

C PBS 1

HWP 45 o

HWP 22 .5 o

1

HWP 22 .5 o

M2

M1 M4

HWP 45 o

DL 1

M3 N− V 2

HWPϑ 1 HWP 22 .5 o

DL 2

2

C PBS 2

D2

N− V3

HWPϑ 2

HWP 22 .5 o

D1

3

C PBS 3 N− V 4

DL 3

HWPϑ 3

C PBS n + 1

DL 4

4

C PBS 4

HWP 22 .5 o

N− V n

HWPϑ n −1

M5

DL n

n HWP 22 .5 o

Fig. 5. Setup of the optimal symmetric 1 ! M EPCCM by which the quantum information from N-V1 is distributed to other N-V centers.

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where the subscript of the photon state represents the path information of the photon. By switching TR1 and TR2, the photon is led to travel along the path TR2–M2 –HWP22.5° – M1 –TR1 and interacts again with N-V1; then the state of the composite system can be expressed as θ ji1 ji2 ji3 ji4 2 θ  jLi1 expiϕ sin ji1 ji2 ji3 ji4 : 2

jRi1 cos

(18)

Next, the photon passes through TR2 to reach C-PBS1 by switching TR2 to transmission status; the jRi1 polarization component is transmitted, while the jLi1 polarization component is reflected. Eventually, all polarization components are collected by C-PBS n  1. Whichever detector clicks, the state of the four N-V centers can be obtained by neglecting an overall phase  θ ji1 cos ji2 ji3 ji4 2 1 θ  p


expiϕ sin j−i2 ji3 ji4 2 3   ji2 j−i3 ji4  ji2 ji3 j−i4  :

19

Thus, by detecting the photon pulse, the optimal cloning for the northern hemisphere is realized deterministically, and we obtained three copies of the initial state on the last three N-V centers. In the case that the input state to be cloned is on the southern hemisphere of the Bloch sphere, an operation σ z  j  1ih1j − j − 1ih−1j should be applied to N-V1 to transform the input state into fcosθ∕2j−i1  eiϕ sinθ∕2ji1 π∕2 < θ ≤ πg. After the similar process, we can obtain three copies of the initial state of N-V1 on the last three N-V centers. We can easily extend the above idea to a 1 ! M M  n − 1 EPCCM as shown in Fig. 5 in a similar way, where n − 1 HWPϑi ip









1; 2;
…; n − 1 should be used with ϑi  1∕2 arcsin1∕ n − ii  1; 2; …; n − 1. As a result, the initial information on N-V1 is distributed to the other n − 1 N-V centers.

FM 

1 jaj2  jbj2  Mjcj2  Mjdj2  θ × cos2 a2  b2  M − 2c2  d2  2  2 2  sin θad  bc  c  d ;

p




θ 2 θ M a  cos r 0  expiϕ sin r 0 − r2 r 0  r; 2 2 8   p




p




1 θ 2 2 M r 0 − M r  4r 0 ; b  expiϕ sin r 0  r 8 2 1 θ c  p




expiϕ sin r 0 − r2 r 0  r; 2 8 M 1 θ d  p




expiϕ sin r 0 − r3 : 2 8 M

(20)

Referring to the experiment with a hybrid diamond– GaP microdisk system with a relatively bad quality, Q ∼ 104 , as reported in Ref. [42], in which the relevant cavity quantum electrodynamics parameters are g; κ; γ∕2π  0.3; 26; 0.0004 GHz, we take γ  6 × 10−4 κ. As shown in Fig. 6, the fidelity of the 1 ! 3 EPCCM increases with increasing g∕κ and decreasing θ. When the coupling between the N-V center and the MTR achieves the condition of strong coupling, i.e., g ≫ κ, γ, fidelity reaches a stable value close to the optimal bound. Even if in the condition of weak coupling, i.e., g < 0.1κ, the scheme can also reach high fidelity. It can be seen in Fig. 7 that as g equals 0.05κ, the fidelities for θ  π∕3, π∕4, and π∕5 are 0.911, 0.953, and 0.971, respectively. Compared with the optimal bounds 0.924, 0.963, 0.978, the fidelity is reduced by 1.39%, 1.05%, and 0.75%, respectively. That means the schemes do not require strong coupling and a high-Q cavity condition. Figure 8 describes the evolution of the cloning fidelity of the 1 ! M EPCCM for the initial state on the northern hemisphere versus g∕κ for different clone numbers M. Because the initial information on the first qubit is distributed to the M clones, the more the clones, the lower the fidelity. As g equals 0.05κ, the fidelities for M15, 30, and 100 are 0.827, 0.804, and 0.778, respectively, which are reduced by 1.1%, 1.0%, and 0.08%, respectively, compared with the optimal bounds 0.838, 0.814, 0.786. Experimentally, the most important problem in all solidstate systems is unavoidable fluctuations of emitters’ properties, such as spectral diffusion. For N-V centers located 0 12

5. INFLUENCE OF DISSIPATION ON THE FIDELITY OF QUANTUM CLONING Now we briefly discuss the dissipative effect. Under the ideal case of strong coupling, i.e., the coupling strength is far greater than the cavity decay rate and dipole decay rate, i.e., g ≫ κ, γ, and the side leakage rate κs is much lower than the cavity decay rate κ, the decoupled and coupled reflection coefficients may be jr 0 ωj  jrωj  1. However, these dissipation factors may inevitably reduce the quality of clones in practice. Under the resonant condition ω0  ωc  ωp and κ ≫ κs , the corresponding fidelity of the 1 ! M EPCCM for the northern hemisphere under the dissipative condition can be calculated as

6 4 5

3

12

1

2

0.9

F 0.8 0.7 0.6 0.5 0.1

0.09

0.08

0.07

0.06

0.05

g

0.04

0.03

0.02

0.01

0

Fig. 6. Evolution of the fidelity of the 1 ! 3 EPCCM under dissipation with γ  6 × 10−4 κ, ϕ  0, and κ s ∕κ  0.

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Vol. 31, No. 10 / October 2014 / J. Opt. Soc. Am. B

1.00

1.00

0.95

0.95 0.90

Fidelity

Fidelity

0.90

2521

3

0.85

4

0.85 3

0.80

0.80

4

0.75

5

5 0.75 0.70

0.00

0.02

0.04

0.06

0.08

0.70

0.10

1.0

0.5

0.0

g Fig. 7. Fidelity of the 1 ! 3 EPCCM versus coupling strength g for different angles θ with γ  6 × 10−4 κ, ϕ  0, and κ s ∕κ  0.

close to the surface, the optical transition lines jump randomly within a Gaussian envelope [43] of width Γinh ; severalgigahertz-wide spectral diffusion can be observed under offresonant illumination [44]. It is widely assumed that the line broadening is due to a fluctuating electrostatic environment. In diamond nanocrystals, it is caused by ionized impurities and charge traps. In Refs. [45–47], the effect of spectral diffusion on the entanglement has been fully demonstrated. It reduces the concurrence of the entanglement. Here, the scheme presented is dependent on the resonance interactions among the N-V center, the probe photon, and the cavity; it is sensitive to frequency matching. Unavoidably, the spectral diffusion caused by the photons emitted from the NV–cavity system reduces the cloning fidelity. Assuming that the random jumps of the optical transition lines of the N-V centers are equivalent to a simultaneous fluctuation of the excited state by Δω, we can rewrite the reflection coefficients in Eq. (5) as r 0 ωp   rωp  

κs − κ ; κs  κ κs − κ2iΔω  γ  4g2 : κs  κ2iΔω  γ  4g2

(21)

Then the fidelity of the proposed 1 ! 3 EPCCM can be obtained as shown in Fig. 9. It can be seen the fidelity reduces with increasing Δω∕κ. We can see in Fig. 9 that the fidelity of the 1 ! 3 EPCCM when Δω∕κ  0.1 is 91.51%, which is reduced by 1.0% compared with the optimal bound 92.47%; however, in Fig. 10, this fidelity has a gradual rising trend with increasing coupling strength. It becomes gradually insensitive

0.5

1.0

Κ Fig. 9. Fidelity of the 1 ! 3 EPCCM when the optical transition frequency changes by Δω for different angles θ and γ  6 × 10−4 κ, g  0.5κ, ϕ  0, and κ s ∕κ  0.

to Δω∕κ with increasing g∕κ. Hence, this reduction in cloning fidelity caused by spectral diffusion can be suppressed by properly increasing the coupling strength g of the cavity, which can be seen in Fig. 10. Additionally, in order to implement our present schemes successfully, practical achievements in experiments with N-V centers and MTRs should be considered. It is necessary to note that the total time for the scheme should be shorter than the spin coherence time [27,28]. For the N-V centers, the relaxation time of the electron spin ranges from approximately milliseconds at room temperature to seconds at low temperatures [28,48,49]. However, with temperatures rising, the radiation spectrum of the N-V centers will have a large broadening and the ZPL emission will be debased. So it is essential to seek a solution to enhance the ZPL emission at higher temperatures. One feasible method is coupling the N-V centers to optical nanocavities and improving the ratio between the Q-factor and cavity mode volume [29]. On the other hand, recent experimental technology can generate 300,000 high-quality single photons within 30 s [50], which ensures that our scheme can be accomplished quickly and nearly deterministically. In addition, some instructive room temperature coherent manipulation of the spin of a single N-V center to achieve quantum information and computation tasks has

gΚ

0.7

0.9 0.8

1

0.6

0.5 0.4 0.3 0.2 0.1

0.90 0 1 0.95

Fidelity

0.85 M 6 M 15 M 30

0.75

0.8 0.75 1

M 100 0.70 0.00

F 0.9 0.85

0.80

0.02

0.04

0.06

0.08

0.10

gΚ Fig. 8. Evolution of the fidelity of the 1 ! M EPCCM based on the number of clones and g∕κ under dissipation with γ  6 × 10−4 κ, ϕ  0, θ  π∕3, and κ s ∕κ  0.

0.8

0.6

0.4

0.2

0

Κ

0.2

0.4

0.6

0.8 1

Fig. 10. Fidelity of the 1 ! 3 EPCCM versus g∕κ and Δω∕κ for γ  6 × 10−4 κ, θ  π∕3, ϕ  0, and κ s ∕κ  0.

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been proposed [51–53]. These works provide the basis for the current schemes.

6. SUMMARY In summary, we have presented deterministic schemes for implementing optimal symmetric 1 ! 2 UQCM and PCCM, and 1 ! M EPCCM among spatially remote N-V centers via a photon input–output process in a solid-state system. These cloning machines are robust even if in the dissipative condition. With development of quantum information technology in solid-state systems, these schemes are promising for use in distributed quantum computation.

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