## Abstract

Here we investigate a physical implementation of the universal quantum controlled phase (CPHASE) gate operation on photonic qubits by using nitrogen vacancy (N-V) centers and microcavity resonators. The quantum CPHASE gate can be achieved by sending the photons through the microcavity and interacting with the N-V center. The proposed scheme can be further used for scalable quantum computation. We show that this technique provides us a deterministic source of cluster state generation on photonic qubits. In this scheme, only single photons and single N-V center are required and the proposed schemes are feasible with the current experimental technology.

© 2013 Optical Society of America

## 1. Introduction

Quantum information processing (QIP) based on solid state system is a dynamically developing research field. The key building block of a quantum processor is quantum gate, which is used to manipulate the quantum state with a high fidelity. During the past decades, optical quantum gates and light matter interfaces based quantum gates have been exploited with linear optics and atomic systems using cavity quantum eletrodynamics. The most attractive quantum gate operations are the controlled qubits operation which was widely implemented in various systems. In 2003, the controlled-not (CNot) gate based on ion traps has been experimentally realized [1]. Later, Li *et al.* realized an all-optical quantum gate using semiconductor quantum dots [2]. In 2005, Zhao *et al.* demonstrated a nondestructive CNot gate for two photonic qubits [3]. In 2010, Isenhower *et al.* presented a CNot gate between two individually addressed neutral atoms which uses Rydberg blockade interactions between neutral atoms held in optical traps [4]. In 2011, the controlled phase gate had been experimentally realized using linear optics [5] and continuous variables [6].

Recently, a promising candidate towards the goal of QIP is provided by using the nitrogen-vacancy (N-V) centers. Theoretically, Yang *et al.* [7] presented an entanglement generation protocol for W state and Bell states by using N-V centers coupled to a whispering-gallery modes (WGM) cavity. Later, they construct quantum memories for quantum computation using the same system [8]. Dayan *et al.* discussed the dynamics of a single atom coupled with a microcavity resonator [9] which indicates that the single atom within the resonator can dynamically control the cavity output. In 2011, Chen *et al.* [10] presented an entanglement generation protocol between distant N-V centers via the coupling to microcavity resonators. Togan *et al.* [11] experimentally realized the entanglement generation process based on N-V centers and microcavities. Stimulated by the advances on the solid state QIP, the idea of entanglement manipulation between the solid states and the photons had been developed [12, 13]. The motivation is to distribute the maximally entanglement between solid state qubits, and to facilitate the future realization of scalable quantum computation.

Quantum logic gates between flying qubits and stationary qubits hold great promise for QIP as flying qubits such as photon are the perfect candidates for fast and reliable long-distance communication while the stationary qubits are suitable for processor and local storage. Here in this study, we present a deterministic scheme to construct universal quantum controlled phase (CPHASE) gate between the flying photonic qubits assisted by N-V centers coupled with microtoroidal cavities. The control qubit and the target qubit of our gates are encoded on the polarization of the moving photons and the gate can be realized with a success probability of 100% in principle. Recent experimental efforts have reported on remarkable progress in the proposed solid-state system. Furthermore, considering the scalability of quantum computation, we provide a further method to implement cluster state generation for the photonic qubits.

## 2. Hybrid universal quantum phase gate based on nitrogen vacancy centers and micro-cavity system

The realization of the CPHASE gate consists of a single N-V center and microcavity coupled system. A fundamental task of the CPHASE gate can be described by the unitary operator
${U}_{j,k}^{CP}={e}^{i\pi |L\u3009\u3008L|\otimes |L\u3009\u3008L|}$, while the assisted N-V center is restored in its initial state, here |*L*〉 represents the left circularly polarization of the photon. The interaction between a single N-V center and a cavity mode can be described by using the input-output process of the Jaynes-Cummings (JC) model [14]. The N-V center is assumed to be a three-level system as shown in the bubble of Fig. 1, with the excited state |*A*〉 and two ground states with the angular momentum |*m _{s}* = ±1〉. The Hamiltonian describes the interaction between the N-V center and the electric cavity field and is given by
$\widehat{H}=\overline{h}{\omega}_{0}|A\u3009\u3008A|+\overline{h}{\omega}_{c,L}{a}_{L}^{\u2020}{a}_{L}+\overline{h}{\omega}_{c,R}{a}_{R}^{\u2020}{a}_{R}+i\overline{h}g\left({a}_{L}{\sigma}_{+}^{+}+{a}_{L}^{\u2020}{\sigma}_{+}^{-}+{a}_{R}{\sigma}_{-}^{+}+{a}_{R}^{\u2020}{\sigma}_{-}^{-}\right)$, here g represents the light-matter interaction strength.
${\sigma}_{\pm}^{-}$ and
${\sigma}_{\pm}^{+}$ are the annihilation and creation operators for the N-V center and the subscripts ± denote the transition in correspondence with the levels

*m*= −1 or

_{s}*m*= +1, respectively.

_{s}*a*

^{−}and

*a*

^{+}are the corresponding annihilation and creation operators for the cavity field. The level transition between |

*A*〉 and |±1〉 can only be excited by the single photon with the angular momentum which obeys the selection rules, for instance, the left circularly polarized photon can only couple with the N-V center in the level |−1〉 and the right circularly polarized one only couples with the N-V center in the level |+1〉.

From the above Hamiltonian, we can obtain the equations governing the dynamics of the system [15]

*ϕ*〉 =

_{out}*exp*(

*iϕ*)|

*ϕ*〉, here the phase shift

_{in}*ϕ*is represented by the reflection coefficient

*r*(

*ω*) =

*â*. And the coefficient

_{out}/â_{in}*r*(

*ω*) can be solved as

*κ*and

*κ*are the cavity decay rate and the cavity leaky rate, respectively, and

_{s}*γ*/2 denotes the decay rate of the N-V center.

*ω*,

_{p}*ω*, and

_{c}*ω*

_{0}represent the frequencies of the input photon, cavity mode and the atomic energy level transitions, respectively. Here we set the resonant conditions with

*ω*=

_{c}*ω*

_{0}=

*ω*, the reflect coefficient

_{p}*r*

_{0}(

*ω*) under the uncoupling case with

*g*= 0 approaches to −1. However, the coupled reflection coefficient

*r*(

*ω*) can be described as As shown in the above equations, there is a phase shift on the output field with respect to the atomic state of the N-V center. If the N-V center is in the state |−〉, we can obtain the following evolution of atom-photon system: |

*ϕ*〉 =

_{out}*r*(

*ω*)|

*L*〉 =

*e*|

^{iϕ}*L*〉 or |

*ϕ*〉 =

_{out}*r*

_{0}(

*ω*)|

*R*〉 =

*e*

^{iϕ0}|

*R*〉, here the parameters

*ϕ*and

*ϕ*

_{0}represent the phase shift determined by the input-output relation. It is obvious that the uncoupled condition will introduce a

*π*phase shift on the input-output state.

In our implementation of the CPHASE gate, we first present a schematic diagram show the principle of CPHASE operation between the single photon and the N-V center in Fig. 1, the control qubits are encoded on the flying qubit (i.e., the two polarization states of a single photon, denoted by |*V*〉 and |*H*〉), while the target qubit is encoded on the N-V center inside an optical microcavity. Our goal is to realize the photon-solid qubits CPHASE gate *U ^{CP}*. This gate enables a controlled phase change process between the input-output single photon and an isolated spin qubit. Suppose the N-V center is initially prepared in the state

*η*|+〉 +

*δ*|−〉, here we denote |±〉 as the state of the N-V center in the level |

*m*= ±1〉. The input photon is prepared in the state

_{s}*α*|

*H*〉 +

*β*|

*V*〉, here |

*H*〉 and |

*V*〉 represent the horizontal and vertical polarization of the photon, respectively. The state of the composite system can be described as

Exploiting the setups shown in Fig. 1, the horizontally polarized input photon will be transmitted by the PBS, otherwise the vertically polarized photon will be reflected by the PBS. Before the photon passes through the cavity, a QWP operation is performed on it which is used to complete the transformation: |*H*〉 → |*L*〉. Then the N-V center and microcavity system will interact with the photon and the composite system will evolve as:

*H*〉, compared with the original state of the two-qubit hybrid system. That is, the quantum circuit shown in Fig. 1 can be used to construct the CPHASE gate by using the photon as the controlling qubit and the N-V center as the target qubit. Following the interaction between the input single photon pulse and the N-V center, the process can be described by the unitary operator

*U*=

^{CP}*e*

^{i}^{π}^{(|}

^{H}^{〉〈}

^{H}^{|⊗|−〉〈−|)}.

## 3. Photonic quantum phase gate and cluster state generation

The framework of the CPHASE gate circuit on the photonic qubits is shown in Fig. 2. The input photon pulses, marked with j and k, are initially prepared in the state: *α*_{1}|*L*〉 + *β*_{1}|*R*〉 and *α*_{2}|*L*〉 + *β*_{2}|*R*〉, here |*α*_{1}|^{2} + |*β*_{1}|^{2} = |*α*_{2}|^{2} + |*β*_{2}|^{2} = 1. And the N-V center is prepared in the superposition state
$\frac{1}{\sqrt{2}}\left(|+\u3009+|-\u3009\right)$. The resonant transition |+〉 → |*A*〉 and |−〉 → |*A*〉 are coupled with the input photon with mode *a _{R}* and

*a*respectively. Now we present the detailed theoretical model of the controlled phase operation on the photonic qubits. The process can be described by the following operations: ${U}_{aj}^{CP}\times R\left(-\frac{\pi}{2}\right)\times {U}_{ak}^{CP}\times R\left(\frac{\pi}{2}\right)\times {U}_{aj}^{CP}$. Here

_{L}*R*(

*θ*) is the single qubit rotation on the N-V center which can be realized by a microwave pulse. The transformation of the rotations obey

*R*(

*θ*)|+〉 =

*cos*(

*θ*/2)|+〉 +

*sin*(

*θ*/2)|−〉 and

*R*(

*θ*)|−〉 = −

*sin*(

*θ*/2)|+〉+

*cos*(

*θ*/2)|−〉. The initial state is denoted by |

*ψ*〉

*⊗ |*

_{j}*ψ*〉

*⊗ |*

_{k}*ϕ*〉

_{N}_{−}

*, here the subscripts*

_{V}*j*and

*k*represent the two input photons. The rotation operations

*R*(

*π*/2) and

*R*(−

*π*/2) denote the single particle rotation on the state of the N-V centers.

The detailed description of the CPHASE gate operation is as follows: at first the j pulse is sent to the CPHASE gate and interacted with the N-V center a and evolve the composite state as

*π*/2 microwave pulse is performed on the N-V center which made a Hadamard operation on the state, the system can be described as

*π*/2 pulse is performed on the N-V center, the system can be described as:

*α*

_{1}

*α*

_{2}|

*L,L*〉 +

*β*

_{1}

*α*

_{2}|

*R,L*〉 +

*α*

_{1}

*β*

_{2}|

*L,R*〉 +

*β*

_{1}

*β*

_{2}|

*R,R*〉.

In our proposed scheme, the N-V center is not required to be measured. The important parameter of the CPHASE gate is the phase shift *ϕ* that the gate imposes on the logical state of the signal qubit according to the state of the controlled qubit. Here we set the phase shift *ϕ* = *π*. In the realistic implementations, the fidelity of the gate operation relies on the key parameters of the system, including the coupling strength, the cavity decay and the decoherence time. Here we numerically simulated the success probability of the gate operation based on realistic experimental results. The relations between the success probability of our CPHASE gate is shown in Fig. 3. As the value of
$g/\sqrt{\gamma \kappa}$ is increased to 2, the success probability of the CPHASE gate is larger than 90%. If there is no cavity leakage, the success probability approaches to 98% if
$g/\sqrt{\gamma \kappa}\approx 5$.

In 2001, Raussendorf and Briegel proposed a scheme for quantum computation using one-qubit measurements and cluster states, called one-way quantum computing (QC) [16]. The key ingredients of one-way QC are the construction of CPHASE gate and the preparation of cluster state. In 2005, Walther *et al.* [17] implemented the one-way QC by experimentally realized four-qubit cluster states encoded in the polarization four photons. Recently, much effort has been devoted to achievement of the CPHASE operations theoretically using linear optics [18, 19], Kerr media [20], atomic cavity system [21], and so on. The CPHASE gate is useful in cluster state generation. In 2004, Nielsen proposed a controlled-Z gate to generate the cluster state using linear optics [22]. In 2007, Louis *et al.* [23] proposed an approach for generating cluster states with weak nonlinearities. In 2010, Lin and He [24] proposed the efficient two-dimensional cluster state generation protocol using cross-Kerr nonlinearities and linear optical elements hybrid system. Also there are many related works on the cluster state generation in various systems, such as quantum dots [25], superconducting qubits [26, 27] and so on.

Exploiting the photonic CPHASE gate, we prepare a single N-V center in the state $\frac{1}{\sqrt{2}}\left(|+\u3009-|-\u3009\right)$ which is coupled to a microcavity. And the photons are prepared in the state $\frac{1}{\sqrt{2}}\left(|L\u3009+|R\u3009\right)$ and are sent to the input port of the CPF. After the first two photons interact with the N-V center, the output state of the composite system becomes

*th*rounds iterations, the final cluster state of the composite system can be described as:

The fidelity of the generated cluster state relies on the high fidelity operation of the CPHASE gate. The key parameters of the gate are the coupling strength *g*, the cavity decay rate *κ* and the decay rate of the N-V center *γ*. We numerically simulated the fidelity of the generated cluster state versus different initial coefficients on the certain reflection coefficients. In Fig. 4 we labeled the fidelity of the generated cluster state and the coefficient
$\left|g/\sqrt{\gamma \kappa}\right|$, the fidelity of 7-qubit cluster state is larger than 0.8 if the coefficient
$g/\sqrt{\gamma \kappa}$ is larger than 5. Here we set the reflection coefficient *κ _{s}* = 0. It is obvious that the efficiency of the scheme relies on the large coupling between the N-V center and the microcavity, also with less decay rate of

*γ*and

*κ*. Actually, the following numerical simulation results show that the CPHASE gate works remarkably well even if g ∼

*κ*. In the implementation of such devices, the coherent coupling between N-V centers and microcavities can be achieved as

*g/κ*= 0.5 and the decay rate is

*γ*= 2

*π*× 10

*MHz*as discussed in [28]. The dynamics of the input-output field theory reveals that the phase shift induced by the atom-cavity interaction depends on the experimental parameters, such as the coupling of N-V centers and microcavities. As discussed in [29], the coherent coupling rate of the N-V center with the microdisk is 28

*MHz*in the near field of a 100nm scale. On the other hand, the protocol is essentially based on the efficient output of photons which implies the large cavity decay rate [9]. In 2008, Dayan

*et al.*designed and experimentally realized a photon turnstile exploiting the atom and microtoroidal resonator coupled system. The photon number at the output can be controlled by the atom dynamically. Also the fidelity of the CPHASE gate operation and the fidelity of the cluster state generation may be effected by the imperfect operation of the RF pulse on the N-V center, which may cause an imperfect Hadamard operation on the state. Of course this will reduce the success probability of the generated cluster state. Moreover, the present model relies on the decoherence time of the spins in N-V centers, which mainly characterized by the spin relaxation time

*T*

_{1}and the dephasing time

*T*

_{2}. According to Refs.[30, 31], the electron spin relaxation time of the N-V centers scaled from microseconds to seconds at low temperature and the dephasing time is approaching to 2ms in an isptopically pure diamond.

## 4. Summary

In summary, we present a robust and realistic scheme to achieve the photonic CPHASE gate operation using N-V centers and linear optical devices. Moreover, the cluster state can be efficiently generated by using photonic qubits based on current experiment techniques. The experimental feasibilities are also discussed which shows us that these protocols have practically applications in quantum information science especially for scalable quantum information processing on solid chips.

## Acknowledgments

This work is supported by the National Fundamental Research Program Grant No. 2010CB923202, the Fundamental Research Funds for the Central Universities, the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University (Grant No. KF201301), and China National Natural Science Foundation Grant No. 61205117.

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