## Abstract

An alternative scheme is proposed for the generation of an *N*-qubit Greenberger-Horne-Zeilinger (GHZ) state with distant nitrogen-vacancy (N-V) centers confined in spatially separated photonic crystal (PC) nanocavities via input-output process of photon. The GHZ state is produced by the phase shift brought by the input-output photon. The certain polarized photon transmitted from a PC nanocavity side-coupled a waveguide can obtain different phase shifts due to the different spin states in diamond N-V centers and the optical spin selection rule. Our calculations show that the proposed scheme can work well with a large cavity damping rate which ensures the efficient output of photon.

© 2012 Optical Society of America

## 1. Introduction

As a striking feature of quantum mechanics, entanglement is well-known today. Over past decades, the generation and engineering of quantum entanglement has attracted much attention because of its extensive applications for fundamental tests of local hidden variable theories [1,2], for high-precision measurements [3], and, in particular, for implementation of quantum communication and computation [4]. Compared with two-partite entangled states, many inequivalent classes of multipartite entangled states, such as W states [5], GHZ states [6] and cluster states [7], cannot be transformed into each other under local operations and classical communication (LOCC) protocols, and their potential applications in quantum information processes have been reported [8–10]. So far, a lots of works have been done for the generation of multiple-partite entangled states with different quantum systems [11–23]. Among the multiple-partite entangled states, GHZ state is usually referred to as “maximally entangled in several senses, e.g., it violates Bell inequalities maximally. In addition, entanglement with spatially separated subsystems is very useful for distributed quantum computation [24]. Recently, many theoretical and experimental works have been proposed for the implementation of GHZ states [16–23]. For example, based on the process of selectively reading qubits, three-qubit GHZ state was observed in an experiment by Roos *et al.* [16]. Xia *et al.* [21] provided a linear optical protocol to generate GHZ state with *N* distant photons based on multiphoton interference, ancillary entangled photon states, and conventional photon detectors. More recently, through photon emission and absorption processes, Lü *et al.* [17] proposed another scheme for the generation of three-qubit GHZ states with three distant atoms trapped individually in three cavities.

In recent years, on the one hand, studies of cavity quantum electrodynamics (CQED) for light-matter interactions inside nanoscale cavities provide an ideal platform for quantum optics. Tremendous progress has been made by coupling single emitters to different nanocavities (for a review, see Ref. [25]). Among them, photonic crystal (PC) nanocavity [26] is most promising due to its extremely high Q factor, ultrasmall mode volume, excellent scalability and ease for low-loss transport of nonclassical states using a tapered waveguide. On the other hand, nitrogen-vacancy (NV) centers in diamond nanocrystal have recently emerged as an excellent test bed for solid-state quantum physics experiments and quantum information processing because they possess long-lived spin triplets at room temperature [27–35]. Combining high-Q PC nanocavities and NV centers represents a promising solid-state CQED system, and attracts much attention both theoretically and experimentally [36–41]. Barth *et al.* [40] have addressed controlled coupling of a single-diamond nanocrystal to a planar PC double-heterostructure cavity. Recent experimental investigations by Wolters *et al.* [41] have developed and demonstrated a scheme to enhance the zero phonon line emission from a single N-V center in a nanodiamond via coupling to a PC cavity.

Based on these achievements, in the present work we proposed an alternative scheme to generate an *N*-qubit GHZ state with remote spin qubits through the input-output process of photon. In Ref. [23], a single-side cavity absorbed a vertically (*V*)-polarized photon and then emitted a horizontally (*H*)-polarized photon, by this way, the atoms in the cavities were entangled in GHZ state. Our proposed scheme, however, works on the weak excited limit. The polarized state of the photon is maintained unchanged after passing through the cavities and only phase shift is brought to the corresponding spin state by the input photon. The major advantages of applying our considered composite PC nanocavity-N-V system over other approaches are as follows:

- Firstly, GHZ state with separate N-V centers is encoded in the electronic spin ground states. The optical control of spin states, long decoherence and the robustness of the spin coherence have enabled the demonstration of basic building blocks for quantum computing even at room temperature [42, 43].
- Secondly, our scheme can work well with the large cavity damping rate, i.e., the low-quality cavity or the so-called bad cavity.
- Thirdly, the composite PC nanocavity-N-V scheme is deterministic and GHZ state can be realized with only one step.

## 2. Physical principles and analytical estimates

The optical coupled system under consideration in this paper is illustrated schematically in Fig. 1. This device is made up of a diamond N-V center, a point-defect PC nanocavity and a line-defect PC waveguide. The negatively charged N-V center is confined in a PC nanocavity with two modes, i.e., left(*L*)-polarization (*σ*_{−}) and right(*R*)-polarization (*σ*_{+}). As shown in Refs. [44, 45], the PC nanocavity, which supports both *σ*_{+}- and *σ*_{−}-polarized modes, can be fabricated experimentally. The N-V center considered is composed of a substitutional nitrogen atom and a adjacent vacancy in diamond lattice. According to the *C*_{3v} symmetry group, an optically transition is allowed between orbital ^{3}*A*_{2} ground state and an orbital ^{3}*E* excited state [46]. The ground states ^{3}*A*_{2} consist of spin triplets (*S* = 1), which are split by 2.88 GHz into the lower level |^{3}*A*_{2}, *m _{s}* = 0〉 and the upper levels |

^{3}

*A*

_{2},

*m*= ±1〉 [47]. There is a likewise split in the excited states

_{s}^{3}

*E*[48]. In this paper, by combining with the two cavity modes, the N-V center are modelled as a Λ-type three-level structure shown in the bubble of Fig. 1, (i.e., |−〉 = |

^{3}

*A*

_{2},

*m*= −1〉, |+〉 = |

_{s}^{3}

*A*

_{2},

*m*= +1〉 and |1〉 = |

_{s}^{3}

*E*,

*m*= 0〉). The |−〉 ⇔ |1〉 and |+〉 ⇔ |1〉 transitions (with the identical transition frequency

_{s}*ω*

_{0}) in the N-V center are resonantly coupled to the right (

*R*)

*σ*

_{+}and left (

*L*)

*σ*

_{−}polarized photons with the coupling strengths

*g*and

_{L}*g*, respectively. For simplicity, the energy of the states |−〉 and |+〉 have been set as zero. Under the rotating-wave approximation (RWA), the total Hamiltonian of the hybrid system can be given by (setting

_{R}*h*̄ = 1) [49–52]

*ω*.

_{c}*γ*and

*κ*are the decay rates of the excited state |1〉 and the cavity damping, respectively.

*a*(

_{k}*ω*) and ${a}_{k}^{\u2020}(\omega )$ are the annihilation and creation operators for the two modes of frequency

*ω*in the waveguide channel, with the commutation relation $\left[{a}_{k}(\omega ),{a}_{k}^{\u2020}({\omega}^{\prime})\right]=\delta \left(\omega -{\omega}^{\prime}\right)$.

Before proceeding further, it is instructive to briefly illuminate the physical picture of the above Hamiltonian operator (1). The first term is the energy of the two PC nanocavity modes *σ*_{+} and *σ*_{−} with frequency *ω _{c}*. The second term stands for the energy of the optical modes with frequency

*ω*in the waveguide channel. The third term describes the coupling of the nanocavity modes to the waveguide continuum. The nanocavity-waveguide coupling strength can be taken as a constant: $\kappa (\omega )=\sqrt{\frac{\kappa}{2\pi}}$ within the first Markov approximation [53–57]. The fourth term is the unperturbed parts of the three-level Λ-type N-V center, which represent the energy of the excited state |1〉. For simplicity, the energy of the ground states |+〉 and |−〉 are set as zero. In the fifth term, the transition |−〉 ⇔ |1〉 of the N-V center is coupled to the nanocavity mode

*C*with a coupling strength

_{R}*g*. At the same time, the transition |+〉 ⇔ |1〉 of the N-V center is coupled to the nanocavity mode

_{R}*C*with a coupling strength

_{L}*g*.

_{L}The coupled system, described by the Hamiltonian (1), has an interesting invariant Hilbert subspace, with the bases |−, 1* _{R}*〉|

*vac*〉, |+, 1

*〉|*

_{L}*vac*〉, |1, 0〉|

*vac*〉, |−, 0〉|

*ω*〉 and |+, 0〉|

_{R}*ω*〉, where in |

_{L}*m*,

*n*〉 (

*m*= −,+,1) denotes the state of the Λ-type three-level N-V center and

*n*denotes the number of photons in the cavity, |

*ω*〉 (

_{k}*k*=

*R*,

*L*) denotes the one-photon Fock state of the waveguide channel mode of frequency

*ω*in the

*k*-polarized modes, and |

*vac*〉 denotes the vacuum state of the waveguide mode. If the initial state of the system is in the state |±, 0〉|

*ω*〉, the evolution the the whole system can be generally described by the wave function

_{k}By making use of the Hamiltonian (1) and the well-known Schrödinger equation $i\overline{h}\frac{\partial}{\partial t}|\psi (t)\u3009=\widehat{H}|\psi (t)\u3009$, we can obtain the time evolution of the amplitudes for the PC nanocavity modes, PC waveguide modes and N-V center in diamond lattice as follows

*t*=

*t*

_{0}.

Now, by substituting
${d}_{k}^{a}(t)$ from Eq. (6) into Eq. (3) as well as taking the decay rate *κ* into account (which can be obtained by following the established procedures of the Weisskopf-Wigner approximation [53–55]), we can obtain

Finally, we perform the Fourier transformations
$F(\omega )=\frac{1}{\sqrt{2\pi}}{\int}_{-\infty}^{+\infty}F(t){e}^{i\omega t}dt$ on the amplitudes
${d}_{k}^{c}$ and *d*_{1} in the linear Eq. (5) and Eq. (7), respectively. Combing with the standard input-output relation
${a}_{k}^{\mathit{out}}={a}_{k}^{\mathit{in}}+\sqrt{\kappa}{d}_{k}^{c}$ [54, 55], we carry out some algebraic calculations and can get the amplitude of the output pulse
${a}_{k}^{\mathit{out}}$ with the resonant condition *ω* = *ω*_{0} = *ω _{c}* as

*k*,

*k*̄} ∈ {

*R*,

*L*} and

*k*≠

*k*̄. If the N-V center is in the initial state |−〉(|+ 〉) and the input pulse is a

*L*(

*R*)-polarized photon, the cavity is in resonance with the input photon and uncoupled to the N-V center. Therefor we can obtain the transmission coefficient from Eq. (9) as when

*g*= 0 and

_{k}*g*= 0.

_{k̄}However, if the initial state of N-V center is |−〉(|+ 〉), the input photon with *R*(*L*)-polarized will resonantly drive the transition |−〉 ⇔ |1〉 (|+〉 ⇔ |1〉). When
$4{g}_{k}^{2}\gg \kappa \gamma $, we can get the transmission coefficient

*T*(

*ω*)| = 1 reveals the fact that the input photon leaks out of the nanocavity without being absorbed by the cavity mode. The above cases can be summarized as follows

*R*〉(|

*L*〉) denotes the input photon in

*R*(

*L*)-polarized state.

## 3. Generation of GHZ entangled state

In this section, we begin to describe how to realize *N*-qubit GHZ state with distant N-V centers via the input-output process. First of all, we address the realization of three-qubit GHZ state in this hybrid PC nanocavity-N-V system. Then such an approach is extend to implement *N*-qubit GHZ state on our scheme. The concrete process is shown in Fig. 2. Three N-V centers are located at three PC nanocavity individually and the nanocavities are identically coupled to the waveguide. All N-V centers are initialized to be in the superposition state
$\frac{1}{\sqrt{2}}\left(|-\u3009+|+\u3009\right)$, and we input a single-photon pulse in an equal superposition of horizontal (H) and vertical (V) polarizations, i.e.,
$\frac{1}{\sqrt{2}}\left(|H\u3009+|V\u3009\right)$. The |*H*〉 component is reflected by the polarization beam splitter (PBS) and then goes through the half-wave plate (HWP), which interchanges the polarized photons (|*H*〉 ⇔ |*V*〉). On the other hand, the |*V*〉 component passes through the PBS and changes to *R*-polarized state after going through the first quarter-wave plate (QWP), then enters the first nanocavity. In the first nanocavity, the |*R*〉-polarized photon interacts with the N-V center and leads to the transition |*R*〉 (|−〉 + |+〉) → |*R*〉 (|−〉 − |+〉). Then the same component |*R*〉 is transmitted and goes into the second nanocavity. The similar case will occur in other nanocavities. The second QWP changes *R*-polarized photon into *V*-polarized state. At last, these two *V*-polarized components from two different paths are mixed by a beam splitter (BS) at the output and which-path information is also erased. We set
$|{0}_{+}\u3009=\frac{1}{\sqrt{2}}\left(|-\u3009+|+\u3009\right)$ and
$|{0}_{-}\u3009=\frac{1}{\sqrt{2}}\left(|-\u3009-|+\u3009\right)$. Summing up the above evolution process is as follows

*N*-qubit GHZ state with distant N-V centers.

## 4. Analysis and discussion

In this paper, the transmission amplitude and phase shift of the output photon play important roles during the process of generating a deterministic GHZ state. To consider the effect of light transmission on our scheme, we have plotted the amplitude [i.e., the absolute value of complex transmission coefficient |*T*(*ω*)|] and phase shift [i.e., the phase of complex transmission coefficient *ϕ* = *arg*(*T*(*ω*))] of the transmission coefficient *T*(*ω*) in Fig. 3. With *ω*_{0} = *ω _{c}* and

*κ*= 1000

*γ*, Fig. 3(a) shows the amplitude of the transmission coefficient |

*T*(

*ω*)| as a function of (

*ω*−

_{c}*ω*)/

*γ*for

*g*= 0 and

_{k}*g*= 500

_{k}*γ*. In the case of no interaction between the nanocavity mode and N-V center (i.e.,

*g*= 0 and

_{k}*g*= 0), the input photon leaks out the nanocavity without absorbtion and experiences a perfect transmission in the whole frequency regime. In contrast, due to the strong resonant coupling between the nanocavity mode and N-V center, the transmission coefficient is nearly unity under the condition of ${g}_{k}^{2}\ge 25\kappa \gamma $. In addition, there is an interesting phenomena that the transmission coefficient has two valleys. The two valleys are caused by the large vacuum splitting which leads to the shifting of the energy levels of the PC nanocavity [49]. And there is a large detuning between the input photon and the dressed nanocavity modes. As shown in Fig. 3(b), for the no interaction case, i.e.,

_{k̄}*g*= 0, the phase shift

_{k}*ϕ*of the transmitted photon is ±

*π*at the cavity resonance

*ω*=

*ω*. Once the frequency of the input photon

_{c}*ω*deviates the resonant point, the phase shift will reduce to zero rapidly. As to the case of ${g}_{k}^{2}\ge 25\kappa \gamma $, the phase shift has a similar splitting, as shown in Fig. 3(b). However, the phase shift is zero at

*ω*=

*ω*and varies quickly from zero to ±

_{c}*π*when we increase or decrease the frequency of the input pulse from the resonant point.

We have also examined the effect of the coupling strength *g _{k}* on the transmission amplitude |

*T*(

*ω*)| in Fig. 4(a). The transmission amplitude |

*T*(

*ω*)| increases as the coupling strength

*g*increases. The transmission amplitude is nearly unity and the requirement on

_{k}*g*is loosen. The present scheme only needs $4{g}_{k}^{2}\gg \kappa \gamma $ but does not need

_{k}*g*≫

_{k}*κ*,

*γ*. In fact, we have chosen

*g*= 500

_{k}*γ*<

*κ*= 1000

*γ*in the above analysis, therefore our proposed scheme can work well even if the PC nanocavity is bad. As shown in Fig. 4(b), the influence of the cavity damping

*κ*on the amplitude of the photon transmission is negligible. When the frequency of the input photon satisfies the resonant condition

*ω*=

*ω*=

_{c}*ω*

_{0}, the smaller the value of the cavity damping

*κ*is, the higher the transmission amplitude |

*T*(

*ω*)| is.

In the present work, three identical N-V centers are individually confined in three identical PC nanocavities and the input pulse passes through the nanocavities one after the other. The ideal GHZ state will be realized only if the shape of the pulse is maintained unchanged. We set a Gaussian shape for the input pulse with *f _{in}*(

*t*) ∝

*exp*[−(

*t*−

*T*/2)

^{2}/(

*T*/5)

^{2}], where

*t*varies from 0 to

*T*and

*T*is the period of the pulse [56]. With

*T*= 10/

*γ*, in Fig. 5 we plot the input and output pulse profiles |

*f*(

_{in}*t*)| and |

*f*(

_{out}*t*)| as a function of time

*t*/

*γ*for the given system parameters in Fig. 2. All the pulse shapes of the output optical field are basically indistinguishable from the input pulse. Different from other works [50,56], we have set $4{g}_{k}^{2}\gg \gamma \kappa $ and

*κ*≫

*γ*, which ensure that the shapes of the output and input pulses closely match.

It should be pointed out that the implementation of GHZ state dependents on the effective transmission of the input photon from the PC nanocavity. The above analysis in Fig. 3 and Fig. 4 shows that the frequency detuning, the coupling strength and the cavity damping rate have little effect on the transmission amplitude |*T*(*ω*)|. However, the change of frequency detuning will effect strictly the phase shift *ϕ* of the transmitted photon. GHZ state can be generated successfully only if the phase shift is maintained as zero or *π* for different ground states. Therefore, the frequency resonant condition *ω _{c}* =

*ω*is required for the realization of GHZ state. On the other hand, the shape mismatching of input and output pulses will result in the reduction of fidelity. We have taken the Gaussian pulse as example and plotted the shapes of the output pulse for the cases of N-V ground states |−〉 and |+〉 in Fig. 5, respectively. In our proposed scheme, the two conditions $4{g}_{k}^{2}\gg \gamma \kappa $ and

*κ*≫

*γ*ensure very good overlap between the shapes of the input and output pulses, i.e.,based on our scheme, GHZ state can be generated with high fidelity.

Before ending this section, we would like to consider the experimental feasibility of our scheme. Firstly, the identical energy level configuration is required by the generation of the GHZ state with high fidelity in our scheme. Each PC nanocavity contains only one N-V center, thus, we can adjust the energy interval by an external laser [57] or by applying different magnetic fields to induce fine structure splitting of the N-V centers [58]. Note that the N-V center transition frequency increases with increasing temperature [33]. Secondly, in our scheme, we choose *γ* = 2*π* × 10 MHz, *κ* = 2*π* × 10 GHz and *g _{k}* = 2

*π*× 5 GHz, which are available in current experiments [35, 59, 60]. Thirdly, the present scheme is essentially based on the efficient output of photons which implies the large cavity decay rate, i.e., the bad cavity [49, 61]. In the above analysis, our scheme can work well with the large cavity damping. Fourthly, in our scheme, we have ignored the effect of the intrinsic cavity loss (

*κ*) into non-waveguide channels on the generation of GHZ state. According to Refs. [61–63], the condition of

_{i}*κ*≫

*κ*can be satisfied by the parameter optimization and the intrinsic cavity loss

_{i}*κ*can be discarded in the present scheme. We also have reconsidered the intrinsic decay rates

_{i}*κ*of the PC nanocavity. However, under the condition that an external loss rate

_{i}*κ*is about ten times higher than an intrinsic loss rate

*κ*, we find that the shape, number and location of peaks in the transmission spectra have no change, except for some quantitative differences in peak heights (not shown here). Another reason for ignoring intrinsic cavity loss into non-waveguide channels is due to its simplicity in mathematical treatment, which permits us to obtain simple analytical expressions, and its transparency in the physical explanation of the results obtained. Finally, it should be point out that in the present study we only focus on scattering light in the forward direction. As a matter of fact, the PC nanocavities side-coupled to the waveguide typically exhibit standing waves that can scatter both in the forward and backward directions. This model is of course oversimplified. A realistic quantitative description would need to include the scatter light in the backward direction. Nevertheless, it has already been possible to minimize light reflections (i.e., scattering light in the backward direction) by using inverse tapers at the end of the strip waveguides nowadays. Details on the fabrication process are given in Refs. [64–68]. Therefore, we believe that the present model can provide a quantitative illustration for the generation of the GHZ states.

_{i}## 5. Conclusion

To summarize, we have put forward a new scheme for the generation of the GHZ states with distant N-V centers in spatially separate PC nanocavities via the input-output process of photon. The scheme proposed here is deterministic and is promising for generating an *N*-qubit GHZ state with the current techniques.

## Acknowledgment

We would like to thank Professor X. X. Yang for her encouragement and helpful discussion. This research was supported in part by the National Natural Science Foundation of China under Grants No. 11004069, No. 10975054 and No. 91021011, by the Doctoral Foundation of the Ministry of Education of China under Grant No. 20100142120081, by the National Basic Research Program of China under Contract No. 2012CB922103.

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