## Abstract

We investigate the entanglement generation between two nitrogen-vacancy (NV) centers in diamond nanocrystal coupled to a high-Q counterpropagating twin whispering-gallery modes (WGMs) of a microtoroidal resonator. For looking into the degree and dynamics of the entanglement, we calculate the concurrence using the microscopic master equation approach. The influences of the coupling strength between the WGMs (or the size of the two spherical NV centers), the distance between two NV centers, the frequency detuning between the NV center and microresonator, and the initial state of the system on the dynamics of concurrence are discussed in detail. It is found that the maximum entanglement between the two NV centers can be created by properly adjusting these controllable system parameters. Our results may provide further insight into future solid-state cavity quantum electrodynamics (CQED) system for quantum information engineering.

© 2013 Optical Society of America

## 1. Introduction

Quantum entanglement has played an important role in quantum information processing including quantum teleportation [1–4], quantum computation [5–8], and quantum networks [9–12]. In the past years, numerous schemes have been proposed to generate entanglement in many quantum systems such as photons [13], ions [14–16] and superconducting qubits [17, 18], etc. Among these quantum systems, cavity quantum electrodynamics (CQED) [19–21] system provides one of the most promising and qualified candidates for quantum information processing [12]. In particular, the whispering-gallery mode (WGM)-type microcavity [22–26] has attracted much attention due to its highly confined ultrasmall mode volume *V* in the order of the cubic wavelength and ultrahigh quality *Q*-factor (i.e., a high *Q/V* ratio), enabling strong temporal and spatial confinement of photons. Unlike the standing modes in a conventional Fabry-Perot (FP) cavity, WGMs are a kind of travelling modes. In other words, one of the prominent properties of WGMs is that the microcavity supports twin modes, clockwise (cw) and counterclockwise (ccw) propagating modes with a degenerate frequency and the same field distribution function. Moreover, these two counterpropagating WGMs couple to each other with a strength due to the scattering of imperfection.

On the other hand, since the first report of optically detected magnetic resonance on the single diamond nitrogen-vacancy (NV) center occurred in 1997 [27], NV centers have been considered as an excellent candidate for quantum information processing because of its extremely long electronic spin decoherence time even at room temperature [28–38]. Recently, a combination of NV centers and high-Q WGM microcavities which represents a promising solid-state CQED system has attracted much attention [39–47]. This composite system combines the advantages of NV centers and WGM microcavities. Therefore, this CQED system has been applied in many fields about quantum information processing (QIP). For instance, Xiao et al. proposed a scheme for producing the NOON state in a hybrid photonic-plasmonic resonant structure which consists of *N* identical metal nanoparticle (MNP) and a WGM microcavity [48]. Recently, Chen et al. investigated the entanglement generation between two quantum dots (QDs) through the scattering of the surface plasmons (SPs) in a metal nanowire [49] and a metal nanoring resonator [50]. They showed that the interference between the incident and scattered SP in the resonator with asymmetric couplings to QDs could lead to interesting concurrence dynamics. Yang et al. proposed two schemes to prepare the W state and Bell state with separate NV centers in the diamond nanocrystal-WGM-microsphere system by virtue of Raman transition with the cavity field virtually excited, dark state evolution and adiabatic passage, which is tolerant to ambient noise and experimental parameter fluctuations [51, 52].

Based on these achievements, we study the interaction of twin cw and ccw WGMs of the microtoroidal resonator with two NV centers in a diamond nanocrystal. Using the microscopic master equation approach, we calculate the concurrence *C* for investigating the degree and dynamics of entanglement. We look into the influences of the coupling strength *g* between WGMs (or the radius *R* of the two spherical NV centers because of *g* ∝ *R*^{3}[53]), the detuning Δ between the NV center and microresonator, the distance *d* between two NV centers (or the phase *ϕ* because of *ϕ* = *kd* with *k* being the wave vector), and the initial state *ρ*(0) of the coupled system on the degree and dynamics of entanglement. It is shown that the maximum entanglement between the two NV centers can be achieved at appropriate times through adjusting these controllable system parameters. The interacting systems may serve as a platform to generate quantum entanglement between the two NV centers.

Our adopted NV centers in diamond are similar to atomic media in energy level structure, but this NV center-coupled-to-WGM system is very different from previous atomic systems. Firstly, unlike an atomic gas which has to be cooled to low temperatures of several tens of *μ*K to eliminate the Doppler broadening effect, our obtained results suggest that high entanglement of the two NV centers could be easily observed in experiment because of the long electronic spin decoherence time of NV centers at room temperature. Thus this relaxes the bandwidth and decoherence constraints of the NV systems. For example, Schietinger et al. demonstrated a method to couple one and two stable single NV centers in diamond to an optical microresonator in a controlled way. Their experimental procedure is scalable in order to assemble more complex systems with a well-defined number of constituents. All experiments were performed at room temperature [43]. Secondly, experimental NV-WGM CQED would be more promising than controlling atomic gas due to the solid state of the NV centers. As compared to atomic systems, these NV-WGM CQED systems are compatible with on-chip integration and room-temperature operation. Thirdly, unlike the previous proposed schemes [48–52, 54, 55], we systematically take into account the influences of the coupling strength *g* between WGMs (or the radius *R* of the two spherical NV centers), the distance *d* between the two NV centers and the frequency detuning Δ between the NV center and microresonator on the concurrence *C*. As shown in Ref. [53], the coupling strength *g* between WGMs closely associated with the radius *R* of the NV center in the spherical diamond nanocrystal. With the increase of *R*, the scattering-induced coupling coefficient *g* grows rapidly. The phase interference induced by different distances *d* between the two NV centers is the key difference between the present multi-scatterer case and the previous single-scatterer case where the coupling strength is assumed real. In view of these factors, our further study reveals that the degree and dynamics of entanglement sensitively depend on the radius *R* and the distance *d*. On the other hand, even when the transition frequency of the NV centers is largely deviated from the frequency of both cc and ccw WGMs, the analytical results show that the maximum entanglement can still be achieved in our system, which is useful in real experiments.

The paper is structured as follows. In Section 2, we introduce physical model of a WGM-type microresonator interacting with two NV centers and further give the microscopic master equation of the coupled system. In Section 3, for investigating the degree and dynamics of entanglement, we solve the microscopic master equation and obtain the concurrence of two NV centers with different initial states. In Section 4, we discuss the entanglement dynamics and entanglement degree of the system. Finally, we present conclusions in Section 5.

## 2. Physical model and microscopic master equation

The system under consideration is composed of a microtoridal resonator and two NV centers as shown in Fig. 1. The NV centers are placed close to the surface of a microtoroidal resonator. A strong dipole-coupling between the NV centers and WGMs can occur [51, 52, 55, 56]. The microtoroidal resonator supports two counter-propagating WGMs, denoted as *a _{cw}* and

*a*, respectively. The resonator modes can not only couple with NV centers but also couple with each other. Under the rotating-wave approximations (RWA), the Hamiltonian of the coupled system can be written as (setting

_{ccw}*h*̄ = 1) [53]

*m*,

*m*′) run through cw and ccw modes. The first part

*H*

_{0}[see Eq. (2)] characterizes the free evolution of the NV centers and WGMs. The second part

*H*

_{1}[see Eq. (3)] describes the dipole interactions between the NV centers and WGMs with the coupling strengths

*G*

_{1}and

*G*

_{2}. Note that, we have ignored the subscript “m” for

*G*(

_{jm}*j*= 1, 2;

*m*=

*cw*,

*ccw*) in Eq. (3). In general, a symmetric microresonator supports two counter-propagating WGMs with the degenerate resonant frequency

*ω*and the same field distribution function

_{c}*f*(

*r⃗*). As shown in Refs. [53] and [57], the coupling strengths

*G*depend on the frequency of the cavity modes and the field distribution function of the WGMs. However, they are independent of the NV center’s radius

_{jm}*R*. Consequently, both WGM modes couple with any of the NV centers with the same strength. In view of this,

*G*

_{1}and

*G*

_{2}have no index (“m”) specifying the cw or ccw mode. The third part

*H*

_{2}[see Eq. (4)] describes the scattering induced by the two NV centers into the same (

*m*=

*m*′) or the counterpropagating (

*m*≠

*m*′) quantized WGM fields with the coupling strengths (i.e., the scattering-induced coupling coefficients)

*g*

_{1mm′}and

*g*

_{2mm′}. Specifically, in the presence of NV1, one of the modes, say cw couples to NV1. The scattered light will couple-back to either the cw or the ccw mode. The same is true when the ccw couples to NV2. For simplicity, the two NV centers are assumed to be identical (spherical), i.e., the radius

*R*

_{1}=

*R*

_{2}and the field distribution function

*f*

_{1}(

*r⃗*

_{1}) =

*f*

_{2}(

*r⃗*

_{2}) [57]. Thus, above we have set the same coefficient

*g*

_{1mm′}=

*g*

_{2mm′}=

*g*for all of these coupling processes. Likewise, we have

*G*

_{1}=

*G*

_{2}=

*G*. In Eq. (2) and Eq. (3), we have defined the following electronic operators ${\sigma}_{j}^{z}=|{e}_{j}\u3009\u3008{e}_{j}|-|{g}_{j}\u3009\u3008{g}_{j}|$, ${\sigma}_{j}^{+}=|{e}_{j}\u3008\u3008{g}_{j}|$ and ${\sigma}_{j}^{-}=|{g}_{j}\u3008\u3008{e}_{j}|$ (

*j*= 1, 2), respectively. The symbol

*ω*is the transition frequency of the NV centers and

_{a}*ω*is the frequency of both cc and ccw WGMs.

_{c}*k*is the wave vector of the quantized WGM field with

*k*=

_{cw}*k*and

*k*= −

_{ccw}*k*because cw and ccw modes are travelling modes.

*d*represents the distance between two NV centers along the mode travelling direction. The distance

*d*should be much larger than the wavelength of the WGMs, so the direct interaction of two NV centers can be neglected [51]. The appearance of the phase factors in Eq. (3) and Eq. (4) is the key difference between the present multi-scatterer case and the previous single-scatterer case where the coupling strength

*g*is assumed real. Experimentally, we can use atomic force microscope (AFM) manipulation to controllably position the NV centers [58, 59].

We note that the composite system described by the Hamiltonian *H* has two invariant subspaces. One is ∀_{1} ∈ {|1〉 = |*e*_{1}, *g*_{2}, 0* _{cw}*, 0

*〉, |2〉 = |*

_{ccw}*g*

_{1},

*e*

_{2}, 0

*, 0*

_{cw}*〉, |3〉 = |*

_{ccw}*g*

_{1},

*g*

_{2}, 1

*, 0*

_{cw}*〉, |4〉 = |*

_{ccw}*g*

_{1},

*g*

_{2}, 0

*, 1*

_{cw}*〉}, and the other is ∀*

_{ccw}_{2}∈ {|5〉 = |

*g*

_{1},

*g*

_{2}, 0

*, 0*

_{cw}*〉}, where in |*

_{ccw}*l*

_{1},

*r*

_{2},

*p*,

_{cw}*q*〉,

_{ccw}*l*=

*g*,

*e*denote the state of the first NV center,

*r*=

*g*,

*e*denote the state of the second NV center, and

*p*,

*q*denote the number of photons in the cw and ccw WGMs, i.e., |1

*〉 (|1*

_{cw}*〉) denotes the one-photon Fock state of the cw (ccw) WGM with frequency*

_{ccw}*ω*, |0

_{c}*〉 (|0*

_{cw}*〉) describes the vacuum state of the cw (ccw) WGM. For simplicity, we abbreviate as |*

_{ccw}*l*

_{1},

*r*

_{2},

*p*,

_{cw}*q*〉 ≡ |

_{ccw}*lrpq*〉 in the following. The state |5〉 will not evolve with time since it remains completely decoupled from the interaction described by Hamiltonian

*H*. If the initial state of the coupled system is |1〉, |2〉, |3〉 or |4〉, the evolution of the system will remain in the subspace ∀

_{1}. Such an approach by the one-photon Fock state of the cw (ccw) WGM (i.e., $|{1}_{cw}\u3009={a}_{cw}^{\u2020}|0\u3009$ and $|{1}_{ccw}\u3009={a}_{ccw}^{\u2020}|0\u3009$) is called the single excitation manifold. Use those five bases, we can rewrite the total Hamiltonian

*H*in Eq. (1) in a matrix representation as

*ω*−

_{a}*ω*is the NV-center-resonator detuning.

_{c}There are two main approaches to describe the dynamics of the NV-center-resonator coupled system. The first approach is the so-called phenomenological master equation, given by
$\dot{\rho}\left(t\right)=-i\left[H,\rho \right]+\gamma \left[n\left(\omega \right)+1\right]\left[a\rho {a}^{\u2020}-\frac{1}{2}\left({a}^{\u2020}a\rho +\rho {a}^{\u2020}a\right)\right]+\gamma n\left(\omega \right)\left[{a}^{\u2020}\rho a-\frac{1}{2}\left(a{a}^{\u2020}\rho +\rho a{a}^{\u2020}\right)\right]$, where *ρ* = *ρ*(*t*) is the density matrix of the total system. *γ* is the loss rate of cavity photons, and *n*(*ω*) is the average number of quanta of the reservoir in the mode of frequency *ω*. In the CQED system, the leakage of cavity photons and spontaneous emission of the NV centers are the main sources of dissipation. However, the spontaneous emission of the NV centers is mostly suppressed by the presence of the cavity, and therefore its effect is usually neglected [60, 61].

Different from the phenomenological master equation above, we can adopt the second approach, i.e., the microscopic master equation approach [60, 62], which comes from the original idea on how to describe the system-reservoir interactions in Markovian master equations [63, 64]. Following the standard procedures [65], that is to say, using the Liouville-von Neumann equation and tracing out the environmental degrees of freedom, under the Born-Markov approximation (BMA) and the RWA, one can obtain the microscopic master equation for the density operator *ρ*(*t*) of the NV-center-resonator coupled system in the Schrödinger picture [60, 61, 65]:

*A*, given by

_{m}*A*(

_{m}*ω*̄) = ∑

_{ω̄=λj−λi}|

*ϕ*〉 〈

_{i}*ϕ*|

_{i}*A*|

_{m}*ϕ*〉 〈

_{j}*ϕ*| with ${A}_{m}={a}_{m}+{a}_{m}^{\u2020}$.

_{j}*λ*

_{i(j)}and |

*ϕ*

_{i(j)}〉 are the eigenvalue and the corresponding normalized eigenstate of the Hamiltonian

*H*[see Eq. (5)]. The brace stands for the anti-commutation relationship. From Eq. (6), it can be clearly seen that the microscopic master equation approach considers the jumps between the eigenstates of the system Hamiltonian.

## 3. Formal solution of the microscopic master equation and the concurrence

In a general case, there is not an analytic solution of the matrix representation as given in Eq. (5). Consequently, we provide a formal solution, and then discuss the influences of the coupling strength between WGMs, the detuning between the NV center and microresonator, the distance or the phase between the two NV centers, and the dissipative factors on the entanglement dynamics and entanglement degree of the two NV centers.

Now we assume that the eigenvalue and the corresponding normalized eigenstate of the Hamiltonian *H* in Eq. (5) are *λ _{i}* and |

*ϕ*〉 (

_{i}*i*= 1,⋯,5). Making use of the given five bases in Section 2, the eigenstate |

*ϕ*〉 can be taken to have the following form

_{i}*c*(

_{ij}*i*,

*j*= 1,⋯,4) are real or complex numbers and the single state |

*ϕ*

_{5}〉 = |5〉 corresponds to the ground state of the total system. Under the condition of the single excitation, according to Eq. (7)–Eq. (11), the Davies operators

*A*(

_{m}*ω*̄) can be reduced into the form

*ω*̄

*=*

_{ij}*λ*−

_{j}*λ*(

_{i}*i*≠

*j*,

*i*,

*j*= 1,⋯,5). The specific expressions of

*A*(

_{m}*ω*̄) are given in the Appendix A. In order to simplify the calculation, we also introduce the expressions

Inserting Eq. (12) and Eq. (13)–Eq. (16) into Eq. (6), the microscopic master equation can be reexpressed as

*ρ*(

*t*) can be easily calculated. In the following, we will solve a set of first-order differential equation under different initial conditions. The initial condition can be written in the vector bases |

*i*〉 (

*i*= 1,⋯,4)

For the convenience of calculation, the initial condition (18) can be reexpressed in the dressed-state bases as

From Eq. (7)–Eq. (11), we can arrive at the relationship

*F*is the inverse matrix of the coefficient matrix in Eq. (7)–Eq. (11). Using Eq. (20), we can write the initial condition in the dressed basis. So we can fully solve the microscopic master equation. The derivation of the solution is presented in the Appendix A. Once the microscopic master equation of the system can be completely solved, then the density matrix can be rewritten in the dressed bases as

*ρ*(

_{ij}*t*) = 〈

*ϕ*|

_{i}*ρ*(

*t*)|

*ϕ*〉 is the matrix elements as shown in the Appendix B.

_{j}In order to explore the entanglement dynamics of the two NV centers, we need to project the density matrix *ρ*(*t*) onto the state |00〉 = |0〉* _{cw}*|0〉

*, with the result*

_{ccw}*ρ*(

_{a}*t*) is the state of the system after the projection measurement. From Eq. (7)–Eq. (11), we can obtain the following states

Based on Eq. (22) and Eq. (23)–Eq. (27), we can achieve the result

To investigate the degree of entanglement, we use *ρ _{a}*(

*t*) to calculate the concurrence [66]

*μ*(

_{i}*i*= 1, ⋯, 5) are the square roots of the eigenvalues of

*ρ*with ${\tilde{\rho}}_{a}=\left({\sigma}_{y}\otimes {\sigma}_{y}\right){\rho}_{a}^{*}\left({\sigma}_{y}\otimes {\sigma}_{y}\right)$ in decreasing order, and

_{a}ρ̃_{a}*σ*being the Pauli operator. In our case the concurrence can be easily written as [67] which is a measure to quantify the degree of entanglement between the two NV centers.

_{y}## 4. Entanglement dynamics and entanglement degree of the two NV centers

First of all, we analyze the influence of the coupling strength *g* between WGMs on the entanglement dynamics and entanglement degree of the two NV centers. As shown in Ref. [53], the coupling strength *g* between WGMs depends on the radius *R* of the NV center in the spherical diamond nanocrystal. With the increase of *R*, the scattering-induced coupling coefficient *g* grows rapidly. We can approximately divide the coupling strength *g* into three different regions: (i) *g* is far less than the coupling strength *G* between the NV center and cavity mode for a small diamond nanocrystal; (ii) *g* becomes comparable with *G*; and (iii) *g* exceeds *G* when the size of the diamond nanocrystal is large enough. However, it is worth mentioning that *g* can not far exceed *G* because of the great energy dampings of the WGMs at larger radius *R* of the NV center. In the calculation, we have set the dampings of the WGMs to be equal, i.e., *γ _{cw}* =

*γ*=

_{ccw}*γ*.

Figure 2 shows the WGM-undamped (blue solid curves) concurrence dynamics and the WGM-damped (dashed red curves) concurrence dynamics of the two NV centers as a function of time for three different regions (i) *g* = *G*/10, (ii) *g* = *G*, (iii) *g* = 2*G* and two different phases *kd* = (2*n*+1)*π*/2 and *kd* = *nπ* (*n* should be big enough so that the condition that the distance *d* is much larger than the wavelength of WGMs can be well satisfied). In plotting Fig. 2, we have set the initial state *ρ*(0) = |3〉 〈3| = |*gg*10〉 〈*gg*10| and the resonant interaction of the cavity field with the NV center dipole, i.e., the NV-center-resonator detuning Δ = 0. It can be clearly seen from Fig. 2 that the behavior of the concurrence dynamics is oscillatory between the zero and maximum due to the Rabi oscillations between the electronic states of the NV center and the states of the quantized WGM field. On the one hand, large degree of entanglement, i.e., high entanglement can be produced through the nanocrystal-induced Rayleigh scattering at appropriate times. The entanglement maximum greatly decreases when the coupling strength *g* (or the radius *R* of the NV centers) increases and the oscillation aggravates. On the other hand, the phase interference induced by different *d* leads to variations of the entanglement maximum, oscillation profile and oscillation frequency. The above results can be explained from the perspective of normal modes of the resonator, which are just linear combinations of the cw and ccw modes. In this case, the two NV centers actually couple to a single normal mode of the resonator, i.e.,
$\left({a}_{cw}+{a}_{ccw}\right)/\sqrt{2}$, with the frequency *ω _{c}* + 4

*g*. The system is therefore equivalent to a pair of two-level NV centers coupled to a single resonator mode. For small

*g*, the resonator normal mode is near-resonant from the transition frequency of the NV center. However, for large

*g*, the resonator normal mode is far off-resonant from the transition frequency of the NV center, which destroys the entanglement. Also, this phenomenon can be seen from the concrete expressions of the concurrence

*C*(

*t*) based on the Eq. (30) and Eq. (57). For example, for the cases of Fig. 2(a1) and Fig. 2(b1), the concurrences respectively are

*C*(

*t*) = sin

^{2}(1.4

*Gt*) and $C\left(t\right)=0.68{\text{sin}}^{2}\left(\sqrt{3}Gt\right)$ when

*γ*= 0. Other cases are similar.

In order to more clearly show the influence of the initial state on the concurrence dynamics, firstly we change the initial state as *ρ*(0) = |1〉 〈1| = |*eg*00〉 〈*eg*00| when the phase *kd* = *nπ* is fixed. It can be easily found from Fig. 3 that the behavior of the concurrence dynamics becomes more periodic and the period becomes longer with the increase of the coupling strength *g*. The maximum entanglement (*C* = 1) can occur at appropriate times. In addition, the concurrence dynamics for the undamped (the blue solid curves *γ* = 0) and damped (red dashed curves *γ* = *G*/30) WGMs become nearer as *g* increases. That is to say, to some extent, the coupling strength *g* can compensate for the decay caused by cavity leakage or *g* may enhance the entanglement. This conclusion is similar to the result reported by Jin et al. [68]. The reason of the influence of the coupling strength *g* on the entanglement can be considered in this way. The two NV centers indirectly interact with each other via WGMs. The coupling strength *g* of the two modes indirectly impacts the coupling strength of the two NV centers.

Alternatively, if the initial state is chosen as *ρ*(0) = |2〉 〈2| = |*ge*00〉 〈*ge*00|, the concurrence dynamics of the system is the same as in the initial state *ρ*(0) = |1〉 〈1| = |*eg*00〉 〈*eg*00| because of the equivalence of the two NV centers. It is worth mentioning that the concurrence is equal to zero with *kd* = (2*n* + 1)*π*/2 for two different cases of the initial state: (i) *ρ*(0) = |2〉 〈2| = |*ge*00〉 〈*ge*00| and (ii) *ρ*(0) = |1〉 〈1| = |*eg*00〉 〈*eg*00| (here not shown). These results clearly indicate that the phase interference induced by different *d* plays an important role in the entanglement generation.

Next we consider the initial state is a mixed state, i.e., *ρ*(0) = *ε*|1〉 〈1| + (1 − *ε*)|2〉 〈2| = *ε*|*eg*00〉 〈*eg*00| + (1 − *ε*)|*ge*00〉 〈*ge*00| with *ε* being a real number. For simplicity, in the following discussions we ignore the dampings of the WGMs since they only play the role of decreasing the height of the concurrence. Figure 4 shows that different *ε* can change the degree of the entanglement but can not alter the profile of the concurrence. In other words, the larger *ε*, the stronger entanglement. According to what has been discussed above, we can arrive at the conclusion that an appropriate initial state and large coupled strength can enhance the entanglement between the two NV centers to some extent.

Now we turn to discuss the off-resonance interaction of the cavity field with the NV center dipole, i.e., Δ ≠ 0. In Fig. 5, the concurrence dynamics of the two NV centers as a function of time is plotted for three different detunings Δ. One can observe that the concurrence regularly oscillates due to the Rabi oscillations between the electronic states of the NV center and the states of the quantized WGM field. Also, it can be shown that the maximal value of concurrence decreases rapidly and the oscillation frequency increases with Δ increasing. These results are similar to ones in Fig. 2. If we choose the initial sate as *ρ*(0) = |1〉 〈1| = |*eg*00〉 〈*eg*00| corresponding to Fig. 6, the concurrence dynamics is similar to Fig. 3. The behavior of the concurrence dynamics is irregularly oscillatory between *C* = 0 and *C* = 1. The curves become more rhythmic and the period becomes longer with Δ increasing. To sum up, both the detuning Δ and the initial state *ρ*(0) can influence the height of concurrence. With an appropriate initial state, the maximum entanglement can still be achieved even when Δ is largely deviated from the zero value, which is useful in real experiments.

Finally, we come back to Fig. 2, where we consider the cases that the scattering-induced coupling coefficients take different values, i.e., *g*_{1} ≠ *g*_{2} (or the radius *R*_{1} ≠ *R*_{2}) for clearly showing the influence of the unequal coupling strengths *g _{j}* (

*j*= 1, 2) on the entanglement generation. For the purpose of comparison, we can fix

*g*

_{1}=

*g*=

*G*/10 and properly adjust

*g*

_{2}. The other system parameters used are the same as those in Fig. 2(a1). We define a parameter

*s*to be the ratio of the two coupling strengths, i.e.,

*g*

_{2}=

*sg*

_{1}=

*sg*. Figure 7 shows that the concurrences reduce slowly as the ratio

*s*increases rapidly. Compared with Fig. 2(a1), we find that relative high entanglement between the two NV centers can still be achieved at appropriate times even if the ratio

*s*is far off the unity value.

## 5. Conclusion

In summary, we have investigated the entanglement generation between the two NV centers coupled to high-Q counterpropagating twin WGMs of the microtoroidal resonator. By applying the microscopic master equation approach which describes transitions between the eigenstates of the full Hamiltonian of the system, we calculate the concurrence to look into the degree and dynamics of entanglement. We analyze and discuss in detail the WGM-undamped concurrence dynamics and the WGM-damped concurrence dynamics of the two NV centers as a function of time for different system parameters. It is clearly shown that the maximum entanglement between the two NV centers can be achieved by appropriately adjusting the radius *R* of the two spherical NV centers, the distance *d* between the two NV centers, the detuning Δ between the NV centers and microresonator, and the initial state *ρ*(0) of the coupled system. The research presented here may be useful in view of recent activity aiming at solid-state quantum information engineering and processing applications.

## Appendix A: Solutions of the microscopic master equation

In this appendix, we work out the matrix elements of the microscopic master equation [see Eq. (17)].

At first, we show the operators *A _{m}*(

*ω*̄

*). For*

_{ij}*m*=

*cw*, we obtain

For *m* = *ccw*, the operators *A _{n}*(

*ω*̄

*) correspond to*

_{ij}Using Eq. (17), we can easily obtain the solutions of the set of first order differential equations. Here we notate 〈*ϕ _{i}*|

*ρ*(

*t*)|

*ϕ*〉 =

_{j}*ρ*(

_{ij}*t*), with results as follows

For the initial condition *ρ*(0) = |3〉 〈3| = |*gg*10〉 〈*gg*10|, the density matrix elements *ρ _{ij}*(0) can be respectively written as

*ρ*

_{11}(0) = |

*f*

_{9}|

^{2},

*ρ*

_{22}(0) = |

*f*

_{10}|

^{2},

*ρ*

_{33}(0) = |

*f*

_{11}|

^{2},

*ρ*

_{44}(0) = |

*f*

_{12}|

^{2}, ${\rho}_{21}\left(0\right)={f}_{10}{f}_{9}^{*}$, ${\rho}_{31}\left(0\right)={f}_{11}{f}_{9}^{*}$, ${\rho}_{41}\left(0\right)={f}_{12}{f}_{9}^{*}$, ${\rho}_{32}\left(0\right)={f}_{11}{f}_{10}^{*}$, ${\rho}_{42}\left(0\right)={f}_{12}{f}_{10}^{*}$, ${\rho}_{43}\left(0\right)={f}_{12}{f}_{11}^{*}$, and the rest of the elements is zero [see Eq. (20)]. Here “*” represents complex conjugate.

When the initial state is a mixed state *ρ*(0) = *ε*|1〉 〈1| + (1 − *ε*)|2〉 〈2| = *ε*|*eg*00〉 〈*eg*00| + (1 − *ε*)|*ge*00〉 〈*ge*00|, the density matrix elements *ρ _{ij}*(0) can be respectively written as

*ρ*

_{11}(0) =

*ε*|

*f*

_{1}|

^{2}+ (1 −

*ε*)|

*f*

_{5}|

^{2},

*ρ*

_{22}(0) =

*ε*|

*f*

_{2}|

^{2}+ (1 −

*ε*)|

*f*

_{6}|

^{2},

*ρ*

_{33}(0) =

*ε*|

*f*

_{3}|

^{2}+ (1 −

*ε*)|

*f*

_{7}|

^{2},

*ρ*

_{44}(0) =

*ε*|

*f*

_{4}|

^{2}+ (1 −

*ε*)|

*f*

_{8}|

^{2}, ${\rho}_{21}\left(0\right)=\epsilon {f}_{2}{f}_{1}^{*}+\left(1-\epsilon \right){f}_{6}{f}_{5}^{*}$, ${\rho}_{31}\left(0\right)=\epsilon {f}_{3}{f}_{1}^{*}+\left(1-\epsilon \right){f}_{7}{f}_{5}^{*}$, ${\rho}_{41}\left(0\right)=\epsilon {f}_{4}{f}_{1}^{*}=\left(1-\epsilon \right){f}_{8}{f}_{5}^{*}$, ${\rho}_{32}\left(0\right)=\epsilon {f}_{3}{f}_{2}^{*}+\left(1-\epsilon \right){f}_{7}{f}_{6}^{*}$, ${\rho}_{42}\left(0\right)=\epsilon {f}_{4}{f}_{2}^{*}+\left(1-\epsilon \right){f}_{8}{f}_{6}^{*}$, and ${\rho}_{43}\left(0\right)=\epsilon {f}_{4}{f}_{3}^{*}+\left(1-\epsilon \right){f}_{8}{f}_{7}^{*}$. Note that the rest of the elements is all zero.

## Appendix B: Elements of the concurrence

After performing the measurement, we can obtain the following elements [see Eq. (28)]

*c.c.*” represents complex conjugate.

## Acknowledgment

Part of this work has been supported by the National Natural Science Foundation of China under Grants No. 11004069, No. 11275074 and No. 91021011, by the Doctoral Foundation of the Ministry of Education of China under Grant No. 20100142120081, and by the National Basic Research Program of China under Contract No. 2012CB922103. We gratefully acknowledge encouraging and helpful discussions with Professor X. X. Yang.

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