## Achieving maximum entanglement between two nitrogen-vacancy centers coupling to a whispering-gallery-mode microresonator |

Optics Express, Vol. 21, Issue 3, pp. 3501-3515 (2013)

http://dx.doi.org/10.1364/OE.21.003501

Acrobat PDF (1304 KB)

### 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 OSA

## 1. Introduction

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*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.

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28. T. A. Kennedy, J. S. Colton, J. E. Butler, R. C. Linares, and P. J. Doering, “Long coherence times at 300 K for nitrogen-vacancy center spins in diamond grown by chemical vapor deposition,” Appl. Phys. Lett. **83**, 4190–4192 (2003). [CrossRef]

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39. P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. **95**, 191115 (2009). [CrossRef]

47. P. E. Barclay, C. Santori, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express **17**, 8081–8097 (2009). [CrossRef] [PubMed]

*N*identical metal nanoparticle (MNP) and a WGM microcavity [48

48. Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. H. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A **85**, 031805(R) (2012). [CrossRef]

49. G. Y. Chen, N. Lambert, C. H. Chou, Y. N. Chen, and F. Nori, “Surface plasmons in a metal nanowire coupled to colloidal quantum dots: scattering properties and quantum entanglement,” Phys. Rev. B **84**, 045310 (2011). [CrossRef]

50. G. Y. Chen, C. M. Li, and Y. N. Chen, “Generating maximum entanglement under asymmetric couplings to surface plasmons,” Opt. Lett. **37**, 1337–1339 (2012). [CrossRef] [PubMed]

51. W. L. Yang, Z. Q. Xu, M. Feng, and J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys. **12**, 113039 (2010). [CrossRef]

52. W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. **96**, 241113 (2010) [CrossRef]

*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

53. Y. C. Liu, Y. F. Xiao, B. B. Li, X. F. Jiang, Y. Li, and Q. H. Gong, “Coupling of a single diamond nanocrystal to a whispering-gallery microcavity: photon transport benefitting from Rayleigh scattering,” Phys. Rev. A **84**, 011805(R) (2011). [CrossRef]

*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.

*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

43. S. Schietinger, T. Schröder, and O. Benson, “One-by-one coupling of single defect centers in nanodiamonds to high-Q modes of an optical microresonator,” Nano. Lett. **8**, 3911–3915 (2008). [CrossRef] [PubMed]

48. Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. H. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A **85**, 031805(R) (2012). [CrossRef]

52. W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. **96**, 241113 (2010) [CrossRef]

54. Q. Chen, W. Yang, M. Feng, and J. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A **83**, 054305 (2011). [CrossRef]

55. P. B. Li, S. Y. Gao, H. R. Li, S. L. Ma, and F. L. Li, “Dissipative preparation of entangled states between two spatially separated nitrogen-vacancy centers,” Phys. Rev. A **85**, 042306 (2012). [CrossRef]

*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

53. Y. C. Liu, Y. F. Xiao, B. B. Li, X. F. Jiang, Y. Li, and Q. H. Gong, “Coupling of a single diamond nanocrystal to a whispering-gallery microcavity: photon transport benefitting from Rayleigh scattering,” Phys. Rev. A **84**, 011805(R) (2011). [CrossRef]

*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.

## 2. Physical model and microscopic master equation

51. W. L. Yang, Z. Q. Xu, M. Feng, and J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys. **12**, 113039 (2010). [CrossRef]

52. W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One-step implementation of multiqubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silica microsphere cavity,” Appl. Phys. Lett. **96**, 241113 (2010) [CrossRef]

55. P. B. Li, S. Y. Gao, H. R. Li, S. L. Ma, and F. L. Li, “Dissipative preparation of entangled states between two spatially separated nitrogen-vacancy centers,” Phys. Rev. A **85**, 042306 (2012). [CrossRef]

56. D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Splitting of high-Q mie modes induced by light backscattering in silica microspheres,” Opt. Lett. **20**, 1835–1837 (1995). [CrossRef] [PubMed]

*a*and

_{cw}*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

53. Y. C. Liu, Y. F. Xiao, B. B. Li, X. F. Jiang, Y. Li, and Q. H. Gong, “Coupling of a single diamond nanocrystal to a whispering-gallery microcavity: photon transport benefitting from Rayleigh scattering,” Phys. Rev. A **84**, 011805(R) (2011). [CrossRef]

*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

**84**, 011805(R) (2011). [CrossRef]

57. X. Yi, Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. Gong, “Multiple-Rayleigh-scatterer-induced mode splitting in a high-Q whispering-gallery-mode microresonator,” Phys. Rev. A **83**, 023803 (2011). [CrossRef]

*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

57. X. Yi, Y. F. Xiao, Y. C. Liu, B. B. Li, Y. L. Chen, Y. Li, and Q. Gong, “Multiple-Rayleigh-scatterer-induced mode splitting in a high-Q whispering-gallery-mode microresonator,” Phys. Rev. A **83**, 023803 (2011). [CrossRef]

*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

*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

51. W. L. Yang, Z. Q. Xu, M. Feng, and J. F. Du, “Entanglement of separate nitrogen-vacancy centers coupled to a whispering-gallery mode cavity,” New J. Phys. **12**, 113039 (2010). [CrossRef]

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

58. D. Ratchford, F. Shafiei, S. Kim, S. K. Gray, and X. Li, “Manipulating coupling between a single semiconductor quantum dot and single gold nanoparticle,” Nano. Lett. **11**, 1049–1054 (2011). [CrossRef] [PubMed]

59. J. Merlein, M. Kah, A. Zuschlag, A. Sell, A. Halm, J. Boneberg, P. Leiderer, A. Leitenstorfer, and R. Bratschitsch, “Nanomechanical control of an optical antenna,” Nat. Photonics **2**, 230–233 (2008). [CrossRef]

*H*has two invariant subspaces. One is ∀

_{1}∈ {|1〉 = |

*e*

_{1},

*g*

_{2}, 0

*, 0*

_{cw}*〉, |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.,

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

*ω*−

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

_{c}*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

60. M. Scala, B. Militello, A. Messina, J. Piilo, and S. Maniscalco, “Microscopic derivation of the Jaynes-Cummings model with cavity losses,” Phys. Rev. A **75**, 013811 (2007). [CrossRef]

61. V. Montenegro and M. Orszag, “Creation of entanglement of two atoms coupled to two distant cavities with losses,” J. Phys. B: At. Mol. Opt. Phys. **44**, 154019 (2011). [CrossRef]

60. M. Scala, B. Militello, A. Messina, J. Piilo, and S. Maniscalco, “Microscopic derivation of the Jaynes-Cummings model with cavity losses,” Phys. Rev. A **75**, 013811 (2007). [CrossRef]

62. M. Wilczewski and M. Czachor, “Theory versus experiment for vacuum rabi oscillations in lossy cavities,” Phys. Rev. A **79**, 033836 (2009). [CrossRef]

63. E. B. Davies, “Markovian master equations,” Commun. Math. Phys. **39**, 91–110 (1974). [CrossRef]

*ρ*(

*t*) of the NV-center-resonator coupled system in the Schrödinger picture [60

60. M. Scala, B. Militello, A. Messina, J. Piilo, and S. Maniscalco, “Microscopic derivation of the Jaynes-Cummings model with cavity losses,” Phys. Rev. A **75**, 013811 (2007). [CrossRef]

61. V. Montenegro and M. Orszag, “Creation of entanglement of two atoms coupled to two distant cavities with losses,” J. Phys. B: At. Mol. Opt. Phys. **44**, 154019 (2011). [CrossRef]

*A*, given by

_{m}*A*(

_{m}*ω*̄) = ∑

_{ω̄=λj−λi}|

*ϕ*〉 〈

_{i}*ϕ*|

_{i}*A*|

_{m}*ϕ*〉 〈

_{j}*ϕ*| with

_{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

*H*in Eq. (5) are

*λ*and |

_{i}*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 where the coefficients

_{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 where we have defined

*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

*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)

*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 where

*ρ*(

_{ij}*t*) = 〈

*ϕ*|

_{i}*ρ*(

*t*)|

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

_{j}*ρ*(

*t*) onto the state |00〉 = |0〉

*|0〉*

_{cw}*, with the result where*

_{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

*ρ*(

_{a}*t*) to calculate the concurrence [66

66. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. **80**, 2245–2248 (1998). [CrossRef]

*μ*(

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

*ρ*with

_{a}ρ̃_{a}*σ*being the Pauli operator. In our case the concurrence can be easily written as [67

_{y}67. M. Orszag and M. Hernandez, “Coherence and entanglement in a two-qubit system,” Adv. Opt. Photon. **2**, 229–286 (2010). [CrossRef]

## 4. Entanglement dynamics and entanglement degree of the two NV centers

*g*between WGMs on the entanglement dynamics and entanglement degree of the two NV centers. As shown in Ref. [53

**84**, 011805(R) (2011). [CrossRef]

*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*.

*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.,

*ω*+ 4

_{c}*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

*γ*= 0. Other cases are similar.

*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

68. J. S. Jin, C. S. Yu, P. Pei, and H. S. Song, “Positive effect of scattering strength of a microtoroidal cavity on atomic entanglement evolution,” Phys. Rev. A **81**, 042309 (2010). [CrossRef]

*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.

*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.

*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.

*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.

*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

*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

*A*(

_{m}*ω*̄

*). For*

_{ij}*m*=

*cw*, we obtain

*m*=

*ccw*, the operators

*A*(

_{n}*ω*̄

*) correspond to*

_{ij}*ϕ*|

_{i}*ρ*(

*t*)|

*ϕ*〉 =

_{j}*ρ*(

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

*ρ*(0) = |3〉 〈3| = |

*gg*10〉 〈

*gg*10|, the density matrix elements

*ρ*(0) can be respectively written as

_{ij}*ρ*

_{11}(0) = |

*f*

_{9}|

^{2},

*ρ*

_{22}(0) = |

*f*

_{10}|

^{2},

*ρ*

_{33}(0) = |

*f*

_{11}|

^{2},

*ρ*

_{44}(0) = |

*f*

_{12}|

^{2},

*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*(0) can be respectively written as

_{ij}*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},

## Appendix B: Elements of the concurrence

*c.c.*” represents complex conjugate.

## Acknowledgment

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**OCIS Codes**

(160.2220) Materials : Defect-center materials

(270.5580) Quantum optics : Quantum electrodynamics

(140.3945) Lasers and laser optics : Microcavities

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: December 10, 2012

Revised Manuscript: January 23, 2013

Manuscript Accepted: January 25, 2013

Published: February 4, 2013

**Citation**

Siping Liu, Jiahua Li, Rong Yu, and Ying Wu, "Achieving maximum entanglement between two nitrogen-vacancy centers coupling to a whispering-gallery-mode microresonator," Opt. Express **21**, 3501-3515 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3501

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### References

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