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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3501–3515
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Achieving maximum entanglement between two nitrogen-vacancy centers coupling to a whispering-gallery-mode microresonator

Siping Liu, Jiahua Li, Rong Yu, and Ying Wu  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3501-3515 (2013)
http://dx.doi.org/10.1364/OE.21.003501


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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

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

27. A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Borczyskowski, “Scanning confocal optical microscopy and magnetic resonance of single defect centers,” Science 276, 2012–2014 (1997). [CrossRef]

], 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

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]

38

38. W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and C. H. Oh, “Quantum dynamics and quantum state transfer between separated nitrogen-vacancy centers embedded in photonic crystal cavities,” Phys. Rev. A 84, 043849 (2011). [CrossRef]

]. Recently, a combination of NV centers and high-Q WGM microcavities which represents a promising solid-state CQED system has attracted much attention [39

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

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]

]. 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

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]

]. 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

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]

] and a metal nanoring resonator [50

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]

]. 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

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

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]

].

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 gR3[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]

]), 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.

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

Fig. 1 Schematic illustration of the coupling system composed of a microtoroidal resonator and two two-level NV centers. The microtoroidal cavity supports two counter-propagating WGMs, denoted as acw and accw. 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. The distance between two NV centers is d. The distance d is far enough, so the interaction of two NV centers can be neglected. Experimentally, we can use atomic force microscope manipulation to controllably position the NV centers [58, 59]. The bubble shows energy configuration for each of the NV centers.

We note that the composite system described by the Hamiltonian H has two invariant subspaces. One is ∀1 ∈ {|1〉 = |e1, g2, 0cw, 0ccw〉, |2〉 = |g1, e2, 0cw, 0ccw〉, |3〉 = |g1, g2, 1cw, 0ccw〉, |4〉 = |g1, g2, 0cw, 1ccw〉}, and the other is ∀2 ∈ {|5〉 = |g1, g2, 0cw, 0ccw〉}, where in |l1, r2, pcw, qccw〉, 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., |1cw〉 (|1ccw〉) denotes the one-photon Fock state of the cw (ccw) WGM with frequency ωc, |0cw〉 (|0ccw〉) describes the vacuum state of the cw (ccw) WGM. For simplicity, we abbreviate as |l1, r2, pcw, qccw〉 ≡ |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., |1cw=acw|0 and |1ccw=accw|0) 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
H=(00G1G2000G2eikdG2eikd0G1G2eikdΔ+2gg(1+e2ikd)0G1G2eikdg(1+e2ikd)Δ+2g00000ωa),
(5)
where Δ = ωaωc is the NV-center-resonator detuning.

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 ρ˙(t)=i[H,ρ]+γ[n(ω)+1][aρa12(aaρ+ρaa)]+γn(ω)[aρa12(aaρ+ρaa)], 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

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

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]

].

Different from the phenomenological master equation above, we can adopt the second approach, i.e., the microscopic master equation approach [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]

, 62

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

], which comes from the original idea on how to describe the system-reservoir interactions in Markovian master equations [63

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

, 64

64. E. B. Davies, Quantum Theory of Open System (Academic, London, 1976).

]. Following the standard procedures [65

65. H. P. Breuer and F. Petruccione, The Theory of Open Quantum System (Clarendon, Oxford, 2006).

], 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

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

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]

, 65

65. H. P. Breuer and F. Petruccione, The Theory of Open Quantum System (Clarendon, Oxford, 2006).

]:
ρ˙(t)=i[H,ρ]+ω¯>0,m={cw,ccw}γm(ω¯)×[Am(ω¯)ρ(t)Am(ω¯)12{Am(ω¯)Am(ω¯),ρ(t)}],
(6)
where we have introduced the Davies operators Am, given by Am(ω̄) = ∑ω̄=λjλi |ϕi〉 〈ϕi|Am|ϕj〉 〈ϕj| with Am=am+am. λi(j) and |ϕi(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 λi and |ϕi〉 (i = 1,⋯,5). Making use of the given five bases in Section 2, the eigenstate |ϕi〉 can be taken to have the following form
|ϕ1=c11|1+c12|2+c13|3+c14|4,
(7)
|ϕ2=c21|1+c22|2+c23|3+c24|4,
(8)
|ϕ3=c31|1+c32|2+c33|3+c34|4,
(9)
|ϕ4=c41|1+c42|2+c43|3+c44|4,
(10)
|ϕ5=|5,
(11)
where the coefficients cij (i, j = 1,⋯,4) are real or complex numbers and the single state |ϕ5〉 = |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 Am(ω̄) can be reduced into the form
Am(ω¯ij)=|ϕiϕi|am|ϕjϕj|,
(12)
where we have defined ω̄ij = λjλi (ij, i, j = 1,⋯,5). The specific expressions of Am(ω̄) are given in the Appendix A. In order to simplify the calculation, we also introduce the expressions
γ1=|c13|2γcw(ω¯51)+|c14|2γccw(ω¯51),
(13)
γ2=|c23|2γcw(ω¯52)+|c24|2γccw(ω¯52),
(14)
γ3=|c33|2γcw(ω¯53)+|c34|2γccw(ω¯53),
(15)
γ4=|c43|2γcw(ω¯54)+|c44|2γccw(ω¯54),
(16)

Inserting Eq. (12) and Eq. (13)Eq. (16) into Eq. (6), the microscopic master equation can be reexpressed as
ρ˙(t)=i[H,ρ(t)]+i=14γi[|ϕ5ϕi|ρ(t)|ϕiϕ5|12{|ϕiϕi|,ρ}],
(17)
Equation (17) is the main result that governs the system dynamics. The matrix elements of 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)
ρ(0)=i,j=14i|ρ(0)|j|ij|.
(18)

For the convenience of calculation, the initial condition (18) can be reexpressed in the dressed-state bases as
ρ(0)=i,j=14i|ρ(0)|j|ij|=i,j=14ϕi|ρ(0)|ϕj|ϕiϕj|.
(19)

From Eq. (7)Eq. (11), we can arrive at the relationship
(|1|2|3|4|5)=(f1f2f3f40f5f6f7f80f9f10f11f120f13f14f15f16000001)(|ϕ1|ϕ2|ϕ3|ϕ4|ϕ5)=F(|ϕ1|ϕ2|ϕ3|ϕ4|ϕ5),
(20)
where 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(t)=i,j=14ϕi|ρ(t)|ϕj|ϕiϕj|,
(21)
where ρij(t) = 〈ϕi|ρ(t)|ϕj〉 is the matrix elements as shown in the Appendix B.

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〉ccw, with the result
ρa(t)=00|ρ(t)|00=i,j=14ϕi|ρ(t)|ϕj00|ϕiϕj|00,
(22)
where ρa(t) is the state of the system after the projection measurement. From Eq. (7)Eq. (11), we can obtain the following states
00|ϕ1=c11|eg+c12|ge,
(23)
00|ϕ2=c21|eg+c22|ge,
(24)
00|ϕ3=c31|eg+c32|ge,
(25)
00|ϕ4=c41|eg+c42|ge,
(26)
00|ϕ5=|gg.
(27)

Based on Eq. (22) and Eq. (23)Eq. (27), we can achieve the result
ρa(t)=ρeg,eg|egeg|+ρge,ge|gege|+ρgg,gg|gggg|+ρeg,ge|egge|+ρeg,ge*|geeg|.
(28)

To investigate the degree of entanglement, we use ρa(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]

]
C(t)=C(ρa)=2max{0,μ1μ2μ3μ4},
(29)
where μi (i = 1, ⋯, 5) are the square roots of the eigenvalues of ρaρ̃a with ρ˜a=(σyσy)ρa*(σyσy) in decreasing order, and σy being the Pauli operator. In our case the concurrence can be easily written as [67

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

]
C(t)=C(ρa)=2|ρeg,ge|,
(30)
which is a measure to quantify the degree of entanglement between the two NV centers.

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

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]

], 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., γcw = γccw = 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 = 2G and two different phases kd = (2n+1)π/2 and kd = (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| = |gg10〉 〈gg10| 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., (acw+accw)/2, with the frequency ωc + 4g. 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) = sin2(1.4Gt) and C(t)=0.68sin2(3Gt) when γ = 0. Other cases are similar.

Fig. 2 Concurrence dynamics of the two NV centers versus the dimensionless time Gt for different phases (a1)–(c1) kd = (2n + 1)π/2 and (a2)–(c2) kd = under the initial state ρ(0) = |3〉〈3| = |gg10〉〈gg10| and the detuning Δ = 0. (a1) and (a2): g = G/10; (b1) and (b2): g = G; (c1) and (c2): g = 2G. The blue solid curves represent the damping rate γ = 0 of the WGMs and the red dashed curves correspond to γ = G/30, respectively.

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| = |eg00〉 〈eg00| when the phase kd = 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]

]. 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.

Fig. 3 Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different coupling strengths: (a) g = G/10, (b) g = G, and (c) g = 2G. The other system parameters are chosen as ρ(0) = |1〉〈1| = |eg00〉〈eg00|, kd = , and Δ = 0, respectively. The blue solid curves represent the damping rate γ = 0 of the WGMs and the red dashed curves correspond to γ = G/30.

Alternatively, if the initial state is chosen as ρ(0) = |2〉 〈2| = |ge00〉 〈ge00|, the concurrence dynamics of the system is the same as in the initial state ρ(0) = |1〉 〈1| = |eg00〉 〈eg00| because of the equivalence of the two NV centers. It is worth mentioning that the concurrence is equal to zero with kd = (2n + 1)π/2 for two different cases of the initial state: (i) ρ(0) = |2〉 〈2| = |ge00〉 〈ge00| and (ii) ρ(0) = |1〉 〈1| = |eg00〉 〈eg00| (here not shown). These results clearly indicate that the phase interference induced by different d plays an important role in the entanglement generation.

Fig. 4 Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different initial states ρ(0). In this case, the mixed state is chosen as ρ(0) = ε|1〉 〈1| + (1 − ε)|2〉 〈2| = ε|eg00〉 〈eg00| + (1 − ε)|ge00〉 〈ge00| with ε being a real number. The other system parameters are chosen as Δ = 0, kd = , g = 2G, and γ = 0, respectively.

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| = |eg00〉 〈eg00| 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.

Fig. 5 Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different detunings Δ. The other system parameters are chosen as ρ(0) = |3〉 〈3| = |gg10〉 〈gg10|, kd = , g = G, and γ = 0, respectively.
Fig. 6 Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different detunings Δ. The other system parameters are the same as Fig. 5 except for the initial state ρ(0) = |1〉 〈1| = |eg00〉 〈eg00|.

Finally, we come back to Fig. 2, where we consider the cases that the scattering-induced coupling coefficients take different values, i.e., g1g2 (or the radius R1R2) for clearly showing the influence of the unequal coupling strengths gj (j = 1, 2) on the entanglement generation. For the purpose of comparison, we can fix g1 = g = G/10 and properly adjust g2. 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., g2 = sg1 = 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.

Fig. 7 Concurrence dynamics of the two NV centers versus the dimensionless time Gt for three different proportional coefficients s. The other system parameters are the same as Fig. 2(a1) for γ = 0.

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 Am(ω̄ij). For m = cw, we obtain
Acw(ω¯51=λ1λ5)=c13|ϕ5ϕ1|,
(31)
Acw(ω¯52=λ2λ5)=c23|ϕ5ϕ2|,
(32)
Acw(ω¯53=λ3λ5)=c33|ϕ5ϕ3|,
(33)
Acw(ω¯54=λ4λ5)=c43|ϕ5ϕ4|.
(34)

For m = ccw, the operators An(ω̄ij) correspond to
Accw(ω¯51=λ1λ5)=c14|ϕ5ϕ1|,
(35)
Accw(ω¯52=λ2λ5)=c24|ϕ5ϕ2|,
(36)
Accw(ω¯53=λ3λ5)=c34|ϕ5ϕ3|,
(37)
Accw(ω¯54=λ4λ5)=c44|ϕ5ϕ4|.
(38)

Using Eq. (17), we can easily obtain the solutions of the set of first order differential equations. Here we notate 〈ϕi|ρ(t)|ϕj〉 = ρij(t), with results as follows
ρ11(t)=ρ11(0)eγ1t,
(39)
ρ22(t)=ρ22(0)eγ2t,
(40)
ρ33t=ρ33(0)eγ3t,
(41)
ρ44(t)=ρ44(0)eγ4t,
(42)
ρ55(t)=ρ55(0)+ρ11(0)(1eγ1t)+ρ22(0)(1eγ2t)+ρ33(0)(1eγ3t)+ρ44(0)(1eγ4t),
(43)
ρ21(t)=ρ21(0)e(i(λ1λ2)12(γ1+γ2))t,
(44)
ρ31(t)=ρ31(0)e(i(λ1λ3)12(γ1+γ3))t,
(45)
ρ41(t)=ρ41(0)e(i(λ1λ4)12(γ1+γ4))t,
(46)
ρ51(t)=ρ51(0)e(i(λ1λ5)12γ1)t,
(47)
ρ32(t)=ρ32(0)e(i(λ2λ3)12(γ2+γ3))t,
(48)
ρ42(t)=ρ42(0)e(i(λ2λ4)12(γ2+γ4))t,
(49)
ρ52(t)=ρ52(0)e(i(λ2λ5)12γ2)t,
(50)
ρ43(t)=ρ43(0)e(i(λ3λ4)12(γ3+γ4))t,
(51)
ρ53(t)=ρ53(0)e(i(λ3λ5)12γ3)t,
(52)
ρ54(t)=ρ54(0)e(i(λ4λ5)12γ4)t.
(53)

For the initial condition ρ(0) = |3〉 〈3| = |gg10〉 〈gg10|, the density matrix elements ρij(0) can be respectively written as ρ11(0) = |f9|2, ρ22(0) = |f10|2, ρ33(0) = |f11|2, ρ44(0) = |f12|2, ρ21(0)=f10f9*, ρ31(0)=f11f9*, ρ41(0)=f12f9*, ρ32(0)=f11f10*, ρ42(0)=f12f10*, ρ43(0)=f12f11*, 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|eg00〉 〈eg00| + (1 − ε)|ge00〉 〈ge00|, the density matrix elements ρij(0) can be respectively written as ρ11(0) = ε|f1|2 + (1 − ε)|f5|2, ρ22(0) = ε|f2|2 + (1 − ε)|f6|2, ρ33(0) = ε|f3|2 + (1 − ε)|f7|2, ρ44(0) = ε|f4|2 + (1 − ε)|f8|2, ρ21(0)=εf2f1*+(1ε)f6f5*, ρ31(0)=εf3f1*+(1ε)f7f5*, ρ41(0)=εf4f1*=(1ε)f8f5*, ρ32(0)=εf3f2*+(1ε)f7f6*, ρ42(0)=εf4f2*+(1ε)f8f6*, and ρ43(0)=εf4f3*+(1ε)f8f7*. 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)]
ρeg,eg=ρ11(t)|c11|2+ρ22(t)|c21|2+ρ33(t)|c31|2+ρ44(t)|c41|2+[(ρ21(t)c21c11*+ρ31(t)c31c11*+ρ41(t)c41c11*+ρ32(t)c31c21*+ρ42(t)c41c21*+ρ43(t)c41c31*)+c.c.],
(54)
ρge,ge=ρ11(t)|c12|2+ρ22(t)|c22|2+ρ33(t)|c32|2+ρ44(t)|c42|2+[(ρ21(t)c22c12*+ρ31(t)c32c12*+ρ41(t)c42c12*+ρ32(t)c32c22*+ρ42(t)c42c22*+ρ43(t)c42c32*)+c.c.],
(55)
ρgg,gg=ρ55(0)+ρ11(0)(1eγ1t)+ρ22(0)(1eγ2t)+ρ33(0)(1eγ3t)+ρ44(0)(1eγ4t),
(56)
ρeg,ge=ρ11(t)c11c12*+ρ22(t)c21c22*+ρ33(t)c31c32*+ρ44(t)c41c42*+ρ21(t)c21c12*+ρ21*(t)c11c22*+ρ31(t)c31c12*+ρ31*(t)c11c32*+ρ41(t)c41c12*+ρ41*(t)c11c42*+ρ32(t)c31c22*+ρ32*(t)c21c32*,+ρ42(t)c41c22*+ρ42*(t)c21c42*+ρ43(t)c41c32*+ρ43*(t)c31c42*,
(57)
where “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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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

  1. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature (London)390, 575–579 (1997). [CrossRef]
  2. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.70, 1895–1899 (1993). [CrossRef] [PubMed]
  3. T. Di, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, “Quantum teleportation of an arbitrary superposition of atomic Dicke states,” Phys. Rev. A71, 062308 (2005). [CrossRef]
  4. S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett.85, 2392–2395 (2000). [CrossRef] [PubMed]
  5. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, UK, 2000).
  6. C. H. Bennett and D. P. Divincenzo, “Quantum information and computation,” Nature (London)404, 247–255 (2000). [CrossRef]
  7. J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett.74, 4091–4094 (1995). [CrossRef] [PubMed]
  8. S. B. Zheng, “Nongeometric conditional phase shift via adiabatic evolution of dark eigenstates: a new approach to quantum computation,” Phys. Rev. Lett.95, 080502 (2005). [CrossRef] [PubMed]
  9. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett.78, 3221–3224 (1997). [CrossRef]
  10. T. Pellizzari, “Quantum networking with optical fibres,” Phys. Rev. Lett.79, 5242–5245 (1997). [CrossRef]
  11. L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London)414, 413–418 (2001). [CrossRef]
  12. H. J. Kimble, “The quantum internet,” Nature (London)453, 1023–1030 (2008). [CrossRef]
  13. A. Aspect, “Bell’s inequality test: more ideal than ever,” Nature (London)398, 189–190 (1999). [CrossRef]
  14. R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature (London)453, 1008–1015 (2008). [CrossRef]
  15. A. Sørensen and K. Mølmer, “Quantum computation with ions in thermal motion,” Phys. Rev. Lett.82, 1971–1974 (1999). [CrossRef]
  16. Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett.81, 3631–3634 (1998). [CrossRef]
  17. C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J. Kimble, “Measurement-induced entanglement for excitation stored in remote atomic ensembles,” Nature (London)438, 828–832 (2005). [CrossRef]
  18. M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, “Measurement of the entanglement of two superconducting qubits via state tomography,” Science313, 1423–1425 (2006). [CrossRef] [PubMed]
  19. H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science298, 1372–1377 (2002). [CrossRef] [PubMed]
  20. R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, “Trapped atoms in cavity QED: coupling quantized light and matter,” J. Phys. B: At. Mol. Opt. Phys.38, S551–S565 (2005). [CrossRef]
  21. A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett.98, 193601 (2007). [CrossRef] [PubMed]
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