Entanglement dynamics for three nitrogen-vacancy centers coupled to a whispering-gallery-mode microcavity

Wanlu Song; Wanli Yang; Qiong Chen; Qizhe Hou; Mang Feng

doi:10.1364/OE.23.013734

1. Introduction

Over recent years, increasing interest has focused on the strong coherent interactions between light and matter in the context of cavity quantum electrodynamics (CQED), which is of central concern in various applications in quantum information processing (QIP), quantum computation, and fundamental studies in quantum optics [1–5]. Among the platforms of CQED, the hybrid system consisting of whispering gallery mode (WGM) microcavities (with very small volumes V_m ≤ 100 μm³ [6] and high-Q factors Q ≥ 10⁸ ~ 10¹⁰ [6, 7]) and solid-state-based emitters becomes quite promising thanks to the experimental advance in the fabrication of optical microcavities and in manipulation of solid-state qubits [8–10]. Especially, the nitrogen-vacancy center (NVC) consisting of a substitutional nitrogen atom and an adjacent vacancy in diamond has gained widespread interest since the NVC is one of the promising building blocks for room-temperature quantum computation and becomes one of the excellent candidates for QIP owing to extremely long electronic and nuclear spin lifetimes as well as the capability for coherent manipulation in an optical fashion, such as fast initialization, high-fidelity information storage [11–15] and well qubit readout [16–19].

In contrast to the standing modes in the conventional Fabry-Perot cavity, the WGM cavity [20–28], such as the microtoroidal [29, 30], microcylinders [31], microdisks [32, 33], and microspheres [34, 35], could support two counter-propagating traveling modes (also called propagating twin modes: the clockwise (cw) and counter-clockwise (ccw)) with the degenerate frequency and the same field distribution function, and they could couple to each other with a scattering strength due to the scattering of imperfection, which has been observed experimentally [36–39]. In addition, the WGMs in microcavities travel around the curved boundary and are confined by continuous total internal reflection [40], which enables the NVCs in the vicinity of the cavity to interact with the two WGMs via the evanescent field [41, 42]. Therefore, coherently coupling the NVCs to microcavities or even nanocavities offers predominant conditions for reaching strong coupling regime and provides an excellent platform for QIP applications [20, 43–56]. Experimentally, the efficient coupling of a NVC to a microcavity has been demonstrated [35, 57–60]. Recent experimental progresses about the nanocrystal-microsphere system also provide experimental evidence for strong coupling between the NVCs and the WGM of silica microsphere [57] or polystyrene microsphere [58], respectively. Based on the hybrid NVC-cavity systems [43,44], much research has been recently devoted to the generation of entanglement between separated NVCs by coupling to the same quantized cavity mode using a probabilistic way [53,54], deterministic and robust method [52], the Raman transition method or adiabatic passage technique [48–50, 55], dissipation-assisted steady entanglement [51], and the common mediation of the structured environment [56].

Most of these above-mentioned studies are mostly focused on the dynamics of bipartite entanglement measures between a pair of NVCs. The multiqubit entanglement dynamics of the NVCs has not been extensively explored because the multiqubit dynamics itself is particularly complicated and well-accepted measures for entanglement are still missing [61–64]. However, it was generally agreed that entanglement in the systems with more than two parties exhibits a richer dynamical behavior, and more interesting features could be expected [65–69]. In this sense such studies not only provide information of how entanglement evolves in time but also suggest ways toward practical purposes with entanglement [61, 62]. Here another important issue is how to develop efficient methods for generating highly entangled states among multiqubits. Therefore, it would be interesting to conduct theoretical and experimental studies on the dynamics of hybrid systems exhibiting a high degree of multi-partite entanglement, where the multiqubit-entanglement dynamics could be controlled by adjusting the tunable parameters. Meanwhile, the related question arises as to what extent entanglement generation and dynamics in such a multipartite system can be obtained and controlled. Therefore, it is desirable to investigate the preparation and dynamics of the entangled states among several distant NVCs, which are prerequisites for realization of large-scale spin-based quantum networks [12, 13, 70]. Furthermore, the WGMs carry phase factors, which plays a central role in the multi-scatterer system, and the Rayleigh scattering brings significant influences in the entanglement generation and photon transportation [21, 22]. However, there have not been well-done studies so far on the effect of the phase factors and the Rayleigh scattering on the multiqubit-entanglement dynamics. We will study below the relation between these physical mechanism and the entanglement generation, and explore the possibility to obtain maximal entanglement among all the NVCs through tuning the key parameters related to the Rayleigh scattering.

In the present paper we investigate the multiparticle solid-state CQED system consisting of a high-Q microtoroidal cavity and three separated diamond nanocrystals with each containing a single NVC. The dissipative effects associated with both the WGM and NVCs are taken into account, and the entanglement is characterized by using the concept of the lower bound of concurrence (LBC) [61, 62, 71]. Our results indicate that the maximum LBC among all the NVCs could be achieved through accurately adjusting the tunable parameters d, g and Δ, where d is the distance between the separated NVCs, g is the scattering strength between the twin modes of the WGMs, and Δ is the frequency detuning between the NVC and cavity. We also show that the system displays a series of damped oscillations under various experimental situations, where the dynamics of the system reflects the intricate balance and competition between the Jaynes-Cummings (JC) type of NVC-WGM couplings and the Rayleigh scattering process of the WGMs. Our further study reveals that using a realistic microscopic model combined with our detailed analysis could find a way to extract the optimal experimental parameters for maximal entanglement of the NVCs. In addition, we resort to the perspective of normal modes of the microcavity to clarify the physical picture behind the multiqubit-entanglement dynamics. For our purpose, the exact entanglement dynamics of the NVCs under dissipation is investigated analytically using the method of microscopic master equation [72–76], which agrees well with the numerical simulation result using the phenomenological Markovian master equation [77–79]. Therefore, the present system provides a platform to generate multiparticle quantum entanglement among three or more NVCs embedded in distant diamond nanocrystals, which may be another route toward building a distributed QIP architecture based on the increasingly-developed nanoscale solid-state technology.

The remainder of this paper is organized as follows. We present our system architecture in details in Sec. 2. Sec. 3 is devoted to the dynamics evolution of entanglement among the NVCs using two different methods to solve the quantum master equation employed in our model. Finally, we compare the results of both the approaches and conclude our findings in Sec. 4. Some details of deductions can be found in Appendixes.

2. System Hamiltonian

The system under consideration consists of three separated NVCs located on the surface of a microtoroidal cavity, where the two counter-propagating WGMs a_cw and a_ccw with the degenerate frequency ω_c and decay rate κ_m (m = cw,ccw), could couple to each other through the process of Rayleigh scattering [43, 44], and the position of each NVC can be controlled accurately [39, 58, 80, 81], illustrated in Fig. 1. Each NVC (usually treated as an electron spin-1) is negatively charged with two unpaired electrons located at the vacancy, where the introduction of an external magnetic field B₀ along the quantized symmetry axis of the NVC could lift the degeneracy of the levels |³A,m_s = ±1⟩ with an level splitting D_eg [82]. Here we choose $| A_{2} 〉 = (| E_{-}, m_{s} = + 1 〉 + | E_{+}, m_{s} = - 1 〉) / \sqrt{2}$ (one of the six excited states defined by the method of group theory) as the excited state |e⟩ and the ground state |g⟩ is encoded into the state |³A,m_s=−1⟩ with the orbital state |E₀⟩. For |E_±⟩ and |E₀⟩, the angular momentum projection along the NVC axis are ±1 and 0, respectively. According to the total angular momentum conservation, the dipole transition between the excited state and ground state will be accompanied by absorbing or releasing photons with σ⁺ circular polarization [13, 55, 56] and frequency ω₀. For the j_th NVC, the spontaneous emission rate is Γ_j and the pure dephasing rate is γ_j. In our scheme, the WGM is coupled to the transition |g⟩ ↔ |e⟩ with the coupling strength G_j, and the NVCs are fixed and separated by the distance d_ij (between the i-th NVC and j-th NVC). Taking the rotating wave approximation under the condition ω_c, ω₀ ≫G_j, the Hamiltonian of the whole system can be written as (in units of ħ = 1) [20, 23, 43, 44],

\begin{array}{l} H = H_{0} + H_{1} + H_{2}, \\ H_{0} = \sum_{j = 1}^{3} \frac{ω_{0}}{2} σ_{j}^{z} + ω_{c} a_{c w}^{†} a_{c w} + ω_{c} a_{c c w}^{†} a_{c c w}, \\ H_{1} = \sum_{j = 1}^{3} G_{j} (e^{i k d_{1}_{j}} σ_{j}^{+} a_{c w} + e^{- i k d_{1_{j}}} σ_{j}^{+} a_{c c w}) + H . C ., \\ H_{2} = \sum_{j = 1}^{3} g_{j} (a_{c w}^{†} a_{c w} + a_{c c w}^{†} a_{c c w}) + \sum_{j = 1}^{3} g_{j} (e^{2 i k d_{1_{j}}} a_{c c w}^{†} a_{c w} + H . C) ., \end{array}

where d₁₁ = 0,

σ_{j}^{+} = | e_{j} 〉 〈 g_{j} |

and

σ_{j}^{z} = | e_{j} 〉 〈 e_{j} | - | g_{j} 〉 〈 g_{j} |

are usual Pauli operators for the j-th NVC.

a_{m}^{†} (a_{m})

is the creation (annihilation) operator for the WGM with the wave vector k_cw = k_ccw = k. We have considered the phase factors

e^{\pm i k d_{1_{j}}}

because the interaction between the NVC and field varies when the NVC’s location in the field changes. The Hamiltonian H₀ describes the free evolution of the system consisting of the NVCs and WGMs. The Hamiltonian H₁ describes the NVC-WGM dipole interaction with the coherent coupling strength

G_{\max} = G_{j} = μ {[ω_{c} / (2 ε_{0} ε_{s} V_{c})]}^{1 / 2} f_{c} (\vec{r}) (j = 1, 2, 3)

, where

f_{c} (\vec{r}) = | E (\vec{r}) / E_{\max} |

is the normalized field distribution function of the WGMs, V_c is the quantized volumes of the WGM interacting with the NVC, μ = 2.74 × 10⁻²⁹Cm is the dipole moment of the NVC transition, and ε₀(ε_s) is the electric permittivity of the vacuum (surrounding medium) [43, 44]. The Hamiltonian H₂ describes the diamond nanocrystal-induced scattering into the co-propagating (m = m′) or counter-propagating (m ≠ m′) WGM fields with the strengths g_jmm_′. Note that the scattering of light should be taken into consideration when the diamond nanocrystals are coupled to ultrahigh-Q microcavities, because the size of the diamond nanocrystals is much larger than the neutral atoms/ions used in conventional CQED systems. For a subwavelength nanocrystal, we can model the interaction between WGMs and NVC by the Weisskopf-Wigner semi-QED method [38], in which the electric field of the WGM’s m-mode induces a dipole moment at the location of the n-th NVC in terms of the dipole approximation and the polarization is described by

{\vec{P}}_{n, m}

. The interaction energy

- {\vec{P}}_{n, m} \cdot {\vec{E}}_{n, m^{'}}

between m′ and m, with

{\vec{E}}_{n, m^{'}}

the electric field of the m′-mode at the location of the n-th NVC, can be regarded as the scattering-induced interaction by the n-th NVC. Then the scattering-induced coupling strengths can be calculated by the relation

g_{j m m^{'}} = α f_{c}^{2} (\vec{r}) ω_{c} / (2 V_{c})

in the case of the elastic Rayleigh scattering, where α = 4πR³ (ε_d − ε_s)/(ε_d + 2ε_s) is the polarizability for the spherical diamond nanocrystal with the electric permittivity ε_d = 2.4² and the radius R [43, 44]. Here we regard the NVCs as entities which are individually coupled to the WGM with the identical strengths G_j = G, and the scattering-induced coupling coefficients g_j are also assumed to be equal for simplicity, namely, g_j = g. One can find that the scattering-induced coupling coefficient g grows rapidly with the increase of R, and there is a phase interference induced by different distances d₁₂ (d₁₃) between NVC1 and NVC2 (NVC3) in the present multi-scatterer case, which is different from the single-scatterer case.

Fig. 1 (a) Schematic of the system consisting of three NVCs and a microtoroidal cavity supporting two counter-propagating WGMs a_cw and a_ccw. (b) Level diagram for the j-th NVC, where Δ_j is the frequency detuning of the NVC from the cavity. D_eg = γ_eB₀ is the level splitting induced by an external magnetic field B₀ with γ_e the electron gyromagnetic ratio.

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3. Quantum dynamics of entanglement among three NVCs

In this section, we investigate the dynamics evolution of entanglement among the NVCs, where the main dissipative effects originating from the leakage of the WGM photons, the spontaneous emission and pure dephasing of the excited-state of NVCs are taken into account by two different methods, respectively. One of the methods analytically uses the microscopic master equation [72–76], which goes back to the ideas of Davies on how to describe the system–reservoir interactions in Markovian master equations [77, 78]. This method (called the microscopic case) focuses on the quantum jumps between the eigenstates of the system Hamiltonian rather than the eigenstates of the field-free subsystems. The other (called the phenomenological case) exploring the dynamics of the system numerically uses the phenomenological master equation with a dissipative (Lindblad) term, which is employed as one of the major approaches in quantum optics and QIP (See the Appendix A).

Based on the above-mentioned two methods, we investigate the dynamics of entanglement among three NVCs. As we all know, for a general pure tripartite state $| φ 〉 \in H_{1} \otimes H_{2} \otimes H_{3}$ , the concurrence can be expressed as [83, 84]

C_{3} (| φ 〉) = \sqrt{\frac{1}{3} [3 - Tr (ρ_{1}^{2}) - Tr (ρ_{2}^{2}) - Tr (ρ_{3}^{2})]},

and ρ_i is the reduced density matrix obtained by tracing over the remaining subsystem j and k (i, j, k = 1, 2, 3 and i ≠ j ≠ k). Then it can be extended to the tripartite mixed state ρ by the convex roof [85, 86],

C_{3} (ρ) = \min_{{p_{i}, φ_{i}}} \sum_{i} p_{i} C_{3} (| φ_{i} 〉)

where {|φ_i⟩} denotes the collection of all the possible pure states into which the mixed state ρ can be decomposed as ρ = ∑_i p_i |φ_i⟩ ⟨φ_i| with the positive p_i (normalized already). However, it is indeed a difficult task to find the minimum because the optimization procedure involves a large number of free parameters. What’s worse, no numerical algorithm could guarantee to find the global minimum. Thus, to avoid finding the exact minimum, the lower bound to this measure is available [61–63, 71] with the following definition

{\underline{C}}_{3} (ρ) = \sqrt{\frac{1}{3} \sum_{j = 1}^{6} [{(C_{j}^{12 | 3} (ρ))}^{2} + {(C_{j}^{31 | 2} (ρ))}^{2} + {(C_{j}^{23 | 1} (ρ))}^{2}]},

C_{j}^{k l | m} (ρ) = \max {0, \sqrt{λ_{j, 1}^{k l | m}} - \sum_{n > 1} \sqrt{λ_{j, n}^{k l | m}}},

where

λ_{j, n}^{k l | m}

represents the eigenvalues of the matrix

\tilde{ρ} = ρ (L_{j}^{k l} \otimes σ_{y}^{m}) ρ * (L_{j}^{k l} \otimes σ_{y}^{m})

in a decreasing order with

L_{j}^{k l} (j = 1, 2, \dots 6)

the six generators of the group SO(4) [87] acting on the qubits k,l and

σ_{y}^{m}

the Pauli matrix acting on the qubit m. If

C_{3} (ρ) = 0

, then the state ρ is separable. But, if

{\underline{C}}_{3} (ρ) = 0

, it does not necessarily mean it is separable. Without doubt,

{\underline{C}}_{3} (ρ) > 0

implies an entangled state and a separable state always results in

{\underline{C}}_{3} (ρ) = 0

. We may calculate numerically the LBC

{\underline{C}}_{3} {(ρ)}_{n u m}

for ρ_num (derived from the phenomenological case), and we may also obtain the analytical expression of the LBC for ρ_ana (derived from the microscopic case) written as

{\underline{C}}_{3} {(ρ)}_{a n a} = \max {0, \sqrt{\frac{8}{5} (ρ_{e g g, e g g} ρ_{g e g, g e g} + ρ_{e g g, e g g} ρ_{g g e, g g e} + ρ_{g e g, g e g} ρ_{g g e, g g e})}} .

Here the LBC can be approximately determined by the analytical expression (Eq. (6)) with respect to the three key time-dependent density matrix elements ρ_egg,egg, ρ_gge,gge, and ρ_geg,geg. One can find that the quantum dynamics based on the analytical method agrees well with the numerical simulation by the phenomenological case, as shown later.

More specifically, we investigate the dynamics of entanglement among the NVCs under the resonant case and detuning case, respectively, via the approach of microscopic master equation. Here we consider two different initial states of the system: (1) all the NVCs are in the ground state and one photon is prepared in one of the twin modes, i.e., ψ(0) = |g₁,g₂,g₃⟩|1_cw,0_ccw⟩; (2) the first NVC is in the excited state, and the other two NVCs are in the ground state, without any photons in the WGM, i.e., Ψ(0) = |e₁,g₂,g₃⟩|0_cw,0_ccw⟩.

3.1. The resonant case

The distinct advantages of the present system, such as the individual accessibility and high tunability of the parameters, make this NVC-WGM system an ideal tunable structure, where the scattering-induced coupling rates can be regulated by the distance between the NVCs or the radius of the diamond nanocrystal, and the dipole interaction between the NVC and the WGM can also be adjusted by changing the relative position of the NVC to the microcavity surface. Figure 2 shows the dependence of the LBC dynamics on the distances (d₁₂ and d₁₃) or the phase differences (θ₁₂ = kd₁₂ and θ₁₃ = kd₁₃) between any pair of NVCs in the resonant case (Δ = 0), where we assume that the distance between NVC1 and NVC2 is fixed at $\frac{π}{2} + n π$ or nπ, and the position of NVC3 is flexible. As shown in Fig. 2, the phase interference induced by the spatial distance d_ij between different NVCs leads to profile/frequency variations of the oscillation and the maximum values of the LBC. For a given value of θ₁₂, as shown in Fig. 2, the behavior of the LBC shows periodic evolution with the value of θ₁₃ changing; and for a given value of θ₁₃, the LBC oscillates between zero and maximum due to the Rabi oscillations between the NVC and the quantized WGM field, the amplitude of which is decaying in time arise from the dissipation effects.

Fig. 2 The LBC vs time t and the phase difference θ₁₃ when the system is initially prepared in the state ψ(0). (a) $θ_{12} = \frac{π}{2} + n π$ , (b) θ₁₂ = nπ. The time parameter is dimensionless and we set G = 1. Other parameters are Δ = 0, g = 0.1, Γ = 0.05, γ = 0.01 and κ = 0.03.

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To get a clear picture of how the LBC evolves in the space of phase difference {θ₁₂, θ₁₃}, we plot the time evolution of the LBC in Fig. 3, where an interesting feature is that entanglement of the NVCs shows a periodic oscillation behavior along both the θ₁₂ axis and the θ₁₃ axis, and the period of the maximal value of the LBC is π. Furthermore, the period of the LBC becomes π/2 along the lines of θ₁₂ = (n+1)π/2 or θ₁₃ = (n+1)π/2, due to the fact that each NVC exerts the same influence on the two WGMs when θ₁₂ = (n+1)π/2 or θ₁₃ = (n+1)π/2. Therefore, the values of LBC are strongly dependent on the interposition of the NVCs, and a subwavelength displacement could result in an evident change of the entanglement. From another point of view, this promises the possibility of utilizing the position-sensitive response to realize the desired maximal entanglement of NVCs, i.e., high-degree LBC of the NVCs could be obtained by independently adjusting the inter-qubit distance so that some special values of phase differences can be taken. Figure 3(b) shows that the maximum of the LBC can reach 0.92 with an optimal choice of θ₁₂ = 33π/50 + nπ and θ₁₃ = 7π/25 + nπ, where the three NVCs are placed with almost equal intervals.

Fig. 3 (a) The maximal values of the LBC in the parameter plane of {θ₁₂, θ₁₃} with Δ = 0, G = 1, g = 0.1, Γ = 0.05, γ = 0.01 and κ = 0.03, where the system is initially prepared in the state ψ(0). (b) The zooming-in plot of (a) with one period.

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From the corresponding eigenvectors |ϕ⟩_i of Eq. (10), we see that the time-dependent superposition of five harmonic wave functions ultimately determines the single-excitation dynamics of the system, and thereby the overall dynamics exhibits some complex interferences or competing effects. The remarkable result is the rich entanglement dynamics of the system reflecting the intricate balance and competition between the two different kinds of couplings. One of such couplings is the JC-type interaction between each NVC and WGM with the coupling strength G, and the other is the coupling between the WGMs induced by the Rayleigh scattering of the NVC with the coupling strength g, where these two tunable parameters (G and g) influence the system dynamics in different ways. Therefore the dynamical feature of the LBC can be characterized by two kinds of oscillation behavior with two periods depending on the related coupling strengths. When the system is initially prepared in the state ψ(0), the concrete physical process can be described in more details by following image. The excitation is initially located in the clockwise WGM a_cw: The energy is exchanged between three NVCs and a_cw with a rate G/2π. Then the excitation gradually flows to the counter-clockwise WGM a_ccw with a rate g/2π through the Rayleigh scattering mechanism during the JC oscillation between the NVC and a_cw. At this stage, a similar JC oscillation between the NVC and the mode a_ccw also occurs. As a result, three NVCs achieve a time-dependent entanglement during the system evolution.

An important aspect of the present system is the ability to tune the Rayleigh scattering strength. Figure 4 displays the influence from the Rayleigh scattering on the entanglement dynamics for different initial states ψ(0) and Ψ(0), respectively, where the phase difference is fixed. To better understand the physics underlying these figures, we consider some illustrations on following three different regimes. The first one is called the strong coupling regime with G ≫ g, where the NVC-WGM dipole interaction is much larger than the Rayleigh scattering, implying that the NVC-WGM interface dominates the single-excitation evolution process. Thus the magnitude of large coupling strength G determines the relatively large speed of the energy-exchange between the NVC and the WGM. On the other hand, the excitation will ultimately tunnel between the twin modes a_cw and a_ccw with a relatively smaller speed. The second one is the strong scattering regime with G ≪ g, where the excitation-transfer between the twin modes dominates the evolution process. In between, lies the third one called the competition regime with G ≈ g, where the coupling strength is comparable to the scattering strength. From Fig. 4(a1) one can find that the large degree LBC can be achieved at some appropriate time points when the system is within the strong coupling regime and initialized in the state ψ(0). With the growth of the values of g, as shown in Fig. 4(a2) and Fig. 4(a3), the dynamics will exhibit two remarkable characters: the maximum of LBC gradually decreases and the LBC among the NVCs oscillates faster. The physical picture behind is that the excitation mainly transmits between the two twin modes a_cw and a_ccw of the WGMs in the strong scattering regime. Therefore, both the maximal LBC among the NVCs and the periodic time inevitably reduce when g increases since the major populations with respect to the excitation almost resides in the twin modes, rather than the NVCs. The situation becomes different when the system is prepared in the state Ψ(0). Figures 4(b1–b3) indicate that the scattering process plays a positive role in the generation of entanglement of the NVCs, similar to the conclusions in the previous work [20–22, 24]. The overall dynamics of the system is characterized by a fast oscillation behavior accompanied by a slow oscillation envelope, as shown in Fig. 4(b3).

Fig. 4 The LBC vs time t and the coupling strength g for different initial states of the system (a) ψ(0) and (b) Ψ(0). Other parameters are $θ_{12} = \frac{π}{4} + n π$ , $θ_{13} = \frac{π}{8} + n π$ , Δ = 0, G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03. In the bottom six subgraphs, the curves of the LBC dynamics are plotted in (a1,b1) g = 0.1, (a2,b2) g = 1, (a3,b3) g = 5.

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3.2. The detuning case

We now turn our attention to the entanglement dynamics in the detuning case of Δ ≠ 0. We plot the dependence of the LBC on the phase difference θ₁₃ (in Figs. 5(a,c)), and the scattering-induced coupling strength g (in Figs. 5(b,d)) when the system is initially prepared in the state ψ(0) under different detunings Δ, respectively. Compared with Fig. 2(b), Figures 5(a,c) presents the remaining periodicity of the LBC with respect to θ₁₃. The maximal LBC decreases greatly and the oscillation frequency increases when the value of Δ increases. Moreover, in comparison with Fig. 4(a), there is a slight dropping of the maximal value of the LBC in Figs. 5(b,d) caused by the growth of Δ. Another interesting feature is that the optimal coupling strength g_op corresponding to the maximal LBC augments with the growth of Δ. It implies that the LBC evolution can be optimized by appropriately tailoring those key parameters. This provides a useful and effective way to controlling the dynamics of entanglement among the NVCs.

Fig. 5 The LBC vs time t and the phase difference θ₁₃ under different detunings (a) Δ = 5 and (c) Δ = 10, where θ₁₂ = nπ, G = 1, g = 0.1, Γ = 0.05, γ = 0.01 and κ = 0.03. The LBC vs time t and the coupling strength g under different detunings (b) Δ = 5 and (d) Δ = 10, where $θ_{12} = \frac{π}{4} π$ , $θ_{13} = \frac{π}{8} + n π$ , G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03.

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Figure 6 displays the time-dependent LBC on the detuning Δ under three different regimes with the fixed phase differences θ₁₂ and θ₁₃. In the strong coupling regime in Fig. 6(a), the large degree LBC can be obtained at a specific time, and the increase of the detuning Δ destroys the entanglement of the NVCs. When the coupling strength is comparable to the scattering strength, i.e., G = g, as shown in Fig. 6(b), the intricate balance and competition between these two kinds of couplings are more obvious than the case in Figs. 6(a,c). In the strong scattering regime (Fig. 6(c)), the maximal value of the LBC drops and the optimized values of Δ for the maximal entanglement are increased in comparison with in Figs. 6(a,b) which is consistent with the features in Figs. 5(b,d).

Fig. 6 The LBC vs time t and the detuning Δ for different scattering strengths (a) g = 0.1, (b) g = 1, (c) g = 5. The other parameters are $θ_{12} = \frac{π}{4} + n π$ , $θ_{13} = \frac{π}{8} + n π$ , G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03.

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In order to clarify the relation between the detuning Δ and the coupling strength g as well as the entanglement dynamics itself, we resort to the perspective of normal modes of the micro-cavity, i.e., the use of the normal modes $a_{1} = (a_{c w} + a_{c c w}) / \sqrt{2}$ and $a_{2} = (a_{c w} - a_{c c w}) / \sqrt{2}$ to replace the cavity modes a_cw and a_ccw. Then the total Hamiltonian is transformed to

\begin{matrix} H = \sum_{j = 1}^{3} [ω_{0} σ_{e_{j} e_{j}} + δ_{1} a_{1}^{†} a_{1} + δ_{2} a_{2}^{†} a_{2} + {\tilde{G}}_{1 j} (a_{1}^{†} σ_{j}^{-} + a_{1} σ_{j}^{+})] \\ + \sum_{j = 2}^{3} {\tilde{G}}_{2 j} (a_{2}^{†} σ_{j}^{-} + a_{2} σ_{j}^{+}) + \tilde{g} (a_{1}^{†} a_{2} - a_{2}^{†} a_{1}) \end{matrix}

where

{\tilde{G}}_{i j}

is the effective coherent coupling strength between the i-th normal mode a_i and the j-th NVC with

{\tilde{G}}_{11} = \sqrt{2} G

,

{\tilde{G}}_{12} = \sqrt{2} G \cos (k d_{12})

,

{\tilde{G}}_{13} = \sqrt{2} G \cos (k d_{13})

,

{\tilde{G}}_{22} - i \sqrt{2} G \sin (k d_{12})

, and

{\tilde{G}}_{23} = - i \sqrt{2} G \sin (k d_{13})

. Eq. (7) indicates that three NVCs simultaneously couple to the normal mode with frequency δ₁ = ω_c + 2g [1 + cos² (kd₁₂) + cos² (kd₁₃)], but only NVC2 and NVC3 couple to the normal mode a₂ with frequency δ₂ = ω_c + 2gsin² (kd₁₂) + sin² (kd₁₃). These two normal modes couple to each other with the effective scattering-induced coupling strength

\tilde{g} = i g (\sin (2 k d_{12}) + \sin (2 k d_{13}))

. One can find that the system is equivalent to the three NVCs coupled to a single normal mode a₁ with frequency ω_c + 6g in the case of kd₁₂ = kd₁₃ = nπ since the normal mode a₂ decouples to all NVCs.

In the detuning case, three NVCs couple to the normal mode a₁ with frequency ${δ^{'}}_{1} = ω_{0} - Δ + 2 g [1 + \cos^{2} (k d_{12}) + \cos^{2} (k d_{13})]$ . NVC2 and NVC3 couple to the normal mode a₂ with frequency ${δ^{'}}_{2} = ω_{0} - Δ + 2 g (\sin^{2} (k d_{12}) + \sin^{2} (k d_{13}))$ . When kd₁₂ = kd₁₃ = nπ, the normal mode a₂ decouples to all NVCs and the frequency of the normal mode a₁ becomes ω₀ − Δ+6g. The modes a₁ and a₂ are decoupled. Thus entanglement of the NVCs turns to be maximal once the condition Δ = 6g is well satisfied. To check whether this conclusion is correct, the density plots of the LBC versus Δ and g with kd₁₂ = kd₁₃ = nπ is presented in Fig. 7(a), where the maximal LBC in or near the brightest line Δ = 6g is larger than in other regions. But the situation becomes more complicated if kd₁₂ = kd₁₃ ≠ nπ, where the normal mode a₂ is not decoupled to all NVCs, and thus it is hard to get an accurate relation between Δ and g. Here we plot the time evolution of the LBC under three different cases respectively: (i) $k d_{12} = \frac{π}{2} + n π$ , $k d_{13} = \frac{π}{4} + n π$ , for which the resonance condition between the normal modes a₁₍₂₎ and NVC is Δ = 3g, and the coupling strength between a₁ and a₂ is g. The results can be found in Fig. 7(b). (ii) $k d_{12} = \frac{π}{4} + n π$ , $k d_{13} = \frac{π}{8} + n π$ , where the resonance condition between the normal mode a₁ and the NVCs is Δ = 4.7g, the resonance condition between the normal mode a₂ and NVC2 (NVC3) is Δ = 1.3g, and the coupling strength between a₁ and a₂ is 1.7g. As plotted in Fig. 7(c), the large frequency difference between a₁ and a₂ splits the brightest line into two lines along different direction. (iii) $k d_{12} = \frac{3 π}{4} + n π$ , $k d_{13} = \frac{3 π}{8} + n π$ , where the resonance condition between the normal mode a₁ and NVCs is Δ = 3.3g, the resonance condition between the normal mode a₂ and NVC2 (NVC3) is Δ = 2.7g, and the coupling strength between a₁ and a₂ is −0.3g (See Fig. 7(d)).

Fig. 7 The maximal LBC vs the coupling strength g and the detuning Δ in (a) θ₁₂ = θ₁₃ = nπ, (b) $θ_{12} = \frac{π}{2} + n π$ and $θ_{13} = \frac{π}{4} + n π$ , (c) $θ_{12} = \frac{π}{4} + n π$ and $θ_{13} = \frac{π}{8} + n π$ , (d) $θ_{12} = \frac{3 π}{4} + n π$ and $θ_{13} = \frac{3 π}{8} + n π$ . Other parameters are G = 1, Γ = 0.05, γ = 0.01 and κ = 0.03.

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4. Discussion and conclusion

We first compare the analytical results from the microscopic master equation with the results of numerical calculation using the phenomenological master equation. As shown in the Fig. 8 (a), the LBC dynamics obtained by two methods are almost completely coincident. We further define a new function $δ_{L B C} = {\underline{C}}_{3} {(ρ)}_{a n a} - {\underline{C}}_{3} {(ρ)}_{n u n}$ to visualize the slight difference between these two different methods, as shown in Fig. 8(b). In fact, the numerical simulation agrees well with the calculation result in Sec. 3.

Fig. 8 (a) The LBC vs time t under the condition $θ_{12} = \frac{π}{4}$ , $θ_{13} = \frac{π}{8}$ , Δ = 0, g = 2, where the red-solid (blue-dashed) lines denote the analytical (numerical) results, respectively. (b) The slight difference between the LBC dynamics using these two different methods.

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Next we survey the relevant experimental parameters. Under practical situations, the transition of the NVCs at the wave-length 637 nm has a decay rate with Γ/2π = 13 MHz and a pure dephasing rate with γ/2π = 2 ~ 11 MHz [50, 88]. Due to the cavity coupling that enhances the photonic emission into the zero-phonon line, we consider V_c ~ 200 μm³, and $f_{c} (\vec{r}) = 0.47$ , yielding the maximal coupling strength G_max/2π ≈ 180 MHz [33, 43, 44, 59]. In addition, for a WGM cavity with the quality factor Q = 10⁸, we can obtain the related cavity decay rate κ/2π = ω_c/2πQ = 4.7 MHz, which is about 0.03G_max [43, 44]. For the NVC, the electron spin relaxation time T₁ of the diamond NVC ranges from 6 ms at room temperature to seconds at cryogenic temperature in realistic experiments. The dephasing time T₂ = 350 μs induced by the nuclear spin fluctuation inside the NVC has been reported [89]. A latest experimental progress [90] with isotopically pure diamond sample has demonstrated a longer dephasing time, i.e., T₂ = 2 ms. This implies that the influence from the intrinsic damping and dephasing of the NVCs is possibly negligible in the specially treated NVC-WGM system.

In summary, we have investigated the entanglement dynamics of three NVCs coupled to a WGM cavity supporting counter-propagating twin modes. We have shown that the system displays a series of damped oscillations, which reflects the intricate balance and competition between the JC type NVC-WGM coupling and the WGM Rayleigh scattering. Our study has also provided a way to extracting the optimal experimental parameters for maximal entanglement of the NVCs. Moreover, straightforward extension of our idea to more than three NVCs is possible. The good agreement between the analytical and numerical results made our results credible and convincing. Therefore, the present study are valuable for building the future full-scale quantum information processor based on the increasingly-developed nanoscale solid-state technology.

APPENDIX A: The quantum master equation

In this section we present the details of our treatments for the quantum master equation by both microscopic and phenomenological considerations. The comparison between the two treatments can make our results more credible, and particularly the analytical expressions help further understand the physics in entanglement.

4.1. The microscopic case

We first describe the microscopic case using the method of the microscopic master equation. For this system described by the Hamiltonian H, there exist two invariant subspaces spanned by the following basic vectors

\begin{matrix} \forall_{1} \in {| 1 〉 = | e_{1} g_{2} g_{3} 0_{c w} 0_{c c w} 〉, \\ | 2 〉 = | g_{1} e_{2} g_{3} 0_{c w} 0_{c c w} 〉, \\ | 3 〉 = | g_{1} g_{2} e_{3} 0_{c w} 0_{c c w} 〉, \\ | 4 〉 = | g_{1} g_{2} g_{3} 1_{c w} 0_{c c w} 〉, \\ | 5 〉 = | g_{1} g_{2} g_{3} 0_{c w} 1_{c c w} 〉}, \end{matrix}

\forall_{2} \in {| 6 〉 = | g_{1} g_{2} g_{3} 0_{c w} 0_{c c w} 〉},

where g_j (e_j) denotes the ground (excited) state of the j-th NVC, 0_cw₍_ccw₎ (1_cw₍_ccw₎) denotes the vacuum state (one-photon Fock state) of the cw (ccw) WGM. One can find that the ground state |6⟩ is a dark state, i.e., with no evolution in time. In addition, the system initially in the single-excitation manifold ∀₁ will remain in that subspace. With these basic vectors, one can rewrite the Hamiltonian H in a matrix with less dimensions as follows,

H = (\begin{matrix} - \frac{ω_{0}}{2} & 0 & 0 & G & G & 0 \\ 0 & - \frac{ω_{0}}{2} & 0 & e^{i k d_{12} G} & e^{- i k d_{12} G} & 0 \\ 0 & 0 & - \frac{ω_{0}}{2} & e^{i k d_{13} G} & e^{- i k d_{13} G} & 0 \\ G & e^{- i k d_{12} G} & e^{- i k d_{13} G} & - \frac{3 ω_{0}}{2} + ω_{c} + 3 g & g (1 + e^{- 2 i k d_{12}} + e^{- 2 i k d_{13}}) & 0 \\ G & e^{i k d_{12} G} & e^{i k d_{13} G} & g (1 + e^{2 i k d_{12}} + e^{2 i k d_{13}}) & - \frac{3 ω_{0}}{2} + ω_{c} + 3 g & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{3 ω_{0}}{2} \end{matrix})

Here the corresponding eigenvectors of H can be expressed by $| ϕ_{i} 〉 = \sum_{j = 1}^{5} c_{i j} | j 〉$ with i = 1,2,3,4,5, and |ϕ₆⟩ = |6⟩, and the forms of the coefficients c_ij are presented in Appendix B.

For this NVC-WGM system at zero temperature, the master equation for the density operator ρ(t) is given by

\begin{array}{l} \dot{ρ} (t) = - i [H, ρ] + \sum_{\bar{ω} > 0, m = {c w, c c w}} κ_{m} (\bar{ω}) \times [A_{m} (\bar{ω}) ρ (t) A_{m}^{†} (\bar{ω}) - \frac{1}{2} {A_{m}^{†} (\bar{ω}) A_{m} (\bar{ω}), ρ (t)}] \\ + \sum_{\bar{ω} > 0, n = {1, 2, 3}} Γ_{n} (\bar{ω}) \times [\sum_{n}^{-} (\bar{ω}) ρ (t) \sum_{n}^{+} (\bar{ω}) - \frac{1}{2} {\sum_{n}^{+} (\bar{ω}) \sum_{n}^{-} (\bar{ω}), ρ (t)}] \\ + \sum_{\bar{ω} > 0, n = {1, 2, 3}} γ_{n} (\bar{ω}) \times [\sum_{n}^{z} (\bar{ω}) ρ (t) \sum_{n}^{z} (\bar{ω}) - \frac{1}{2} {\sum_{n}^{z} (\bar{ω}) \sum_{n}^{z} (\bar{ω}), ρ (t)}], \end{array}

where

A_{m} (\bar{ω})

and

\sum_{n}^{-} (\bar{ω})

are the Davies operators given by

A_{m} (\bar{ω}) = | ϕ_{i} 〉 〈 ϕ_{i} | A_{m} | ϕ_{j} 〉 〈 ϕ_{j} |

with

A_{m} = a_{m} + a_{m}^{†}

,

\sum_{n}^{-} (\bar{ω}) = | ϕ_{i} 〉 〈 ϕ_{i} | \sum_{n}^{-} | ϕ_{j} 〉 〈 ϕ_{j} |

with

\sum_{n}^{-} = σ_{n}^{-} + σ_{n}^{+}

and

\sum_{n}^{z} (\bar{ω}) = \sum_{n}^{+} (\bar{ω}) \sum_{n}^{-} (\bar{ω}) - \sum_{n}^{-} (\bar{ω}) \sum_{n}^{+} (\bar{ω})

. And

\bar{ω} = λ_{j} - λ_{i}

must be non-negative with λ_i the eigenvalues of H. The sum on m and n is over all the dissipation channels and the dissipative parameters

κ_{m} (\bar{ω})

,

Γ_{n} (\bar{ω})

and

γ_{n} (\bar{ω})

are the Fourier transform of the correlation functions of the environment [79]. Under the condition of the single excitation, we find that only the terms such as

〈 6 | σ_{1}^{-} | 1 〉

,

〈 6 | σ_{2}^{-} | 2 〉

,

〈 6 | σ_{3}^{-} | 3 〉

,

〈 6 | a_{c w} | 4 〉

,

〈 6 | a_{c c w} | 5 〉

and their Hermitian conjugations are non-zero. Therefore, we can use these Davies operators to simplify the master equation (Eq. (11)) as

\begin{array}{l} \dot{ρ} (t) = - i [H, ρ] + \sum_{i = 1}^{5} {\tilde{Γ}}_{i} [| ϕ_{6} 〉 〈 ϕ_{i} | ρ (t) | ϕ_{i} 〉 〈 ϕ_{6} | - \frac{1}{2} {| ϕ_{i} 〉 〈 ϕ_{i} |, ρ (t)}] \\ + \sum_{i = 1}^{5} {\tilde{γ}}_{i} [(| ϕ_{i} 〉 〈 ϕ_{i} | - | ϕ_{6} 〉 〈 ϕ_{6} |) ρ (t) (| ϕ_{i} 〉 〈 ϕ_{i} | - | ϕ_{6} 〉 〈 ϕ_{6} |) \\ - \frac{1}{2} {(| ϕ_{i} 〉 〈 ϕ_{i} | + | ϕ_{6} 〉 〈 ϕ_{6} |), ρ (t)}], \end{array}

where the definition of the parameters

{\tilde{Γ}}_{i}

and

{\tilde{γ}}_{i} (1, 2, 3, 4, 5)

and some other details of our deductions are presented in Appendix B.

Now we can calculate the time-dependent density matrix of the total system by solving a set of differential equations under the given initial state |ψ(0)⟩. For uniformity, we should transform the initial state into the representation of the Hamiltonian H through the unitary transformation operator $\tilde{U}$ with

\tilde{U} = (\begin{matrix} c_{11} & c_{12} & c_{13} & c_{14} & c_{15} & 0 \\ c_{21} & c_{22} & c_{23} & c_{24} & c_{25} & 0 \\ c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & 0 \\ c_{41} & c_{42} & c_{43} & c_{44} & c_{45} & 0 \\ c_{51} & c_{52} & c_{53} & c_{54} & c_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) .

After tracing out the freedom degree of the WGM and transforming back to the original basis {|e⟩, |g⟩}, we can obtain the reduced density matrix ρ_ana(t) with respect to the three NVCs. The detailed derivation is presented below in Appendix B.

4.2. The phenomenological case

The entanglement dynamics of the NVCs can also be numerically explored using the phenomenological master equation. Different from the microscopic case, the calculation in the phenomenological case is not made in the eigenstates of the system Hamiltonian, but in a much larger space whose dimension grows exponentially with the number of qubits.

In order to model this open (dissipative) system, coupling to the environment (in the limit of weak coupling to a Markovian bath) can be considered in a phenomenological master equation with a dissipative (Lindblad) term on the right-hand side

\dot{ρ} = - i [H_{e f f,} ρ] + ℒ ρ,

where

\begin{matrix} ℒ ρ = \sum_{m = c w, c c w} κ_{m} (a_{m} ρ a_{m}^{+} - \frac{1}{2} a_{m}^{+} a_{m} ρ - \frac{1}{2} ρ a_{m}^{+} a_{m}) \\ + \sum_{j = 1}^{3} Γ_{j} (σ_{j}^{-} ρ σ_{j}^{+} - \frac{1}{2} σ_{j}^{+} σ_{j}^{-} ρ - \frac{1}{2} ρ σ_{j}^{+} σ_{j}^{-}) \\ + \sum_{j = 1}^{3} γ_{j} (σ_{j}^{z} ρ σ_{j}^{z} - \frac{1}{2} σ_{j}^{z} σ_{j}^{z} ρ - \frac{1}{2} ρ σ_{j}^{z} σ_{j}^{z}) . \end{matrix}

Here κ_m is the damping rate of the twin modes of the cavity, and Γ_j and γ_j are the spontaneous emission rate and pure dephasing rate of the j-th NVC respectively. For simplicity, we set κ_cw = κ_ccw = κ, Γ₁ = Γ₂ = Γ₃ = Γ and γ₁ = γ₂ = γ₃ = γ in the following calculation. In Eq. (14), the Hamiltonian H_eff in the interacting picture with the unitary transformation $U = e^{- i H_{0} t}$ reads

\begin{array}{l} H_{e f f} = \sum_{j = 1}^{3} [e^{i Δ t} G_{j} (e^{i k d_{1 j}} σ_{j}^{+} a_{c w} + e^{- i k d_{1 j}} σ_{j}^{+} a_{c c w}) + g_{j} e^{2 i k d_{1 j}} a_{c c w}^{†} a_{c c w} + H . C] \\ + \sum_{j = 1}^{3} g_{j} (a_{c w}^{†} a_{c w} + a_{c c w}^{†} a_{c c w}), \end{array}

where Δ = ω₀ − ω_c is the detuning between the WGM mode and the transition of the NVC.

In order to study the dynamics of entanglement among all the NVCs, we numerically solve Eq. (14) with the given initial state |ψ(0)⟩, and then trace over the twin modes of the cavity to obtain the time-dependent reduced density matrix ρ_num of the NVCs.

APPENDIX B: Analytical solutions to the microscopic master equation

In this appendix, we give detailed deductions for the analytical solutions to the microscopic master equation.

Firstly, we use Eqs. (8) and (9) to rewrite the Hamiltonian H into the matrix form shown in Eq. (10), and we can obtain the corresponding eigenvalues λ₁ = 0, $λ_{i} = e^{- 2 i k d_{12} - 2 i k d_{13}} A_{i - 1}$ , and λ₆ = −ω₀ with i = 2,3,4,5. The corresponding eigenvectors of H (Eq. (10)) can be expressed by $| ϕ_{i} 〉 = \sum_{j = 1}^{5} c_{i j} | j 〉$ and |φ₆⟩ = |6⟩ with the forms of the coefficients c_ij as

\begin{array}{l} c_{11} = e^{- i k d_{13}} (e^{2 i k d_{13}} - e^{2 i k d_{12}}) / (e^{2 i k d_{12}} - 1), c_{12} = e^{i k d_{12} -}^{i k d_{13}} (1 - e^{2 i k d_{13}}) / (e^{2 i k d_{12}} - 1), \\ c_{i 1} = \frac{G {ξ [(e^{2 i k d_{12}} + e^{2 i k d_{13}} - 2) η + Δ A_{i - 1}] + A_{i - 1}^{2}}}{q A_{i - 1}}, \\ c_{i 2} = \frac{e^{i k d_{12}} G {ξ [(e^{- 2 i k d_{12}} + e^{2 i k d_{13}} - 2) η + Δ A_{i - 1}] + A_{i - 1}^{2}}}{q A_{i - 1}}, \\ c_{i 3} = \frac{e^{i k d_{13}} G {ξ [(e^{- 2 i k d_{13}} + e^{2 i k d_{12}} - 2) η + Δ A_{i - 1}] + A_{i - 1}^{2}}}{q A_{i - 1}}, \\ c_{i 4} = - {ξ [3 η - Δ A_{i - 1}] - A_{i - 1}^{2}} / ξ q, c_{14} = c_{15} = c_{16} = c_{i 6} = 0, c_{13} = c_{i 5} = 1, \end{array}

where i = 2,3,4,

ξ = e^{2 i k d_{12} + 2 i k d_{13}}

,

η = e^{2 i k d_{12} + 2 i k d_{13}} G^{2} + g A_{i - 1}

, and

q = (1 + e^{2 i k d_{12}} + e^{2 i k d_{13}}) (e^{2 i k d_{12} + 2 i k d_{13}} G^{2} + g A_{i - 1})

. Here A₁, A₂, A₃, A₄ are the four roots of the equation

x^{4} + a x^{3} + b x^{2} + c x + d = 0,

with a = 2ξ (Δ − 3g), b = ξ² [2εg² − 6G² − 6gΔ + Δ²], c = 2ξ³G² [2εg − 3Δ], d = 2ξ⁴εG⁴ and ε = 3 − cos(2kd₁₂) − cos(2kd₁₃) − cos (2kd₁₂ − 2kd₁₃).

In order to solve the microscopic master equation Eq. (11), we rewrite the master equation with eigenvalues and eigenvectors of the Hamiltonian H. By calculation we obtain

A_{c w} (λ_{i} - λ_{6}) = c_{i 4} | ϕ_{6} 〉 〈 ϕ_{i} |, A_{c c w} (λ_{i} - λ_{6}) = c_{i 5} | ϕ_{6} 〉 〈 ϕ_{i} |,

and

\begin{array}{l} \sum_{1}^{-} (λ_{i} - λ_{6}) = c_{i 1} | ϕ_{6} 〉 〈 ϕ_{i} |, \\ \sum_{2}^{-} (λ_{i} - λ_{6}) = c_{i 2} | ϕ_{6} 〉 〈 ϕ_{i} |, \\ \sum_{3}^{-} (λ_{i} - λ_{6}) = c_{i 3} | ϕ_{6} 〉 〈 ϕ_{i} | . \end{array}

Substituting Eqs. (19)–(20) into Eq. (11), the master equation can be rewritten as Eq. (12) with

\begin{array}{l} {\tilde{Γ}}_{i} = \sum_{j = 1}^{3} | c_{i j} |^{2} Γ_{j} (λ_{i} - λ_{6}) + | c_{i 4} |^{2} γ_{c w} (λ_{i} - λ_{6}) + | c_{i 5} |^{2} γ_{c c w} (λ_{i} - λ_{6}), \\ {\tilde{γ}}_{i} = \sum_{j = 1}^{3} | c_{i j} |^{4} γ_{j} (λ_{j} - λ_{6}) . \end{array}

At this stage, we just need to solve a set of different equations with the initial state ρ(0) = |ψ(0)⟩ ⟨ψ(0)| or ρ(0) = |Ψ(0)⟩ ⟨Ψ(0)| represented in the vector bases |ϕ_i⟩ (i = 1,2,…,6). Then the elements of the density matrix are given by

\begin{array}{l} ρ_{i i} (t) = ρ_{i i} (0) e^{- {\tilde{Γ}}_{i} t}, ρ_{i j} (t) = ρ_{i j} (0) e^{[- 2 i (λ_{i} - λ_{j}) - ({\tilde{Γ}}_{i} + {\tilde{Γ}}_{j} + {\tilde{γ}}_{i} + {\tilde{γ}}_{j})] t / 2}, \\ ρ_{i 6} (t) = ρ_{i 6} (0) e^{[- i (λ_{i} - λ_{6}) - \frac{{\tilde{Γ}}_{i}}{2} - \sum_{j = 1}^{5} \frac{({\tilde{γ}}_{j} + 3 {\tilde{γ}}_{i})}{2}] t}, \\ ρ_{66} (t) = \sum_{j = 1}^{5} ρ_{j j} (0) (1 - e^{- {\tilde{Γ}}_{j} t}) + ρ_{66} (0) . \end{array}

Here the subscripts i and j in Eqs. (19)–(22) are all running from 1 to 5, and the condition i < j should be satisfied because $\bar{ω} = λ_{j} - λ_{i}$ must be non-negative.

Finally, tracing over the twin modes of the cavity, one can obtain the reduced density matrix ρ_ana(t) with respect to three NVCs as

\begin{array}{l} ρ_{a n a} (t) = \sum_{i, j = 1}^{6} 〈 ϕ_{i} | ρ (t) | ϕ_{j} 〉 〈 00 | ϕ_{i} 〉 〈 ϕ_{j} | 00 〉 + 〈 10 | ϕ_{i} 〉 〈 ϕ_{j} | 10 〉 \\ + 〈 01 | ϕ_{i} 〉 〈 ϕ_{j} | 01 〉 + 〈 11 | ϕ_{i} 〉 〈 ϕ_{j} | 11 〉], \end{array}

where i = 1,2,3,4,5,6. Direct calculation yields the reduced density matrix ρ_ana(t) as

\begin{array}{l} ρ_{a n a} (t) = ρ_{e g g, g e g} | e_{1} g_{2} g_{3} 〉 〈 g_{1} e_{2} g_{3} | + ρ_{e g g, g g e} | e_{1} g_{2} g_{3} 〉 〈 g_{1} g_{2} e_{3} | + ρ_{e g g, g g g} | e_{1} g_{2} g_{3} 〉 〈 g_{1} g_{2} g_{3} | \\ + ρ_{g e g, e g g} | g_{1} e_{2} g_{3} 〉 〈 e_{1} g_{2} g_{3} | + ρ_{g e g, g g e} | g_{1} e_{2} g_{3} 〉 〈 g_{1} g_{2} e_{3} | + ρ_{g e g, g g g} | g_{1} e_{2} g_{3} 〉 〈 g_{1} g_{2} g_{3} | \\ + ρ_{g g e, e g g} | g_{1} g_{2} e_{3} 〉 〈 e_{1} g_{2} g_{3} | + ρ_{g g e, g e g} | g_{1} g_{2} e_{3} 〉 〈 g_{1} e_{2} g_{3} | + ρ_{g g e, g g g} | g_{1} g_{2} e_{3} 〉 〈 g_{1} g_{2} g_{3} | \\ + ρ_{g g g, e g g} | g_{1} g_{2} g_{3} 〉 〈 e_{1} g_{2} g_{3} | + ρ_{g g g, g e g} | g_{1} g_{2} g_{3} 〉 〈 g_{1} e_{2} g_{3} | + ρ_{g g g, g g e} | g_{1} g_{2} g_{3} 〉 〈 g_{1} g_{2} e_{3} | \\ + ρ_{e g g, e g g} | e_{1} g_{2} g_{3} 〉 〈 e_{1} g_{2} g_{3} | + ρ_{g e g, g e g} | g_{1} e_{2} g_{3} 〉 〈 g_{1} e_{2} g_{3} | \\ + ρ_{g g e, g g e} | g_{1} g_{2} e_{3} 〉 〈 g_{1} g_{2} e_{3} | + ρ_{g g g, g g g} | g_{1} g_{2} g_{3} 〉 〈 g_{1} g_{2} g_{3} |, \end{array}

where

ρ_{e g g, e g g} = \sum_{i, j = 1}^{5} c_{i 1} c_{j 1}^{*} ρ_{i j}

,

ρ_{g e g, g e g} = \sum_{i, j = 1}^{5} c_{i 2} c_{j 2}^{*} ρ_{i j}

,

ρ_{g g e, g g e} = \sum_{i, j = 1}^{5} c_{i 3} c_{j 3}^{*} ρ_{i j}

,

ρ_{e g g, g e g} = \sum_{i, j = 1}^{5} c_{i 1} c_{j 2}^{*} ρ_{i j}

,

ρ_{e g g, g g e} = \sum_{i, j = 1}^{5} c_{i 1} c_{j 3}^{*} ρ_{i j}

,

ρ_{g e g, g g e} = \sum_{i, j = 1}^{5} c_{i 2} c_{j 3}^{*} ρ_{i j}

,

ρ_{e g g, g g g} = \sum_{i = 1}^{5} c_{i 1} ρ_{i 6}

,

ρ_{g e g, g g g} = \sum_{i = 1}^{5} c_{i 2} ρ_{i 6}

,

ρ_{g g e, g g g} = \sum_{i = 1}^{5} c_{i 3} ρ_{i 6}

, and

ρ_{g g g, g g g} = ρ_{66} + \sum_{i, j = 1}^{5} (c_{i 4} c_{j 4}^{*} + c_{i 5} c_{j 5}^{*}) ρ_{i j}

.

Acknowledgments

W.L.Y. thanks Prof. Yun-Feng Xiao, Prof. Qiongyi He, and Dr. Zhangqi Yin for enlightening discussions. This work is supported partially by the National Fundamental Research Program of China under Grants No. 2012CB922102 and No. 2013CB921803, by the National Natural Science Foundation of China under Grants No. 11274351, No. 11104326 and No. 11274352.

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Entanglement dynamics for three nitrogen-vacancy centers coupled to a whispering-gallery-mode microcavity

Abstract

1. Introduction

2. System Hamiltonian

3. Quantum dynamics of entanglement among three NVCs

3.1. The resonant case

3.2. The detuning case

4. Discussion and conclusion

APPENDIX A: The quantum master equation

4.1. The microscopic case

4.2. The phenomenological case

APPENDIX B: Analytical solutions to the microscopic master equation

Acknowledgments

References and links

Cited By

Figures (8)

Equations (24)

Optics Express