Optical bistability and four-wave mixing with a single nitrogen-vacancy center coupled to a photonic crystal nanocavity in the weak-coupling regime

Jiahua Li; Rong Yu; Chunling Ding; Ying Wu

doi:10.1364/OE.22.015024

1. Introduction

In recent years, cavity quantum electrodynamics (CQED) studies light-matter interactions inside a resonator, which provides an ideal platform for quantum optics and few-photon nonlinear optics [1]. Tremendous progress has been made by coupling single quantum dipole emitters to different cavities [2–5]. Among them, nanoscale photonic crystal (PC) cavities may be promising due to its highly confined ultrasmall mode volume V in the order of the qubic wavelength and ultrahigh quality Q-factor (i.e., a high Q/V ratio) [6–9]. Moreover, waveguides can be easily incorporated into PCs, thus complex chip-scale systems of coupled PC waveguides and cavities can be produced [10–12]. Hence these PC waveguide-coupled nanocavity systems are compatible with large scale integration for the development of complex devices on-a-chip.

On the other hand, nitrogen-vacancy (NV) centers consisting of a substitutional nitrogen atom and an adjacent vacancy have recently emerged as an excellent test bed for solid-state quantum physics experiments and quantum information processing because they possess a combination of the excellent ground-state spin coherence, the NV’s level structure, and the stability of the diamond host lattice [13–21]. In nano-size diamond, the NV centers can be integrated into various other systems, for example, plasmonic elements, cells, or planar PC structures [22–24]. Combining high-Q PC cavities and NV centers in diamond represents a promising solid-state CQED system, and attracts much attention. Several attempts have been pursued to couple NV centers to cavities such as microsphere resonators [25–27], microtoroids [28–32], microdisk [33], or PC nanocavities [34–44] both theoretically and experimentally. For example, recent experimental investigations by Sar et al. have demonstrated deterministic coupling of single NV centers to high-quality PC nanocavities [34]. Barth et al. have addressed controlled coupling of a single-diamond nanocrystal to a planar PC double-heterostructure cavity [36]. Wolters et al. have developed a scheme to enhance the zero phonon line emission from a single NV center in a nanodiamond via coupling to a PC nanocavity [37]. Yang et al. have investigated quantum state transfer and quantum correlation in a composite system consisting of two NV centers embedded in two spatially separated single-mode nanocavities in a planar PC [38, 42].

Among CQED platforms, the response of the single quantum dipole emitter (e.g., NV defect center) coupled to a PC nanocavity via a nearby photonic waveguide is generally described by the Heisenberg-Langevin equations. As shown in [45–47], the Heisenberg-Langevin equations are nonlinear, and it is very difficult to get an analytic solution to these equations. The weak-excitation approximation is widely used in these CQED research, where the dipole emitter will remain mostly in its ground state, i.e., σ̂₁₁(t) ≈ 1 and σ̂₂₂(t) ≈ 0 with σ̂₁₁ and σ̂₂₂ being the population operators of the emitter ground and excited states. By assuming this so-called low-excitation regime (no more than one photon in the system), we can approximate 〈σ̂_z(t)〉 ≈ −1/2 for all time (i.e., we can substitute σ̂_z(t) with its average value of −1/2), and thus linearize the operator equation (see Eqs. (13), (14) and (15) below). Here, σ̂_z(t) = [σ̂₂₂(t) − σ̂₁₁(t)]/2 is the operator of population inversion between the ground state |1〉 and the excited state |2〉 (see Fig. 1). The Heisenberg-Langevin equations are reduced to a set of linear equations. Such a procedure by ignoring the nonlinear nature of the quantum emitter has been commonly adopted in many previous studies [30, 32, 48–60]. However, when optical driving power is intermediate (not too low and not too high), the weak-excitation approximation fails and the nonlinear terms need to be considered (see Fig. 2 below).

Fig. 1 Schematic structure of optical coupeld system, which is composed of a two-level NV center, a single-mode PC nanocavity and a nearby photonic waveguide serving for in- and outcoupling of light into the nanocavity. The single-mode PC nanocavity containing the NV center is evanescently coupled to the row defect waveguide with the coupling strength κ_e. The |1〉 ⇔ |2〉 transition of the NV center is coupled to the mode ĉof the PC nanocavity with the coupling strength g_cav. κ_i is the intrinsic loss rates of the nanocavity mode excluding coupling to the NV center and the waveguide. A bichromatic field consisting of a cw control laser and a cw probe laser is injected into the waveguide via grating couplers (see Refs. [67, 68]). S_in and S_out denote the input and the output field in the waveguide. The bubble shows the detailed structure of quantized energy levels and the coupling scheme of the cavity mode for the two-level NV center. |1〉 and |2〉 denote the ground and excited states of the NV center, respectively. A small red sphere shows the position of the NV center.

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Fig. 2 The steady-state population inversion σ̄_z as a function of optical driving power P_c for four different values of Δ₂. The other system parameters used for the simulation are chosen as g_cav/2π = 2.25 GHz, κ_i/2π = 1.6 GHz, κ_e/2π = 8 GHz, γ_spon/2π = 13 MHz, and δ = 0. Curves A–D are for Δ₂ = 0, 100, 140, and 180 MHz, respectively.

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In the present work, using the perturbation method we take into account the nonlinear terms such as higher-order moments −2g_cavĉσ̂_z, g_cavĉ^†σ̂₁₂, and g_cavĉσ̂₂₁, where g_cav is the coupling strength between the NV center and the cavity mode and ĉis the annihilation operator of the PC cavity mode, and study nonlinear optical response of such a hybrid system in the continuous-wave-operation regime. We find that these nonlinear terms can give rise to some interesting phenomena of the PC cavity-waveguide system in the experimentally achievable parameter range, such as optical bistability (OB) for the population inversion and the nonlinear degenerate four-wave mixing (FWM) process. Besides, the OB and FWM works in the weak-coupling regime where the coupling strength between the NV center and the cavity is smaller than the cavity decay rate and this condition allows more a practical parameter range for solid-state materials on a compact integrated nanophotonic platform.

It should be pointed out that many investigations of the OB and FWM effects in various CQED systems have been reported (for a review, see [61]). The OB of a Fabry-Perot resonator was analyzed experimentally [62] and theoretically [63] in the context of an ensemble of laser cooled atoms, where the input-output response of the atom-cavity system is measured. However, the unusual dependence of the population inversion related to the OB and degenerate FWM effect on optical driving power was not considered before based on the weak coupling between a single NV center and a PC nanocavity. Additionally, we analyze the nanophotonic platform for possible experimental realization of these effects in the continuous-wave-operation regime. We note that the bichromatic driving of a solid-state cavity QED system has been proposed and analyzed by the Vučković group [64–66] from different perspectives. However, in these studies, they looked experimentally and theoretically into the problem of the spectral features from the cavity emission.

The rest of the paper is organized into four parts as follows. In Sec. II, we present the theoretical model and establish the corresponding evolution equations for the cavity and the NV center. In Sec. III, we discuss in details optical bistable behaviors of the papulation inversion in the steady-state case of the coupled NV center-cavity system. In Sec. IV, under the driving of a strong control field and a weak probe field, we analyze the nonlinear four-wave mixing process using the perturbation method. Finally, we conclude in Sec. V.

2. Proposed model and evolution equations

2.1. Hamiltonian

A schematic description of optical coupled system under investigation is illustrated schematically in Fig. 1. In the present scheme, an NV center is confined in a single-mode PC nanocavity, and the nanocavity is side-coupled to an optical waveguide serving for in- and outcoupling of light into the nanocavity, with a geometry similar to the one reported in Refs. [34,35,67]. Here, the nanocavity is formed by imbedding spatial point defects into a PC platform with the cavity frequency tunable by changing the geometrical parameters of the defects. The cavity is assumed to have a single mode that couples only to the forward propagating fields [48, 49]. The waveguide modes are formed by row defects. The NV center is a point defect in the diamond lattice, which consists of a substitutional nitrogen atom (N) plus a vacancy (V) in an adjacent lattice site. The NV center addressed in this study is negatively charged with two unpaired electrons located at the vacancy, usually treated as electron spin-1 system [13]. The spin-spin interaction leads to an energy splitting D_gs = 2.88 GHz between the ground states of electronic spin triplet |³A₂, m_s = 0〉 and |³A₂, m_s = ±1〉 (the quantum number m_s is the electron-spin projection along the NV axis). The optical transitions in the NV center are spin-conserving. Thus, we can choose the electronic states |³A₂, m_s = 0〉 and |³E, m_s = 0〉 of the NV center as the ground state |1〉 and the excited state |2v, respectively. Using the two-dimensional (2D) planar photonic crystal structure, the initial input field, which we call the driving field, is guided by the waveguide in the crystal plane. The |1〉 ⇔ |2〉 transition of the NV center is coupled to the cavity mode with resonance frequency ω_cav and coupling strength g_cav (also called vacuum Rabi frequency). For dipole-type NV center-field mode interaction inside the cavity, this coupling strength is related to the transition dipole moment μ₂₁ and the location r of the NV center and effective cavity mode volume V_eff, etc. The Hamiltonian of a cavity-coupled NV center in the presence of a driving field is given by [67, 68]

ℋ = \bar{h} ω_{N V} {\hat{σ}}_{22} + \bar{h} ω_{cav} {\hat{c}}^{†} \hat{c} + i \bar{h} g_{cav} (\hat{c} {\hat{σ}}_{21} - {\hat{c}}^{†} {\hat{σ}}_{12}) + i \bar{h} \sqrt{κ_{e}} [S_{in} (t) {\hat{c}}^{†} - S_{in}^{*} (t) \hat{c}],

where the rotating-wave approximation (RWA) and the electric-dipole approximation (EDA) have been made. In the above equation, the first, second and third terms account for the energy of the coupled cavity-NV center system and the fourth term represents the driving of the cavity by an external laser field [67]. In the derivation of the Hamiltonian, the energy of the ground state |1〉 is set as zero to allow for simplicity. h̄ω_NV is the energy of the electronic state |2〉, that is to say, ω_NV is the frequency of the NV center’s optical transition between the ground state |1〉 and the excited state |2〉. The symbols σ̂_mn = |m〉〈n| (m, n = 1, 2) for m ≠ n, are the electronic transition or projection operators between the states |m〉 and |n〉 and σ̂_mm = |m〉〈m| (m = 1, 2) represent the electronic population operators involving the levels of the NV center [see also the bubble of Fig. 1]. ĉand ĉ^† are the bosonic annihilation and creation operators of the cavity mode. The parameter κ_e is the coupling rate between the cavity and the forward propagating mode of the waveguide, and S_in(t) is the driving laser propagating in the waveguide. For the driving laser field, these two cases are considered individually in the following.

Transforming the Hamiltonian (1) into the rotating frame at the frequency of the input laser field ω_l by using [46, 47]

ℋ_{free} = \bar{h} ω_{l} ({\hat{σ}}_{22} + {\hat{c}}^{†} \hat{c}),

U (t) = e^{- i ℋ_{free} t / \bar{h}} = e^{- i ω_{l} t ({\hat{σ}}_{22} + {\hat{c}}^{†} \hat{c})},

\begin{array}{l} ℋ_{rot} & = & U^{†} (t) ℋ U (t) - i U^{†} (t) \frac{\partial U (t)}{\partial t} \\ = & U^{†} (t) (ℋ - ℋ_{free}) U (t), \end{array}

we can derive the resulting effective Hamiltonian as follows:

\begin{array}{l} ℋ_{rot} & = & \bar{h} Δ_{1} {\hat{σ}}_{22} + \bar{h} Δ_{2} {\hat{c}}^{†} \hat{c} + i \bar{h} g_{cav} (\hat{c} {\hat{σ}}_{21} - {\hat{c}}^{†} {\hat{σ}}_{12}) \\ + i \bar{h} \sqrt{κ_{e}} [S_{in} (t) e^{i ω_{l} t} {\hat{c}}^{†} - S_{in}^{*} (t) e^{- i ω_{l} t} \hat{c}], \end{array}

where Δ₁ = ω_NV − ω_l and Δ₂ = ω_cav − ω_l are respectively the detunings of the NV center resonance frequency ω_NV and the PC cavity resonance frequency ω_cav from the laser field ω_l.

Case (i): When we consider a monochromatic continuous-wave (cw) driving field with the carrier frequency ω_p and the field amplitude s_p, i.e., a cw probe field S_in(t) = s_pe^−iω_pt, we can choose ω_l = ω_p (in a frame rotating at the probe frequency ω_p). In this case, the effective Hamiltonian can be expressed by

ℋ_{rot} = \bar{h} Δ_{1} {\hat{σ}}_{22} + \bar{h} Δ_{2} {\hat{c}}^{†} \hat{c} + i \bar{h} g_{cav} (\hat{c} {\hat{σ}}_{21} - {\hat{c}}^{†} {\hat{σ}}_{12}) + i \bar{h} \sqrt{κ_{e}} (s_{p} {\hat{c}}^{†} - s_{p}^{*} \hat{c}),

with Δ₁ = ω_NV − ω_p and Δ₂ = ω_cav − ω_p. In the above, s_p is the field amplitude of the monochromatic driving laser propagating in the waveguide, which is normalized to a photon flux at the input of the cavity and directly related to the power propagating in the waveguide by

P_{p} = \bar{h} ω_{p} s_{p}^{2}

.

Case (ii): When we consider a bichromatic cw driving field, i.e., a control field and a probe field S_in(t) = s_ce^−iω_ct + s_pe^−iω_pt, where ω_c (ω_p) are the carrier frequency of the control (probe) field and s_c (s_p) is the field amplitude, we can choose ω_l = ω_c (in a frame rotating at the control frequency ω_c). Consequently, under the above transformation, the resulting Hamiltonian can be given by

\begin{array}{l} ℋ_{rot} & = & \bar{h} Δ_{1} {\hat{σ}}_{22} + \bar{h} Δ_{2} {\hat{c}}^{†} \hat{c} + i \bar{h} g_{cav} (\hat{c} σ_{21} - {\hat{c}}^{†} {\hat{σ}}_{12}) \\ + i \bar{h} \sqrt{κ_{e}} [(s_{c} + s_{p} e^{- i Ω t}) {\hat{c}}^{†} - H . C .], \end{array}

where H.C. represents the Hermitian conjugate. We have defined Δ₁ = ω_NV − ω_c, Δ₂ = ω_cav − ω_c, and Ω = ω_p − ω_c, respectively. Note that, if δ = ω_NV − ω_cav denoting the NV center-cavity detuning, then Δ₁ = Δ₂ + δ. In the above, s_c and s_p are the field amplitudes of the bichromatic driving laser propagating in the waveguide, which are normalized to a photon flux at the input of the cavity and directly related to the power propagating in the waveguide by

P_{c} = \bar{h} ω_{c} s_{c}^{2}

and

P_{p} = \bar{h} ω_{p} s_{p}^{2}

, respectively. Unlike case (i) above, the time factors in Eq. (7) cannot be eliminated.

2.2. Heisenberg-Langevin equations

Our analysis is based on the Heisenberg-Langevin equations that are derivable from the Hamiltonian of Eq. (7). Including losses in both the cavity and NV center, as well as cavity excitation, we apply the Heisenberg-Langevin formalism to attain the Heisenberg-Langevin equations of motion as follows

\frac{d \hat{c}}{d t} = - g_{cav} {\hat{σ}}_{12} - (i Δ_{2} + κ_{i} / 2 + κ_{e} / 2) \hat{c} + \sqrt{κ_{e}} (s_{c} + s_{p} e^{- i Ω t}) + {\hat{f}}_{c},

\frac{d {\hat{σ}}_{11}}{d t} = γ_{spon} {\hat{σ}}_{22} - g_{cav} \hat{c} {\hat{σ}}_{21} - g_{cav} {\hat{c}}^{†} {\hat{σ}}_{12} + {\hat{f}}_{11},

\frac{d {\hat{σ}}_{22}}{d t} = - γ_{spon} {\hat{σ}}_{22} + g_{cav} {\hat{c}}^{†} {\hat{σ}}_{12} + g_{cav} \hat{c} {\hat{σ}}_{21} + {\hat{f}}_{22},

\frac{d {\hat{σ}}_{12}}{d t} = - (i Δ_{1} + γ_{spon} / 2) {\hat{σ}}_{12} - 2 g_{cav} \hat{c} {\hat{σ}}_{z} + {\hat{f}}_{12},

where σ̂_z = (σ̂₂₂ − σ̂₁₁) /2 stands for the operator of the population inversion. κ_i is the cavity intrinsic decay rate, which is related to the intrinsic quality factor Q_i by κ_i = ω_cav/Q_i. κ_e is the waveguide-cavity coupling rate, which is related to the coupling quality factor Q_e by κ_e = ω_cav/Q_e [10–12]. The total cavity decay rate (cavity linewidth) is κ = κ_i +κ_e. γ_spon is the decay rate of the NV center. The operators f_c, f̂₁₁, f̂₂₂, and f̂₁₂ are the quantum noise operators with 〈f̂_c〉 = 0, 〈f̂₁₁〉 = 0, 〈f̂₂₂〉 = 0, and 〈f̂₁₂〉 = 0. In this work, we are interested in the mean response of the coupled system [69], so the operators can be reduced to their expectation values, i.e., c(t) ≡ 〈ĉ(t)〉, c^*(t) ≡ 〈ĉ^†(t)〉, σ₁₁(t) = 〈σ̂₁₁(t)〉, σ₂₂(t) = 〈σ̂₂₂(t)〉, σ_z(t) = 〈σ̂_z(t)〉, σ₁₂(t) = 〈σ̂₁₂(t)〉, and

σ_{12}^{*} (t) = 〈 {\hat{σ}}_{21} (t) 〉

. In these situations, we reduce the operator equations to the mean value equations and drop the above quantum noise terms because 〈f̂_c〉 = 0, 〈f̂₁₁〉 = 0, 〈f̂₂₂〉 = 0, and 〈f̂₁₂〉 = 0. The Heisenberg-Langevin equations then become

\frac{d c}{d t} = - g_{cav} σ_{12} - (i Δ_{2} + κ_{i} / 2 + κ_{e} / 2) c + \sqrt{κ_{e}} (s_{c} + s_{p} e^{- i Ω t}) .

\frac{d σ_{11}}{d t} = γ_{spon} σ_{22} - g_{cav} c σ_{12}^{*} - g_{cav} c^{*} σ_{12},

\frac{d σ_{22}}{d t} = - γ_{spon} σ_{22} + g_{cav} c^{*} σ_{12} + g_{cav} c σ_{12}^{*},

\frac{d σ_{12}}{d t} = - (i Δ_{1} + γ_{spon} / 2) σ_{12} - 2 g_{cav} c σ_{z} .

The derivation of Eqs. (13)–(15) uses the well-known mean-field (or so-called factorization) assumption 〈ÂB̂〉 = 〈Â〉〈B̂〉 [69]. Note that, Eqs. (13) and (14) can be incorporated and rewritten as

d σ_{z} / d t = - γ_{spon} (σ_{z} + 1 / 2) + g_{cav} c^{*} σ_{12} + g_{cav} c σ_{12}^{*}

. This set of coupled equations are ordinary nonlinear differential equations of complex functions instead of operators, and describe the time evolution of the cavity-NV center coupled system.

3. The bistable behavior of the population inversion in the steady-state case

Following standard methods from quantum optics, under the condition that the control laser field is much stronger than the probe laser field, we can use the perturbation method to deal with Eqs. (12)–(15). The control laser field provides a steady-state solution (c̄, σ̄₁₂, and σ̄_z) of the coupled system with respect to the probe laser field, while the probe laser field is treated as the perturbation of the steady state. In order to linearize the dynamics of the cavity-NV center system, the total solution of the intracavity field, the coherence of the NV center, and the corresponding population inversion under both the control field and the probe field can be written as the forms c = c̄ + δc, σ₁₂ = σ̄₁₂ + δσ₁₂, and σ_z = σ̄_z + δσ_z around the steady state values (c̄, σ̄₁₂, σ̄_z). Here δc, δσ₁₂, and δσ_z describe the fluctuations around the steady state values.

In the following, the steady state without the perturbation will be studied. The steady-state solution of Eqs. (12)–(15), in which all time derivatives vanish and s_p → 0, can be obtained as

\bar{c} = \frac{F_{1} \sqrt{κ_{e} s_{c}}}{F_{1} F_{2} - 2 g_{cav}^{2} {\bar{σ}}_{z}},

{\bar{σ}}_{z} = \frac{- \frac{1}{2} γ_{spon}}{γ_{spon} + 2 g_{cav}^{2} (\frac{1}{F_{1}} + \frac{1}{F_{1}^{*}}) {| \bar{c} |}^{2}},

{\bar{σ}}_{12} = \frac{- 2 g_{cav}}{F_{1}} \bar{c} {\bar{σ}}_{z},

where F₁ = iΔ₁ + γ_spon/2 and F₂ = iΔ₂ + κ_i/2 +κ_e/2.

Since the above coupled equations (16) and (17) are cubic in σ̄_z, the system may be characteristic of the OB for a certain parameter range. To demonstrate our numerical results, we briefly demonstrate the experimentally accessible parameters in this hybrid structure. We focus our attention on the dipole transition |1〉 ⇔ |2〉 with a zero phonon line at λ = 637 nm of the diamond NV center. The maximal coupling strength $g_{cav}^{\max}$ between the PC cavity mode and the NV center can be calculated by the relation $g_{cav}^{\max} = | \vec{E} (r) / {\vec{E}}_{\max} | \sqrt{3 π c^{3} γ_{spon} / (2 ω_{c}^{2} n_{e} n_{c} V_{m})}$ [39,40], where V_m is the volume of the nanocavity electromagnetic mode, γ_spon the spontaneous decay rate of the excited state |2〉 of the NV center, and c the light speed in free space. n_c and n_e are the PC nanocavity and diamond nanocrystal refractive indexes, respectively. |E⃗ (r)/E⃗_max| stands for the normalized electric field strength at the NV center’s location r. Experimental evidence involving coupling waveguides to similar nanocavities have shown efficiencies of 90% [70]. Experimentally, the coupling parameter κ_e can be controlled in fabrication, for example, by tuning the waveguide-nanocavity gap [2–4]. The design with a Q₀ factor of 1.5×10⁵ (Q₀ here is an order of magnitude less than the value reported from [40], therefore κ_i = ω_c/Q₀ = 1.6 GHz) and a mode volume V_m of ∼ 0.52(λ/n_c)³ ∼ 4.4 × 10⁻²¹ m³ with n_c ≃ 3.3 and n_e ≃ 2.4 can reach the weak-coupling CQED regime (g_cav, κ_i, κ_e, γ_spon)/2π = (2.25, 1.6, 8, 0.013) GHz [40]. In the figures of this paper, unless otherwise stated, we will always consider the above-mentioned parameters.

By solving Eqs. (16) and (17) numerically, in Fig. 2 we displays the steady-state value of the population difference σ_z as a function of optical driving power P_c for four different values of the detuning Δ₂ = 0, 100, 140, and 180 MHz when the NV center and cavity frequencies is resonant δ = 0. It is clearly shown that the system exhibits bistable behavior when the driving power is large enough. In this case, σ̄_z has three real roots. The largest and smallest roots are stable, and the middle one is unstable, which corresponds to the blue dashed lines in Fig. 2. It is easy to see from Fig. 2 that the bistable threshold is increased gradually and the area of the hysteresis loop becomes narrower as Δ₂ increases. In Fig. 3, we show the influence of the cavity-NV center coupling strength g_cav on the behavior of OB, while keeping all other parameters fixed. It is found that, with increasing g_cav from 1.75, 2.25, 2.75, to 3.25 GHz, the threshold of OB increases progressively and the area of the hysteresis loop becomes wider. In Fig. 4, we show the influence of the cavity-NV center detuning δ on the behavior of OB. The bistable threshold is increased gradually and the area of the hysteresis loop becomes narrower as δ increases from 0, 80, 160, to 190 MHz, which is very similar to Fig. 2.

Fig. 3 The steady-state population inversion σ̄_z as a function of optical driving power P_c for four different values of g_cav. The other system parameters for the simulation are chosen as κ_i/2π = 1.6 GHz, κ_e/2π = 8 GHz, γ_spon/2π = 13 MHz, Δ₁ = Δ₂ = 100 MHz and δ = 0. Curves A–D are for g_cav = 1.75, 2.25, 2.75, and 3.25 GHz, respectively.

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Fig. 4 The steady-state population inversion σ̄_z as a function of optical driving power P_c for four different values of δ. The other system parameters for the simulation are chosen as g_cav/2π = 2.25 GHz, κ_i/2π = 1.6 GHz, κ_e/2π = 8 GHz, γ_spon/2π = 13 MHz, and Δ₂ = 0. Curves A–D are for δ = 0, 80, 160, and 190 MHz, respectively.

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From Figs. 2–4, we can also note that the value of the population difference σ_z is confined between −1/2 and 0. For the case of low excitation power, the approximation σ_z ≈ −1/2 is valid. In this case, the nonlinear nature of the NV center can be neglected safely. Therefore, at low excitation power this approximation matches the actual output quantitatively and successfully explains a lot of theoretical and experimental observations in the linear regime. However, with increasing the driving power P_c, this model fails completely, as the approximation σ_z ≈ −1/2 becomes invalid. For sufficiently high driving power, however, one can approximate σ_z → 0, and Eq. (15) reduces to dσ₁₂/dt = −(iΔ₁ +γ_spon/2)σ₁₂. In view of these factors, one needs to retain the dynamics of the σ_z term in the Heisenberg-Langevin equation when the monochromatic driving field is intermediate (not too low and not too high).

Similar bistable steady-state behavior of the intracavity photon number has been presented recently in a cavity optomechanical system with a Bose-Einstein condensate [71] or a quantum-degenerate Fermi gas [72]. Our OB system here is totally different from the conventional cavity optomechanical system. In conventional cavity optomechanics, the photon number inside the cavity is determined by the driving laser. In other words, the driving laser field directly affects the coupling between the cavity photons and the mechanical oscillator. However, for the single NV center coupled to the PC cavity, the population inversion of the NV center is dependent on the driving field, which can take the value between −1/2 and 0 via the interaction between the NV center and the cavity mode. The NV center-cavity coupling is independent on the driving field. On the other hand, this OB is a new addition to the family of OB in typical atomic systems in that there are two possible steady states for a single input filed. While the OB in a laser-driven atomic medium has been suggested as a mechanism for an all-optical switching, similarly our OB can provide a candidate for a controlled switching device, since its population inversion for the lower stable branch and the upper stable branch can be manipulated by the driving laser. However, it should be pointed out that different points exist between them. The OB in the atomic medium mainly refers to the absorption or dispersion-type OB in the input-output intensity, while our OB refers to the bistable behavior of the population inversion.

4. The nonlinear four-wave mixing process under the driving of a strong control field and a weak probe field

Now we turn to consider the perturbation made by the probe field. By substituting c = c̄+ δc, σ₁₂ = σ̄₁₂ + δσ₁₂, and σ_z = σ̄_z + δσ_z into Eqs. (12)–(15) and retaining only first order terms in the small quantities δc, δσ₁₂, and δσ_z, we then obtain

\frac{d δ c}{d t} = - g_{cav} δ σ_{12} - (i Δ_{2} + κ_{i} / 2 + κ_{e} / 2) δ c + \sqrt{κ_{e}} s_{p} e^{- i Ω t},

\frac{d δ σ_{z}}{d t} = - γ_{spon} δ σ_{z} + g_{cav} \bar{c^{*}} δ σ_{12} g_{cav} δ c^{*} {\bar{σ}}_{12} + g_{cav} \bar{c} δ σ_{12}^{*} + g_{cav} δ c {\bar{σ}}_{12}^{*},

\frac{d δ σ_{12}}{d t} = - (i Δ_{1} + γ_{spon} / 2) δ σ_{12} - 2 g_{cav} \bar{c} δ σ_{z} - 2 g_{cav} δ c {\bar{σ}}_{z} .

Next we solve the problem for the driving field (in the rotating frame) δs_in = s_pe^−iΩt. For a given Ω = ω_p − ω_c, we use the ansatz

δ c = c_{+} e^{- i Ω t} + c_{-} e^{i Ω t},

δ σ_{z} = σ_{z +} e^{- i Ω t} + σ_{z -} e^{i Ω t},

δ σ_{12} = σ_{12 +} e^{- i Ω t} + σ_{12 -} e^{i Ω t} .

If sorted by rotation terms e^±iΩt, this yields six equations about c₊, c₋, σ_z+, σ_z−, σ₁₂₊, and σ₁₂₋ as follows

D_{1} σ_{12 -} + 2 g_{cav} \bar{c} σ_{z -} + 2 g_{cav} c_{-} {\bar{σ}}_{z} = 0,

g_{cav} σ_{12 -} + D_{2} c_{-} = 0,

g_{cav} σ_{12 +} + D_{3} c_{+} - \sqrt{κ_{e}} s_{p} = 0,

D_{4} σ_{12 +} + 2 g_{cav} \bar{c} σ_{z +} + 2 g_{cav} c_{+} {\bar{σ}}_{z} = 0,

- (γ_{spon} - i Ω) σ_{z +} + g_{cav} \bar{c^{*}} σ_{12 +} + g_{cav} c_{-}^{*} {\bar{σ}}_{12} + g_{cav} \bar{c} σ_{12 -}^{*} + g_{cav} c_{+} {\bar{σ}}_{12}^{*} = 0,

- (γ_{spon} + i Ω) σ_{z -} + g_{cav} \bar{c^{*}} σ_{12 -} + g_{cav} c_{+}^{*} {\bar{σ}}_{12} + g_{cav} \bar{c} σ_{12 +}^{*} + g_{cav} c_{-} {\bar{σ}}_{12}^{*} = 0,

where D₁ = iΔ₁ + iΩ + γ_spon/2, D₂ = iΔ₂ + iΩ + κ_i/2 + κ_e/2, D₃ = iΔ₂ − iΩ + κ_i/2 + κ_e/2, and D₄ = iΔ₁ − iΩ + γ_spon/2, respectively. c̄, σ̄₁₂, and σ̄_z are decided by Eqs. (16)–(18). It is difficult to obtain the analytical solutions of the above equations (25)–(30) except under very special conditions and hence we will resort to numerical solutions of them.

The output field transmitted through the coupled system can be obtained using the standard input-output relation $S_{out} (t) - S_{in} (t) = \sqrt{κ_{e}} c$ [73, 74] for the propagating field, where S_out (t) is the output field amplitude. We have

\begin{array}{l} S_{out} (t) & = & (s_{c} + \sqrt{κ_{e}} \bar{c}) e^{- i ω_{c} t} + (s_{p} + \sqrt{κ_{e}} c_{+}) e^{- i (ω_{c} + Ω) t} + \sqrt{κ_{e}} c_{-} e^{- i (ω_{c} - Ω) t} \\ = & \underset{Control field}{\underset{︸}{(s_{c} + \sqrt{κ_{e}} \bar{c}) e^{- i ω_{c} t}}} + \underset{Probe field}{\underset{︸}{(s_{p} + \sqrt{κ_{e}} c_{+}) e^{- i ω_{p} t}}} + \underset{Degenerate FWM field}{\underset{︸}{\sqrt{κ_{e}} c_{-} e^{- i (2 ω_{c} - ω_{p}) t}}}, \end{array}

with the coefficients

s_{c} + \sqrt{κ_{e}} \bar{c}

,

s_{p} + \sqrt{κ_{e}} c_{+}

, and

\sqrt{κ_{e}} c_{-}

being the optical responses at the control frequency ω_c, the probe frequency ω_p, and the generated frequency 2ω_c − ω_p, respectively. The transmission of the probe laser field is defined as

t_{p} = 1 + \sqrt{κ_{e}} c_{+} / s_{p}

. Some previous works have used |t_p|² to study dipole-induced transparency (DIT) [48]. Also, it is easy to see from Eq. (31) that the output field contains two input components (the control field ω_c and the probe field ω_p) and one new degenerate FWM component at frequency ω_f = 2ω_c − ω_p [75]. Then, the normalized FWM intensity in terms of the probe field, which is also called the FWM efficiency, can be defined as

I_{FWM} = {| \frac{\sqrt{κ_{e}} c_{-}}{s_{p}} |}^{2} .

In what follows, we present a discussion about the intensity of the FWM. In Fig. 5, we plot the normalized intensity of the generated FWM field as a function of the detuning Δ₂ between the cavity field and the control field for P_c = 30 pW and P_p = 3 pW, respectively. It can be seen that there is a peak value located at Δ₂ = 0, which is caused by the effect of DIT [48]. At the same time, there is a transparency window in the transmission spectrum of the probe field as shown in Fig. 5. The normalized intensity of the generated FWM field versus the power of the control driving field P_c is shown in Fig. 6. We observe that, due to the nonlinear nature of the NV center, the intensity of the generated FWM first increases to a maximum point, then decreases rapidly with P_c increasing. Finally it approaches to a zero steady-state value for sufficiently high driving P_c.

Fig. 5 The normalized FWM intensity versus the detuning Δ₂. The other system parameters for the simulation are chosen as P_c = 30 pW, P_p = 3 pW, g_cav/2π = 2.25 GHz, Ω/2π = 1 GHz, κ_i/2π = 1.6 GHz, κ_e/2π = 8 GHz, γ_spon/2π = 13 MHz, and δ = 80 MHz, respectively.

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Fig. 6 The normalized FWM intensity versus optical driving power P_c. The other system parameters for the simulation are chosen as P_p = 3 pW, g_cav/2π = 2.25 GHz, Ω/2π = 1 GHz, κ_i/2π = 1.6 GHz, κ_e/2π = 8 GHz, γ_spon/2π = 13 MHz, Δ₂ = 0, and δ = 80 MHz, respectively.

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The reason for the above results can be interpreted by analyzed the nonlinearity of the system in the following three steps:

(i) For the low-power driving field (no more than one photon in the system), the NV center will remain mostly in its ground state |1〉, and we can approximate σ̄_z → −1/2 as can be verified in Figs. 2–4. In this case, the nonlinear nature of the NV center is negligible and can be neglected.
(ii) However, with increasing the driving power, the nonlinear nature of the NV center gradually boosts up and needs to be taken into account [45]. Furthermore, the approximation σ̄_z → −1/2 becomes invalid.
(iii) For sufficiently high driving power, we can approximate σ̄_z → 0. Similarly, the nonlinear nature of the NV center is negligible. According to Eqs. (16)–(18), we have the results σ̄₁₂ = 0 and $\bar{c} = \sqrt{κ_{e} s_{c}} / F_{2}$ . Inserting them into Eqs. (25)–(30) and performing some algebra, it is found that the expression c₋ = 0 ⇒ I_FWM = 0 which is in good agreement with Fig. 6 for sufficiently high driving power P_c.

Before ending this section, we would like to address more technical aspects about the NV center in this hybrid system. We consider a simplified model for the NV center consisting of a ground-state level (|1〉) and an excited-state level (|2〉). For a more general treatment, the NV center has a relatively complicated structure of excited states [13], which includes six excited states defined by the method of group theory as $| A_{1} 〉 = \frac{1}{\sqrt{2}} (| E_{-}, m_{s} = + 1 〉 - | E_{+}, m_{s} = - 1 〉)$ , $| A_{2} 〉 = \frac{1}{\sqrt{2}} (| E_{-}, m_{s} = + 1 〉 + | E_{+}, m_{s} = - 1 〉)$ , |E_x〉 = |X, m_s = 0〉, |E_y〉 = |Y, m_s = 0〉, $| E_{1} 〉 = \frac{1}{\sqrt{2}} (| E_{-}, m_{s} = - 1 〉 - | E_{+}, m_{s} = + 1 〉)$ , and $| E_{2} 〉 = \frac{1}{\sqrt{2}} (| E_{-}, m_{s} = - 1 〉 + | E_{+}, m_{s} = + 1 〉)$ with |E₊〉, |E₋〉 being orbital states with angular momentum projection ±1 along the NV axis, and |X〉, |Y〉 being orbital states with zero projection of angular momentum. The spin-orbit interaction splits the pair (|A₁〉, |A₂〉) from the others by at least about 5.5 GHz. The spin-spin interaction increases the energy gap and produces a gap of 3.3 GHz between |A₁〉 and |A₂〉. At the same time, the ground states |³A₂, m_s = 0, ±1〉 are associated with the orbital state |E₀〉 with zero projection of angular momentum (for simplicity, the spatial part of the wavefunction is not explicitly written). The optical transitions in NV centers are spin-conserving, but could change electronic orbital angular momentum depending on the photon polarization [20]. That is to say, spin-conserving transitions between the ground state and six different excited states can be driven optically. In the limit of zero strain, the |A₂〉 state is robust due to the stable symmetric properties, and decays to the ground-state sublevels |³A₂, m_s = −1〉 and |³A₂, m_s = +1〉 with radiation of σ⁺ and σ⁻ circular polarizations, respectively. On the one hand, as we all know, only when the frequency of the cavity mode and the transition frequency of the NV center between two levels are same (resonance) or close (near resonance), a strong coherent interaction of the cavity mode with the NV center appears. Therefore, in the case of a single-mode cavity field, the NV center can be regarded as two-level system owing to a large level space. On the other hand, by properly choosing the polarization of the cavity mode according to selection rules [20], the other levels can be well decoupled. Therefore, we believe that the present model can provide a qualitative illustration of the observable OB and FWM properties.

5. Conclusions

In summary, we have theoretically investigated OB and FWM effects occurring in a hybrid optical system composed of a PC nanocavity, a single NV center and a PC waveguide by taking into account nonlinear terms. It is shown that, in the weak-coupling regime and the experimentally available parameter range, the tunable bistable behavior of the population inversion and the efficient generation of the FWM signal can be achieved due to the nonlinearity of the system. The results obtained here may be useful for gaining further insight into the properties of solid-state CQED systems and find applications in an all-optical wavelength converter and switch in a PC platform.

This photonic device combines the advantages of photons (low decoherence rates and high velocities), waveguides (low-loss and long-range photon transfer with a controllable group velocity), NV centers (long electronic spin decoherence time at room temperature), and PC cavities (small volumes and high-quality Q-factors). In particular, the great attraction of the solid-based CQEDs is that they can be monolithically fabricated and integrated into the large-scale arrays. Finally, it is worth pointing out that the present theory can also be applied to other similar systems, such as a single quantum emitter (e.g., atom, molecule, or quantum dot) positioned close to a microtoroidal resonator [2, 3] with the whispering-gallery-mode fields propagating inside the resonator. Therefore, we believe that our proposal is feasible in experimental realizations and deserves to be tested with the currently available technology.

Acknowledgments

J.L. gratefully acknowledges Xiaoxue Yang for useful advice and discussion. This research was supported in part by the National Natural Science Foundation (NNSF) of China (Grants No. 11104210, No. 11375067, and No. 11275074), the National Basic Research Program of China (Contract No. 2012CB922103), and by the Doctoral Foundation of the Ministry of Education of China (Grant No. 20134103120005).

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Optical bistability and four-wave mixing with a single nitrogen-vacancy center coupled to a photonic crystal nanocavity in the weak-coupling regime

Abstract

1. Introduction

2. Proposed model and evolution equations

2.1. Hamiltonian

2.2. Heisenberg-Langevin equations

3. The bistable behavior of the population inversion in the steady-state case

4. The nonlinear four-wave mixing process under the driving of a strong control field and a weak probe field

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (32)

Optics Express