1 Introduction

Over the last few years, there has stimulated tremendous interests in the study of induced-laser quantum coherence and interference in quantum optics and atomic physics. Examples include electromagnetically induced transparency (EIT) [1–4] and related topics [5, 6]. A wide variety of novel phenomena and promising applications have been predicted and explored in recent years. Because of the EIT suppression of the linear absorption and the resulting enhancement of the nonlinear susceptibilities, nonlinear optical phenomena can be observed at low light intensities and near atomic resonant frequencies [5], which may lead to interesting practical applications in nonlinear optical devices. As a method to investigate quantum coherent control and interference, the relative phase of the applied laser fields has been extensively employed, which is usually termed as phase control, in several important processes in atomic, molecular, and solid-state systems [7–16]. Particularly in the closed-loop configuration, the relative phase becomes very important as it can well control the linear and nonlinear optical properties of atomic, molecular, and solid media dramatically [7, 8, 12–15]. For instance, studies involving phase control of coherent population trapping (CPT) [7], EIT [8], absorption and dispersion [12], as well as cross-phase modulation (XPM) [15], etc., have been reported both theoretically and experimentally in a closed-loop atomic system.

Several approaches for optical switching [17–31] have been proposed and demonstrated theoretically and experimentally based on quantum coherence and interference. Harris and Yamamoto [18] suggested the scheme of photon switching by quantum interference, in which switching of a single photon by another is achievable. Yan et al. [21] and Braje et al. [23] demonstrated this scheme; Chen et al. [24] reported switching at the light level of about one photon per atomic cross section. Schmidt and Ram [25] investigated all-optical wavelength converter and switching based on linear absorption modulation by EIT in a three-state system. Wu et al. [26] investigated optical switching via tunable Fano-type interference in asymmetric quantum wells (QWs). Recently, using a two-photon absorption scheme in an alkali-metal vapor cell, Yavuz [27] suggested a technique where a strong laser beam switches off another laser beam of different wavelength in femtosecond time scales.

On the other hand, the nitrogen-vacancy (NV) center in diamond nanocrystal have recently emerged as a particularly strong candidate for solid-state quantum physics experiments and quantum information processing because they possess a long electronic spin decoherence time at room temperature [32–47]. Owing to the potential key roles of NV center in quantum information and quantum computing, many properties and interesting phenomena about NV center are researched and discussed recently. Santori et al. [48] demonstrated coherent population trapping in single NV center in diamond under optical excitation, the results show that all-optical control of single spins is possible in diamond. Fuchs et al. [49] investigated spin coherence during optical excitation of a single NV center in diamond nanocrystal at room temperature using Ramsey experiments. Their measurements indicate the process fidelity is 0.87 ± 0.03 and extrapolation to the moment of optical excitation is ≈0.95. Wang and Dobrovitski [50] study the applicability of the time-optimal bang-bang control designed for spin 1/2 to the rotation of the electron spin of a NV center in diamond. They find that the bang-bang control protocol decreases the rotation time by 20–25% in comparison with the traditional oscillating sinusoidal driving. More recently, Du et al. [51] realize continuous-wave dynamical decoupling (CWDD) and quantum gate operation in a single NV center in diamond. The coherence time of the NV center is prolonged by about 20 times with CWDD and the performance of the quantum gate of a given duration is greatly improved compared to the same quantum manipulation without CWDD.

Based on these achievements, we investigate theoretically phase-controlled absorption-gain and optical switching characteristics in a diamond NV center. The three-level NV centers are coupled by the small-signal probe field and the control laser field together with the microwave field, and thus the closed-loop configuration can be formed. To describe the transient behavior of the coupled system, we go beyond the steady-state approximation and simultaneously solve the coupled Bloch–Maxwell equations for the NV center and the probe field on numerical grids in time and space. We find that the continuous-wave input probe field is switched on and off as the relative phase of the applied field is periodically modulated, i.e., the switching status can be tuned by the relative phase of the applied field. Our study and the system are similar to atomic vapors in energy level structure, but our work is very different from previous atomic system. First of all, our numerical results for phase-controlled absorption-gain properties and optical switching of NV center are experimentally observed at low temperature because of their long electronic spin decoherence time, not as the atomic gas has to decreased temperature to several tens of μK to eliminate the Doppler broadening effect. Thus this relaxes the bandwidth and decoherence constraints of NV systems. Secondly, our study of controlling the diamond NV center with laser field and microwave field is more convenient than controlling atomic gas experimentally due to the solid state of NV center. Thirdly, compared with the previous proposed schemes [17–31], the goal of the present paper is to control the propagation dynamics of a weak probe signal in pulsed regime under phase-modulation operations through the above-mentioned closed-loop three-level ∇-type NV center. To the best of our knowledge, the phase-controlled light switching at low-light intensities via a microwave-driven closed-loop NV center has rarely been investigated so far.

The outline of this paper is organized into five parts as follows. In Sect. 2, we establish physical model under study and derive the Hamiltonian of the system and the evolution equations of the electron operators, then we analyze nonlinear optical effects in the steady-state regime, i.e., the probe absorption-gain behaviors in such a three-level closed-loop NV center. In Sect. 3, to describe the transient system behavior we go beyond the steady-state approximation and solve numerically the coupled equations for the NV center and the probe field and further discuss the possibility of phase-controlled light switching. In Sect. 4, we give a possible experimental realization of our proposed scheme. Finally, our main conclusions are summarized in Sect. 5.

2 Description of NV center system and phase-dependent optical absorption-gain response

Schematic description of optical coupled system under investigation is illustrated schematically in Fig. 1. 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 we described in this paper is negatively charged with two unpaired electrons located at the vacancy, usually treated as electron spin-1. As a result, the spin–spin interaction leads to the energy splitting D gs = 2.88 GHz and D es = 1.42 GHz between m s  = 0 and m s  =  ±1 in the triplet ground state 3 A and in the triplet excited state 3 E [43, 52–56]. The NV center has a relatively complicated structure of excited states, where the individual excited state levels can be resolved at low temperature as shown in Refs. [43, 52]. The triplet excited state 3 E is associated with a broadband photoluminescence emission with the resonant zero phonon line (ZPL) of 637 nm (1.945 eV), which allows optical detection of individual NV defects using confocal microscopy. By combining two external laser fields with a microwave field, one can model the NV center as a ∇-type three-level structure, i.e., a ground state \(|1\rangle\) and two excited states \(|2\rangle\) and \(|3\rangle. \) Specifically, we label \(|^{3}A,m_{\rm s}=0\rangle, \) \(|E2\rangle=b|^{3}E_{2}, m_{s}=+1\rangle-a|^{3}E_{2}, m_{s}=0\rangle\) and \(|E1\rangle=a|^{3}E_{2}, m_{s}=+1\rangle+b|^{3}E_{2}, m_{s}=0\rangle\) (|a|2 + |b|2 = 1) as \(|1\rangle, \) \(|2\rangle\) and \(|3\rangle, \) respectively. \(|E1\rangle\) and \(|E2\rangle\) are the mixed spin sublevels of the excited state 3 E under the condition that there is some strain in the diamond crystal. A weak probe field with Rabi frequency \(2\Upomega_{\rm p}\) (amplitude E p and carrier frequency ω p, which is called as probe and denoted by “p”) is applied to the transition \(|1\rangle\leftrightarrow|3\rangle\) of frequency ω 31. The transition \(|1\rangle\leftrightarrow|2\rangle\) of frequency ω 21 is driven by a strong continuous-wave control laser with Rabi frequency \(2\Upomega_{\rm c}\) (amplitude E c and carrier frequency ω c, which is called as control and denoted by “c”). An assisting microwave-driving field with Lamor frequency \(2\Upomega_{\rm m}\) (amplitude B m and carrier frequency ω m, which is called as microwave and denoted by “m”) is used to resonantly couple upper two-folded levels, i.e., \(|2\rangle\) and \(|3\rangle\) through an allowed magnetic dipole transition [52] and to further form the closed-loop configuration. The decay rates from the excited state \(|3\rangle\) to the ground state \(|1\rangle\) and from the excited state \(|2\rangle\) to the ground state \(|1\rangle\) are γ 31 and γ 21, respectively. It should be pointed out that the dephasing decay rate of coherence (or the decoherence rate) among states \(|2\rangle\) and \(|3\rangle, \) denoted by γ dph23 , is quite small because the corresponding transition is non-electric-dipole allowed in our considered model [46, 48].

Fig. 1
figure 1

a Structure of the NV color center in the diamond lattice, consisting of a substitutional nitrogen (N) and a neighboring vacancy (V). b Energy diagram of electronic and spin levels for the NV center in diamond nanocrystal. Both the ground state (3A) and the excited state (3E) are spin triplets, with a zero-field splitting (D gs = 2.88 GHz, D es = 1.42 GHz) between the m s  = 0 and the m s  =  ±levels. Normally, optical transitions within a NV center are spin-conserving. However, such a set of excited states are highly sensitive to strain. For a wide range of strain parameters it is possible to obtain a ∇-type system with a common ground state coupled to two excited states due to factors such as strain which reduce the symmetry and primarily modify the excited states [46, 48]. For example, the mixed spin sublevels of the excited state 3E can be taken as \(|2\rangle=b|^{3}E_{2}, \, m_{s}=+1\rangle-a|^{3}E_{2}, \, m_{s}=0\rangle, \) \(|3\rangle=a|^{3}E_{2}, \ m_{s}=+1\rangle+b|^{3}E_{2}, \ m_{s}=0\rangle\) (|a|2 + |b|2 = 1) if there is some strain in the diamond crystal. The transitions \(|1\rangle\leftrightarrow|3\rangle, \) \(|1\rangle\leftrightarrow|2\rangle\) and \(|2\rangle\leftrightarrow|3\rangle\) within the NV center can interact with a weak probe laser with carrier frequency ω p and amplitude E p, a control laser with carrier frequency ω c and amplitude E c, and a microwave-driving field with carrier frequency ω m and amplitude B m (the transition \(|2\rangle\leftrightarrow|3\rangle\) is electric dipole forbidden transition while magnetic dipole allowed). The levels are labeled as \(|1\rangle, \) \(|2\rangle, \) and \(|3\rangle ,\) respectively. The transitions \(|1\rangle\rightarrow|2\rangle\rightarrow|3\rangle\rightarrow|1\rangle\) owns a closed-loop three-level configuration. \(\Updelta_{\rm p}\) and \(\Updelta_{\rm c}\) are the frequency detunings of the corresponding probe and control optical fields, see text for more details

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In the interaction picture, with the rotating-wave approximation (RWA) and the dipole approximation (DA), according to the spirit of Refs. [57, 58] via choosing the proper free Hamiltonian, the resulting 3 × 3 interaction Hamiltonian describing the NV center-field interaction for the system under study can be expressed as (taking \(\hbar=1\))

$$ {\mathcal{H}}_{\rm int}= \left[\begin{array}{lll} 0 &\qquad -\Upomega_{\rm c}^{*}& \qquad -\Upomega_{\rm p}^{*}\\ -\Upomega_{\rm c} & \qquad \Updelta_{\rm c} & \qquad-\Upomega_{\rm m}^{*}{\text{e}}^{-i\phi}\\ -\Upomega_{\rm p} & \qquad -\Upomega_{\rm m}{\text{e}}^{i\phi} & \qquad \Updelta_{\rm p} \end{array}\right], $$
(1)

where we have taken the ground level \(\left|1\right\rangle\) as the energy origin for the sake of convenience. \(\Upomega_{n} (n=p, c, m)\) are one-half Rabi and Lamor frequencies for the respective transitions, i.e., \(\Upomega_{\rm p}=\mu_{31}E_{\rm p}/\left(2\hbar\right), \quad \Upomega_{\rm c}=\mu_{21}E_{\rm c}/\left(2\hbar\right), \quad {\rm and} \ \Upomega_{\rm m}=\mu_{32}B_{\rm m}/\left(2\hbar\right), \) with \(\mu_{kl}=\vec{\mu}_{kl}\cdot\vec{e}_{L}\) denoting the dipole moment for the transition between levels \(\left\vert k\right\rangle\) and \(\left\vert l\right\rangle (\vec{e}_{L}\) is the unit polarization vector of the corresponding laser field). It should be noted that another commonly used definition of the Rabi frequency is: e.g., \(\Upomega_{\rm p}=\mu_{31}E_{\rm p}/\hbar, \) which differs by a factor of two from the definition used here. The frequency detunings \(\Updelta_{\rm p}\) and \(\Updelta_{\rm c}\) is, respectively, given by \(\Updelta_{\rm p}=\omega_{31}-\omega_{\rm p}\) and \(\Updelta_{\rm c}=\omega_{21}-\omega_{\rm c}. \) The symbol ϕ is the relative phase, it is only relevant to the initial phases of the applied fields and is defined by ϕ = ϕ p − ϕ c − ϕ m. In the above derivation process of the Hamiltonian operator (1), we have assumed that the central frequencies of these three optical fields satisfy ω p = ω c + ω m in order to make the density matrix equations explicitly time independent, thus the relationship \(\Updelta_{\rm p}=\Updelta_{\rm c}\) can be automatically obtained and vice versa. In this regard, all discussions followed is under the condition of \(\Updelta_{\rm p}=\Updelta_{\rm c}. \)

To proceed further, we would like to address two points as follows:

Firstly, it is quite obvious from the general structure of Fig. 1 that two possible pathways from state \(|1\rangle\) to state \(|3\rangle\) exists: the direct one \(|1\rangle\buildrel{E_{\rm p}} \over {\longrightarrow}|3\rangle\) and the indirect one \(|1\rangle\buildrel{E_{\rm c}} \over {\longrightarrow}|2\rangle\buildrel{B_{\rm m}} \over {\longrightarrow}|3\rangle. \) The role of the relative phase ϕ on the nonlinear optical effect in such a closed-loop three-level system can be explained from quantum interference induced by these two pathways. As a result, the relative phase ϕ can be used as a control parameter to investigate the nonlinear optical behaviors.

Secondly, according to Eq. (1), the relative phase is placed on the microwave-driving field. In fact, the phase dependence in a closed-loop three-level NV center can be imparted to anyone of the applied fields. It is evident that the phase dependence would vanish provided that no microwave-driving field is applied. In order to account for the effect of the relative phase on the dynamic propagation, we will keep the initial phases of the probe and control laser fields fixed and modulate the initial phase of the microwave-driving field.

To give a full description of the dynamical evolution of the closed-loop NV center under study, by adopting the standard procedures we can easily obtain the following optical Bloch equations about density matrix elements ρ ij from Eq. (1) and \(\frac{\partial\rho}{\partial t}=-i\left[\mathcal{H}_{int}, \rho\right]+\Uplambda\rho: \)

$$ \frac{\partial\rho_{11}}{\partial t}=\gamma_{21}\rho_{22}+\gamma_{31}\rho_{33}+i\Upomega_{\rm p}^{*}\rho_{31}-i\Upomega_{\rm p}\rho_{13} +i\Upomega_{\rm c}^{*}\rho_{21}-i\Upomega_{\rm c}\rho_{12}, $$
(2)
$$ \frac{\partial\rho_{22}}{\partial t}=-\gamma_{21}\rho_{22}+i\Upomega_{\rm c}\rho_{12}-i\Upomega_{\rm c}^{*}\rho_{21} +i\Upomega_{\rm m}^{*}e^{-i\phi}\rho_{32}-i\Upomega_{\rm m}e^{i\phi}\rho_{23}, $$
(3)
$$ \frac{\partial\rho_{33}}{\partial t}=-\gamma_{31}\rho_{33}+i\Upomega_{\rm p}\rho_{13}-i\Upomega_{\rm p}^{*}\rho_{31} +i\Upomega_{\rm m}e^{i\phi}\rho_{23}-i\Upomega_{\rm m}^{*}e^{-i\phi}\rho_{32}, $$
(4)
$$ \frac{\partial\rho_{12}}{\partial t}=\left(i\Updelta_{\rm c}-\frac{\gamma_{21}}{2}\right)\rho_{12}+i\Upomega_{\rm p}^{*}\rho_{32} +i\Upomega_{\rm c}^{*}\left(\rho_{22}-\rho_{11}\right)-i\Upomega_{\rm m}e^{i\phi}\rho_{13}, $$
(5)
$$ \frac{\partial\rho_{13}}{\partial t}=\left(i\Updelta_{\rm p}-\frac{\gamma_{31}}{2}\right)\rho_{13}+i\Upomega_{\rm p}^{*}\left(\rho_{33}-\rho_{11}\right) +i\Upomega_{\rm c}^{*}\rho_{23}-i\Upomega_{\rm m}^{*}e^{-i\phi}\rho_{12}, $$
(6)
$$ \frac{\partial\rho_{23}}{\partial t}=\left[i\left(\Updelta_{\rm p}-\Updelta_{\rm c}\right)-\frac{\gamma_{21}+\gamma_{31}}{2}-\gamma_{23}^{dph}\right]\rho_{23}+i\Upomega_{\rm c}\rho_{13} -i\Upomega_{\rm p}^{*}\rho_{21}\\ +i\Upomega_{\rm m}^{*}e^{-i\phi}\left(\rho_{33}-\rho_{22}\right), $$
(7)

where the overdots represent the derivative with respect to time t and closure of this NV center requires that \(\sum\nolimits_{j=1}^{3}\rho_{jj}=1 \hbox{ and } \rho_{ij}=\rho_{ji}^{*}. \)

By a straightforward semiclassical analysis, the above matrix elements can be used to calculate the total complex susceptibility χ p of the probe laser field for the \(|1\rangle\leftrightarrow|3\rangle\) transition, i.e.,

$$ \chi_{\rm p}=\frac{{\mathcal{N}}\left|\mu_{31}\right|^{2}\rho_{31}}{2\hbar\varepsilon_{0}\Upomega_{\rm p}} =\frac{{\mathcal{N}}\left|\mu_{31}\right|^{2}}{2\hbar\varepsilon_{0}}\chi'_{\rm p}, $$
(8)

where \(\chi'_{\rm p}=\frac{\rho_{31}}{\Upomega_{\rm p}}. \) \(\mathcal{N}\) is the electron number density in the NV center and \(\varepsilon_{0}\) is permittivity in free space, respectively. As a result, the absorption-gain coefficient for the probe laser field coupled to the \(|1\rangle\leftrightarrow|3\rangle\) transition is directly proportional to the imaginary part Im\(\left(\chi'_{\rm p}\right). \)

In the following, we begin with investigating the absorption-gain properties of such a weak probe field by numerically solving the above density matrix equations (2)–(7) in the steady state by means of some nice Matlab-7.1 codes. Note that, in this paper, all involving parameters are scaled by \(\gamma_{31},\) which should be in the order of MHz for diamond NV center.

In Fig. 2, we plot probe absorption-gain coefficient Im(χ′p) as a function of the probe laser detuning \(\Updelta_{\rm p}\) with four different values of the relative phase \(\phi\) in the steady-state case. By comparing the curves in Fig. 2, it is clearly shown that, for four different relative phase ϕ, the exhibition of the probe absorption-gain profile is quite different. Adjusting the relative phase ϕ from 0 to 3π/2, the probe absorption-gain coefficient has a convert from zero value, positive value to negative value at the line center of the probe transition \(\Updelta_{\rm p}=0,\) correspondingly the transparency (see point B in Fig. 2), the absorption (see point A) and the gain (see point C) of the weak probe field can be realized in this case. Specifically, for the case that ϕ = 0 (solid curve), the probe absorption can be completely suppressed and the NV center becomes transparent to the weak probe field at the line center \(\Updelta_{\rm p}=0. \) While for the case of ϕ = π/2 (dashed curve), the probe field will be absorbed for all detunings and an absorption peak appears in the central part of the spectrum. For the case that ϕ = π (dotted curve), the transparency of the NV medium for the probe field can be observed again at the line center \(\Updelta_{\rm p}=0. \) When the relative phase ϕ is further increased up to 3π/2, namely, ϕ = 3π/2 (dash-dotted curve), the gain of the probe field appears obviously for all detunings and a gain peak appears in the central part of the spectrum. According to what has been analyzed above, we can arrive at the conclusion that the response of the small probe signal can be changed from the transparency (zero-absorption), the high absorption to the large gain by modulating the relative phase ϕ appropriately.

Fig. 2
figure 2

Probe absorption Im(χ′p) as a function of the probe detuning \(\Updelta_{\rm p}\) for four different values of the relative phase ϕ. The parameters are chosen as \(\Upomega_{\rm p}=0.2\gamma_{31}, \ \Upomega_{\rm c}=4\gamma_{31}, \ \Upomega_{\rm m}=1.5\gamma_{31}, \ \Updelta_{\rm c}=\Updelta_{\rm p}, \ \gamma_{21}=\gamma_{31}, \) and \(\gamma_{23}^{dph}=0.01\gamma_{31}, \) respectively

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In order to get a deeper insight into the dependence of the probe signal absorption-gain spectra on the relative phase ϕ, the probe absorption-gain coefficient Im(χ′p) as a function of relative phase \(\phi\) is plotted in Fig. 3. From the figure, we can find interesting and useful phenomena: (1) for some values of the relative phase ϕ = kπ (k is integer here and hereinafter) such as ϕ = 0 and π, complete transparency can be achieved; (2) for ϕ = 2kπ + π/2 such as ϕ = π/2 the peak value of the probe absorption occurs; and (3) for ϕ = 2kπ + 3π/2 such as \(\phi=3\pi/2\) the peak value of the probe gain can appear. These are in good agreement with the results obtained from Fig. 2. As a result, a proper change in the relative phase ϕ is enough to cause the system from complete transparency to high absorption [(1)\(\rightarrow\)(2)] or from high absorption to large gain [(2)\(\rightarrow\)(3)], so the considered three-level closed-loop NV system can be used for the phase switching of the probe absorption and/or amplification.

Fig. 3
figure 3

Probe absorption Im(χ′p) as a function of the relative phase ϕ. The parameters are chosen as \(\Upomega_{\rm p}=0.2\gamma_{31}, \ \Upomega_{\rm c}=4\gamma_{31}, \ \Upomega_{\rm m}=1.5\gamma_{31}, \ \Updelta_{\rm c}=0, \ \Updelta_{\rm p}=0, \ \gamma_{21}=\gamma_{31}, \) and \(\gamma_{23}^{dph}=0.01\gamma_{31}, \) respectively

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3 Phase-controlled light switching beyond the steady-state approximation

In this section, we turn to our main interest, the propagation dynamics of a light pulse, and investigate the switching characteristics beyond the steady-state approximation by numerically solving the coupled Bloch–Maxwell equations for the NV center and the probe laser field simultaneously on numerical grids as functions of space and time.

In order to study the propagation properties of probe laser field E p(z, t) along the z axis in the NV center, the following Maxwell’s wave equation in the slowly-varying-envelope approximation (SVEA) is required

$$ \frac{\partial E_{\rm p}(z,t)}{\partial z}+\frac{1}{\rm c}\frac{\partial E_{\rm p}(z,t)}{\partial t}=i\frac{\omega_{\rm p}}{2c\varepsilon_{0}}{\mathcal{P}}_{\rm p}(z, t), $$
(9)

where c is the light speed in free space. \(\mathcal{P}_{\rm p}(z,t)\) is the macroscopic coherent polarization in the \(|1\rangle\leftrightarrow|3\rangle\) transition, and is given by \(\mathcal{P}_{\rm p}\left(z, t\right)=\mathcal{N}\mu_{13}\rho_{31}. \)

Then, working in the local (retarded) frame where ξ = z and τ = t − z/c, according to ∂/∂ z = ∂/∂ ξ − 1/c∂/∂τ and ∂/∂ t = ∂/∂τ, the optical Bloch density matrix equations (2)–(7) for the NV center ρ ij (ξ, τ) and the Maxwell’s wave equation (9) for the time-dependent probe laser field \(\Upomega_{\rm p}(\xi,\tau)\) through the medium can be rewritten as

$$ \frac{\partial\rho_{11}}{\partial\tau}=\gamma_{21}\rho_{22}+\gamma_{31}\rho_{33} +i\Upomega_{\rm p}^{*}\rho_{31}-i\Upomega_{\rm p}\rho_{13}+i\Upomega_{\rm c}^{*}\rho_{21}-i\Upomega_{\rm c}\rho_{12}, $$
(10)
$$ \frac{\partial\rho_{22}}{\partial\tau}=-\gamma_{21}\rho_{22} +i\Upomega_{\rm c}\rho_{12}-i\Upomega_{\rm c}^{*}\rho_{21}+i\Upomega_{\rm m}^{*}e^{-i\phi}\rho_{32}-i\Upomega_{\rm m}e^{i\phi}\rho_{23}, $$
(11)
$$ \frac{\partial\rho_{33}}{\partial\tau}=-\gamma_{31}\rho_{33} +i\Upomega_{\rm p}\rho_{13}-i\Upomega_{\rm p}^{*}\rho_{31}+i\Upomega_{\rm m}e^{i\phi}\rho_{23}-i\Upomega_{\rm m}^{*}e^{-i\phi}\rho_{32}, $$
(12)
$$ \frac{\partial\rho_{12}}{\partial\tau}=\left(i\Updelta_{\rm c}-\frac{\gamma_{21}}{2}\right)\rho_{12} +i\Upomega_{\rm p}^{*}\rho_{32} +i\Upomega_{\rm c}^{*}\left(\rho_{22}-\rho_{11}\right)-i\Upomega_{\rm m}e^{i\phi}\rho_{13}, $$
(13)
$$ \frac{\partial\rho_{13}}{\partial\tau}=\left(i\Updelta_{\rm p}-\frac{\gamma_{31}}{2}\right)\rho_{13}+i\Upomega_{\rm p}^{*}\left(\rho_{33} -\rho_{11}\right)+i\Upomega_{\rm c}^{*}\rho_{23} -i\Upomega_{\rm m}^{*}e^{-i\phi}\rho_{12}, $$
(14)
$$ \frac{\partial\rho_{23}}{\partial\tau}=\left[i\left(\Updelta_{\rm p}-\Updelta_{\rm c}\right) -\frac{\gamma_{21}+\gamma_{31}}{2}-\gamma_{23}^{dph}\right]\rho_{23} +i\Upomega_{\rm c}\rho_{13}-i\Upomega_{\rm p}^{*}\rho_{21}\\ +i\Upomega_{\rm m}^{*}e^{-i\phi}\left(\rho_{33}-\rho_{22}\right), $$
(15)
$$ \frac{\partial \Upomega_{\rm p}}{\partial(\alpha\xi)}=i\gamma_{31}\rho_{31}, $$
(16)

above the propagation constants on the left-hand side (LHS) of Eq. (16) is given by \({\alpha=\frac{\mathcal{N}\omega_{\rm p}\left|\mu_{13}\right|^2}{4\hbar\varepsilon_{0}c\gamma_{31}}. }\) Solving the coupled Bloch equations (10)–(15) and Maxwell equation (16) for NV center and field is our starting point and central task, which governs the propagation dynamics of the probe laser field in the closed-loop NV center.

In what follows, we have made a numerical simulation starting from Eqs. (10)−(16) on a space-time grid by a fourth-order Runge–Kutta method combining a standard finite-difference approach for both the initial condition that electrons start in the ground state \(|1\rangle\) and the boundary condition that the initial injected probe laser field is a continuous wave signal at the beginning of the NV center. These detailed results are shown in Figs. 4, 5, 6, 7.

Fig. 4
figure 4

a Time evolution of a continuous-wave probe field at the exit of the medium ξ = 20/α under the variation of the relative phase ϕ(τ). b The relative phase ϕ(τ) versus time is a cosine-type wave ranged from 0 to π/2: \(\phi(\tau)=0.25\pi\cos(2\pi f\tau)+0.25\pi. \) Other parameters are \(\Upomega_{\rm p}=0.2\gamma_{31},\ \Upomega_{\rm c}=4\gamma_{31}, \ \Upomega_{\rm m}=1.5\gamma_{31},\ f=0.05\gamma_{31}, \ \Updelta_{\rm p}=0, \ \Updelta_{\rm c}=0,\gamma_{21}=\gamma_{31}, \) and γ dph23  = 0.01γ 31, respectively

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Fig. 5
figure 5

a Time evolution of a continuous-wave probe field at the exit of the medium ξ = 20/α under the variation of the relative phase ϕ(τ). b The relative phase ϕ(τ) versus time is a nearly-square wave with smooth rising and falling edges ranged from 0 to π/2: \(\phi(\tau)=0.5\pi\{1-0.5\tanh[0.4(\tau-10)] +0.5\tanh[0.4(\tau-35)]-0.5\tanh[0.4(\tau-60)]+0.5\tanh[0.4(\tau-85)]\}\). Other parameters are \(\Upomega_{\rm p}=0.2\gamma_{31}, \ \Upomega_{\rm c}=4\gamma_{31}, \ \Upomega_{\rm m}=1.5\gamma_{31}, \ \Updelta_{\rm p}=0, \ \Updelta_{\rm c}=0,\ \gamma_{21}=\gamma_{31}, \) and γ dph23  = 0.01γ 31, respectively

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Fig. 6
figure 6

a Time evolution of a continuous-wave probe field at the exit of the medium ξ = 20/α under the variation of the relative phase ϕ(τ). b The relative phase ϕ(τ) versus time is a cosine-type wave ranged from π/2 to 3π/2: \(\phi(\tau)=0.5\pi\cos(2\pi f\tau)+\pi. \) Other parameters are \(\Upomega_{\rm p}=0.2\gamma_{31},\ \Upomega_{\rm c}=4\gamma_{31}, \ \Upomega_{\rm m}=1.5\gamma_{31},\ f=0.05\gamma_{31}, \ \Updelta_{\rm p}=0, \ \Updelta_{\rm c}=0, \ \gamma_{21}=\gamma_{31}, \) and γ dph23  = 0.01γ 31, respectively

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Fig. 7
figure 7

a Time evolution of a continuous-wave probe field at the exit of the medium ξ = 20/α under the variation of the relative phase ϕ(τ). b The relative phase ϕ(τ) versus time is a nearly-square wave with smooth rising and falling edges ranged from π/2 to 3π/2: \(\phi(\tau)=\pi\{1.5-0.5\tanh[0.4(\tau-10)] +0.5\tanh[0.4(\tau-35)]-0.5\tanh[0.4(\tau-60)]+0.5\tanh[0.4(\tau-85)]\}.\) Other parameters are \(\Upomega_{\rm p}=0.2\gamma_{31}, \ \Upomega_{\rm c}=4\gamma_{31}, \ \Upomega_{\rm m}=1.5\gamma_{31}, \ \Updelta_{\rm p}=0, \ \Updelta_{\rm c}=0,\ \gamma_{21}=\gamma_{31}, \) and γ dph23  = 0.01γ 31, respectively

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Figure 4a plots the transmitted probe light intensity at the output as a function of time when the relative phase ϕ(τ) versus time is a cosine-type wave ranged from 0 to π/2: \(\phi(\tau)=0.25\pi\cos(2\pi f\tau)+0.25\pi\) as shown in Fig. 4b, which can be produced by a cosine voltage applied to an electro-optic modulator (EOM) [22]. Figure 5a plots the transmitted probe light intensity as a function of time when the relative phase ϕ(τ) versus time is a nearly-square wave with smooth rising and falling edges ranged from 0 to π/2, i.e., \(\phi(\tau)=0.5\pi\{1-0.5\tanh[0.4(\tau-10)] +0.5\tanh[0.4(\tau-35)]-0.5\tanh[0.4(\tau-60)]+0.5\tanh[0.4(\tau-85)]\}\) as shown in Fig. 5b. It is clearly shown from Figs. 4 and 5 that the periodic switching of the probe light transmission versus the relative phase ϕ can be formed efficiently in an anti-synchronous way with respect to the periodic phase change and the switching pattern is reversed through the π/2 phase reversal. The transmitted probe light either in the “open” or “close” position is seen to exhibit much the same profile. These results can be well explained in terms of the above-described behavior of the probe signal found in Figs. 2 and 3 of susceptibility, where absorption or gain are obtained depending on the relative phase value.

Figure 6a plots the transmitted probe light intensity as a function of time when the relative phase ϕ(τ) versus time is a cosine-type wave ranged from π/2 to 3π/2, which is generated by a cosine voltage applied to the EOM: \(\phi(\tau)=0.5\pi\cos(2\pi f\tau)+\pi\) as shown in Fig. 6b. Figure 7a plots the transmitted probe light intensity as a function of time when the relative phase ϕ(τ) versus time is a nearly-square wave with smooth rising and falling edges ranged from π/2 to 3π/2: \(\phi(\tau)=\pi\{1.5-0.5\tanh[0.4(\tau-10)] +0.5\tanh[0.4(\tau-35)]-0.5\tanh[0.4(\tau-60)]+0.5\tanh[0.4(\tau-85)]\}\) as shown in Fig. 7b. As shown in these figures, our results exhibit an obvious switching behavior in a synchronous way with respect to the periodic phase modulation, i.e., the phase control of the switching for the probe light transmission due to the quantum interference induced by the two transition pathways mentioned above. When the Maxwell wave equation is added to the Bloch equations, the variations of the susceptibility with the mentioned phase are manifested in the propagated transmitted field, so that if the relative phase varies in time following a periodic (cosine-type or nearly square-type) function, the transmitted intensity presents the same type of periodic variations in transmission–absorption, according to the value of the phase.

4 Possible experimental realization in this model

Before ending this section, let us briefly discuss the possible experimental realization of our proposed scheme of Fig. 1 by means of a single NV center, external cavity diode lasers (ECDL), and tunable microwave source, which are given as follows:

  1. 1.

    For NV centers, there is a relatively complicated structure of excited states, which includes six excited states [43, 52–56]. The optical transitions in NV centers are spin-conserving. However, non-spin-conserving optical transitions occur in some cases because a set of excited states are highly sensitive to strain in the diamond crystal. For a wide range of strain parameters it is possible to obtain a ∇-type system with a common ground state coupled to two excited states due to factors such as strain which reduce the symmetry and primarily modify the excited states [46, 48, 52, 56]. For example, if the states \(|^{3}E_{2}, m_{s}=+1\rangle\) and \(|^{3}E_{2}, m_{s}=0\rangle\) are mixed under strain, the resulting spin sublevels will be \(|E1\rangle=a|^{3}E_{2}, m_{s}=+1\rangle+b|^{3}E_{2}, m_{s}=0\rangle\) and \(|E2\rangle=b|^{3}E_{2}, m_{s}=+1\rangle-a|^{3}E_{2}, m_{s}=0\rangle\) with |a|2 + |b|2 = 1. So the transitions \(|^{3}A_{2}, m_{s}=0\rangle\leftrightarrow|^{3}E_{2}, m_{s}=-1\rangle\) and \(|^{3}A_{2}, m_{s}=0\rangle\leftrightarrow|^{3}E_{2}, m_{s}=+1\rangle\) will become allowed through either of the two states \(|E1\rangle\) and \(|E2\rangle. \) As a result, the closed-loop configuration can be in principle formed if and only if the excited states \(|^{3}E_{2}, m_{s}=+1\rangle\) and \(|^{3}E_{2}, m_{s}=-1\rangle\) are replaced by two excited states (e.g., \(|E_{x}\rangle\) and \(|E_{y}\rangle\)) in the six excited states defined by group theory in Ref. [43]. The energy-level scheme presented in Fig. 1b is only valid for the strained case. The excited states are split by their orbital components into two spin triplets. As reported in Refs. [55, 56], under strain, the non-spin-conserving transitions can be enhanced through the spin–orbit interaction. Alternatively, the individual excited state levels can be resolved at low temperature as shown in Refs. [43, 52].

    Based on these facts, we consider, for instance, the NV center on the transitions \(|E1\rangle\leftrightarrow|^{3}A_{2}\rangle\hbox{ and } |E2\rangle\leftrightarrow|^{3}A_{2}\rangle\) as a possible candidate under strain. A detailed coupling diagram and corresponding arrangements of experimental apparatus are shown in Fig. 1b. Specifically, the designated states and the decay rates can be chosen as follows: \(|1\rangle=|^{3}A_{2},\;m_{s}=0\rangle, |2\rangle=|E2\rangle=b|^{3}E_{2},\;m_{s}=+1\rangle-a|^{3}E_{2},\; m_{s}=0\rangle,\;|3\rangle=|E1\rangle=a|^{3}E_{2},\; m_{s}=+1\rangle+b|^{3}E_{2},\;m_{s}=0\rangle\) and the spontaneous decay rate from the state \(|2\rangle\) to \(|1\rangle\) and the state \(|2\rangle\) to \(|1\rangle\)can be taken as \(\gamma_{31}=\gamma_{21}\simeq15\) MHz (derived from [43]), respectively. In this case, the two coherent laser radiations at \(\omega_{\rm p} \hbox{ and } \omega_{\rm c}\) are side bands created by modulating the frequency of a laser tuned to the \(|^{3}A_{2}, m_{s}=0\rangle \hbox{ to } |E1\rangle \; (|E2\rangle)\) transitions (ZPL at 637 nm).

  2. 2.

    A moderate power narrow-band continuous-wave GHz radiation from a GHz microwave generator is applied to drive the magnetic dipole transition between the mixed spin sublevels \(|2\rangle=|E2\rangle=b|^{3}E_{2}, m_{s}=+1\rangle-a|^{3}E_{2}, m_{s}=0\rangle\) and \(|3\rangle=|E1\rangle=a|^{3}E_{2}, m_{s}=+1\rangle+b|^{3}E_{2}, m_{s}=0\rangle\) with the strain splitting ω 23∼10.6 GHz [56]. The magnetic dipole moment μ 32 of the microwave transition is ∼2μ B [52], where μ B = 9.27401 × 10−24 J/T is the Bohr magneton. For a given value of Lamor frequency, the amplitude of the microwave field can be deduced by the relationship \(\Upomega_{\rm m}=\frac{\mu_{32}B_{\rm m}}{2\hbar}, \) then the power of the microwave field can be deduced. For example, in the case of \(\Upomega_{\rm m}=1.5\gamma_{31}\) in our numerical estimation, the amplitude of the microwave field is \(B_{\rm m}=\frac{2\hbar\Upomega_{\rm m}}{\mu_{32}}\simeq2.6\times10^{-4}\) T, then the power of the microwave field is \(I=\frac{1}{2}\frac{\rm c}{\mu_{0}}\left|B_{\rm m}\right|^{2}\simeq806.9\) W/cm2. So it is feasible for this NV center to be driven by the microwave field in experiment.

  3. 3.

    Before interacting with the NV center, the probe field E p and the control field E c pass through an acoustic–optic modulator (AOM) and can be controlled individually. For the above two fields (probe and control) and this microwave-driving field, the relative phase ϕ can be controlled by phase locking between the GHz microwave generator and the AOM.

5 Conclusion

To summarize, we have examined the phase-controlled absorption-gain spectra and light switching in the three-level closed-loop NV center under different dynamic conditions. The system interacts with the weak probe field, the strong control field and the microwave field. By numerically solving the coupled Bloch–Maxwell equations for NV center and optical field simultaneously in space and time, we find that the continuous-wave input probe field can be switched on and off when the relative phase of the applied field is externally varied periodically in time. According to our analysis, these interesting phenomena, such as light gain and switching status, should be observable at low temperature in realistic experiments by using NV center in diamond nanocrystal, appropriate diode lasers and microwave source. The research presented in the paper may be useful in view of recent activity aiming at a solid-state quantum register.