Laser-polarization-dependent and magnetically controlled optical bistability in diamond nitrogen-vacancy centers

Physics Letters A 377 (2013) 2621–2627

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Physics Letters A www.elsevier.com/locate/pla

Laser-polarization-dependent and magnetically controlled optical bistability in diamond nitrogen-vacancy centers Duo Zhang a , Rong Yu b , Jiahua Li c,e,∗ , Chunling Ding d , Xiaoxue Yang c a

School of Electrical and Electronic Engineering, Wuhan Polytechnic University, Wuhan 430023, People’s Republic of China School of Science, Hubei Province Key Laboratory of Intelligent Robot, Wuhan Institute of Technology, Wuhan 430073, People’s Republic of China c School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China d School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China e Key Laboratory of Fundamental Physical Quantities Measurement of Ministry of Education, Wuhan 430074, People’s Republic of China b

a r t i c l e

i n f o

Article history: Received 3 April 2013 Received in revised form 31 July 2013 Accepted 1 August 2013 Available online 7 August 2013 Communicated by R. Wu Keywords: Diamond nitrogen-vacancy centers Optical ring cavity Optical bistability Elliptically polarized light

a b s t r a c t We explore laser-polarization-dependent and magnetically controlled optical bistability (OB) in an optical ring cavity filled with diamond nitrogen-vacancy (NV) defect centers under optical excitation. The shape of the OB curve can be significantly modified in a new operating regime from the previously studied OB case, namely, by adjusting the intensity of the external magnetic field and the polarization of the control beam. The influences of the intensity of the control beam, the frequency detuning, and the cooperation parameter on the OB behavior are also discussed in detail. These results are useful in real experiments for realizing an all-optical bistate switching or coding element in a solid-state platform. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Control of light by light is one of the most active research topics in quantum optics and nonlinear optics because of its potential application in optical transistors, all-optical communication, quantum information and optical computing. In the past few years, alloptical switching, and all-optical storage devices based on optical bistability (OB) in various quantum systems have been extensively investigated both theoretically and experimentally [1–10]. For example, Rosenberger et al. [4] discussed a theoretical model of OB in a two-level atom with a single mode field. Xiao group [5,6] demonstrated theoretically and experimentally the realization of OB in a three-level atomic system confined in an optical ring cavity. Harshawardhan and Agarwal [3] researched coherent control of OB using electromagnetic-field-induced transparency and quantum interferences, and demonstrated the possibility of control-fieldinduced multistability in two-level systems. Li et al. [8,9] studied the behavior of OB in semiconductor quantum well systems with tunneling-induced interference. Wu et al. [10] proposed a scheme for realizing OB and optical multistability (OM) in a double two-

*

Corresponding author at: School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China. Tel.: +86 2787557477; fax: +86 2787557477. E-mail addresses: [email protected], [email protected] (J. Li). 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.08.004

level atomic system and displayed the transition from OB to OM or vice versa by adjusting the relative phase between the control and probe fields. Recently, Sheng et al. [11] experimentally observed OM in an optical ring cavity containing three-level Λ-type Doppler-broadened rubidium atoms. Furthermore, some other researches have shown that the squeezed state field [12–14], the spontaneously generated coherence [15–18], the atomic cooperation parameter [19], the phase fluctuation [20,21], and the intensity of the microwave field [22] play a crucial role in controlling the bistable threshold intensity and the hysteresis loop. On the other hand, the nitrogen-vacancy (NV) centers in diamond nanocrystal have emerged as particularly strong candidates for solid-state quantum physics experiments and quantum information processing because they possess a long electronic spin decoherence time at room temperature, single-shot spin detection, subnanosecond spin control, and efficient quantum state transfer between electron and nearby nuclear spins [23–48]. Owing to the potential key roles of diamond NV centers in solid-state quantum information and quantum computing, many properties and interesting phenomena about them are researched and discussed recently. Fuchs et al. [32] 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 the extrapolation to the moment of optical excitation is ≈ 0.95. Yang et al. [34] investigated the dynamics of a laser-driven and dissipative system consisting

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of the two NV centers embedded in two spatially separated single-mode nanocavities in a planar photonic crystal, and analyzed the relevant entanglement dynamics in the presence of decoherence. Wang and Dobrovitski [44] studied the applicability of the time-optimal bang–bang control designed for spin 1/2 to the rotation of the electron spin of an 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. Santori et al. [47] demonstrated, for the first time, a coherent population trapping (CPT) in a single NV center in diamond under optical excitation, the results show that all-optical control of single spins is possible in diamond. More recently, Du et al. [48] realized 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. However, to the best of our knowledge, no related theoretical or experimental work has been carried out to realize OB in diamond NV centers, which motivates the current work. Based on these achievements, here we investigate the control of OB in diamond NV centers driven by an elliptically polarized coherent field and an external magnetic field inside a unidirectional ring cavity. Various parameters can be adjusted to change the bistable behavior, including a magnetic field. The physical mechanism of this type of OB is that the cavity field experiences third-order Kerr nonlinearities when it passes through an ensemble of NV centers in diamond, which can be coherently manipulated by an elliptically polarized control field. In the present system there is only one cavity field, since the control field (not on resonance in the cavity) is only used to prepare the coherent medium and generate a large Kerr nonlinearity. Our proposed scheme is mainly based on Refs. [47,49,50], but our work is drastically different from those works. First and foremost, we are mainly interested in showing the control of the OB behavior in an NV center system via the different parameters. Secondly, the intensity of external magnetic field can be used to control OB, which makes our scheme much more practical than the other schemes due to the magnetic field is more easily available and more effective control compared with an extra laser field. Thirdly, we only need one single elliptically polarized control field to couple two electric dipole transitions simultaneously and the polarization-dependent phase difference between the two circularly polarized components of the control field can control OB effectively. Naturally, the combination of an elliptically polarized laser field and an external magnetic field offers us further flexibility to manipulate OB. The Letter is organized as follows. In Section 2, we present the theoretical model and establish the corresponding equations for the ring cavity input–output relation. In Section 3, we give a detailed analysis and explanation for the behaviors of OB. In Section 4, we provide a possible experimental realization of our proposed scheme. Finally, our main conclusions are summarized in Section 5. 2. Description of physical model and cavity input–output relation We consider NV color centers in diamond consist of a substitutional nitrogen atom (N) plus a vacancy (V) in an adjacent lattice site as shown in Fig. 1(a), which is negatively charged with two unpaired electrons located at the vacancy, usually treated as electron spin-1 [51–53]. The spin–spin interaction leads to the energy splitting D gs = 2.88 GHz between the ground levels |3 A , ms = 0 and |3 A , ms = ±1 as depicted in Fig. 1(b). Meanwhile, the degeneracy of the ground sublevels |3 A , ms = ±1 can be lifted by employing an external static magnetic field B along

the quantized symmetry axis of diamond NV centers, which induces a Zeeman splitting 2 B . We label |3 A , ms = 0, |3 A , ms = −1 and |3 A , ms = +1 as |0, |1 and |2, respectively. The NV center has a relatively complicated structure of excited states [33], which includes six excited states defined by the method of group theory as | A 1  = √1 (| E − , ms = +1 − | E + , ms = −1), 2

| A2 =

√1

2

(| E − , ms = +1 + | E + , ms = −1), | E x  = | X , ms = 0,

| E y  = |Y , ms = 0, | E 1  = |E2 =

√1

2

(| E − , ms = −1 − | E + , ms = +1), and

√1 (| E − , m s 2

= −1 + | E + , ms = +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. Spin-conserving transitions between the ground state and six different excited states can be driven optically. Under moderate transverse strain, level anticrossings in the lower excited-state orbital (Ex) mix electron spin projection, permitting optical transitions from both ground-state ms = 0 and ms = ±1 manifolds [47,49], the state |3 can be considered as a mixed state of two excited states | E x  = | X , ms = 0 and | A 2  = √1 (| E − , ms = 2

+1 + | E + , ms = −1), i.e., |3 = α | A 2  + β| E x  with |α |2 + |β|2 = 1. The state |3 decays to the ground-state sublevels |1 and |2 with

radiation of σ + and σ − circular polarizations, respectively. While the state | E x  can be coupled to the ground state |0 with linear polarization [33,34]. Now we couple the transition |3 ↔ |0 with a linearly polarized probe field with a carrier frequency ω p and one-half Rabi frequency Ω p = μ30 E p /(2h¯ ), where E p is the amplitude of the probe field and μ30 is the electric dipole moment for the transition |3 ↔ |0. The electric dipole transitions from the excited state |3 to the two ground states |1 and |2 are driven simultaneously by an elliptically polarized control field with a carrier frequency ωc . The elliptically polarized control field can be regarded as a combination of the right- and left-circularly polarized components [50], which can be obtained by using a quarter-wave plate (QWP). An initial vertically polarized control  beam with intensity I 0 and electric field amplitude E 0 =

2I 0 εo c ,

where εo is the permittivity of free space and c the speed of light, becomes elliptically polarized after passing through the QWP that has been rotated by an angle θ (the polarization-dependent parameter), so the polarized control beam can be decomposed E into E c = E + σ + + E − σ − , where E + = √0 (cos θ + sin θ)ei θ and E E − = √0 (cos θ − sin θ)e−i θ . Here, 2

2

σ + and σ − are the unit vectors

of the right- and left-circularly polarized basis, respectively. When θ = 0 and π /2, we have E + = E − , that is, the control beam is linearly polarized. When θ = π /4 (3π /4), we have E − = 0 (E + = 0), that is, the control beam is right-handed circularly (left-handed circularly) polarized. The QWP can change the strengths and phase difference of the two electric field components. Then, the one-half Rabi frequencies become Ωc + = μ31 E + /(2h¯ ) = Ωc (cos θ + sin θ)ei θ and Ωc − = μ32 E − /(2h¯ ) = Ωc (cos θ − sin θ)e−i θ , here we assume μ31 = μ32 = μ (μ denotes the electric dipole √ moment between the corresponding transitions) and Ωc = μ E 0 /(2 2h¯ ). Under the rotating-wave approximation (RWA) and electricdipole approximation (EDA), the interaction Hamiltonian for our system can be written as [54–56] (taking h¯ = 1)

ˆ I =  p |33| + ( p − c +  B )|22| + ( p − c −  B )|11| H   − Ω p |30| + Ωc+ |31| + Ωc− |32| + H.c. ,

(1)

where H.c. means Hermitian conjugation, the notations c = ω31 −  B − ωc = ω32 +  B − ωc and  p = ω30 − ω p are the detunings of the electronic transitions from the corresponding laser frequencies.

 B is the Zeeman shift of levels |1 and |2 in the presence of an external magnetic field (see Fig. 1(b)). Then, by the standard

D. Zhang et al. / Physics Letters A 377 (2013) 2621–2627

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Fig. 1. (a) Structure of the NV color center in the diamond lattice, consisting of a substitutional nitrogen (N) and a neighboring vacancy (V) [52,53]. (b) A schematic diagram of an NV center interacting with an elliptically polarized control field. A right(left)-hand circularly polarized component Ωc + (Ωc − ) of the elliptically polarized control field induces the transition |3 ↔ |1 (|3 ↔ |2). The state |3 can be considered as a mixed state of two excited states | A 2  = √1 (| E − , ms = +1 + | E + , ms = −1) and 2

| E x  = | X , ms = 0 under strain (i.e., |3 = α | A 2  + β| E x  with |α |2 + |β|2 = 1), |3 A , ms = 0, |3 A , ms = −1 and |3 A , ms = +1 are labeled as |0, |1 and |2, respectively. A linearly polarized probe field drives the transitions |3 and |0. The symbol c is the frequency detunings of the control field, see text for details. By applying a uniform magnetic field along the quantized symmetry axis of the NV center, the degeneracy among the ground states |1 and |2 is lifted, where an amount  B stands for the Zeeman shift. D gs = 2.88 GHz is the zero-field splitting between the ground state sublevels |3 A , ms = 0 and |3 A , ms = ±1 (|3 A , ms = ±1 are degenerate at zero magnetic field) of the NV center. (c) A unidirectional ring cavity containing three mirrors (M 1 , M 2 and M 3 ) and diamond NV center sample of length L. E Ip and E Tp are the incident and transmitted fields, respectively. E c is the control field which is non-circulating in the ring cavity. Before entering the cavity, the vertically polarized beam E c becomes elliptically polarized after passing through the QWP that has been rotated by an angle θ , and the elliptically polarized control light can be decomposed into two components Ωc+ and Ωc− .

approach, the time-dependent density matrix equations of motion can be written as

are due primarily to longitudinal optical (LO) phonon emission events at low temperature. The total decay rates γi j (i = j ) are

ρ˙22 = i Ωc∗− ρ32 − i Ωc− ρ23 + Γ2 ρ33 ,

(2)

given by

ρ˙11 = i Ωc∗+ ρ31 − i Ωc+ ρ13 + Γ1 ρ33 ,

(3)

ρ˙00 = i Ω p∗ ρ30 − i Ω p ρ03 + Γ0 ρ33 ,

(4)





ρ˙31 = −i (c +  B ) − γ31 ρ31 − i Ωc+ ρ33 + i Ω p ρ01 + i Ωc+ ρ11 + i Ωc− ρ21 ,   ρ˙32 = −i (c −  B ) − γ32 ρ32 − i Ωc− ρ33 + i Ω p ρ02 + i Ωc+ ρ12 + i Ωc− ρ22 ,

(5)

(6)

ρ˙30 = [−i  p − γ30 ]ρ30 − i Ω p ρ33 + i Ω p ρ00 + i Ωc+ ρ10 + i Ωc− ρ20 , (7)   ∗ ρ˙20 = −i ( p − c +  B ) − γ20 ρ20 − i Ω p ρ23 + i Ωc− ρ30 , (8)

ρ˙21 = [−i2 B − γ21 ]ρ21 − i Ωc+ ρ23 + i Ωc∗− ρ31 , 



(9)

ρ˙10 = −i ( p − c −  B ) − γ10 ρ10 − i Ω p ρ13 + i Ωc∗+ ρ30 ,

(10)

ρi j = ρ ∗ji and the carrier conservation condition ρ00 + ρ11 + ρ22 + ρ33 = 1. The population decay rates and dephastogether with

ing decay rates are added phenomenologically in the above density matrix equations [57–60]. The population decay rates from the excited state |3 to the ground states | j  denoted by Γ j ( j = 0, 1, 2),

dph dph dph γ = γ /2 + γ30 , γ31 = γ /2 + γ31 , γ32 = γ /2 + γ32 ,  dph dph =γ γ20 = γ20 and γ21 = γ21 , where γ = i Γi and 30 dph 10 ,

γ10 γidph is the dephasing decay rate of the quantum coherence of the j

|i  ↔ | j  transition [57–60]. Note that in the following numerical calculations, the choices of the system parameters are based on the result of Ref. [47]. Now, we put the NV center sample in a unidirectional ring cavity as shown in Fig. 1(c). For simplicity, we assume that the mirror M 3 is the perfect reflector, and the intensity reflection and transmission coefficients of the mirrors M 1 and M 2 are R and T (with R + T = 1), respectively. The total electromagnetic field can be written as E = E p e −i ω p t + E c e −i ωc t + c.c., where “c.c.” means the complex conjugation. As we know, the probe field circulates in the ring cavity while the control field does not circulate in the ring cavity. So, the dynamics of the probe field in the optical cavity is governed by Maxwell’s equation, under slowly varying envelope approximation (SVEA), which is given by

∂Ep ∂Ep ωp +c =i P (ω p ), ∂t ∂z 2ε0

(11)

where ε0 and c are the permittivity of free space and the speed of light in vacuum, respectively. P (ω p ) is the slowly oscillating term of the induced polarization in the transition |3 ↔ |0, and is determined by P (ω p ) = N μ03 ρ30 , where μ03 = μ∗30 and N is the number density of the NV center sample.

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For a perfectly tuned cavity, the cavity boundary conditions are

√

E p ( L ) = E Tp / T , E p (0) =

√

(12)

T E Ip + R E p ( L ),

(13)

where L is the length of the NV center sample. E p (0) and E p ( L ) are the intracavity field amplitudes at z = 0 and z = L, respectively. E Ip and E Tp are the input and output field amplitudes via the mirrors M 1 and M 2 . The second term on the right-hand side of Eq. (13) describes a feedback mechanism due to the mirrors, which is essential to give rise to OB, that is to say, there is no OB when R = 0. Under the steady-state and mean-field limits [61], using the cavity boundary conditions Eqs. (12) and (13), then normalizing the input and output fields by letting y =

μ03 E Ip h¯

√

T

can get the ring cavity input–output relation as

y = 2x − iC ρ30 (x), N ω L |μ |

and x =

μ03 E Tp 2h¯

√

T

, we

(14) 2

where C = 2ph¯ ε cT03 is the cooperation parameter defined for OB. 0 It is worthwhile pointing out that the second term on the righthand side of Eq. (14), namely, f (x) ≡ −iC ρ30 (x), is vital for OB to take place. In the case of diamond NV centers, f (x) is a linear and quadratic polynomial in x (the explicit expression of f (x) is so complicated that it will not be presented here). This form of f (x), suggesting Kerr nonlinearities and complicated dependence of absorption and dispersion on various system parameters including the intensity of the external magnetic field (B or  B because  B ∝ B), the polarization and intensity of the control beam (θ and Ωc ), the frequency detuning ( p and c ), and the cooperation parameter (C ), is responsible for OB observed in this four-level NV center system. As a consequence, we can control and manipulate OB by adjusting these system parameters under certain conditions.

Fig. 2. (a) Output field |x| versus input field | y | for five different values of the magnetic field intensity B. Other parameters are c =  p = 0, θ = 0, Ωc = γ , and C = 300, respectively. (b) Plot of the absorptive curves versus the external magnetic field. Other parameters are the same as in panel (a) except for Ω p = 0.5γ .

3. Results and discussion In the following, we solve the density matrix equations (2)–(10) numerically in the steady-state condition and the ring cavity input–output relation (14) to get the result for OB under the assumption of specific parameters. For simplicity, in our numerical calculation, all involved parameters are reduced to dimensionless units by scaling γ , which should be in the order of MHz for diamond NV centers. Followed by the paper  of Santori et al. [47], we select the system parameters as γ = i Γi = 2π × 13.4 MHz, Γ0 = 0.8γ , Γ1 = Γ2 = 0.1γ , γ30 = γ31 = γ32 = 2.2γ , and γ10 = γ20 = γ21 = 0.1γ . In this approach, when the Zeeman shift  B is scaled by γ , according to the relationship 2 B = γe B, then the magnetic field strength B should be in units of the combined constant γc = 2γ /γe , where γe = 2.8 × 1010 Hz/T is the electron gyromagnetic ratio of NV centers. In addition, we assume the Rabi frequency Ω p is real. Next, we show that OB can be realized in the present NV center system, and discuss the influences of the corresponding system parameters on the behaviors of the OB, as shown in Figs. 2–6. First of all, we will analyze the influence of the applied magnetic field B on the shape of the OB curve. It is easy to see from Fig. 2(a) that the bistable threshold is dramatically increased as the intensity of the external magnetic increases, and then decreases gradually. In order to gain deeper insight into the above phenomena, we plot the absorption coefficients of the probe field Im(ρ30 ) (see Fig. 2(b)) versus the intensity of the external magnetic field. When the magnetic field B is absent (i.e., B = 0), the levels |1 and |2 are the same, the absorption of the NV center medium is suppressed maximally. With the increase of the strength of the magnetic field, the level splitting between |1 and

|2 is enhanced, the quantum interference between the two quantum paths |1 → |3 → |0 and |2 → |3 → |0 is reduced, which increases the absorption of probe field and makes it harder for the intracavity field to reach saturation. While when the external magnetic field B increases to a certain value (B is around 2γc ), the absorption of the probe field reaches a saturation value, the bistable threshold reaches to the maximum. When B further increases, the bistable threshold decreases slowly. Consequently, the threshold of OB first increases dramatically with the increase of the external magnetic field and then decreases gradually. In Fig. 3(a), we show the influence of the frequency detuning of the probe field  p on the behavior of OB, while keeping all other parameters fixed. It is clearly shown that, with increasing  p from 0 to 0.3γ , the threshold of OB decreases progressively. While when the frequency detuning continues to increase, the bistable threshold increases greatly and the area of the hysteresis loop becomes wider. The reason for the above results can be qualitatively interpreted as follows. The gradual increasing of the frequency detuning  p from 0 to 0.6γ lead to the corresponding change of the absorption for the probe laser field. The probe absorption first decreases to the minimum and then increases rapidly, as shown in Fig. 3(b), which makes the bistable threshold value reduces gradually and then increases obviously. In what follows, we will analyze the dependence of OB on the polarization-dependent phase difference θ between the two circularly polarized components of the control laser field, i.e., the ellipticity of the control laser field. In Fig. 4(a), we present numerical results for the steady state of the output field amplitude |x| as a function of the input field amplitude | y | for twelve different values of the phase difference θ . It is clearly shown that, the

D. Zhang et al. / Physics Letters A 377 (2013) 2621–2627

Fig. 3. (a) Output field |x| versus input field | y | for four different values of the probe-field frequency detuning  p . Other parameters are c = 0, B = 0.3γc , θ = 0, Ωc = γ , and C = 300, respectively. (b) Probe absorption Im(ρ30 ) as a function of the probe-field frequency detuning  p . Other parameters are the same as in panel (a) except for Ωc = Ω p = γ .

curves of the cavity input–output field vary periodically with the polarization-dependent phase difference θ between the two components of the control field. In order to investigate the physical nature of the above phenomenon, we plot the absorption spectrum of the probe field against the phase difference θ in Fig. 4(b). When the phase difference θ = 0, we have the result E + = E − , that is, the control beam is linearly polarized and the intensity of rightcircularly component is the same with the left-circularly component. The absorption of the probe field is larger and the OB threshold value is bigger accordingly. When the phase difference changes from 0 to 2π /9, the control beam becomes right-handed elliptically polarized and the absorption of the probe field decreases progressively. As a result, the OB threshold value decreases gradually and the area of the hysteresis cycle becomes narrower. While when phase difference θ = π /4, E − = 0 and the Rabi frequency Ωc− = 0 (i.e., right-handed circularly polarized control field), the system is similar to a typical Λ-type three-level atom model, the electromagnetically induced transparency (EIT) occurs easily and the absorption of the probe field is equal to zero. Consequently, the OB disappears accordingly. On increasing θ further from 5π /18 (i.e., right-handed elliptically polarization) to π /2 (i.e., linearly polarization), the absorption of the probe field is enhanced from 0 to maximum, the threshold value of the OB increases considerably and the area of the hysteresis cycle becomes broader. When the phase difference continues to increase from π /2 to π , that is, linearly polarized control field becomes a left-handed elliptically (circularly) polarization, and then continues to change into linearly polarization, the process described above is repeated periodically. Specifically, the absorption spectra of the probe field is

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Fig. 4. (a) Output field |x| versus input field | y | for twelve different values of the polarization-dependent phase difference θ . Other parameters are c =  p = 0, B = 0.3γc , Ωc = γ , and C = 300, respectively. (b) Probe absorption Im(ρ30 ) as a function of the polarization-dependent phase difference θ . Other parameters are the same as in panel (a) except for Ωc = γ and Ω p = 0.5γ .

Fig. 5. Output field |x| versus input field | y | for four different values of the controlfield intensity Ωc . Other parameters are c =  p = 0, B = 0.3γc , θ = 0, and C = 300, respectively.

strongly dependent on the laser-polarization-dependent phase θ and also is a periodical function of the relative phase θ with a period π . Furthermore, it can be seen that the relations [Im(ρ30 )]|θ = [Im(ρ30 )]|π /2−θ = [Im(ρ30 )]|3π /2−θ = [Im(ρ30 )]|π /2+θ hold for the probe absorption via solving Eqs. (2)–(8) (not shown here), which can well explain the numerical results of Fig. 4(a). As mentioned above, the polarization-dependent phase difference between the two circularly polarized components of the control beam can effectively influence the optical response of the NV center system.

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Fig. 6. Output field |x| versus input field | y | for four different values of the cooperation parameter C . Other parameters are c =  p = 0, B = 0.3γc , Ωc = γ , and θ = 0, respectively.

Therefore, the behavior of OB can be tuned by appropriately adjusting the polarization-dependent phase difference θ . Finally, the effects of the control field intensity Ωc and the cooperation parameter C on the OB behavior are depicted in Figs. 5 and 6, respectively. It is easy to see from Fig. 5 that the increasing intensity of the control field leads to the decrease of bistable threshold, which can be easily explained that an increasingly coherent control field will modify the absorption for the probe field on the transition |3 to |0 and affect the Kerr nonlinearity of diamond NV centers, which leads to the change of the OB behavior. So, we can control the threshold value simply by adjusting the intensity of the coherent control field Ωc . In Fig. 6, we show the dependence of OB on the cooperation parameter C . From N ω L |μ |2

the term C = 2ph¯ ε cT03 , we can see that the cooperation param0 eter C is directly proportional to the number density N of NV center sample, and the increase of the number density will enhance the absorption of the sample, which accounts for the raise of the bistable threshold when the cooperation parameter becomes larger. 4. Possible experimental realization Before ending this section, let us briefly discuss the possible experimental realization of our proposed scheme in Fig. 1 by means of diamond NV centers, an external magnetic field and an elliptically polarized control field, which are given as follows: (1) For diamond NV centers, there is relatively complicated structure of excited states, which includes six excited states [33, 62–64]. 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. Recent works have suggested that, for a wide range of strain parameters, it is possible to obtain a Λ-type four-level system with multiple ground states coupled to a common excited state even at zero magnetic field due to factors such as strain which reduce the symmetry and primarily modify the excited states [38,47,49]. The level diagram in Fig. 1(b) is appropriate for the strained case. The excited states are split by their orbital components into two spin triplets. As shown in Refs. [51,65], under strain, the non-spinconserving 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. [33,66]. We consider, for instance, the NV center on the transitions |3 ↔ |3 A , ms = 0 or |3 A , ms = ±1 as a possible candidate [26, 33,47]. A detailed coupling diagram and corresponding arrangements of experimental apparatus are shown in Fig. 1(b). Specif-

ically, the designated states and the decay rates can be chosen as follows: The state |3 can be considered as a mixed state√of two excited states | A 2  = (| E − , ms = +1 + | E + , ms = −1)/ 2 and | E x  = | X , ms = 0 under strain (i.e., |3 = α | A 2  + β| E x  with |α |2 + |β|2 = 1), |1 = |3 A , ms = −1, |2 = |3 A , ms = +1 and |0 = |3 A , ms = 0, respectively. In this case, the wavelength of elliptically polarized control field is 637 nm, and the right(left)-circularly polarized component of elliptically polarized control field is used to couple the transition |3 ↔ |1 (|3 ↔ |2). (2) We briefly address the requirement of the magnetic field, supplied by a permanent magnet. We can employ an external magnetic field B along the quantized symmetry axis of the NV center to induce a Zeeman shift  B . For example, when taking the intensity of the magnetic field B = 4γc in our numerical calculations, according to the relationship γc = 2γ /γe , where γe = 2.8 × 1010 Hz/T is the electron gyromagnetic ratio [67] and γ = 2π × 13.4 MHz is the spontaneous decay rate of diamond NV centers [47], the magnetic field strength B 2.4 × 10−2 T = 240 G is needed and can be easily fulfilled in the experiment. (3) The control field components Ωc + and Ωc − can be obtained through the following ways. Before entering the unidirectional ring cavity, the control beam passes through a half-wave plate (HWP) followed by a QWP [50]. The HWP makes the control beam vertically polarized, and the QWP has been rotated by an angle θ to control the polarization of the incoming vertically polarized control beam. After passing through the QWP, the control beam becomes elliptically polarized light which contains two components Ωc + and Ωc − . 5. Conclusion In summary, we have proposed a new scheme for realizing OB in an NV center system controlled by an elliptically polarized coherent field and an external magnetic field inside an optical ring cavity. We find that the intensity of the external magnetic field and control laser field, the polarization of the control laser field, the frequency detuning of the probe field, as well as the cooperation parameter can affect the OB behavior dramatically, which can be used to manipulate efficiently the bistable threshold intensity and the hysteresis loop. Our calculations also provide a guideline for optimizing and controlling the optical switching process in NV center system. We hope that our results may be helpful in real experiments for realizing an alloptical bistate switching or coding element in a solid-state platform. Acknowledgements We would like to thank Professor Ying Wu for his encouragement and helpful discussion. This research was supported in part by the National Natural Science Foundation of China under Grants No. 11004069, No. 11275074, and No. 10975054, by the Doctoral Foundation of the Ministry of Education of China under Grant No. 20100142120081, and by the National Basic Research Program of China under Contract No. 2012CB922103. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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