Transparency and tunable slow and fast light in a nonlinear optomechanical cavity

We investigate theoretically the optical response of the output field and the tunable slow and fast light in a nonlinear optomechanical cavity with a degenerate optical parametric amplifier (OPA) and a higher order excited atomic ensemble. Studies show that the higher-order-excitation atom which is similar to the degenerate OPA that acts as a nonlinear medium, induces an additional dip in absorption spectrum of the probe field. The coherence of the mechanical oscillator leads to split the peak in absorption in the probe field spectrum so that the phenomenon of optomechanically induced transparency (OMIT) is generated from the output probe field. In particular, the presence of nonlinearities with the degenerate OPA and the higher order excited atoms can affect significantly the width of the transparency windows, providing an additional flexibility for controlling optical properties. Furthermore, in the presence of the degenerate OPA, the optical-response properties for the probe field become phase-sensitive so that a tunable switch from slow to fast light can be realized.

In addition, we know the nonlinear optical effect of the optomechanical system can be obviously enhanced by adding a degenerate optical parametric amplifier (OPA). The degenerate OPA placed in the optomechanical system can increase the effective optomechanical coupling between the movable mirror and the cavity field, which results from increase of the photon numbers in the cavity. Some researches about the influences of the degenerate OPA on the propagation of the probe field are reported in refs 28,46,47, where the optical properties of the output field, i.e., the width of OMIT, can be controlled easily by adjusting the pump amplitude of the degenerate OPA. On the other hand, when the degenerate OPA is included, the quantum interference effect between the probe field and the generated anti-Stokes field depends strongly on the phase of the degenerate OPA so that the optical-response properties for the probe field will become phase-sensitive. As so far, the response of nonlinear optomechanical cavity with an excited atomic ensemble to a weak probe light is not reported under the condition of the existence of the degenerate OPA. In this paper, we investigate the properties of absorption and dispersion of the probe field propagating the optomechanical system including low-or high-excitation atomic medium and a degenerate OPA. Also, we contrast the role of the nonlinearities of the higher-order excitation of atoms with one of the degenerate OPA in the optical properties of the output field. Further, OMIT behavior in absorption of the probe field generated through the optomechanical coupling is discussed in detail, which is influenced by the higher order excitation of the atoms and the degenerate OPA. We also discuss in detail how to control the switch from slow to fast light of the output probe field by adjusting the phase of the field driving the degenerate OPA. The result suggests that the phase-sensitive interference effects have potential applications to provide new tools for controlling and engineering light propagation.

Optomechanical Model and Hamiltonian
As shown schematically in Fig. 1, the model that we consider is a nonlinear optomechanical system, where a degenerate OPA and N identical two-level atoms (with transition frequency ω b and decay rate γ b ) are placed in a Fabry-Férot cavity with length L consisting of one fixed mirror and one movable mirror. Specifically, we consider a quadrupole transition between the 6s 2 1 S 0 ground state |g〉 and the 6s6p 3 P 1 excited state |e〉 of atomic barium for the two-level atoms 48,49 . The transition wavelength between the two states in atomic barium and the decay rate of the excited state to the ground state are λ = 791 nm and γ b = 47 kHz, respectively 48 . Further, we consider a realistic optical cavity, where the intracavity photon leakage can be occurred through the input and the movable mirror 33,50,51 . We assume the decay rates of two cavity mirrors are equal without loss of generality, i.e., κ 0 = κ 1 , where κ 0 and κ 1 are the decay rates of the cavity field through the input and the movable mirrors, respectively.
The cavity mode with frequency ω c is driven by a classical control field with frequency ω f and amplitude ε c as well as a weak probe field with frequency ω p and amplitude ε p , which then exerts an optical radiation pressure on the movable cavity mirror. Moreover, the system is pumped by an additional laser beam with coupling coefficient β to produce parametric amplification 28,46,47 . In general, the movable mirror is treated as a quantum-mechanical harmonic oscillator with resonance frequency ω m , effective mass m and damping rate γ m . Then, the total Hamiltonian of the system can be written as 15,28,43 Figure 1. Schematic of the setup studied in the paper. The optomechanical cavity contains N identical twolevel atoms and a degenerate OPA. The cavity mode is driven by a strong input laser field and a weak probe field through the left cavity mirror. The nonlinear crystal is pumped by an additional laser beam to produce parametric amplification.
, respectively. P c and P p are the laser powers.
For simplification, we use the Holstein-Primakoff transformation 54 , which is defined as , to rewrite Hamiltonian (0), where B and B † satisfy the fundamental commutation relation [B, B † ] = 1 54 . In general, in the low-excitation limit 〈 B † B〉 /N ≪ 1 for the atoms, the Holstein-Primakoff transformation becomes . In the present system, we assume that the low-excitation condition is broken slightly for the atomic ensemble so that the higher orders of the Holstein-Primakoff transformation should be included 45,55 , i.e.,

System Dynamics and Equation of Motion
For a detailed analysis of the system, we consider photon losses in the cavity through the input mirror with decay rate κ 0 and the movable mirror with decay rate κ 1 ; and Brownian noise acting on the mirror as well as decays associated with the atoms. Based on the Hamiltonian in Eq. (2), the quantum dynamics of the system can be described by the following Heisenberg-Langevin equation 33 : where a in 0 , a in 1 and B in are the input vacuum noise operators with zero mean value corresponding to the input mirror, the movable mirror and the atomic ensemble, which are fully characterized by the nonzero correlation , respectively 56 . ξ(t) is the Brownian stochastic force with zero mean value and its correlation function is characterized by where k B is the Boltzmann constant and T is the temperature of the reservoir related to the movable mirror. In order to studying the effect of the higher order excited atomic ensemble and the degenerate OPA on the optical properties of the output field in the optomechanical system, we need to investigate the motional equations of the quantum fluctuations around the mean values. Further, the mean values at steady state for the movable mirror, atomic ensemble and cavity field can be obtained from Eq. (3) by setting all time derivatives to 0. These are found to be is the total cavity decay rate. It is seen from Eq. (4) that the steady-state values of the system depend strongly on the nonlinear gain G A and the phase θ. Thus, the combine of the nonlinear optics and optomechanics may be used to control the optical properties of the output field.
To this end, one can split each operator in Eq. (3) into the steady-state mean value at the fixed point and a small quantum fluctuation, i.e., O = O s + δO(O = a, B, p, q). Further, we consider that there are a large number of photons in the cavity, i.e., |a s | ≫ 1, so that all the higher terms (δoδo) in Eq. (3) can be neglected. Then, the quantum Langevin equations for the fluctuations can be written as In order to solve Eq. (5), we use the ansatz δO = O + e −iδt + O − e iδt . Substituting this ansatz into Eq. (5), we can obtain the following solution of interest for the response of the cavity: 2 . The expression (6) and the steady-state values in Eq. (4) help us investigate the component of the output field oscillating at the probe frequency ω p , which relates to the higher order excitation of the atomic ensemble |b s | 2 as well as the degenerate OPA. In addition, the expression of a − is not necessary since this describes the four-wave mixing with frequency ω p − 2ω f for the driving field and the weak probe field. We can calculate the response of the system to all frequencies detected by the output field via the standard input-output theory 58 Using the above input-output relation and the ansatz δa = a + e −iδt + a − e iδt , we can express the mean value of the output field as It is noted that the second term on the right-hand side of Eq. (7) corresponds to the response of the whole system to the weak probe field at frequency ω p . Thus, we can examine the total output field at the frequency ω p by defining an amplitude of the rescaled output field corresponding to the weak probe field as where we have removed the constant term. The real and imaginary parts of the output probe field account for in-phase and out-phase quadratures of the output field spectra and can be written as , which describe the absorption and dispersion of the whole system to the weak probe field, respectively. In general, the modification of the probe response and the phenomenon of the transparency can be generated by the coupling the atoms or the mechanical motion with cavity field 41,42 . In the present optomechanical system, we investigate in detail the generation of dips in absorption induced by the atom-field and optomechanical couplings as well as the degenerate OPA.
Moreover, in the region of the narrow transparency window the rapid phase dispersion ϕ(ω p ) = arg(E out ) can cause the group delay expressed as τ = .
φ ω A positive group delay with τ g > 0 corresponds to slow light propagation and a negative group delay with τ g < 0 corresponds to fast light propagation 30,32,59 . In the following section, we also investigate theoretically a tunable switch from slow to fast light in the nonlinear optomechanical cavity by adjusting the nonlinear gain of the OPA and the phase of the field driving the OPA.

Results
In this section, we numerically evaluate the values of phase quadratures Re(E out ) and Im(E out ) and quantify the slow and fast light effects through the corresponding output field a + . Further, in order to demonstrate the phenomenon of transparency and the group delay of the probe field in the system, we select the accessible parameters in optomechanical systems 43,60 , i.e., the wavelength of the driving field λ f ≈ 791 nm, the total cavity length L = 0.001 m, the total cavity decay rate κ = 2π × 215 × 10 3 Hz and κ 0 /κ = 0.5, the frequency of the moving mirror ω m = 2π × 947 × 10 3 Hz, the mechanical factor Q = ω m /γ m = 6700, and the mass of the oscillating mirror m = 25 ng. In addition, we choose the parameters of atoms, i.e., the number of the atoms N = 10 6 , the atom-field coupling strength g = 2π × 2.3 × 10 2 Hz and the decay rate of atom γ b = 2π × 7.5 × 10 3 ≈ 47000 Hz 43,48 . We also consider that the cavity is driven on its red sideband, i.e., Δ = ω m . The atom-field detuning Δ b and the driving strength of the optical cavity ε c can be calculated in terms of the excitation number of the atoms |b s | 2 and the steady-state value b s . Transparency effect. We first consider that the low-excitation condition of atoms is satisfied, i.e., |b s | 2 ≪ 1.
Moreover, the degenerate OPA and the optomechanical coupling between the cavity field and the movable mirror are also removed or turned off, i.e., G A = 0 and G 0 = 0. In this case, the expression of a + in Eq. (6) can be simplified as It is found that the denominator of the response function is quadratic in δ so that the coherent coupling between the atomic collective mode and the optical mode can lead to a dip in absorption obtained as δ ω .  0 13 m and therefore the generation of the transparency behavior which is shown in Fig. 2(a), where we show the absorption Re(E out ) of the output field as a function of δ/ω m with b s = − 0.01 + 0.04i; the corresponding driving strength ε c = 35.12κ, atom-field detuning Δ b = 0.43κ and excitation number of atoms |b s | 2 ≈ 0.002. We can see from Fig. 2(a) that the positions of two peaks in absorption appear at δ  0 and δ ω  m , where the left peak in absorption results from the atom-field coupling 42 .
In order to compare with the above situation, we now explore the effect of the degenerate OPA on the atom-field coupled system. Correspondingly, the expression of the response of a + in Eq. (6) is simplified as The denominator of the response function is quadruplicate in δ. Then, under the condition of certain parameters, we can obtain more dips in absorption. Indeed, when the degenerate OPA is included in the system, i.e., G A = 0.7κ and θ = 3π/2 in Fig. 2(a), an additional dip in absorption appears near the left peak in absorption, i.e., δ ω − .  0 04 m , which results from the coupling between the degenerate OPA and the cavity field. Thus, we can obtain two transparent windows. Figure 2(b) depicts the dispersion shapes of the output field in the absence and presence of the degenerate OPA.
The additional dip in absorption that is similar to the one induced by the degenerate OPA can be generated by the higher order excitation of the atomic ensemble. In this case of presence of the higher order excited atomic ensemble and absence of the degenerate OPA and the optomechanical coupling, the amplitude of the output field [Eq. (6)] can be simplified as 3 2 G 1 A 4 are the terms related to the higher order of the Holstein-Primakoff transformation. One can clearly observe from Eq. (10) and Eq. (11) that the higher order coupling terms of the atoms play a role of nonlinear medium similar to the degenerate OPA for the generation of the transparency behavior.
In Fig. 3(a), we show the absorption Re(E out ) and dispersion Im(E out ) of the output field in the absence of the degenerate OPA as a function of δ/ω m , where we select b s = − 0.25 + 0.40i so that the excitation number of the atomic ensemble |b s | 2 = 0.22, the driving strength of the cavity ε c = 267.5κ and the atom-field detuning Δ b = 0.20κ. We see from Fig. 3(a) that for the selected system parameters there exists indeed an additional dip in absorption at δ ω − .  0 04 m which results from the higher order excitation of the atomic ensemble, corresponding to the terms of α 1,2,3,4 . Further, the higher order terms related to the |b s | 2 influence slightly the position of the dip in absorption at δ ω .  0 13 m as well as the peaks in absorption; such as, the dip in absorption is attained at δ ω .  0 10 m . In particular, when the low-excitation condition of atoms breaks slightly, a large driving strength i.e., ε c = 267.5κ but a relatively small atom-field detuning i.e., Δ b = 0.20κ can be applied to help observe the transparency behavior in the atom-field coupled system. In contrast, the low-excitation limit of atoms requires a small driving of the cavity but a relatively large atom-field detuning. For example, in the low-excitation limit, i.e., |b s | 2 = 0.002, the driving strength ε c = 35.12κ and the atom-field detuning Δ b = 0.43κ could be used [see Fig. 2]. Figure 3(b) demonstrates the transparency behavior in absorption of output probe field in the presence of the optomechanical coupling between the cavity field and the movable mirror when the degenerate OPA and the higher order excitation of the atomic ensemble are both included, i.e., G A = 0.7κ Hz and b s = − 0.25 + 0.40i. In this case, |b s | 2 = 0.22, ε c = 140.78κ and Δ b = 0.21κ. We can easily find from Fig. 3(b) that there exist three dips in absorption, where the peak in absorption at δ ω  m splits and therefore the optomechanically induced transparency (OMIT) appears due to the coupling between the cavity field and the movable mirror. This is because in general the strongest optomechanical coupling obtains at δ ω  m so that the probe beam interferes destructively with the anti-Stokes field generated by the mechanical oscillator and therefore a transparency window appears in the probe transmission spectrum 24 . Moreover, the positions of the left side of the two dips in absorption are δ ω − .  0 04 m and δ ω .  0 11 m , respectively, which depends on the atom-field coupling and the degenerate OPA. We can study the effect of the degenerate OPA on the transparency behaviors in the nonlinear optomechanical system with a higher order excited atomic medium. In Fig. 4(a), we show the absorption Re(E out ) in the presence of the OPA as a function of δ/ω m with different G A 's. Other parameter values are the same as in Figs 2 and 3. From Fig. 4(a), we see clearly that the larger the nonlinear gain of the degenerate OPA, the wider the OMIT windows. This is because the photon number in the cavity increases with increasing G A [see Eq. (4)]. Therefore, the effective optomechanical coupling G 0 a s can be increased by adding a degenerate OPA inside the optical cavity, which leads to widen the OMIT window. This suggest that the nonlinear gain of the degenerate OPA can be used to control the optical-response properties in the nonlinear optomechanical system and the width of OMIT window. Further, in the presence of both the higher order excitation of the atoms and the degenerate OPA, the left dip in absorption shallows, i.e., G A = 0.5κ corresponds to the blue dashed line in Fig. 4(a). This means that the roles of the higher order nonlinearity of atoms and the degenerate OPA cancel each other for generating the dip in absorption in the output field. When the nonlinear gain of the OPA becomes larger, i.e., G A = 1.0κ in Fig. 4(a), the degenerate OPA plays a determined role for the transparency of the probe field and therefore the left transparency window reappears.
We now discuss the important role of excitation of the atomic ensemble in the properties of the output field and OMIT. In Fig. 4(b), we show the absorption Re(E out ) as a function of δ/ω m with different b s 's in the absence of the degenerate OPA. We consider the three cases with b s = − 0.25 + 0.40i, b s = − 0.25 + 0.30i and b s = − 0.25 + 0.20i. The corresponding excitation numbers are calculated as |b s | 2 = 0.22, |b s | 2 = 0.15 and |b s | 2 = 0.10, respectively. In addition, the corresponding atom-field detunings are Δ b = 0.20κ, Δ b = 0.22κ and Δ b = 0.23κ, respectively, and driving strengths of the optical cavity are ε c = 267.50κ, ε c = 191.99κ and ε c = 137.23κ, respectively. From Fig. 4(b), we see that the increase of the excitation number of the atoms widens the OMIT window. This is because when the low-excitation condition of atoms is broken slightly, the effective optomechanical coupling strength is enhanced by the increase of the photon number, which depends strongly on the higher order excitation number of atoms. In addition, the additional dip in absorption shallows so that the left transparency behavior disappears when the excitation number decreases. These results can be applied to determinate the excitation number of atomic ensemble and its important role in the properties of the output field.
Tunable slow and fast light. As mentioned in previous section, the optical response of the system to the weak probe field can be described by the group delay. In Fig. 5, the group delay of the output field at the frequency of the probe field is plotted as a function of δ/ω m with different G A 's and θ's (a) and different b s 's (b). Here we consider the driving of the cavity field is so large that the low-excitation condition of atoms is broken slightly. For example, in Fig. 5(a) we still select b s = − 0.25 + 0.4i and therefore |b s | 2 = 0.22. This leads to generate an additional dip in absorption at δ ≈ 0 even in the absence of the degenerate OPA [see the red line in Fig. 4(b)]. In Fig. 5(b), we always remove the degenerate OPA, i.e., G A = 0 and focus justly on the effect of the higher-order excitation of atoms on the optical properties of the system. Other parameter values are the same as in Fig. 4.
We see clearly from Fig. 5(a) that in the absence of the degenerate OPA, a positive group delay of the output field is obtained at δ ≈ 0 or δ ≈ ω m , which corresponds to the slow light effect of the output probe field and results from the enhancement of the optical transparency induced by the higher order excitation of the atoms and the optomechanical coupling, respectively [see the red line in Fig. 5(a)]. When the degenerate OPA is included in the system, i.e., G A = 1.5κ and θ = π/2, the group delay at δ ≈ 0 can be turned to negative value and therefore the output probe field contains fast light. In contrast, the positive group delay at δ ≈ ω m is decreased due to the effect of the degenerate OPA. Moreover, when the degenerate OPA with the phase θ = 3π/2 is included, the group delays at δ ≈ 0 and δ ≈ ω m can be decreased due to the influence the degenerate OPA. In Fig. 5(b), we see that in the absence of the degenerate OPA, the group delay of the output field is always positive with different steady-state excitation number. Thus, only the phenomenon of slow light can be appeared. In Fig. 6, we show the group delays τ g at δ ≈ 0 and δ ≈ ω m , respectively, as a function of the phase θ/π with different G A 's. In Fig. 6(a), we can see that in the absence of the degenerate OPA, i.e., G A = 0 Hz, the group delay with δ ≈ 0 induced by the higher order excitation of the atoms is positive with τ g = 2.2μs and the transmitted probe field contains slow light. In the presence of the degenerate OPA, i.e., G A = 0.3κ, the positive group delay is not a monotonous function of the phase and a peak appears in the intermediate value of the phase. When the nonlinear gain of the OPA becomes larger, i.e., G A = 0.6κ, the group delay can be negative corresponding to the fast light within a finite interval of the phase around θ = 0.88π, whose position moves in the positive direction with increasing G A for the selected range of the parameters [see the dash-dot and dot lines in Fig. 6(a)]. Physically, when the degenerate OPA is added inside the atom-field coupled system, the quantum interference effect between the probe field and the anti-Stokes field generated by the atoms is related directly to the phase of the degenerate OPA by the coupling coefficient G 4 [see Eq. (6) or Eq. (10)]. Therefore, the optical-response properties for the probe field become phase-sensitive so that a tunable switch from slow to fast light can be realized by adjusting the phase of the degenerate OPA. In Fig. 6(b), we can see that in the absence of the degenerate OPA, i.e., G A = 0, the group delay with δ ≈ ω m induced by the OMIT is τ g = 54.5μs, which is much greater than that induced by the higher order atomic excitaiton. When the degenerate OPA is included, i.e., G A = 0.5κ, the effect of the degenerate OPA on the optomechanical system leads to the increase in the group delay for the intermediate value of the phase. In particular, when the nonlinear gain of the OPA becomes larger, i.e., G A = 2.5κ, a negative group delay and therefore the fast light appears in a small interval of the phase around θ = 0.36π. Similarly, the tunability of the group delay induced by the OMIT results from the quantum interference effect between the probe field and the anti-Stokes field generated by the mechanical oscillator, which depends strongly on the phase of the degenerate OPA. These results suggest that the phase-sensitive interference effect can be used to control the light propagation from slow to fast light of the transmitted probe field.

Discussion
In conclusion, we have studied a nonlinear optomechanical cavity with a degenerate OPA and a higher order excited atomic ensemble, which is driven by pump and probe laser fields. We derive the expression of the response of probe field in the system and demonstrate numerically the optical properties of the output field with experimentally accessible parameters. It is shown that there exist three dips in absorption in the output field in the presence of a higher order excited atomic medium, which induced by the optomechanical and atom-field couplings as well as the higher order excited terms related to the atom-field coupling, respectively. Further, the OMIT behavior in absorption of probe field can be generated through the optomechanical coupling between the cavity field and the movable mirror. We show that the higher order excitation of the atoms and the degenerate OPA can significantly affect the width of these transparency windows, which can be applied to determinate the excitation number of atoms and the important roles of nonlinear media in the optical properties of the output field. In particular, we find that the higher order excitation of the atoms and the degenerate OPA play a similar role of nonlinear medium in the generation of the additional transparency behavior in the probe field spectrum. We also discuss in detail the change in the magnitude of the group delay as well as a tunable switch from slow to fast light of the output probe field, where the group delay tunability is mainly due to the phase of the degenerate OPA. The phase-sensitive interference effects in the optical-response properties have potential applications to provide new tools for controlling and engineering light propagation.