Temporal rocking in a nonlinear hybrid optomechanical system

Xiaotian Zhang; Jiteng Sheng; Haibin Wu

doi:10.1364/OE.26.006285

1. Introduction

Cavity optomechanics [1, 2] studies the physics of a mechanical object coupled to a cavity field via radiation pressure. The simplest and most commonly used system is a driven high finesse Fabry-Perot (FP) cavity with a movable end-mirror as a mechanical oscillator. Aiming to manipulate massive objects at the quantum level [3–5] and develop ultra-sensitive sensors [6,7], the coherent control of the center-of-mass motion of mechanical oscillators has been achieved on very impressive levels in the past decades. The ground-state cooling of mechanical modes has been realized in different optomechanical systems [8–11]. The sensitivity of displacement measurement has been greatly improved with a precision down to the standard quantum limit. Recently, sophisticated techniques for manipulating the mechanical oscillators and the cavity field have brought them to the quantum regime. Some important experiments include, for example, the observation of radiation pressure shot noise [12] and the optomechanical squeezing of light field [13,14].

Currently, the hybrid optomechanical systems have attracted a lot of attention due to the versatility to integrate different fantastic physical systems with the bare optomechanical setups. Among these systems, the most promising way to achieve fully reliable quantum control is to use the hybrid atomic optomechanical systems [15–24]. This is mainly due to the large coherent time and the well-controlled quantum toolbox at hand for atoms. Along this direction, the collective motion of ultracold rubidium atoms inside a FP cavity has been used to observe the signatures of shot-noise radiation pressure fluctuations [25]. The Bose-Einstein condensation of the atoms has been used to as a mechanical oscillator [26, 27]. The degenerate Fermi gas integrated into the optomechanical systems has been proposed [28–30]. Another motivation to implement such hybrid atomic optomechanical systems is that there exists a strong analogy between quantum optomechanics and nonlinear optics. Hence, many optomechanical effects can be directly mapped onto the well-known optical effects. The optical parametric amplifier inside a cavity has been proposed to considerably enhance the cooling of the micromechanical mirror [31]. The Kerr nonlinearity in the optomechanics cavity has been investigated to inhibit the normal mode splitting (NMS) due to the photon blockade and the large nonlinearity, which is essentially detrimental to the optomechanical cooling and will increase the effective temperature of the mechanical oscillators [32]. All the above phenomena are investigated with constant cavity driving or mechanical parameters. Recently, driving optomechanical systems with a mildly amplitude-modulated light field has been proposed to achieve large degrees of squeezing of oscillators without classical feedback or squeezed input light [33]. The new cooling resonances can be determined by the modulated frequency of mechanical oscillator [34]. Other quantum effects, such as entanglement and discord, can be greatly enhanced with parametric driving [35]. In nonlinear optics, an intriguing concept called rocking can convert a phase-invariant self-oscillatory system into a phase-bistable one [36]. Rocking has been experimentally realized in nonlinear optical systems [37] and nonlinear electronic circuits [38,39], and has been proposed to generate phase-bistable Kerr cavity solitons [40].

In this paper, we exploit the temporal rocking effect in a nonlinear hybrid optomechanical system. The optomechanical interactions between a light field and a mechanical oscillator (membrane) with a Kerr nonlinear medium inside a Fabry-Perot cavity are theoretically explored. The rocking states of the intracavity field and the membrane emerge with a large and fast amplitude-modulated driving field. Moreover, the rocking states are very robust and they exist even at zero average cavity driving. Due to the temporal rocking, the continuous phase symmetry of the cavity field is broken down to a discrete one. Each rocking state shows phase-bistable with exact π phase difference. The NMS and the optomechanical cooling of the mechanical oscillator are also investigated. The response of the optomechanical field is greatly modified by the temporal modulation of the driving field. The NMS and intracavity photon numbers can be controlled by the rocking parameter. The ground-state cooling can be achieved with the optimal rocking parameters.

The paper is organized as follows. In Sec. II we describe the theoretical model and derive the quantum Langevin equations. The steady-state solutions of the system are analyzed. In Sec. III we study the rocking and the phase-bistable states. In Sec. IV we calculate the spectrum of fluctuations in the position of the membrane and investigate the rocking enhanced NMS. In Sec. V we show how the membrane can be effectively cooled by the temporal rocking. Sec. VI serves as the conclusion and perspective.

2. Model

We consider a hybrid optomechanical system with a membrane coupled to a cavity field mediated by an intracavity Kerr nonlinear medium as sketched in Fig. 1. The Kerr nonlinearity can be realized with a highly detuned dispersive two-level atomic system or quantum dots. The Hamiltonian of the system can be written as

H = ℏ ω_{c} a^{†} a + ℏ ω_{m} b^{†} b + ℏ g a^{†} a (b + b^{†}) + ℏ η a^{† 2} a^{2} + H_{d},

where a is the intracavity field and ω_c is the cavity resonant frequency. The mechanical mode of the membrane is described by a pair of canonical bosonic operator b and b^† with the vibrational frequency ω_m. g is the optomechanical interaction strength. η represents the nonlinear coupling strength between the Kerr medium and the cavity field. For simplicity and without loss of generality, we assume that η is real here. H_d is the driving term with a temporally modulated field. H_d = iħa^†(E₀ + ϵ cos(Ωt)) exp(−iω_pt) + h.c., where E₀ is the constant driving amplitude with a real value, ϵ is the amplitude of the modulation, Ω is the frequency of the modulation, and ω_p is the frequency of the driving light field. The equations of motion in a frame rotating at the frequency of the laser field are derived as

\begin{array}{l} \dot{a} = (i Δ_{c} - κ) a - i g a (b + b^{†}) - 2 i η a^{†} a^{2} + E (t) + \tilde{α}, \\ \dot{b} = (- i ω_{m} - γ_{m}) b - i g a^{†} a + \tilde{ξ}, \end{array}

where ∆_c = ω_p −ω_c is the cavity detuning and E (t) = E₀ + ϵ cos(Ωt. κ and γ_m are the decay rates for the cavity field and the membrane, respectively.

\tilde{α}

and

\tilde{ξ}

are the quantum noise operators with the correlations

〈 \tilde{α} (t) {\tilde{α}}^{†} (t^{'}) 〉 = 2 κ δ (t - t^{'})

and

〈 \tilde{ξ} (t) {\tilde{ξ}}^{†} (t^{'}) 〉 = 2 γ_{m} (n_{i} + 1) δ (t - t^{'})

, respectively, where n_i is the initial thermal occupancy of the membrane oscillator.

Fig. 1 Schematic diagram of a hybrid optomechanical system with a movable membrane and Kerr nonlinear medium in an optical cavity. The optomechanical cavity is driven by an amplitude-modulated light field, E_driving = (E₀ + ϵ cos(Ωt)) exp(−iω_pt (E₀ ≡ 0 for the study of rocking). M represents the membrane. M1 and M2 are the cavity mirrors. a is the cavity field.

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For the rocking mode studied here, we consider the case of large and fast modulation, i.e., ϵ ≫ E₀ and Ω ≫ ω_m, ∆_c, κ, γ_m, g, η. The operators, a, b can be treated with the few-mode expansion [40] and written formally as a → a + a_±e ^±ⁱ^Ω^t, and b → b + b_±e ^±ⁱ^Ω^t, respectively, where all amplitudes are slowly varying. Substituting these expressions into Eq. (2) and neglecting the higher harmonics, a set of coupled equations is given by

\begin{array}{l} \dot{a} = (i Δ - κ) a - i g a (b + b^{†}) - 2 i η a^{†} a^{2} - i χ a^{†} + E_{0} + \tilde{α}, \\ \dot{b} = (- i ω_{m} - γ_{m}) b - i g (a^{†} a + C) + \tilde{ξ}, \end{array}

where ∆ = ∆_c − 2χ and χ = 2ηC. C ≡ ϵ²/(2Ω²) is defined as the rocking parameter. It is clear that the temporal modulation and the nonlinearity induce a shift of the cavity resonant frequency. More importantly, there exists a phase-sensitive term, i χa^†, in the cavity field equation, which is responsible to the phase-bistable states. Both a and −a are the solutions of the equations and obey the same dynamical evolution.

Setting all the time derivatives to zero and $〈 \tilde{α} 〉 = 0, 〈 \tilde{ξ} 〉 = 0$ , the steady-state solutions of the mechanical mode and the intracavity field are given by

b_{s} = - i g (| a_{s} |^{2} + C) / (i ω_{m} + γ_{m}),

| a_{s} |^{2} = \frac{{(M - χ)}^{2} + κ^{2}}{{(M^{2} + κ^{2} - χ^{2})}^{2}} | E_{0} |^{2},

where M ≡ ∆ + (2g²/ω_m − 2η)|a_s|² + 2g²C/ω_m. The intracavity intensity |a_s|² satisfies a fifth-order polynomial equation and therefore it should have five solutions in principle. When the rocking parameter C = 0, Eq. (5) becomes a third-order polynomial equation and the typical bistable states appear as the pumping intensity |E₀|² is larger than a threshold, as shown in Fig. 2(a). The dependence of |a_s|² on |E₀|² for C = 5 is plotted in Fig. 2(b), which shows the multistabilities of the intracavity field. Interestingly, when |E₀| = 0, there are two stable nonzero intracavity states for C ≠ 0, as shown in Fig. 2(b). We call these two states as the rocking states.

Fig. 2 The intracavity intensity |a_s|² as a function of |E₀|² for (a) C = 0 and (b) C = 5. The parameters are (a) ∆ = 1.0 ω_m, χ = 0 and (b) ∆ = 2.0 ω_m, χ = 1.0 ω_m. Other parameters are g²/ω_m = 0.02ω_m, η = 0.1 ω_m, and κ = 0.1 ω_m.

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3. Temporal rocking states

We will focus on the rocking states in the following. When |E₀|² = 0, the cavity field can be expressed as a_s exp(iϕ) and their solutions are obtained as

| a_{s} |_{\pm}^{2} = \frac{- Δ - (2 g^{2} / ω_{m} - 4 η) C \pm \sqrt{χ^{2} - κ^{2}}}{2 g^{2} / ω_{m} - 2 η},

2 ϕ_{\pm} = \arg [(\pm \sqrt{χ^{2} - κ^{2}} - i κ) / χ],

which is similar as in [40], Valcarcel et al. It is clear that the cavity intensity becomes bistable resulting from the phase sensitive gain in Eq. (3). Two rocking states

| a_{s} |_{\pm}^{2}

and

| a_{s} |_{-}^{2}

emerge, and their phases exactly differ by ϕ₊ + ϕ₋ = π/2. Correspondingly, the position of the membrane

q_{s} \equiv \sqrt{ℏ / m ω_{m}} Re (b_{s})

is also bistable and given by

q_{s \pm} = \sqrt{\frac{ℏ g^{2} ω_{m}}{m}} \frac{Δ - χ \mp \sqrt{χ^{2} - κ^{2}}}{(2 g^{2} / ω_{m} - 2 η) (ω_{m}^{2} + γ^{2})} .

The steady-state position of the mechanical oscillator as functions of C and ∆ ω_m is shown in Fig. 3(a). The parameters region of the existence of the rocking states is plotted in Fig. 3(b). In region III, when C < κ/2η, there is no steady-state. In the region I, two rocking states q_s₊ and q_s₋ exist. In the region II, only rocking state q_s₊ exists.

Fig. 3 (a) The steady-state position q_s of the mechanical oscillator as functions of C and ∆/ω_m for two rocking states. (b) The parameters region of the existence of the rocking states. The parameters are κ = 0.1 ω_m, g²/ω_m = 0.04 ω_m, η = 0.05 ω_m, and χ = 2.4 ω_m.

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In order to better understand the phase-bistable states and the rocking states of the system, the parametric plot of the cavity field quadratures Re(a_s) and Im(a_s) as a function of the rocking parameter C is illustrated in Fig. 4. The red and blue solid curves denote two rocking states. The red and blue dashed curves are the corresponding phase-bistable states for each rocking state, respectively. The phase diagram shows the interesting pattern as increasing the rocking parameter. It is clear that the phase difference is π for the bistable states of each rocking state, and the sum of the phases is π/2 for two rocking states, as shown in Fig. 4.

Fig. 4 The parametric plot of the cavity field quadratures Re(a_s) and Im(a_s) as a function of the rocking parameter C. The red and blue solid curves are the rocking states. The dashed curves are the corresponding phase-bistable states. The parameters are Δ = −1.0 ω_m, κ = 0.03 ω_m, g = 0.01 ω_m, η = −0.005 ω_m, γ_m = 10⁻⁸ ω_m, respectively.

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4. Normal mode splitting

Linearizing Eq. (3) near the steady-state solutions by mapping $o \to \bar{o} + o (t) (o = a, a^{†}, b, b^{†})$ , where ō is the mean value and 〈o (t)〉 = 0, we get the fluctuation equations as

\begin{array}{l} \dot{b} = (- i ω_{m} - γ_{m}) b - i (G a^{†} + G^{*} a) + \tilde{ξ}, \\ \dot{a} = (i Δ_{e f f} - κ) a - i G (b + b^{†}) - i χ^{'} a^{†} + \tilde{α}, \end{array}

where ∆_eff = M − 2η|a_s|²,

χ^{'} = χ + 2 η a_{s}^{2}

and G = ga_s. The phase sensitive terms (−i χ′a^† and Ga^†) appear in the equations of the cavity field and the oscillator’s momentum, therefore they can enhance the optomechanical interaction and facilitate the cooling. Equation (9) can be also expressed in a compact form

\dot{O} = M O + ξ

, with M being a 4 × 4 time-independent matrix, and with O and ξ being the column vectors of the fluctuations and the noise sources, respectively. The spectrum S (ω) of the mechanical oscillator can be solved as

S (ω) = \frac{2}{| d (ω) |^{2}} [γ_{m} n_{i} + \frac{| G |^{2} κ}{| h (ω) |^{2}} [{(ω - Δ_{e f f} - 2 η | a_{s} |^{2} - χ c o s (2 ϕ))}^{2} - {(κ - χ s i n (2 ϕ))}^{2}]],

where h(ω) = −8η|a_s|² M – ω² – 2iωκ and d(ω) = i(ω_m – ω)+ γ_m + 4i |G|² M/h (ω).

Using the conditions of the rocking states as given in Eq. (7), Eq. (10) can be simplified as

S (ω) = \frac{2}{| d (ω) |^{2}} [γ_{m} n_{i} + \frac{| G |^{2} κ}{| h (ω) |^{2}} [{(ω + 4 η | a_{s} |^{2})}^{2} + 4 κ^{2}]] .

When the rocking parameter C = 0, Eq. (11) is reduced to the similar form as in [31], Huang et al. In Eq. (11), the first term is the thermal contribution from the mechanical oscillator and the second is the coherent interaction between the cavity field and the oscillator, i.e. the radiation pressure. For the weak optomechanical interaction, the spectrum is essentially that of a simple harmonic oscillator of Brownian motion with the absence of the radiation pressure. When the photon enhanced coupling G is dominated and larger than the decays, the strongly coupling regime is achieved and leads to the so-called NMS, which has been widely studied in cavity optomechanics. Including the nonlinearity in such systems, the NMS could be inhibited due to the photon blockade mechanism [32]. Here we show that rocking can be used as a new route to control the NMS. In Fig. 5, we show the plots of S(ω) as a function of the dimensionless frequency ω/ω_m for two rocking states q₊ and q₋ at different rocking parameters C. For the state q₊, the mode splitting is hard to be observed at small C. As the rocking parameter C increases, NMS becomes apparent, accompanying the mode broadening, as shown in Fig. 5(a). The amplitude and splitting of two peaks can be fully controlled by the single parameter C. The splitting just exists in certain parameters space. For the very large C, the splitting disappears. For the state q₋, there is no mode splitting at C = 10, as shown in Fig. 5(b).

Fig. 5 The spectra S(ω) as a function of ω/ω_m for two rocking states q₊ (a) and q₋ (b). The parameters are ∆ = −1.0 ω_m, κ = 0.03 ω_m, g = 0.01 ω_m, η = −0.005 ω_m, γ_m = 10⁻⁸ω_m. In (a), the blue, red, and black curves are for the rocking parameters C = 60, C = 50, and C = 38, respectively. In (b), the rocking parameter C = 10.

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5. The optomechanical cooling

Now we turn attention to the optomechanical cooling in such a system. For the case of no mode-splitting and assume that the frequency shift by the nonlinearity and rocking is small, the final mechanical mode occupancy is given by

n_{f} = \frac{1}{γ_{e f f}} (γ_{m} n_{i} + \frac{| G |^{2} κ [{(ω_{m} + 4 η | a_{s} |^{2})}^{2} + 4 κ^{2}]}{{(8 η | a_{s} |^{2} M + ω_{m}^{2})}^{2} + 4 ω_{m}^{2} κ^{2}}),

where γ_eff is the effective cooling rate and given by

γ_{e f f} = | γ_{m} - \frac{8 | G |^{2} M ω_{m} κ}{{(8 η | a_{s} |^{2} M + ω_{m}^{2})}^{2} + 4 ω_{m}^{2} κ^{2}} | .

γ_eff is strongly dependent on the rocking parameter C. By tuning C the effective temperature of the mechanical oscillator can be greatly decreased. Under certain conditions, the ground-state cooling can be achieved. To better illustrate such cooling effect, we now take a practical optomechanical system as an example. The membrane vibrational frequency is ω_m = 2π × 2 MHz, quality factor Q_m = ω_m/γ_m ≈ 10⁸ and an effective mass 1 ng [41]. The thermal temperature is chosen to be 10 K with initial occupancy n_i = 10⁴. The cavity decay is κ = 2π × 60 kHz. The optomechanical coupling strength and the nonlinearity are taken as G = 2π × 20 kHz and η = −2π × 1 kHz, respectively. The final mode occupancy n_f as a function of the rocking parameter C is shown in Fig. 6, where all the parameters are scaled with ω_m for convenience. The solid blue curves are the results from Eq. (12) and the dashed red curves are the numerical calculations. The dashed blue and black line represent the ground-state mode number n_f = 1 and the initial mode number n_i = 10⁴, respectively. It is clear that the cooling rate is different for two rocking states. When C is small, the cooling rates for two rocking states are close and they have almost the same cooling effect. For the large C, the effective cooling rate for the rocking state q₊ is greatly enhanced. Therefore, the membrane can be more easily cooled to its ground state. For the very large C, the intracavity intensity approaches to zero and the final occupancy n_f correspondingly goes back to its initial occupancy. There exists an optimal rocking parameter C for the mechanical cooling. Since there is NMS at the large rocking parameters C for the rocking state q₊, there is some difference between the numerical calculation and analytical result. The final occupancy for numerical calculation is increased due to the heating from the nonlinearity and strongly interaction, which is consistent with [32], Kumar et al.

Fig. 6 The final mode occupancy n_f as a function of the rocking parameter C for the rocking states. The parameters are ∆ = −1.0 ω_m, κ = 0.03 ω_m, g = 0.01 ω_m, η = −0.005 ω_m, γ_m = 10⁻⁸ω_m. The solid blue curves are the results from Eq. (12) and the dashed red curves are for the numerical calculations. The dashed blue and black lines represent n_f = 1 and n_i = 10⁴, respectively.

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6. Conclusion

In conclusion, a hybrid optomechanical system, combining the membrane and intracavity Kerr nonlinearity, has been investigated in the large temporal modulation. The temporal rocking could cause the non-threshold multistabilities for the cavity field and the position of the membrane. It is interesting that each stable state has two degenerate phases. The optomechanical interaction and the radiation pressure can be greatly enhanced by the phases and rocking parameters. For the experimental parameters close to those of recently performed experiments [41], the rocking parameters can induce and control the mode-splitting. For large rocking parameters, the ground-state cooling of the mechanical oscillator can be implemented. Such optomechanical systems with temporal rocking can find many useful applications in all-optical switching and manipulation of the quantum states.

In addition, many other potential investigations can be envisaged. First, the system extended to spatial rocking is straightforward, where the cavity driving field is spatially modulated [40]. The optomechanical system with nonlinearity should have complicated spatial structure and patterns, which is possible to study the optomechanical solitons. Second, another important generalization is to study the stochastic resonance [42] of optomechanics in such a system. Since the phase and intensity bistable states emerge under certain conditions, the input signal is amplified. Therefore the signal-to-noise ratio can be greatly improved and shows the resonance even at the existence of the external noises. Finally, our calculations can be used to investigate the quantum correlations in the regime of the spatial and temporal rocking. The quantum effect, such as squeezing, in such systems should be controllable and enhanced [35].

Funding

National Key Research and Development Program of China (2017YFA0304201); National Natural Science Foundation of China (NSFC) (11734008, 11374101, 91536112, 11621404, 11704126); Shanghai Subject Chief Scientist (17XD1401500); Shanghai Sailing Program (17YF1403900).

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Temporal rocking in a nonlinear hybrid optomechanical system

Abstract

1. Introduction

2. Model

3. Temporal rocking states

4. Normal mode splitting

5. The optomechanical cooling

6. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (13)

Optics Express