Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system

Wen-ju Gu; Gao-xiang Li

doi:10.1364/OE.21.020423

1. Introduction

Quantum effects of a movable mirror coupled to a cavity field via radiation pressure have many potential applications, such as in quantum information processing (QIP) as well as in ultra-sensitive force measurements and in tests of fundamental quantum effects at mesoscopic or macroscopic scales [1–7]. An important resource in optomechanics is the generation of non-classical states of motion around the ground state [8–10]. Especially, achieving squeezed states of a mechanical oscillator, in which the variance of one quadrature of motion is below zero-point value, can enable ultrahigh precision detection at or even below the standard quantum limit [11–14]. Recently, via exploiting the quantum nature of the mechanical interaction between laser light and a membrane mechanical resonator embedded in an optical cavity, the optomechanical squeezing of light has been experimentally demonstrated [15, 16]. These technique developments may offer the possibility to achieve squeezed states of the mechanical oscillator. By now, different schemes for generating squeezed state in optomechanics have been proposed. For example, Clerk et al. [17] investigated the possibility of generation of a squeezed state via back-action evading measurements, Huang and Agarwal [18] presented that an optical parametric amplifier inside a cavity can considerably improve the cooling of the micromechanical mirror by radiation pressure and Ian et al. [19] showed the squeezing of quadrature variance of a mirror assisted by the optical parametric process which is realized by an atomic gas when regarded as a cavity dielectric, and Jähne et al. [20] investigated the creation of squeezed states of a movable mirror by feeding a squeezed light onto the cavity via the radiation pressure under the assumption of the resolved-sideband limit, etc. In contrast, a different idea without the need of any feedback or control but just by appropriately exploiting time-periodic driving in an optomechanical system, is studied to achieve large degrees of squeezing of a mechanical micromirror [21, 22].

The modulation-assisted driving can give rise to interesting effects and modifications on the quantum dynamics [23, 24], and the employment of the driving has been concerned with the cavity optomechanics in which the motion of a mechanical oscillator modulates the cavity resonance frequency (dispersive coupling). Alternately, the cavity optomechanical system in the sense that the width of the cavity resonance depends on the mechanical displacement (dissipative coupling), was shown to give rise to remarkable quantum noise interference effects which dramatically relax requirements for cooling to the ground state without sideband resolution by Elste et al. [25] in microwave domain and by Xuereb et al. [26] in optical domain, respectively. Further, the predicted strong cooling effect outside the resolved-sideband regime will reduce the influence of the thermal noise on the mechanical motion, since the thermal noise will decrease the squeezing of the oscillator. Therefore, it may offer the possibility to obtain the squeezed state of the movable mirror around the ground state in a relatively “hot” environment via implementing the periodic modulations on the dissipative optomechanics.

In this paper, we investigate the generation of squeezed state of the mirror motion in a dissipative optomechanical system, in which the cavity field is driven by an amplitude mildly-modulated light. We obtain the master equation for the cavity-mirror system following the general reservoir theory based on the density operator approach, in which the reservoir variables are adiabatically eliminated by using the reduced density operator for the system in the interaction picture [27]. We first derive the analytical results for the first moments of quadratures in a perturbation method, which are in a good agreement with the asymptotic numerical configurations. Then we carry out the full numerical calculations of the variances of the quantum quadratures around the classical orbits, and find that the squeezed state of the movable mirror is achievable with a much low phonon number occupancy when the modulation frequency is twice the oscillating frequency of the mirror. Finally, we derive the master equation of the mirror by adiabatically eliminating the cavity field due to the weak-coupling strength between them with the experimentally realizable parameters, and the analytical results agree with the numerical results. Moreover, the obtained squeezing can be robust against the thermal noise because of the strong cooling effect in the non-sideband-resolved regime arising from interference of quantum noise.

The paper is arranged as follows: in Sec. 2, the periodically-modulated dissipative optomechanical system is introduced. In Sec. 3, the semiclassical phase orbits are analyzed. In Sec. 4, the full quantum dynamics around the classical orbits is investigated. In Sec. 5, the explicit motion equation for the mirror is obtained via adiabatically eliminating the cavity field and in the last the conclusions are driven.

2. The periodically-modulated dissipative optomechanical system

We consider a dissipative optomechanical system consisted of an effective Fabry-Pérot interferometer [26, 28, 29], sketched in Fig. 1. The movable mirror ℳ oscillates along the x-axis with the frequency ω_m and couples to a cavity mode with the resonant frequency ω_a via the dispersive and dissipative couplings, which corresponds to the shifts of the cavity’s resonant frequency and damping rate respectively due to its mechanical motion. The full Hamiltonian is a sum of the free cavity Ĥ_c, free movable mirror Ĥ_m, free reservoir field Ĥ_R, cavity-reservoir interaction Ĥ_c-R and cavity-mirror interaction Ĥ_int Hamiltonians (h̄ = 1):

\begin{matrix} \hat{H} = {\hat{H}}_{c} + {\hat{H}}_{m} + {\hat{H}}_{R} + {\hat{H}}_{c - R} + {\hat{H}}_{int}, \\ {\hat{H}}_{c} = ω_{a} {\hat{a}}^{†} \hat{a}, {\hat{H}}_{m} = ω_{m} {\hat{b}}^{†} \hat{b}, {\hat{H}}_{R} = \int d ω ω {\hat{a}}_{ω}^{†} {\hat{a}}_{ω}, \\ {\hat{H}}_{c - R} = i \sqrt{κ_{c} / π} \int d ω ({\hat{a}}_{ω}^{†} \hat{a} - {\hat{a}}^{†} {\hat{a}}_{ω}), \\ {\hat{H}}_{int} = g_{0} [α {\hat{a}}^{†} \hat{a} + i β \sqrt{L / (2 π c)} \int d ω ({\hat{a}}_{ω}^{†} \hat{a} - {\hat{a}}^{†} {\hat{a}}_{ω})] (\hat{b} + {\hat{b}}^{†}) . \end{matrix}

The operators â and b̂ are the annihilation operators of cavity and phonon modes, and â_ω describes the continuous modes of the optical reservoir coupled to the cavity mode. The cavity field dissipates at the rate κ_c without the motion of the mirror. The parameters α (dispersive) and β (dissipative) respectively indicate the cavity frequency’s (ω_a) and damping rate’s (κ_c) linear dependence on the small displacement x̂₀ with

{\hat{x}}_{0} = x_{zp} ({\hat{b}}^{†} + \hat{b}) / \sqrt{2}

, where x_zp is the zero-point motion amplitude of the movable mirror. The effective length of the interferometer is L. This optomechanical setup can realize the strong effective dissipative coupling strength which is enhanced by a factor of the cavity amplitude, even in the order of cavity linewidth in the absence of dispersive coupling, i.e. α = 0, as described in [26]. Our main purpose is to stress on the dissipative coupling between the cavity field and the mirror motion and we assume α = 0 that is realizable in the following.

Fig. 1 Sketch of the effective Fabry-Pérot interferometer coupled to the cavity mode via the dispersive and dissipative couplings. The cavity field is pumped by a periodically-modulated laser.

Download Full Size | PDF

The dispersively and dissipatively coupled optomechanical system has been investigated to cool the mechanical oscillator to its ground state in both the microwave and optical domains in the Heisenberg-Langevin approach [25, 26, 30]. In this paper, we present the dynamics of the cavity-mirror system following the density operator approach, in which the reservoir variables can be adiabatically eliminated by using the reduced density operator for the system. The optical reservoir has two contributions on the cavity field: the c-number part 〈â_ω〉 and the random fluctuation part δâ_ω. The c-number component is given by periodically-modulated driving field, which is in the form

〈 {\hat{a}}_{ω} 〉 = \sqrt{2 π} {\bar{a}}_{in} [1 + ε cos (Ω t)] exp (- i ω_{R} t) .

The carrier frequency is ω_R and its input amplitude is represented by

{\bar{a}}_{in} = \sqrt{P_{R} / \bar{h} ω_{R}}

with P_R the input power. Two mild modulation sidebands, equally placed on both sides of the carrier frequency, are separated by a modulation frequency Ω = 2π/τ, in which τ > 0 is the modulation period. The ratio of the modulation amplitude to ā_in is ε, which is less than 1. The noise operator δâ_ω obeys the correlation relations

〈 δ {\hat{a}}_{ω} δ {\hat{a}}_{ω^{'}}^{†} 〉 = δ (ω - ω^{'})

and

〈 δ {\hat{a}}_{ω}^{†} δ {\hat{a}}_{ω} 〉 = 0

of the vacuum reservoir.

Following the general reservoir theory [31], the system-reservoir interaction is given by 𝒱̂(t) = Ĥ_c-R(t) + Ĥ_int(t) in the interaction picture with respect to the sum of free Hamiltonians Ĥ_c, Ĥ_m, and Ĥ_R. By tracing over the reservoir coordinates under the Born-Markov approximation, the reduced density operator ρ̂_s for the cavity-mirror system is governed by

{\dot{\hat{ρ}}}_{s} = - i {Tr}_{R} [\hat{𝒱} (t), {\hat{ρ}}_{s} (t) \otimes {\hat{ρ}}_{R} (t_{i})] - {Tr}_{R} \int_{t_{i}}^{t} [\hat{𝒱} (t), [\hat{𝒱} (t^{'}), {\hat{ρ}}_{s} (t), \otimes {\hat{ρ}}_{R} (t_{i})]] d t^{'} .

By substituting the c-number component in Eq. (2) and the correlation functions of noise operator into Eq. (3), now the motion equation for the density operator ρ̂_s can be obtained as

{\dot{\hat{ρ}}}_{s} = - i [{\hat{H}}_{0}, {\hat{ρ}}_{s}] + {\hat{ℒ}}_{1} {\hat{ρ}}_{s} + {\hat{ℒ}}_{2} {\hat{ρ}}_{s},

where the Hamiltonian Ĥ₀ is given by

{\hat{H}}_{0} = - Δ {\hat{a}}^{†} \hat{a} + ω_{m} {\hat{b}}^{†} \hat{b},

with

Δ = ω_{R} - ω_{a} + δ ω {[1 + λ (\hat{b} + {\hat{b}}^{†}) / \sqrt{2}]}^{2}

the effective detuning of the cavity’s resonant frequency from the carrier frequency, in which

λ = g_{0} β \sqrt{L / κ_{c} c}

characterizes the single-photon level dissipative coupling strength between the mirror and the cavity field, and

δ ω = \frac{1}{2 π} 𝒫 \int_{0}^{\infty} \frac{d ω κ_{c} (ω)}{ω - ω_{a}}

is Lamb shift [32]. Commonly, we may neglect it because it is small and define Δ = ω_R − ω_a. The Liouvillian operators ℒ̂₂ and ℒ̂₁ contain the dissipations of the cavity and phonon modes and the interaction between them, which are expressed in the form

\begin{array}{l} {\hat{ℒ}}_{1} {\hat{ρ}}_{s} = - i [i \sqrt{2} (1 + ε cos (Ω t) (\bar{a_{in}^{*}} \hat{C} - {\bar{a}}_{in} {\hat{C}}^{†}), {\hat{ρ}}_{s})], \\ {\hat{ℒ}}_{2} {\hat{ρ}}_{s} = 2 \hat{C} {\hat{ρ}}_{s} {\hat{C}}^{†} - {\hat{C}}^{†} \hat{C} {\hat{ρ}}_{s} - {\hat{ρ}}_{s} {\hat{C}}^{†} \hat{C}, \end{array}

with a composite operator

\hat{C} = \sqrt{κ_{c}} [1 + λ (\hat{b} + {\hat{b}}^{†}) / \sqrt{2}] \hat{a}

of phonon and cavity operators.

3. Analysis of semiclassical phase space orbits

It is very difficult to achieve the full solution for the master equation (4) because of the nonlinear terms and periodic coefficients. We will take the strategy discussed in [22] in the following, where we first calculate the phase space orbits of the first moments of quadratures and then linearize the quantum dynamics around the asymptotic quasiperiodic orbits.

We introduce the phase and amplitude quadratures for the cavity and mechanical modes,

\begin{array}{l} \hat{x} = (\hat{a} + {\hat{a}}^{†}) / \sqrt{2}, & \hat{y} = (\hat{a} - {\hat{a}}^{†}) / i \sqrt{2}, \\ \hat{q} = (\hat{b} + {\hat{b}}^{†}) / \sqrt{2}, & \hat{p} = (\hat{b} - {\hat{b}}^{†}) / i \sqrt{2} . \end{array}

The first moments for these quadratures are denoted by Ō(t) = 〈Ô(t)〉, where Ô = x̂, ŷ, p̂, q̂. In addition, the movable mirror also undergoes the Brownian motion and the resulting dissipation is described by the Liouvillian operator ℒ̂_mρ̂_s, which reads [33]

{\hat{ℒ}}_{m} {\hat{ρ}}_{s} = - i \frac{γ_{m}}{2} [\hat{q}, {\hat{p}, {\hat{ρ}}_{s}}] + γ_{m} [{\bar{n}}_{th} + \frac{1}{2}] [\hat{q}, [\hat{q}, {\hat{ρ}}_{s}]] .

Here γ_m is the friction coefficient, n̄_th is the mean thermal phonon number, and {...,...} is the anticommutator.

From the master equation (4) with the damping term in Eq. (9), we obtain a group of nonlinear differential equations for the first moments under the factorisation assumption such as $〈 \hat{x} \hat{q} 〉 = \bar{x q}$ , which is expressed as

\begin{array}{l} \bar{\dot{q}} = ω_{m} \bar{p}, \\ \bar{\dot{p}} = - ω_{m} \bar{q} - γ_{m} \bar{p} + λ \sqrt{2} E (1 + ε cos Ω t) \bar{y}, \\ \bar{\dot{x}} = - Δ \bar{y} - κ_{c} \bar{x} {(1 + λ \bar{q})}^{2} - \sqrt{2} E (1 + ε cos Ω t) (1 + λ \bar{q}), \\ \bar{\dot{y}} = Δ \bar{x} - κ_{c} \bar{y} {(1 + λ \bar{q})}^{2}, \end{array}

with γ_m connected to the movable mirror’s quality factor Q by the relation γ_m = ω_m/Q, and the driving amplitude

E = \sqrt{2 κ_{c}} {\bar{a}}_{in}

. In this paper we will follow the experimentally realizable parameters [26]: mechanical oscillator effective mass m = 100 ng, frequency ω_m = 2π × 103 kHz, and quality factor Q = ω_m/(2γ_m) = 2 × 10⁶, such that the zero-point fluctuation is

x_{zp} = \sqrt{\bar{h} / (2 m ω_{m})} ≃ 1

fm; driving wavelength λ_c = 1064 nm, effective cavity length L = 7.5 cm, cavity damping rate κ_c = 2π × 196 kHz, and λ = 0.5 × 10⁻⁶. We suppose the initial mean thermal phonon number n̄_th = 10³ and choose the input power of carrier component equal to P_R = 10 mW, where the coherent driving amplitude becomes

E = \sqrt{2 κ_{c}} {\bar{a}}_{in} = 2 π \times 58

GHz. The ratio of the modulation sidebands’ amplitude to the carrier’s amplitude ε is equal to 0.3 and the modulation frequency Ω is about twice the natural frequency ω_m of the movable mirror. We can numerically solve the Eqs. (10) with the above parameters, and the results are shown by the thin blue solid curves in Fig. 2. One can find that the mean values of each quadrature evolve towards an asymptotic periodic orbit with the same periodicity 2π/Ω of the applied modulation.

Fig. 2 Phase space trajectories of the first moments of the movable mirror and cavity light modes. Numerical simulations for t ∈ [0, 100τ] (thin blue solid lines) and analytical approximations of the asymptotic orbits (green dashed lines) from t = 99τ to t = 100τ. The plots are obtained with the parameters shown in the text.

Download Full Size | PDF

When the system is far away from optomechanical instabilities, approximate analytic solutions for these first moments of quadratures can be derived by applying a perturbation method in λ [21, 22]. Thus, we expand the variables Ō(t) into a power series

\bar{O} (t) = \sum_{j = 0}^{\infty} {\bar{O}}_{j} (t) λ^{j} .

We will obtain a chained set of recursive differential equations for the variables Ō_j(t) by substituting the series into Eqs. (10), which is presented in appendix A. The differential equations for one set of variables Ō_j(t) form a linear inhomogeneous system with τ-periodic coefficients for each j. The asymptotic solution of the variables Ō_j(t) is stable and will have the same periodicity of the modulation [34] due to the existence of the damping. Then, we can expand the variables into Fourier series

{\bar{O}}_{j} (t) = \sum_{n = - \infty}^{\infty} {\bar{O}}_{n, j} exp (in Ω t) .

In this study, we find that it is sufficient to approximate the asymptotic mean values by truncating the series to the terms with indexes j = 0, 1, 2, 3 and n = −1, 0, 1. The obtained analytical approximations for these first moments are in a good match with the numerical asymptotic quasiperiodic orbits, which are indicated by the green dashed lines in Fig. 2. These analytical approximations are convenient to proceed to linearization on the full quantum dynamics, which can also show a desirable accuracy for the dynamics. The explicit calculations for Ō_n,j are presented in detail in appendix A.

4. The full quantum dynamics around the classical orbits

The condition of the large mean values of first moments |Ō(t)| ≫ 1 makes it implementable to treat the system as quantum fluctuations evolving around their classical orbits. Therefore, it is valid to proceed to linearization on the full quantum dynamics by assuming that each operator in the system can be written as a sum of its classical value and a small fluctuation with zero-mean value [21, 22, 35],

\hat{a} = \bar{a} (t) + \hat{d}, \hat{b} = \bar{b} (t) + \hat{f},

where

\bar{a} (t) = [\bar{x} (t) + i \bar{y} (t)] / \sqrt{2}

and

\bar{b} (t) = [\bar{q} (t) + i \bar{p} (t)] / \sqrt{2}

are the mean values of annihilation operators of the cavity and phonon modes. For later convenience, we define two quadratures

\hat{Q} = (\hat{f} + {\hat{f}}^{†}) / \sqrt{2}

and

\hat{P} = (\hat{f} - {\hat{f}}^{†}) / i \sqrt{2}

with zero-mean values 〈Q̂〉 = 0, 〈P̂〉 = 0, corresponding to the quadratures of quantum noise component of phonon field.

In this shift presentation, the quantum behavior of cavity-mirror system is governed by contributions respectively induced by the motion of uncoupled cavity and mirror modes, and the interaction between them, which is determined by

\frac{d}{d t} {\hat{ρ}}_{s} = {\hat{ℒ}}^{d} {\hat{ρ}}_{s} + {\hat{ℒ}}^{f} {\hat{ρ}}_{s} + {\hat{ℒ}}^{d - f} {\hat{ρ}}_{s} .

Here the motion of the uncoupled cavity mode is determined by the Liouvillian operator

{\hat{ℒ}}^{d} {\hat{ρ}}_{s} = i [Δ {\hat{d}}^{†} \hat{d}, {\hat{ρ}}_{s}] + κ_{c} {(1 + λ \bar{q} (t))}^{2} (2 \hat{d} {\hat{ρ}}_{s} {\hat{d}}^{†} - {\hat{d}}^{†} \hat{d} {\hat{ρ}}_{s} - {\hat{ρ}}_{s} {\hat{d}}^{†} \hat{d}),

in which the dissipation rate is mildly periodically-modulated because of the relation λ|q̄(t)| ≪ 1. The Liouvillian operator for the uncoupled mirror is given by

{\hat{ℒ}}^{f} {\hat{ρ}}_{s} = - i [\frac{ω_{m}}{2} ({\hat{P}}^{2} + {\hat{Q}}^{2}), {\hat{ρ}}_{s}] + {\hat{𝒟}}_{f} {\hat{ρ}}_{s} + {\hat{ℒ}}_{m} {\hat{ρ}}_{s},

with a phonon damping term

{\hat{𝒟}}_{f} {\hat{ρ}}_{s} = κ_{c} {| \bar{a} (t) |}^{2} λ^{2} (2 \hat{Q} {\hat{ρ}}_{s} \hat{Q} - {\hat{Q}}^{2} {\hat{ρ}}_{s} - {\hat{ρ}}_{s} {\hat{Q}}^{2})

relevant to the cavity variables and the intrinsic damping ℒ̂_mρ̂_s, the form of which is unchanged in Eq. (9) just by replacing q̂, p̂ with Q̂, P̂, i.e.,

{\hat{ℒ}}_{m} {\hat{ρ}}_{s} = - i \frac{γ_{m}}{2} [\hat{Q}, {\hat{P}, {\hat{ρ}}_{s}}] + γ_{m} ({\bar{n}}_{th} + \frac{1}{2})

[Q̂, [Q̂, ρ̂_s]]. The interaction between the cavity and phonon fields is described by

{\hat{ℒ}}^{d - f} {\hat{ρ}}_{s} = [λ E (1 + ε cos Ω t) (\hat{d} - {\hat{d}}^{†}) \hat{Q}, {\hat{ρ}}_{s}] + {2 λ κ_{c} \bar{a} (t) (1 + λ \bar{q} (t)) (\hat{Q} {\hat{ρ}}_{s} {\hat{d}}^{†} - {\hat{d}}^{†} \hat{Q} {\hat{ρ}}_{s}) + H . c .} .

Obviously, the effective coupling strengths between the cavity and phonon modes λE(1 + ε cos Ωt) and 2λ κ_cā(t)(1 +λq̄(t)) are also τ-periodic.

Now we can see that the quantum dynamics of the system will evolve to a zero-mean Gaussian state, for which the second moments dominate. Accordingly, we write the second moments of the cavity-mirror system in a vector form

X (t) = {(〈 {\hat{d}}^{†} \hat{d} 〉, 〈 {\hat{d}}^{2} 〉, 〈 {\hat{d}}^{† 2} 〉, 〈 \hat{d} \hat{Q} 〉, 〈 {\hat{d}}^{†} \hat{Q} 〉, 〈 \hat{d} \hat{P} 〉, 〈 {\hat{d}}^{†} \hat{P} 〉, 〈 {\hat{Q}}^{2} 〉, 〈 {\hat{P}}^{2} 〉, 〈 \hat{Q} \hat{P} 〉)}^{T},

where T denotes the transpose of the vector. The motion of the second moments X(t), which is derived from the master equation (14), obeys the equation

\frac{d}{d t} X (t) = \bar{M} (t) X (t) + C (t) .

The explicit forms of time-dependent coefficient matrix M̄(t) and inhomogeneous term C(t) are shown in appendix B. The coefficient matrix M̄(t) and term C(t) are both τ-periodic by substituting the approximate values which indeed well approach the asymptotic periodic orbits, and they can also be written as a sum of harmonically oscillating terms

\bar{M} (t) = \sum_{n = - 2}^{2} M_{n} exp (in Ω t), C (t) = \sum_{n = - 2}^{2} C_{n} exp (in Ω t) .

In the long time limit, X(t) will be also τ-periodic and can be written in the Fourier series

X (t) = \sum_{n = - \infty}^{\infty} X_{n} exp (in Ω t) .

It is sufficient to truncate the sum in Eq. (22) to n = ±2 to calculate the variances of the quantum fluctuations around the classical orbits for this weak modulation. This is because the calculations on the second moments of the variables are adequate up to the first sidebands ±Ω at the condition of Ω ∼ 2ω_m, where the dynamics resembles the effect of parametric amplification because the two-phonon average 〈f̂²〉 is nonzero in the rotating wave approximation, leading to the squeezing of the mechanical mode. To be more precise, we present the numerical calculations for X(t) by expanding to the second sidebands ±2Ω. Detailed calculations are shown in appendix B.

In Fig. 3, we plot the time-dependent variances of the mirror position and momentum operators, which are in the form 〈ΔQ̂²〉 = 〈Q̂²〉 − 〈Q̂〉² and 〈ΔP̂²〉 = 〈P̂²〉 − 〈P̂〉², respectively. Here 〈Q̂〉 = 0 and 〈P̂〉 = 0 are adopted. The mirror motion is squeezed if either 〈ΔQ̂²〉 or 〈ΔP̂²〉 is less than 1/2 [36]. The initial mean thermal phonon number is 10³ and we compare the cases of with and without modulations (ε = 0). For ε = 0, the system becomes the dissipative optomechanics with the monochrome driving [22, 25, 30], where the mirror is allowed to cool down to its ground state at 2Δ = ω_m via a destructive interference of quantum noise. The mirror is transferred to a vacuum state and the variances of the mirror position and momentum operators are nearly equal to 0.5 because the thermal noise has little influence on the mirror motion due to the strong cooling effect outside the resolved-sideband regime. The results are shown by the red dashed lines in Fig. 3. For ε = 0.3, the phase and amplitude quadratures get periodically squeezed over time in an opposite phase with the well-chosen Ω around twice the frequency ω_m because of the cavity-induced energy shift of the oscillator. Therefore, via appropriately modulating the input light field, quantum squeezing of the quadratures of the mirror motion is achievable in the dissipative optomechanics without the need of any feedback or control. Meanwhile, the motion is closely around its ground state. Such ideas to generate the squeezed state of mechanical resonator have been investigated in the dispersive coupling regime [21, 22, 37, 38]. But a sufficiently strong nonlinearity to overcome the losses and the initial state close to the ground state are required, because the thermal occupation number will decrease the squeezing. The achieved squeezed state here is robust against the thermal noise because the strong cooling effect without requiring the cavity to be in the so-called good cavity limit κ_c ≪ ω_m in this dissipative optomechanics, which arises from the interference of quantum noise, can relax the requirement of much low initial temperature.

Fig. 3 Variances of the mirror position and momentum operators with the initial mean thermal phonon number n̄_th = 10³. The numerical results in the periodically-modulated model are shown by blue solid lines (with the well-chosen modulation frequency Ω around twice the frequency ω_m because of the cavity-induced energy shift of the oscillator), and the numerical results in the nonmodulated model are shown by the red dashed lines for t ∈ [100τ, 102τ].

Download Full Size | PDF

We should point out that the effective Hamiltonian Ĥ_int in Eq. (1) that describes the interaction between the motion of the micromirrror and the electric field is derived in the regime where the displacement x̂₀ of the membrane from its average position is small on the scale of a wavelength, i.e., in the Lamb-Dicke regime. The Hamiltonian for the radiation-pressure inter-action in linear order of x̂₀. Our calculation contains all the terms in the Eqs. (7), and the terms in the second order of λ is remaining. We may take into account of the quadratic mechanical position coupling between the motion of the mirror and the cavity field that is much smaller compared with linear mechanical position coupling in Lamb-Dicke regime in order to obtain a consistent result. The system-reservoir interaction becomes

\begin{array}{l} \hat{𝒱} (t) = i \sqrt{\frac{κ_{c}}{π}} {1 + \frac{λ}{\sqrt{2}} [\hat{b} exp (- i ω_{m} t) + {\hat{b}}^{†} exp (i ω_{m} t)] + \frac{λ^{(2)}}{2} {[\hat{b} exp (- i ω_{m} t) + {\hat{b}}^{†} exp (i ω_{m} t)]}^{2}} \\ \times \int d ω {{\hat{a}}_{ω}^{†} \hat{a} exp [- i (ω_{a} - ω) t] - {\hat{a}}^{†} {\hat{a}}_{ω} exp [i (ω_{a} - ω) t]}, \end{array}

where λ⁽²⁾ ∝ λ² characterizes the quadratic mechanical position coupling strength with two phonons involved. By tracing over the reservoir coordinates under the Born-Markov approximation, the composite operator in Eq. (7) becomes

\hat{C} = \sqrt{κ_{c}} [1 + λ / \sqrt{2} (\hat{b} + {\hat{b}}^{†}) + λ^{(2)} / 2 {(\hat{b} + {\hat{b}}^{†})}^{2}] \hat{a} .

We can verify that the effects caused by the quadratic mechanical position coupling on the mean values of quadratures are negligible because of the much small value of λ⁽²⁾. When we investigate the quantum dynamics around the mean values of the quadratures and proceed to linearization on the full quantum dynamics up to the second order of λ, an extra term with respect to Eq. (14) is

\begin{array}{l} {\hat{ℒ}}_{ext} {\hat{ρ}}_{s} = λ^{(2)} E (1 + ε cos Ω t) [(\bar{a} - \bar{a^{*}}) {\hat{Q}}^{2} + 2 \bar{q} (\hat{d} - {\hat{d}}^{†}) \hat{Q}, {\hat{ρ}}_{s}] \\ + 2 κ_{c} λ^{(2)} {\bar{q}}^{2} (2 \hat{d} {\hat{ρ}}_{s} {\hat{d}}^{†} - {\hat{d}}^{†} \hat{d} {\hat{ρ}}_{s} - {\hat{ρ}}_{s} {\hat{d}}^{†} \hat{d}) + {4 κ_{c} λ^{(2)} \bar{a q} (\hat{Q} ρ_{s} {\hat{d}}^{†} - {\hat{d}}^{†} \hat{Q} {\hat{ρ}}_{s}) + H . c .}, \end{array}

which is not in a Lindblad form. The extra term is much smaller compared with the Eqs. (15), (17) and (18) because it is in a higher order of λq̄, which is much smaller 1. We can infer that the quantum effects caused by caused by the quadratic mechanical position coupling are also negligible here.

5. The explicit motion equation for mirror via adiabatically eliminating the cavity variable

In the shift representation, the effective coupling strengths between the cavity fields and the movable mirror denoted by λE and 2λ κ_c|ā(t)| in Eq. (18) are about 2π × 29 kHz and 2π × 50 kHz with the experimentally realizable parameters, which are much smaller than the cavity damping rate κ_c. Thus, the cavity variable arrives at the steady state much faster than the mirror variable, and the motion equation of the reduced density operator for the movable mirror can be obtained by adiabatically eliminating the cavity variable in the master equation (14).

In order to see the main contributions on the motion equation of mirror, we first simplify the obtained semiclassical variables in appendix A. For the cavity variable, we find that when we retain the result to the zeroth order in λ, this reduced result can approximately represent the asymptotic orbits. Now we introduce the classical values of the phase and amplitude quadratures for the cavity mode $\bar{x} (t) = [\bar{a} (t) + {\bar{a}}^{*} (t)] / \sqrt{2}$ and $\bar{y} (t) = [\bar{a} (t) - {\bar{a}}^{*} (t)] / i \sqrt{2}$ calculated from the mean value of the annihilation operator ā(t), which is

\bar{a} (t) = \frac{E}{i Δ - κ_{c}} + \frac{ε E / 2 exp (i Ω t)}{i (Δ - Ω) - κ_{c}} + \frac{ε E / 2 exp (- i Ω t)}{i (Δ + Ω) - κ_{c}} .

In Fig. 4, it is shown that the mean values of the quadratures from Eq. (23), which is indicated by the red dotted line, can approximately approach the asymptotic orbit indicated by the blue solid line, which is also equal to the numerical results up to the order of λ³ as discussed in Sec. 3. The mean values of the mirror quadratures q̄(t) and p̄(t) are an order of magnitude lower than the cavity variables x̄(t) and ȳ(t) shown by the numerical values in Fig. 2 because they are nonzero up to the first order in λ. Further, the relation λ|q̄(t)| ≪ 1, which can be also seen from Fig. 2, makes q̄(t) negligible when it appears in Eqs. (15) and (18). In the following, we will display the feasibility of above simplifications on the first moments by comparing the quantum dynamics of the mirror motion governed by the analytical master equation with the full numerical results in Fig. 3.

Fig. 4 Comparison between the analytical approximations of phase orbits for cavity variables of the zeroth order in λ (red dashed line) and the numerical asymptotic orbit (blue solid line) which is also equal to the corrections up to λ³, for t ∈ [99τ, 100τ].

Download Full Size | PDF

Now we will neglect the term q̄(t) and take the value of ā(t) in Eq. (23). Then, by adiabatically eliminating the cavity variable and making use of the second-order perturbation method with respect to the effective coupling strengths λE and 2λ κ_cā(t) in the interaction picture of the Hamiltonian ω_mf̂^†f̂, the motion equation of the reduced density operator for the movable mirror ρ̂_f is given by

\frac{d}{d t} {\hat{ρ}}_{f} = {Tr}_{d} \int_{t_{0}}^{t} {\hat{ℒ}}^{d - f} (t) {\hat{ℒ}}^{d - f} (t^{'}) {\hat{ρ}}_{d} (t_{0}) \otimes {\hat{ρ}}_{f} (t) d t^{'} + {\hat{𝒟}}_{f} {\hat{ρ}}_{f} + {\hat{ℒ}}_{m} {\hat{ρ}}_{f},

where ρ̂_d(t₀) is the steady-state density operator for cavity field, governed by the Liouvillian operator in Eq. (15), 𝒟̂_f is the superoperator in the same form as Eq. (17), and ℒ̂_mρ̂_f describes the intrinsic damping of the mirror motion.

In the following, we define δ = Ω − 2ω_m and assume δ ≪ (Δ, ω_m, Ω) to compensate for the cavity-induced energy shift of the phonon mode. If we neglect the oscillating terms with frequencies ±ω_m, ±2ω_m,..., the resulting motion equation for the mirror is given by

\frac{d}{d t} {\hat{ρ}}_{f} = - i [{\hat{ℋ}}_{f}, {\hat{ρ}}_{f}] + {\hat{𝒦}}_{f} {\hat{ρ}}_{f} .

The Hamiltonian ℋ̂_f is the optically-induced energy shift for the movable mirror and given by

{\hat{ℋ}}_{f} = ω_{f}^{s} {\hat{f}}^{†} \hat{f},

and the shift frequency of optical spring effect [39] is

\begin{array}{l} ω_{f}^{s} = \frac{1}{2} λ^{2} E^{2} {\sum_{j = 1, 2} {(- 1)}^{j + 1} [θ_{j} (Δ, ω_{m}) + θ_{j} (Δ, - ω_{m}) + \frac{ε^{2}}{4} (θ_{j} (Δ - Ω, ω_{m}) + θ_{j} (Δ + Ω, ω_{m}) \\ + θ_{j} (Δ - Ω, - ω_{m}) + θ_{j} (Δ + Ω, - ω_{m}))]}, \end{array}

with functions

θ_{1} (Δ, ω_{m}) = (Δ + ω_{m}) / [{(Δ + ω_{m})}^{2} + κ_{c}^{2}]

and

θ_{2} (Δ, ω_{m}) = 2 κ_{c}^{2} (2 Δ + ω_{m}) / {(Δ^{2} + κ_{c}^{2}) [{(Δ + ω_{m})}^{2} + κ_{c}^{2}]}

. The term 𝒦̂_fρ̂_f is relevant to the cavity-assisted dissipation and the squeezing of the mirror motion, which is given by

\begin{matrix} {\hat{𝒦}}_{f} {\hat{ρ}}_{f} = [A (ω_{m}) + \frac{γ_{m}}{2} ({\bar{n}}_{th} + 1)] (\hat{f} {\hat{ρ}}_{f} {\hat{f}}^{†} - {\hat{f}}^{†} \hat{f} {\hat{ρ}}_{f}) + [A (- ω_{m}) + \frac{γ_{m}}{2} {\bar{n}}_{th}] ({\hat{f}}^{†} {\hat{ρ}}_{f} \hat{f} - \hat{f} {\hat{f}}^{†} {\hat{ρ}}_{f}) \\ + B_{1} exp (i δ t) (2 \hat{f} {\hat{ρ}}_{f} \hat{f} - {\hat{f}}^{2} {\hat{ρ}}_{f} - {\hat{ρ}}_{f} {\hat{f}}^{2}) + C_{1} exp (i δ t) [{\hat{ρ}}_{f}, {\hat{f}}^{2}] + H . c .. \end{matrix}

The coefficients are

\begin{matrix} A (ω_{m}) & = \frac{1}{2} κ_{c} λ^{2} E^{2} F (Δ, ω_{m}), \\ B_{1} & = \frac{1}{4} λ^{2} E^{2} \frac{ε}{2} [h (Δ, ω_{m}) + h^{*} (Δ, - ω_{m}) + h (Δ - Ω, ω_{m}) + h^{*} (Δ + Ω, - ω_{m}) - 2 \tilde{Δ}], \\ C_{1} & = \frac{1}{4} λ^{2} E^{2} \frac{ε}{2} [h (Δ, ω_{m}) - h^{*} (Δ, - ω_{m}) + h (Δ - Ω, ω_{m}) - h^{*} (Δ + Ω, - ω_{m})], \end{matrix}

with the functions

\begin{matrix} F (Δ, ω_{m}) & = f (Δ, ω_{m}) + \frac{ε^{2}}{4} [f (Δ - Ω, ω_{m}) + f (Δ + Ω, ω_{m})], \\ f (Δ, ω_{m}) & = \frac{{(2 Δ + ω_{m})}^{2}}{(Δ^{2} + κ_{c}^{2}) [{(Δ + ω_{m})}^{2} + κ_{c}^{2}]}, h (Δ, ω_{m}) = \frac{{(Δ - i κ_{c})}^{2}}{Δ^{2} + κ_{c}^{2}} \frac{i (Δ + ω_{m}) + κ_{c}}{{(Δ + ω_{m})}^{2} + κ_{c}^{2}}, \\ \tilde{Δ} & = \frac{κ_{c}}{(i Δ - κ_{c}) [i (Δ + Ω) + κ_{c}]} + \frac{κ_{c}}{(i Δ + κ_{c}) [i (Δ - Ω) - κ_{c}]} . \end{matrix}

Note that if we carefully choose the detuning δ to fulfill the relation

δ = 2 ω_{f}^{s}

, which will compensate for the cavity-induced energy shift, the motion equation for the mirror is governed by the superoperator 𝒦̂_f and without the oscillating frequency δ in the rotating frame at frequency

ω_{f}^{s}

. We will obtain a better squeezing of the mirror motion. The motion equations of expectation values for the quadratic operators with the above relation becomes

\begin{array}{l} \frac{d}{d t} ({\hat{f}}^{†} \hat{f}) = - 2 [A (ω_{m}) - A (- ω_{m}) + γ_{m} / 2] 〈 {\hat{f}}^{†} \hat{f} 〉 + 2 [C_{1} 〈 {\hat{f}}^{2} 〉 + c . c .] + 2 A (- ω_{m}) + γ_{m} {\bar{n}}_{th}, \\ \frac{d}{d t} {〈 {\hat{f}}^{†} 〉}^{2} = - 2 [A (ω_{m}) - A (- ω_{m}) + γ_{m} / 2] 〈 {\hat{f}}^{† 2} 〉 + 4 C_{1} 〈 {\hat{f}}^{†} \hat{f} 〉 - 2 (B_{1} - C_{1}) . \end{array}

In general, the heating rate is approximately equal to A(−ω_m), in which f (Δ, −ω_m) plays a main role due to the small value of ε. When the detuning fulfills the relation Δ = ω_m/2, there is f (Δ, −ω_m) = 0, which is just the optimal condition for the ground-state cooling of a mechanical oscillator appearing in the dissipative optomechanics with monochromic driving [25, 26]. The heating rate becomes proportional to ε²,

A (- ω_{m}) = κ_{c} λ^{2} E^{2} \frac{ε^{2}}{8} [f (Δ - Ω, - ω_{m}) + f (Δ + Ω, - ω_{m})],

and the cavity-induced damping rate κ_m is

κ_{m} = A (ω_{m}) - A (- ω_{m}) + γ_{m} / 2,

which is still dominated by the cooling rate A(ω_m) also because of the small value of ε. Therefore, κ_m is nearly equal to the result obtained in the monochromically-driven dissipative optomechanical system, in which the strongly enhanced cooling effect outside the resolved-sideband regime arising from the Fano interference of quantum noise is achievable. In contrast, for the squeezing of mechanical oscillator motion in the dispersive coupling regime, the laser’s central frequency should be tuned to drive the red sideband of the cavity eigenmode, i.e. Δ = −ω_m, the good cavity limit κ_c ≪ ω_m is necessary [21, 22], and the initial mean thermal phonon number should be much small in order to obtain the desired squeezed state because the thermal noise can degrade the squeezing [38,40]. In this periodically-modulated dissipative optomechanical system, the strong cooling effect can guarantee the squeezed state robust against the thermal noise and it may offer the probability to realize the squeezing of the mirror motion in a relatively “hot” environmental temperature.

The full solution of Eq. (31) in the long time limit can be easily calculated, and then the variance of momentum operator ${〈 Δ \hat{P} 〉}^{2} = - \frac{1}{2} 〈 {[\hat{f} exp (- i ω_{m} t) - {\hat{f}}^{†} exp (i ω_{m} t)]}^{2} 〉 + \frac{1}{2} {〈 [\hat{f} exp (- i ω_{m} t) - {\hat{f}}^{†} exp (i ω_{m} t)] 〉}^{2}$ can be also obtained, in which 〈f̂〉 = 0 is adopted. The result is shown by the red dashed line in Fig. 5. The analytical result approximately agrees with the numerical result which is indicated by the blue solid line. Thus the result validates the perturbation method we employ.

Fig. 5 Analytical approximation of the variance of the mirror momentum operator (red dashed line) compared with the numerical result (blue solid line) for t ∈ [100τ, 102τ].

Download Full Size | PDF

We now turn to analyze the squeezed state with the numerical values, which are n_st = 〈f̂^†f̂〉 = 0.065 and 〈f̂²〉 = −0.15 −0.05i with the above experimentally realizable parameters. The final phonon number in the mirror motion is very close to zero, thus the mirror is near its motional ground state. Meanwhile, the mirror motion is squeezed if 〈ΔP̂²〉 < 1/2. When we expand the expression of 〈ΔP̂²〉, we will obtain the relation

〈 {\hat{f}}^{†} \hat{f} 〉 - \frac{1}{2} [〈 {\hat{f}}^{2} 〉 exp (- i 2 ω_{m} t) + 〈 {\hat{f}}^{† 2} 〉 exp (i 2 ω_{m} t)] < 0 .

Now by defining 〈f̂²〉 = |〈f̂²〉|exp(−iθ), 〈f̂^†2〉 = |〈f̂²〉|exp(iθ) with θ the phase angle, Eqs. (34) is simplified as

〈 {\hat{f}}^{†} \hat{f} 〉 - \frac{1}{2} | 〈 {\hat{f}}^{2} 〉 | {exp [i (2 ω_{m} t + θ)] + exp [- i (2 ω_{m} t + θ)]} < 0 .

When 2ω_mt ∈ [2kπ, 2(k + 1)π] for some integer k, the minimum value of the left-hand side of the relation is 〈f̂^†f̂〉 − |〈f̂²〉| at some t. Thus, if n_st = 〈f̂^†f̂〉 < |〈f̂²〉|, the mirror motion is squeezed. It again demonstrates that the periodic modulation can favor the generation of quantum effects in the dissipative optomechanical systems. But it should note that the mirror is in a mixed squeezed state because the relation |〈f̂²〉| = [n_st (n_st + 1)]^1/2 needed for ideal squeezed state is not hold here. It is partly because that there exist two conflicting squeezing processes: one behaves like a parametric amplifier and the other behaves like an injection of squeezed noise shown in Eq. (28). These two processes both contribute to the squeezing of the mirror motion, but the directions from these two processes do not coincide with each other. This leads to a destructive interference for squeezing [41, 42].

In the last, we discuss the squeezing of the mirror motion with the increased coupling strength, where the destructive interference of quantum noise becomes not complete. As a result, the cavity will no longer appear to the oscillator as an effective zero-temperature bath, giving rise to nonzero limiting temperature [25, 26]. We follow the procedure of the numerical calculation in Sec. 4 to investigate the variance of the momentum operator 〈ΔP̂²〉 with the increased input power P_R while the other parameters are unchanged, and the numerical results are shown in Fig. 6. It shows that in the dissipative optomechanics, the complete destructive interference of quantum noise can help to achieve a better squeezing of the mirror motion. With the increased coupling strength, the photonic excitation of the cavity field will preclude the complete destructive interference and induce an extra thermal phonon excitation, which may decrease the squeezing of the mirror motion.

Fig. 6 Numerical calculations of the variance of the mirror momentum operator with the following set of input powers P_R = 10 mW (blue solid line), P_R = 20 mW (red dashed line), P_R = 40 mW (green dash-dotted line), P_R = 80 mW (black dotted line) for t ∈ [100τ, 102τ].

Download Full Size | PDF

6. Conclusions

In summary, we present the generation of squeezed state of the movable mirror motion via exploiting the periodically-modulated driving on the dissipative optomechanical system. The quantum dynamics of the system around the classical orbits are investigated first with a numerical method and then in an analytical approach via adiabatically eliminating the cavity field and employing the second-order perturbation method, and these two results are in a good agreement with each other. Both the results show that by tuning the carrier frequency of the driving fields at the Fano resonance and the modulation frequency twice the oscillating frequency of the mirror, squeezed state of the mirror motion around its motional ground state is achievable. Further, the strong cooling effect outside the resolved-sideband regime arising from the Fano resonance of quantum noise makes the squeezed state robust against the thermal noise, which may offer the possibility to realize the squeezing of the mirror motion in a relatively “hot” environmental temperature.

A. The perturbation calculations of first moments

In this section, we derive the analytical solutions for the first moments of the quadratures by applying the perturbation method in λ. This originates a chained set of equations, and by truncating a finite number of orders j ⩽ j_max, we will achieve desirable results. In the paper, we find that the finally obtained results are in a good match with the asymptotic numerical results when j_max = 3, with the experimentally realizable parameters presented in the main text,.

The variables Ō_j(t) form a set of linear differential equations with τ-periodic inhomogeneous terms for each j. These variables can be expanded by the Fourier series in Eq. (12). By substituting the expansions into the nonlinear differential equations (10), at zeroth order in λ we have a group of equations.

\begin{matrix} \sum_{n} ({\bar{\dot{q}}}_{n, 0} + in Ω {\bar{q}}_{n, 0}) exp (in Ω t) & = & ω_{m} \sum_{n} {\bar{p}}_{n, 0} exp (in Ω t), \\ \sum_{n} ({\bar{\dot{p}}}_{n, 0} + in Ω {\bar{p}}_{n, 0} + γ_{m} {\bar{p}}_{n, 0}) exp (in Ω t) & = & - ω_{m} \sum_{n} {\bar{q}}_{n, 0} exp (in Ω t), \\ \sum_{n} ({\bar{\dot{x}}}_{n, 0} + in Ω {\bar{x}}_{n, 0} + κ_{c} {\bar{x}}_{n, 0}) exp (in Ω t) & = & - Δ \sum_{n} {\bar{y}}_{n, 0} exp (in Ω t) - \sqrt{2} E (1 + ε cos Ω t), \\ \sum_{n} ({\bar{\dot{y}}}_{n, 0} + in Ω {\bar{y}}_{n, 0} + κ_{c} {\bar{y}}_{n, 0}) exp (in Ω t) & = & Δ \sum_{n} {\bar{x}}_{n, 0} exp (in Ω t) . \end{matrix}

Since the forcing term is purely oscillating at frequencies 0 and ±Ω, the first moment of each variable is a sum of oscillating terms at frequencies 0 and ±Ω in the long time limit, i.e., Ō₀ = Ō_0,0 + Ō_1,0 exp(iΩt) + Ō_−1,0 exp(−iΩt). Thus, n takes the values 0, ±1. Note that the mean values of the mirror quadratures in zeroth order q̄₀ and p̄₀ are zero indicated by the first two equations in Eqs. (39).

For the differential equations of the variables in first order of λ, with use of the zeroth-order results we obtain a group of equations.

\begin{matrix} \sum_{n} ({\bar{\dot{q}}}_{n, 1} + in Ω {\bar{q}}_{n, 1}) exp (in Ω t) & = & ω_{m} \sum_{n} {\bar{p}}_{n, 1} exp (in Ω t), \\ \sum_{n} ({\bar{\dot{p}}}_{n, 1} + in Ω {\bar{p}}_{n, 1} + γ_{m} {\bar{p}}_{n, 1}) exp (in Ω t) & = & - ω_{m} \sum_{n} {\bar{q}}_{n, 1} exp (in Ω t) + \sqrt{2} E (1 + ε cos Ω t) {\bar{y}}_{0}, \\ \sum_{n} ({\bar{\dot{x}}}_{n, 1} + in Ω {\bar{x}}_{n, 1} + κ_{c} {\bar{x}}_{n, 1}) exp (in Ω t) & = & - Δ \sum_{n} {\bar{y}}_{n, 1} exp (in Ω t), \\ \sum_{n} ({\bar{\dot{y}}}_{n, 1} + in Ω {\bar{y}}_{n, 1} + κ_{c} {\bar{y}}_{n, 1}) exp (in Ω t) & = & Δ \sum_{n} {\bar{x}}_{n, 1} exp (in Ω t) . \end{matrix}

The forcing term now becomes

\sqrt{2} E (1 + ε cos Ω t) {\bar{y}}_{0}

, and its oscillating frequencies are at ±2Ω, ±Ω, 0. Similarly, the solution is a sum of terms at these oscillating frequencies, i.e.,

{\bar{O}}_{1} (t) = \sum_{n = - 2}^{2} {\bar{O}}_{n, 1} exp (in Ω t)

, in which the time-independent coefficients Ō_n,1 can be calculated in Eqs. (40). At the first order, the cavity quadratures x̄₁ and ȳ₁ are zero shown by the 3rd and 4th equations.

The differential equations for the variables in the second order of λ can be also achieved in the following with the use of the zeroth- and first- order results, which are given by

\begin{matrix} \sum_{n} ({\bar{\dot{q}}}_{n, 2} + in Ω {\bar{q}}_{n, 2}) exp (in Ω t) & = & ω_{m} \sum_{n} {\bar{p}}_{n, 2} exp (in Ω t), \\ \sum_{n} ({\bar{\dot{p}}}_{n, 2} + in Ω {\bar{p}}_{n, 2} + γ_{m} {\bar{p}}_{n, 2}) exp (in Ω t) & = & - ω_{m} \sum_{n} {\bar{q}}_{n, 2} exp (in Ω t), \\ \sum_{n} ({\bar{\dot{x}}}_{n, 2} + in Ω {\bar{x}}_{n, 2} + κ_{c} {\bar{x}}_{n, 2}) exp (in Ω t) & = & - Δ \sum_{n} {\bar{y}}_{n, 2} exp (in Ω t) - \sqrt{2} E (1 + ε cos Ω t) {\bar{q}}_{1} - 2 κ_{c} {\bar{x}}_{0} {\bar{q}}_{1}, \\ \sum_{n} ({\bar{\dot{y}}}_{n, 2} + in Ω {\bar{y}}_{n, 2} + κ_{c} {\bar{y}}_{n, 2}) exp (in Ω t) & = & Δ \sum_{n} {\bar{x}}_{n, 2} exp (in Ω t) - 2 κ_{c} {\bar{y}}_{0} {\bar{q}}_{1} . \end{matrix}

Then the forcing terms are

\sqrt{2} E (1 + ε cos Ω t) {\bar{q}}_{1}

, 2κ_cx̄₀q̄₁ and 2κ_cȳ₀q̄₁, and the oscillating frequencies are ±3Ω, ±2Ω, ±Ω, 0. The mirror quadratures q̄₂ and p̄₂ are zero. The mean values of the cavity quadratures are the sums of oscillating terms, i.e.,

{\bar{x}}_{2} (t) ({\bar{y}}_{2} (t)) = \sum_{n = - 3}^{3} {\bar{x}}_{n, 2} ({\bar{y}}_{n, 2}) exp (in Ω t)

, in which the time-independent coefficients are obtained from the 3rd and 4th equations in Eqs. (41). Following the same procedure, the equations of the variables in the third order of λ are given by

\begin{matrix} \sum_{n} ({\bar{\dot{q}}}_{n, 3} + in Ω {\bar{q}}_{n, 3}) exp (in Ω t) & = & ω_{m} \sum_{n} {\bar{p}}_{n, 3} exp (in Ω t) \\ \sum_{n} ({\bar{\dot{p}}}_{n, 3} + in Ω {\bar{p}}_{n, 3} + γ_{m} {\bar{p}}_{n, 3}) exp (in Ω t) & = & - ω_{m} \sum_{n} {\bar{q}}_{n, 3} exp (in Ω t) + \sqrt{2} E (1 + ε cos Ω t) {\bar{y}}_{2}, \\ \sum_{n} ({\bar{\dot{x}}}_{n, 3} + in Ω {\bar{x}}_{n, 3} + κ_{c} {\bar{x}}_{n, 3}) exp (in Ω t) & = & - Δ \sum_{n} {\bar{y}}_{n, 3} exp (in Ω t) - \sqrt{2} E (1 + ε cos Ω t) {\bar{q}}_{2} - 2 κ_{c} {\bar{x}}_{0} {\bar{q}}_{2}, \\ \sum_{n} ({\bar{\dot{y}}}_{n, 3} + in Ω {\bar{y}}_{n, 3} + κ_{c} {\bar{y}}_{n, 3}) exp (in Ω t) & = & Δ \sum_{n} {\bar{x}}_{n, 3} exp (in Ω t) - 2 κ_{c} {\bar{y}}_{0} {\bar{q}}_{2} . \end{matrix}

Now oscillating frequencies of the forcing terms are ±4Ω, ±3Ω, ±2Ω, ±Ω, 0, and the final results for these variables are in the form

{\bar{O}}_{3} (t) = \sum_{n = - 4}^{4} {\bar{O}}_{n, 3} exp (in Ω t)

.

The perturbation results for the mean values of the quadratures are given by $\bar{O} (t) = \sum_{j = 0}^{3} \sum_{n = - 4}^{4} {\bar{O}}_{n, j} (t) λ^{j} exp (in Ω t)$ up to λ³. We numerically show that by truncating the oscillating frequencies to n = ±1, the resulting values

\bar{O} (t) = \sum_{j = 0}^{3} \sum_{n = - 1}^{1} {\bar{O}}_{n, j} (t) λ^{j} exp (in Ω t)

are in a good approximation with the asymptotic numerical orbits in Fig. 2. Physically, this is because higher sidebands well fall outside the cavity bandwidth, nΩ > κ_c, leading to negligible driving effects on the cavity field. Based on these approximate values of first moments, we can investigate the full quantum dynamics of the optomechanical system by assuming small quantum fluctuations around their classical values.

B. The numerical calculation of the second moments X(t)

The motion equation for second moments X(t) can be derived from the master equation (14), and written in a matrix form in Eq. (20). The coefficient matrix M̄(t) is

\bar{M} (t) = (\begin{matrix} - 2 ξ_{1} & 0 & 0 & ξ_{3}^{*} & ξ_{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 {\tilde{ξ}}_{1} & 0 & 2 ξ_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 {\tilde{ξ}}_{1}^{*} & 0 & 2 ξ_{3}^{*} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\tilde{ξ}}_{1} & 0 & ω_{m} & 0 & ξ_{3} & 0 & 0 \\ 0 & 0 & 0 & 0 & {\tilde{ξ}}_{1}^{*} & 0 & ω_{m} & ξ_{3}^{*} & 0 & 0 \\ ξ_{2} & - ξ_{2} & 0 & - ω_{m} & 0 & {\tilde{ξ}}_{1} - γ_{m} & 0 & 0 & 0 & ξ_{3} \\ - ξ_{2} & 0 & ξ_{2} & 0 & - ω_{m} & 0 & {\tilde{ξ}}_{1}^{*} - γ_{m} & 0 & 0 & ξ_{3}^{*} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 ω_{m} \\ 0 & 0 & 0 & 0 & 0 & - 2 ξ_{2} & 2 ξ_{2} & 0 & - 2 γ_{m} & - 2 ω_{m} \\ 0 & 0 & 0 & - ξ_{2} & ξ_{2} & 0 & 0 & - ω_{m} & ω_{m} & - γ_{m} \end{matrix}),

and the inhomogeneous term C(t) is

C (t) = {(\begin{matrix} 0, & 0, & 0, & 0, & 0, & - i ξ_{3}, & 0, & - i ω_{m}, & i ω_{m} + 2 κ_{c} λ^{2} {| \bar{a} (t) |}^{2} + 2 γ_{m} ({\bar{n}}_{th} + 1 / 2), & i γ_{m} / 2 \end{matrix})}^{T},

with the variables ξ₁ = κ_c [1 + λq̄(t)]², ξ̃₁ = iΔ −ξ₁, ξ₂ = iλE(1 + ε cos Ωt), and ξ₃ = iξ₂ −2κ_cλā(t)[1 + λq̄(t)]. By writing the matrix M̄(t) and vector C(t) as a sum of harmonically oscillating terms in Eqs. (21) and truncating Fourier series of X(t) to n = ±2 in Eq. (22), we obtain a chained set of equations.

\begin{matrix} {\dot{X}}_{0} & = M_{0} X_{0} + M_{1} X_{- 1} + M_{- 1} X_{1} + M_{2} X_{- 2} + M_{- 2} X_{2} + C_{0}, \\ {\dot{X}}_{1} + i Ω X_{1} & = M_{0} X_{1} + M_{1} X_{0} + M_{2} X_{- 1} + M_{- 1} X_{2} + C_{1}, \\ {\dot{X}}_{- 1} + i Ω X_{- 1} & = M_{0} X_{- 1} + M_{- 1} X_{0} + M_{- 2} X_{1} + M_{1} X_{- 2} + C_{- 1}, \\ {\dot{X}}_{2} + i 2 Ω X_{2} & = M_{0} X_{2} + M_{2} X_{0} + M_{1} X_{1} + C_{2}, \\ {\dot{X}}_{- 2} + i 2 Ω X_{- 2} & = M_{0} X_{- 2} + M_{- 2} X_{0} + M_{- 1} X_{- 1} + C_{- 2} . \end{matrix}

The steady-state solutions for X_±2 are obtained by solving the last two equations, which are expressed in the form

X_{\pm 2} = - \frac{M_{\pm 1} X_{\pm 1} + M_{\pm 2} X_{0} + C_{\pm 2}}{M \mp 2 i Ω I},

in which I is the 10 × 10 identity matrix. By substituting the results into the first three equations in Eqs. (46), the steady-state solutions for X_0,±1 can be calculated by writing the equations in a matrix form

\begin{matrix} (\begin{matrix} M_{2} & M_{1} - M_{- 1} \frac{M_{2}}{M_{0} - 2 i Ω I} & M_{0} - i Ω I - M_{- 1} \frac{M_{1}}{M_{0} - 2 i Ω I} \\ M_{0} + i Ω I - M_{1} \frac{M_{- 1}}{M_{0} + 2 i Ω I} & M_{- 1} - M_{1} \frac{M_{- 2}}{M_{0} + 2 i Ω I} & M_{- 2} \\ M_{1} - M_{2} \frac{M_{- 1}}{M_{0} + 2 i Ω I} & M_{0} - M_{2} \frac{M_{- 2}}{M_{0} + 2 i Ω I} - M_{- 2} \frac{M_{2}}{M_{0} - 2 i Ω I} & M_{- 1} - M_{- 2} \frac{M_{1}}{M_{0} - 2 i Ω I} \end{matrix}) \\ \times (\begin{matrix} X_{- 1} \\ X_{0} \\ X_{1} \end{matrix}) = (\begin{matrix} - C_{1} + M_{- 1} \frac{C_{2}}{M_{0} - 2 i Ω I} \\ - C_{- 1} + M_{1} \frac{C_{- 2}}{M_{0} + 2 i Ω I} \\ - C_{0} + M_{2} \frac{C_{- 2}}{M_{0} + 2 i Ω I} + M_{- 2} \frac{C_{2}}{M_{0} - 2 i Ω I} \end{matrix}) . \end{matrix}

Then, with use of the solutions in Eq. (48), X_±2 in Eq. (47) can be also achieved. Finally, the second moments of the variables in the cavity-mirror system are given by

X (t) = \sum_{n = - 2}^{2} X_{n} exp (in Ω t),

from which we will get the variances of the mirror position and momentum operators.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants No. 11074087 and No. 61275123), the Nature Science Foundation of Wuhan City (Grant No. 201150530149), and the National Basic Research Program of China (Grant No. 2012CB921602).

References and links

1. K. Stannigel, P. Rabl, A. S. Sorensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105, 220501 (2010). [CrossRef]

2. V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. L. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011). [CrossRef] [PubMed]

3. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107, 063601 (2011). [CrossRef] [PubMed]

4. B. He, “Quantum optomechanics beyond linearization,” Phys. Rev. A 85, 063820 (2012). [CrossRef]

5. E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509514 (2012). [CrossRef]

6. Y. Wu, M. C. Chu, and P. T. Leung, “Dynamics of the quantized radiation field in a cavity vibrating at the fundamental frequency,” Phys. Rev. A 59, 3032–3037 (1999). [CrossRef]

7. I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys. 8393–397 (2012). [CrossRef]

8. A. Nunnenkamp, K. Borkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107, 063602 (2011). [CrossRef] [PubMed]

9. T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett. 110, 193602 (2013). [CrossRef] [PubMed]

10. G. Z. Cohen and M. Di Ventra, “Reading, writing, and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID,” Phys. Rev. B 87, 014513 (2013). [CrossRef]

11. M. T. Jaekel and S. Reynaud, “Quantum limits in interferometric measurements,” Europhys. Lett. 13, 301–306 (1990). [CrossRef]

12. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009). [CrossRef]

13. Y. Wu and C. Côté, “Quadrature-dependent bogoliubov transformations and multiphoton squeezed states,” Phys. Rev. A 66, 025801 (2002). [CrossRef]

14. M. F. Bocko and R. Onofrio, “On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress,” Rev. Mod. Phys. 68, 755–799 (1996). [CrossRef]

15. A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezing of light via refection from a silicon micromechanical resonator,” arXiv: 1302.6179v2 (2013).

16. T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” arXiv: 1306.1268v2 (2013).

17. A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008). [CrossRef]

18. S. Huang and G. S. Agarwal, “Enhancement of cavity cooling of a micromechanical mirror using parametric interactions,” Phys. Rev. A 79, 013821 (2009). [CrossRef]

19. H. Ian, Z. R. Gong, Yu-xi Liu, C. P. Sun, and Franco Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A 78, 013824 (2008). [CrossRef]

20. K. Jähne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009). [CrossRef]

21. A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86, 013820 (2012). [CrossRef]

22. A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103, 213603 (2009). [CrossRef]

23. B. Rogers, M. Paternostro, G. M. Palma, and G. De Chiara, “Entanglement control in hybrid optomechanical systems,” Phys. Rev. A 86, 042323 (2012). [CrossRef]

24. P. Doria, T. Calarco, and S. Montangero, “Optimal control technique for many-body quantum dynamics,” Phys. Rev. Lett. 106, 190501 (2011). [CrossRef] [PubMed]

25. F. Elste, S. M. Girvin, and A. A. Clerk, “Quantum noise interference and backaction cooling in cavity nanomechanics,” Phys. Rev. Lett. 102, 207209 (2009). [CrossRef] [PubMed]

26. A. Xuereb, R. Schnabel, and K. Hammerer, “Dissipative optomechanics in a Michelson–Sagnac interferometer,” Phys. Rev. Lett. 107, 213604 (2011). [CrossRef]

27. W. J. Gu, G. X. Li, and Y. P. Yang, “Generation of squeezed states in a movable mirror via dissipative optomechanical coupling,” Phys. Rev. A 88, 013835 (2013). [CrossRef]

28. K. Yamamoto, D. Friedrich, T. Westphal, S. Gobler, K. Danzmann, K. Somiya, S. L. Danilishin, and R. Schnabel, “Quantum noise of a Michelson–Sagnac interferometer with a translucent mechanical oscillator,” Phys. Rev. A 81, 033849 (2010). [CrossRef]

29. D. Friedrich, H. Kaufer, T. Westphal, K. Yamamoto, A. Sawadsky, F. Y. Khalili, S. L. Danilishin, S. Gobler, K. Danzmann, and R. Schnabel, “Laser interferometry with translucent and absorbing mechanical oscillators,” New J. Phys. 13, 093017 (2011). [CrossRef]

30. T. Weiss, C. Bruder, and A. Nunnenkamp, “Strong-coupling effects in dissipatively coupled optomechanical systems,” New J. Phys. 15, 045017 (2013). [CrossRef]

31. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997). [CrossRef]

32. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1990).

33. C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlarg, 2000).

34. G. Teschl, Ordinary Differential Equations and Dynamical Systems, http://www.mat.univie.ac.at/gerald.

35. L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012). [CrossRef]

36. X. G. Wang and B. C. Sanders, “Relations between bosonic quadrature squeezing and atomic spin squeezing,” Phys. Rev. A 68, 033821 (2003). [CrossRef]

37. M. J. Woolley, A. C. Doherty, G. J. Milburn, and K. C. Schwab, “Nanomechanical squeezing with detection via a microwave cavity,” Phys. Rev. A 78, 062303 (2008). [CrossRef]

38. M. P. Blencowe and M. N. Wybourne, “Quantum squeezing of mechanical motion for micron-sized cantilevers,” Physica B 280, 555–556 (2000). [CrossRef]

39. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008). [CrossRef]

40. P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004). [CrossRef]

41. G. S. Agarwal, “Interferences in parametric interactions driven by quantized fields,” Phys. Rev. Lett. 97, 023601 (2006). [CrossRef] [PubMed]

42. G. X. Li, H. T. Tan, and M. Macovei, “Enhancement of entanglement for two-mode fields generated from four-wave mixing with the help of the auxiliary atomic transition,” Phys. Rev. A 76, 053827 (2007). [CrossRef]

Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system

Abstract

1. Introduction

2. The periodically-modulated dissipative optomechanical system

3. Analysis of semiclassical phase space orbits

4. The full quantum dynamics around the classical orbits

5. The explicit motion equation for mirror via adiabatically eliminating the cavity variable

6. Conclusions

A. The perturbation calculations of first moments

B. The numerical calculation of the second moments X(t)

Acknowledgments

References and links

Cited By

Figures (6)

Equations (49)

Optics Express