Quantum squeezing in a modulated optomechanical system

Zhu-Cheng Zhang; Yi-Ping Wang; Ya-Fei Yu; Zhi-Ming Zhang

doi:10.1364/OE.26.011915

1. Introduction

Cavity optomechanics [1, 2], since it was put forward, has been a very hot research field. A prototypical cavity optomechanical system consists of a Fabry-Pérot cavity with one movable end mirror, which couples mechanical freedom with optical freedom through radiation pressure and provides a powerful research platform for a wide variety of theoretical and experimental investigations. For example, a considerable quantum effects have been found and confirmed including quantum squeezing [3–21], quantum ground-state cooling of the mechanical mirror [22–24], entanglement between light and the mirror [25], photon anti-bunching [26,27] and so on.

Quantum squeezing, as a typical quantum effect, is an important resource for many applications, such as gravitational wave detection [28, 29], quantum limited displacement sensing [30,31] and even biological measurements [32]. It is well known that optomechanical system, in its steady state, can mimic a Kerr medium [3] and the fluctuations in the system can give rise to optical quadrature squeezing termed the ponderomotive squeezing [3–5]. Many schemes for creating the ponderomotive squeezing in different physical systems have been proposed. For example, Hichem Eleuch et al. theoretically and experimentally investigated the ponderomotive squeezing in semiconductor microcavities [6–8]. Moreover, Marino et al. experimentally generated the ponderomotive squeezing by using classical intensity noise in optomechanical system [9], and Brooks et al. also experimentally demonstrated that the backaction of the motion of an ultracold atomic gas on the cavity light field produces the ponderomotive squeezing [10]. Besides ponderomotive squeezing, many schemes for creating quantum squeezing of the moving mirror in different physical systems have been proposed. For example, Jahne et al. theoretically demonstrated the strong squeezing of the mechanical oscillator via injecting a nonclassical light [11], and the experimental squeezing of a nanomechanical object has been achieved by Almog et al. through a nonlinear Duffing resonator [12]. Moreover, the mechanical resonator can also be squeezed by coupling it to an auxiliary nonlinear system, such as a superconducting quantum interference device loop [13], a Cooper-pair box circuit [14], or an optical cavity containing an atomic medium [15].

In this paper, we propose a scheme for generating quantum squeezing including ponderomotive squeezing and mechanical squeezing in an optomechanical system, where the radiation-pressure coupling and mechanical spring constant are modulated periodically [33–36]. In particular, the mechanical parametric amplification was theoretically and experimentally demonstrated by Rugar et al. and Szorkovszky et al. through the periodic modulation of the mechanical spring constant [33–35]. Moreover, together with the sinusoidal modulation of the radiation-pressure coupling, the resonant nonlinear photon-phonon interaction enhanced to the single-photon strong-coupling regime was also theoretically demonstrated through mechanical parametric amplification [36].

Besides applying the modulated optomechanical system, we assume that the mechanical resonator is coupled to a squeezed vacuum reservoir. It could be realized in principle by introducing an ancillary cavity mode, which has been deeply investigated in a great deal of studies [36–39], so we do not discuss it further in this paper. Our scheme is different from Ref. [16], in which the resonator is coupled to a thermal reservoir, so that the degree of squeezing will be affected by the thermal reservoir temperature even if the system can achieve a strong and robust squeezing. Moreover, our scheme is also different from Ref. [17], in which the cavity mode is coupled to a squeezed vacuum reservoir but the degree of squeezing is still affected by the effective phonon noise. Comparing with previous schemes for generating quantum squeezing, the main novelties of our scheme are the resonant radiation-pressure interaction can be enhanced remarkably by the modulation-induced mechanical parametric amplification, and the effective phonon noise can be suppressed completely by introducing a squeezed vacuum reservoir. Thus, under the conditions of enhanced radiation-pressure interaction and suppressed phonon noise, a controllable quantum squeezing can be achieved by modulating the amplitude of the mechanical parametric driving, which allows us to find the optimal quantum squeezing in the modulated optomechanical system.

This paper is organized as follows: In Sec. 2, we describe the model and solve the dynamical equation of the system. The ponderomotive squeezing and the mechanical squeezing are displayed in Sec. 3. Finally, we summarize our main results in Sec. 4.

2. Model and dynamical equation

As shown in Fig. 1, our system consists of an optomechanical cavity with modulated radiation-pressure coupling and mechanical spring constant, pumped by a laser with frequency ω_p and amplitude $ϵ_{p} = \sqrt{2 κ P_{p} / ℏ ω_{p}}$ . Here P_p is the power of the pump field, and κ is the decay rate of the cavity field. A mechanical parametric amplification is induced due to the periodic modulation of the mechanical spring constant at 2ω_d. Then, in the rotating frame with the frequency (ω_d + ω_p) and under the rotating wave approximation, the total Hamiltonian of the system can be written as (ħ = 1) [33–36]

H_{t} = Δ_{c} c^{†} c + Δ_{m} b^{†} b + \frac{1}{2} ϵ_{b} (b^{2} + b^{† 2}) - \frac{1}{2} g_{0} c^{†} c (b + b^{†}) + i ϵ_{p} (c^{†} - c),

where c(b) and c^†(b^†) are the annihilation and creation operators of the cavity (mechanical) mode, respectively. Δ_c = ω_c − ω_p (Δ_m = ω_m − ω_d) is the frequency detuning with the optical and mechanical resonant frequencies ω_c and ω_m. ϵ_b characterizes the amplitude of the mechanical parametric driving and g₀ is the single-photon optomechanical coupling.

Fig. 1 Sketch of the system. A cavity mode pumped by a laser at the red-detuned mechanical sideband, couples to a mechanical resonator with a modulating coupling strength g(t) = g₀ cos (ω_dt), in addition, the mechanical resonator is parametrically driven by modulating its spring constant, i.e., k(t) = k₀ − k_r cos(2ω_dt).

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In particular, the quadratic part of H_t can be diagonalized with the squeezing transformation, S(r_d)H_tS^†(r_d), where S(r_d) = exp[r_d(b² − b^†2)] is the squeezing operator and $r_{d} = \frac{1}{4} \ln \frac{Δ_{m} + ϵ_{b}}{Δ_{m} - ϵ_{b}}$ is the squeezing amplitude. By using the relation

S (r_{d}) b S^{†} (r_{d}) = b \cosh (r_{d}) + b^{†} \sinh (r_{d}),

we obtain

\begin{array}{l} H_{e f f} = S (r_{d}) H_{t} S^{†} (r_{d}) \\ = Δ_{c} c^{†} c + ω_{m}^{'} b^{†} b - g^{'} c^{†} c (b + b^{†}) + i ϵ_{p} (c^{†} - c), \end{array}

where

ω_{m}^{'} = Δ_{m} / \cosh (2 r_{d})

is the transformed mechanical frequency, hence the coherent cavity driving on the red-detuned mechanical sideband corresponds to

Δ_{c} = ω_{m}^{'}

in our system. And

g^{'} = \frac{1}{2} g_{0} e^{r_{d}}

is the transformed optomechanical coupling, which can be exponentially enhanced with the increase of r_d. Specifically, a large value of r_d can be obtained by adjusting the system parameter ϵ_b to approaching ∆_m, which is feasible with current technologies [40, 41]. The Heisenberg-Langevin equations can be derived as

\dot{c} = - (κ + i Δ_{c}) c + i g^{'} c (b + b^{†}) + ϵ_{p} + \sqrt{2 κ} c_{i n},

\dot{b} = - (γ_{m} + i ω_{m}^{'}) b + i g^{'} c^{†} c + \sqrt{2 γ_{m}} b_{i n},

where κ and γ_m are the decay rates of the cavity and mechanical modes. c_in and b_in are the Langevin noise operators for the cavity and mechanical modes, respectively.

Here we assume that the mechanical resonator is coupled to a squeezed vacuum reservoir with the center squeezing parameter r_b and reference phase Φ_e, and the cavity is coupled to a vacuum reservoir. Thus, both noise operators have zero mean, that is, 〈c_in〉 = 〈b_in〉 = 0, and their only nonzero correlation functions are

〈 c_{i n} (t) c_{i n}^{†} (t^{'}) 〉 = δ (t - t^{'}),

〈 b_{i n} (t) b_{i n}^{†} (t^{'}) 〉 = (N_{e f f} + 1) δ (t - t^{'}),

〈 b_{i n}^{†} (t) b_{i n} (t^{'}) 〉 = N_{e f f} δ (t - t^{'}),

〈 b_{i n} (t) b_{i n} (t^{'}) 〉 = M_{e f f}^{†} δ (t - t^{'}),

〈 b_{i n}^{†} (t) b_{i n}^{†} (t^{'}) 〉 = M_{e f f} δ (t - t^{'}),

where N_eff and M_eff represent the effective thermal noise and the two-phonon correlation interaction [42,43] with the expressions of

\begin{array}{l} N_{e f f} = \sinh^{2} (r_{e}) \cosh^{2} (r_{d}) + \sinh^{2} (r_{d}) \cosh^{2} (r_{e}) + \\ \frac{1}{2} \cos (Φ_{e}) \sinh (2 r_{e}) \sinh (2 r_{d}), \end{array}

\begin{array}{l} M_{e f f} = [\cosh (r_{e}) \sinh (r_{d}) + e^{i Φ_{e}} \sinh (r_{e}) \cosh (r_{d})] \times \\ [\cosh (r_{e}) \cosh (r_{d}) + e^{i Φ_{e}} \sinh (r_{e}) \sinh (r_{d})] . \end{array}

As shown in Fig. 2, the effective thermal noise N_eff is plotted as a function of the squeezing parameters r_e and reference phase Φ_e for different values of the amplitude ϵ_b (note that the amplitude of M_eff has the same behavior as N_eff, and is not plotted here). From the figures, we can easily observe that the effective thermal noise N_eff will increase exponentially (periodically) when the squeezing parameter r_e (phase Φ_e) deviates from the ideal parameters (r_e = r_d and Φ_e = ±nπ (n = 1, 3, 5, …)). That is, the effective thermal noise and the two-photon correlation interaction can be suppressed completely (i.e., N_eff = M_eff = 0) under the ideal parameter conditions, which shows that the squeezed vacuum reservoir, coupled to mechanical resonator, corresponds to an effective vacuum reservoir. Based on our scheme, it is necessary to suppress the effective thermal noise and two-phonon correlation interaction to observe the ponderomotive squeezing and mechanical squeezing as shown in the following sections.

Fig. 2 The effective thermal noise N_eff is plotted as a function of (a) the squeezing parameters r_e, (b) the reference phase Φ_e for different values of the amplitude ϵ_b, where ϵ_b = 3999.50κ (solid black curve), ϵ_b = 3999.65κ (dashed blue curve), ϵ_b = 3999.80κ (dotted red curve), ϵ_b = 3999.95κ (dot-dashed green curve). Other parameters are ∆_m = 4000κ and κ = 0.1MHz.

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By using o = o_s + δo (o = c, b), Eqs. (4)–(5) can be divided into the steady parts and the fluctuation ones. Substituting the division forms into Eqs. (4)–(5), we can obtain the following steady-state solutions of the system operators

c_{s} = \frac{ϵ_{p}}{i Δ + κ},

b_{s} = \frac{g^{'} {| c_{s} |}^{2}}{ω_{m}^{'} - i γ_{m}},

where

Δ = Δ_{c} - g^{'} (b_{s} + b_{s}^{*})

is the effective detuning.

As shown in Fig. 3, in our system, the strong red-detuned driving on the cavity generates large steady-state amplitudes in both the cavity and mechanical modes. For example, with a pump field amplitude ϵ_p ≈ 2000κ, |c_s|² ≈ 10 can ensure the validity of our following assumptions for linearization. Thus, the Heisenberg-Langevin equations for the fluctuations δo (o = c, b) can be given by

δ \dot{c} = - (κ + i Δ) δ c + i g^{'} c_{s} (δ b + δ b^{†}) + \sqrt{2 κ} c_{i n},

δ \dot{b} = - (γ_{m} + i ω_{m}^{'}) δ b + i g^{'} (c_{s}^{*} δ c + c_{s} δ c^{†}) \sqrt{2 γ_{m}} b_{i n} .

Fig. 3 The steady-state amplitudes |c_s| and |b_s| vs pump amplitude ϵ_p. The parameters are κ = 0.1MHz, γ_m = 0.01κ, g₀ = 0.005κ, ∆_m = 4000κ, ϵ_b = 3999.95κ, $Δ_{c} = ω_{m}^{'}$ .

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We rewrite the above equations in the compact form as

\dot{μ} (t) = A μ (t) + ν (t),

with the column vector of fluctuations in the system being µ^T = δc, δc^†, δb, δb^†) and the column vector of noise being

ν^{T} = (\sqrt{2 κ} c_{i n}, \sqrt{2 κ} c_{i n}^{†}, \sqrt{2 γ_{m}} b_{i n}, \sqrt{2 γ_{m}} b_{i n}^{†})

. The matrix A is given by

A = (\begin{matrix} - (κ + i Δ) & 0 & i g^{'} c_{s} & i g^{'} c_{s} \\ 0 & - (κ - i Δ) & - i g^{'} c_{s}^{*} & - i g^{'} c_{s}^{*} \\ i g^{'} c_{s}^{*} & i g^{'} c_{s} & - (γ_{m} + i ω_{m}^{'}) & 0 \\ - i g^{'} c_{s}^{*} & - i g^{'} c_{s} & 0 & - (γ_{m} - i ω_{m}^{'}) \end{matrix}) .

The system is stable and reaches its steady state when all of the eigenvalues of A have negative real parts. The stability conditions can be derived by applying the Routh-Hurwitz criterion [44], yielding the following two nontrivial conditions on the system parameters,

4 γ_{m} κ {{[{(γ_{m} + κ)}^{2} + Δ^{2}]}^{2} + 2 [{(γ_{m} + κ)}^{2} - Δ^{2}] ω_{m}^{' 2} + ω_{m}^{' 4}} + 16 {(γ_{m} + κ)}^{2} Δ ω_{m}^{'} {g^{'}}^{2} {| c_{s} |}^{2} > 0,

(Δ^{2} + κ^{2}) (γ_{m}^{2} + ω_{m}^{' 2}) - 4 Δ ω_{m}^{'} g^{' 2} {| c_{s} |}^{2} > 0,

which will be considered to be satisfied from now on.

3. Quantum squeezing

3.1. The ponderomotive squeezing

In this section, we investigate the squeezing properties of the transmitted field, which is accessible to experiment and useful for practical applications. It is well known that fluctuations of the electric field are more convenient to measure in the frequency domain than in the time domain experimentally. Therefore, by using the definition of the Fourier transform for an operator o = (δc, δb, c_in, b_in),

o (ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} o (t) e^{- i ω t} d t, o^{†} (ω) = {[o (- ω)]}^{†},

the correlations of noise operators in the frequency domain can be defined as

〈 c_{i n} (ω) c_{i n}^{†} (ω^{'}) 〉 = δ (ω + ω^{'}),

〈 b_{i n} (ω) b_{i n}^{†} (ω^{'}) 〉 = (N_{e f f} + 1) δ (ω + ω^{'}),

〈 b_{i n}^{†} (ω) b_{i n} (ω^{'}) 〉 = N_{e f f} δ (ω + ω^{'})

〈 b_{i n} (ω) b_{i n} (ω^{'}) 〉 = M_{e f f}^{*} δ (ω + ω^{'}),

〈 b_{i n}^{†} (ω) b_{i n}^{†} (ω^{'}) 〉 = M_{e f f} δ (ω + ω^{'}) .

After solving the matrix Eq. (17) in the frequency domain, we obtain

δ c (ω) = ζ_{1} (ω) c_{i n} + ζ_{2} (ω) c_{i n}^{†} + ζ_{3} (ω) b_{i n} + ζ_{4} (ω) b_{i n}^{†},

with

ζ_{1} (ω) = \frac{\sqrt{2 κ} [Θ_{1} ({g^{'}}^{2} {| c_{s} |}^{2} + Θ_{2} Λ_{2}) - Θ_{2} {g^{'}}^{2} {| c_{s} |}^{2}]}{d (ω)}

ζ_{2} (ω) = \frac{\sqrt{2 κ} (Θ_{1} - Θ_{2}) {g^{'}}^{2} c_{s}^{2}}{d (ω)},

ζ_{3} (ω) = \frac{i \sqrt{2 γ_{m}} Θ_{2} Λ_{2} g^{'} c_{s}}{d (ω)}

ζ_{4} (ω) = \frac{i \sqrt{2 γ_{m}} Θ_{1} Λ_{2} g^{'} c_{s}}{d (ω)}

d (ω) = Θ_{1} [{g^{'}}^{2} {| c_{s} |}^{2} (Λ_{1} - Λ_{2}) + Θ_{2} Λ_{1} Λ_{2}] + Θ_{2} {g^{'}}^{2} {| c_{s} |}^{2} (Λ_{2} - Λ_{1}),

where Θ_1,2 = γ_m − i (ω ∓ ω_m) and Λ_1,2 = κ − i (ω ∓ ∆).

By using the results above and the input-output relation [45], $δ c_{o u t} = \sqrt{2 κ} δ c - c_{i n}$ , we can obtain the intensity spectrum of the transmitted field given by

S_{o u t} (ω) = 〈 δ c_{o u t}^{†} (ω) δ c_{o u t} (ω) 〉,

and the stationary squeezing spectrum of the transmitted field given by [3,4,18,19]

S_{θ} (ω) = 〈 δ X_{θ}^{o u t} (ω) δ X_{θ}^{o u t} (ω) 〉,

where

δ X_{θ}^{o u t} (ω) = e^{- i θ} δ c_{o u t} (ω) + e^{i θ} δ c_{o u t}^{†} (ω)

with θ being the measurement phase angle. Since

[δ X_{θ}^{o u t}, δ X_{θ + π / 2}^{o u t}] = 2 i

, the quadrature squeezing occurs when

〈 δ X_{θ}^{o u t} (ω) δ X_{θ}^{o u t} (ω) 〉 < 1

, i.e., S_θ (ω) < 1. Then, the squeezing spectrum can be put in the form

S_{θ} (ω) = 1 + 2 B_{c^{†} c} (ω) + [e^{- 2 i θ} B_{c c} (ω) + c . c],

where

〈 δ c_{o u t} (ω) δ c_{o u t} (ω^{'}) 〉 = δ (ω + ω^{'}) B_{c c} (ω),

〈 δ c_{o u t}^{†} (ω) δ c_{o u t} (ω^{'}) 〉 = δ (ω + ω^{'}) B_{c^{†} c} (ω) .

As the degree of squeezing depends on the direction of the measurement of the quadratures, we choose the optimal phase angle θ_opt by solving dS_θ(ω)/d_θ = 0, then we obtain

e^{2 i θ_{o p t}} = \pm \frac{B_{c c} (ω)}{| B_{c c} (ω) |} .

We need to choose the solution with a negative sign as it minimizes the spectrum function, therefore, we have

S_{o p t} (ω) = 1 + 2 B_{c^{†} c} (ω) - 2 | B_{c c} (ω) |,

where

\begin{array}{l} B_{c^{†} c} (ω) = 2 κ [{| ζ_{2} (- ω) |}^{2} + {| ζ_{3} (- ω) |}^{2} N_{e f f} + ζ_{3}^{*} (- ω) ζ_{4} (ω) M_{e f f} + \\ ζ_{4}^{*} (- ω) ζ_{3} (ω) M_{e f f}^{*} + {| ζ_{4} (- ω) |}^{2} (N_{e f f} + 1)], \end{array}

\begin{array}{l} B_{c c} (ω) = 2 κ [ζ_{1} (ω) ζ_{2} (- ω) + ζ_{3} (ω) ζ_{3} (- ω) M_{e f f}^{*} + ζ_{3} (- ω) ζ_{4} (- ω) (N_{e f f} + 1) + \\ ζ_{4} (ω) ζ_{3} (- ω) N_{e f f} + ζ_{4} (ω) ζ_{4} (- ω) M_{e f f} - \frac{ζ_{2} (ω)}{\sqrt{2 κ}}] . \end{array}

Note that under the ideal parameter conditions r_e = r_d and Φ_e = ±nπ (n = 1, 3, 5, …, N_eff = M_eff = 0. Thus, $B_{c^{†} c} (ω)$ and B_cc(ω) simplify to

B_{c^{†} c} (ω) = 2 κ [{| ζ_{2} (- ω) |}^{2} + {| ζ_{4} (- ω) |}^{2}],

B_{c c} (ω) = 2 κ [ζ_{1} (ω) ζ_{2} (- ω) + ζ_{3} (ω) ζ_{4} (- ω) - \frac{ζ_{2} (- ω)}{\sqrt{2 κ}}],

respectively.

The parameters used in this work are experimentally realizable with current technologies. The single-photon optomechanical coupling strength we choose is g₀ = 0.005κ, however, for the required optomechanical coupling strength (i.e., $g^{'} = \frac{1}{2} g_{0} e^{r_{d}}$ ), our scheme is applicable in principle to use a smaller g₀ by employing a larger squeezing parameter r_d in the critical parameter regime ϵ_b → ∆_m [40,41]. Moreover, our scheme is general and applicable to the optomechanical system in the optical wave range [46] or electromechanical system in the microwave range [47]. To implement our scheme, we can choose the ultrahigh-Q toroid microcavity [48] as an example and the system parameters are analogous to Ref. [36]: the cavity damping rate κ = 0.1MHz, the mechanical damping rate γ_m = 0.01κ, the single-photon optomechanical coupling g₀ = 0.005κ, the frequency detuning ∆_m = 4000κ and the pump field amplitude ϵ_p = 2150κ. It should be pointed out that ω_m ≫ κ, i.e., the system operates in the resolved sideband regime.

At first, we analyze the intensity spectrum of the transmitted field. As shown in Eq. (3), due to the periodic modulation of the radiation-pressure coupling and mechanical spring constant, the mechanical frequency ω_m and the single-photon optomechanical coupling g₀ are modified to the amplitude ϵ_b-dependent frequency $ω_{m}^{'}$ and coupling g′, respectively. These modifications affect the spectrum of the transmitted field reflected from the oscillating mirror, that is, some changes would occur in the amplitude and position of the spectral peaks. As shown in Fig. 4, we plot the intensity spectrum of the transmitted field S_out(ω) as a function of the normalized frequency $(ω - ω_{c}) / ω_{m}^{'}$ for different values of the amplitude ϵ_b. From the curves, with the increase of the amplitude ϵ_b, we can easily observe that the amplitude of the intensity spectrum increases remarkably. Moreover, the splitting phenomenon occurs in the intensity spectrum, which also becomes more and more obvious with the increase of amplitude ϵ_b. Specifically, the splitting phenomenon can be analyzed by d(ω) in Eq. (32) quantitatively, whose real and imaginary parts give the position and width of the spectral peaks, respectively. Thus, the explicit expressions for the frequencies at which spectral peaks occur can be derived as

ω_{1} = \pm \sqrt{\frac{z_{1} - \sqrt{z_{1}^{2} + z_{2}}}{2}},

ω_{2} = \pm \sqrt{\frac{z_{1} + \sqrt{z_{1}^{2} + z_{2}}}{2}},

where

z_{1} = γ^{2} + ω_{m}^{' 2} + κ^{2} + Δ^{2} + 4 γ κ,

z_{2} = 16 Δ ω_{m}^{'} + g^{' 2} {| c_{s} |}^{2} - 4 (Δ^{2} + κ^{2}) (γ^{2} + ω_{m}^{' 2}) .

Fig. 4 The intensity spectrum of the transmitted field is plotted as a function of normalized frequency $(ω - ω_{c}) / ω_{m}^{'}$ for different values of the amplitude ϵ_b. Other parameters are the same as in Fig. 3.

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From above equations, we can easily find that the shift in the spectral peaks strongly depends on the amplitude ϵ_b, which is clearly manifested for higher values of ϵ_b as shown in Fig. 4. As the enhanced and splitting spectral features of the transmitted field are observed, it is evident that one can also expect the same to follow for the squeezing spectrum.

Now we focus on how the amplitude ϵ_b affects the squeezing degree of the transmitted field. As shown in Fig. 5, we plot the squeezing spectrum S_opt (ω) as a function of the normalized frequency $(ω - ω_{c}) / ω_{m}^{'}$ for different values of the amplitude ϵ_b. As can be seen from figures, the transmitted field exhibits squeezing with the degree of squeezing strongly relying on the amplitude ϵ_b. Moreover, a considerable amount of squeezing at frequencies (ω₁, ω₂) is observed as expected.

Fig. 5 The squeezing spectrum of the transmitted field is plotted as a function of normalized frequency $(ω - ω_{c}) / ω_{m}^{'}$ for different values of the amplitude ϵ_b. Other parameters are the same as in Fig. 3.

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The spectral features of the squeezing spectrum S_opt (ω) can be analyzed as follows: As shown in Eq. (3), the effective optomechanical coupling can be exponentially enhanced in the modulated optomechanical system, hence our system can be seen as an enhanced Kerr medium [3–5]. Further, as shown in Fig. 2, in our system, the radiation pressure force noise can completely dominate the effects of thermal noise under the ideal parameters. Thus, under the conditions of enhanced radiation-pressure interaction and suppressed phonon noise, the amount of squeezing of the transmitted field is ultimately enhanced remarkably.

In order to observe the features of quantitative change when deviating from the ideal parameters, the squeezing spectrum S_opt (ω₁) is shown as a function of the squeezing parameter r_e and reference phase Φ_e as shown in Fig. 6 (note that S_opt (ω₂) has the same behavior as S_opt(ω₁) in our numerical calculations, and is not plotted here). From the figures, it is clearly shown that the optimal squeezing occurs in the vicinity of the ideal parameters, in which the thermal noise for the mechanical mode is completely suppressed. Thus, the above results show that a squeezed vacuum environment with appropriate squeezing parameter r_e and reference phase Φ_e is required to observe the ponderomotive squeezing, which provides us an effective method for controlling the degree of squeezing and finding the optimal squeezing.

Fig. 6 The squeezing spectrum is plotted as a function of (a) the squeezing parameter r_e, (b) the reference phase Φ_e. Other parameters are the same as in Fig. 3.

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3.2. The mechanical squeezing

Next, we analyze the squeezing properties of the mechanical resonator in the same picture, H_eff = S(r_d) H_tS^†(r_d), by evaluating the variances of the transformed quadrature operators, X₊ = S(r_d) (δb^† + δb) S^†(r_d) and X₋ = S(r_d)[i(δb^† − δb)] S^†(r_d), given by

〈 Δ X_{\pm}^{2} 〉 = [1 + 2 〈 δ b^{†} δ b 〉 \pm (〈 δ b^{2} 〉 + 〈 δ b^{† 2} 〉) \mp {(〈 δ b^{†} 〉 \pm 〈 δ b 〉)}^{2}] e^{\pm 2 r_{d}} .

Thus, the mechanical squeezing takes place when the variances satisfy the condition that either $〈 Δ X_{+}^{2} 〉 < 1$ or $〈 Δ X_{-}^{2} 〉 < 1$ .

From Eqs. (15)–(16), we can obtain the linearized Hamiltonian, and under the rotating wave approximation, it can be written as

H_{e f f, L} = Δ δ c^{†} δ c + ω_{m}^{'} δ b^{†} δ b - g^{'} c_{s} δ c^{†} δ b - g^{'} c_{s}^{*} δ c δ b^{†} .

In the new rotating frame, the Heisenberg-Langevin equations for the fluctuations can be given by

δ \dot{c} = - (κ + i Δ) δ c + i g^{'} c_{s} δ b + \sqrt{2 κ} c_{i n},

δ \dot{b} = - (γ_{m} + i ω_{m}^{'}) δ b + i g^{'} c_{s}^{*} δ c + \sqrt{2 γ_{m}} b_{i n} .

We can solve the above equations and obtain the analytical result for the fluctuation δb(t) using the Laplace transform [17,20,21],

\begin{array}{l} δ b (t) = f_{1} (t) δ c (0) + \sqrt{2 κ} \int_{0}^{t} f_{1} (t - t^{'}) c_{i n} (t^{'}) d t^{'} + \\ f_{2} (t) δ b (0) + \sqrt{2 γ_{m}} \int_{0}^{t} f_{2} (t - t^{'}) b_{i n} (t^{'}) d t^{'}, \end{array}

where

f_{1} (t) = \frac{i g^{'} c_{s}^{*} [e^{(u - v) t / 2} - e^{- (u + v) t / 2}]}{u},

f_{2} (t) = \frac{[u + v - 2 (γ_{m} + i ω_{m}^{'})] e^{(u - ν) t / 2} + [u + v - 2 (κ + i Δ)] e^{- (u + v) t / 2}}{2 u},

u = \sqrt{{[γ_{m} - κ - i (Δ - ω_{m}^{'})]}^{2} - 4 {g^{'}}^{2} {| c_{s} |}^{2}},

v = γ_{m} + κ + i (Δ + ω_{m}^{'}) .

Moreover, we assume that the cavity and the mechanical modes are initially in the vacuum state, that is,

〈 δ c^{†} (0) δ c (0) 〉 = 0, 〈 δ b^{†} (0) δ b (0) 〉 = 0.

Then, using Eq. (52) and taking into account the correlations of noise operators in the time domain, we can obtain the steady-state variances after some tedious calculations [17, 20, 21],

{〈 Δ X_{+}^{2} 〉}_{s s} = [1 + 4 γ_{m} N_{e f f} η_{1} \pm 2 γ_{m} M_{e f f}^{*} η_{2} \pm 2 γ_{m} M_{e f f} η_{2}^{*}] e^{\pm 2 r_{d}},

where

η_{1} = \frac{κ [{(γ_{m} + κ)}^{2} + {(Δ - ω_{m}^{'})}^{2}] + (γ_{m} + κ) {g^{'}}^{2} {| c_{s} |}^{2}}{2 γ_{m} κ [{(γ_{m} + κ)}^{2} + {(Δ - ω_{m}^{'})}^{2}] + 2 {(γ_{m} + κ)}^{2} {g^{'}}^{2} {| c_{s} |}^{2}},

η_{2} = \frac{(κ + i Δ) [γ_{m} + κ + i (Δ + ω_{m}^{'})] + {g^{'}}^{2} {| c_{s} |}^{2}}{2 [(κ + i Δ) (γ_{m} + ω_{m}^{'}) + {g^{'}}^{2} {| c_{s} |}^{2}] [γ_{m} + κ + i (Δ + ω_{m}^{'})]} .

From the above expressions, it is apparent that it can only be present in ${〈 Δ X_{-}^{2} 〉}_{s s}$ if there is any squeezing in mechanical resonator. Further, the steady-state variance ${〈 Δ X_{-}^{2} 〉}_{s s}$ crucially depends on the squeezed vacuum reservoir. We confirm this behavior by plotting the steady-state variance ${〈 Δ X_{-}^{2} 〉}_{s s}$ as a function of the squeezing parameter r_e and reference phase Φ_e as shown in Fig. 7. From the figures, we can easily observe that a considerable amount of squeezing occurs in the mechanical resonator even if the squeezing parameter r_e is very small, i.e., r_e ≪ r_d. Moreover, similar to the ponderomotive squeezing discussed above, higher squeezing in the reservoir doesn’t lead to better squeezing of the mechanical resonator and it would lead to the vanishment of squeezing instead, but we can achieve the optimal squeezing in the vicinity of the ideal parameters.

Fig. 7 The steady-state variance ${〈 Δ X_{-}^{2} 〉}_{s s}$ is plotted as a function of (a) the squeezing parameter r_e, (b) the reference phase Φ_e. Other parameters are the same as in Fig. 3.

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4. Conclusion

In conclusion, we have theoretically investigated the squeezing properties of the cavity and the mechanical modes in an optomechanical system, where the radiation-pressure coupling and the mechanical spring constant are modulated periodically. We have shown that the resonant radiation-pressure interaction can be enhanced remarkably by the modulation-induced mechanical parametric amplification. Moreover, the effective phonon noise can be suppressed completely by introducing a squeezed vacuum reservoir. Specifically, a controllable quantum squeezing is demonstrated by modulating the amplitude of the mechanical parametric driving, then the optimal quantum squeezing can be found in the modulated optomechanical system. This work presents an alternative approach to generate quantum squeezing with current experimental technologies, which can play an important role in the ultrahigh precision measurement and other related fields.

Funding

National Natural Science Foundation of China (Grant Nos. 11574092, 61775062, 61378012, 91121023, and 60978009); National Basic Research Program of China (Grant No. 2013CB921804); the Innovation Project of Graduate School of South China Normal University (2017LKXM090).

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Quantum squeezing in a modulated optomechanical system

Abstract

1. Introduction

2. Model and dynamical equation

3. Quantum squeezing

3.1. The ponderomotive squeezing

3.2. The mechanical squeezing

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (60)

Optics Express