Squeezing transfer of light in a two-mode optomechanical system

Ling Liu; Bang-Pin Hou; Xiao-Hui Zhao; Bei Tang

doi:10.1364/OE.27.008361

1. Introduction

A canonical optomechanical system (OMS) is composed by a fixed mirror and a movable mirror which is coupled to the optical field in the cavity by the radiation pressure [1–3]. The OMS has attracted much attention due to its many important applications in quantum optics and information science, such as in the stationary entanglement [4], the cooling of mechanical oscillators to their quantum ground states [5–8], the detection of ultrahigh precision [9,10], the demonstration of the quantum nonlinearities [11,12], the generation of the macroscopic quantum superposition [13], and optomechanically induced transparency (OMIT) [14–17].

In particular, the squeezings in the optomechanical system have attracted researchers’ attention. It is well known that the squeezed light, which was proposed nearly three decades ago, has less fluctuation in one quadrature than that in a coherent state at the expense of increased fluctuation in the other quadrature [18,19]. The squeezing plays an important role in improving the identification sensitivity of gravitational-wave detection [20] and in continuous-variable information processing [21]. The squeezings in the optomechanical systems have been studied extensively. These investigations are mainly devoted to the mechanical and optical squeezings in the optomechanical system. For example, large degrees of squeezing of a mechanical micromirror in periodically amplitude-modulated optomechanical systems were achieved [22]. The squeezing of the nanoresonator motion in a system composed of a parametrically driven nanomechanical resonator capacitively coupled to a microwave cavity was studied [23]. The generation and detection of the measurement-induced squeezing of a cantilever in optomechanics were studied by using back-action free single-quadrature detection [24]. The momentum quadrature squeezing of a waveguide by injecting a quantum field and laser into the resonator through the waveguide in a dissipative optomechanical system was investigated [25]. Also, the quadrature squeezing of the mechanical oscillator in an optomechanical system can be created by using an field with a large detuning to drive the cavity [26]. The generation of the two-mode squeezed vacuum of the mechanical oscillators in the two-mode optomechanical system was discussed [27,28].

Besides the mechanical squeezing in the optomechanical system was studied, the squeezing of the cavity field in an optomechanical system has been investigated. Firstly, the generation of the output quadrature squeezing, i.e. ponderomotive squeezing produced in an optomechanical system was analysed in Refs. [29,30]. Subsequently, the phase-noise power spectrum of the output light in a cavity with a movable mirror subject to quantum Brownian motion was studied by using the quantum Langevin approach [31]. The quadrature squeezing of the cavity output field, which is induced by the combined effects of dispersive and dissipative couplings, in an optomechanical system was theoretically demonstrated [32]. And, the generation of a two-mode broadband squeezed light was investigated in the cascading double-cavity optomechanical systems [33,34]. In a quadratically coupled optomechanical system, the squeezing of the mechanical oscillator and that of the optical output field were discussed [35,36].

Correspondingly, the experimental investigations of the squeezing in the optomechanical system have been demonstrated. The ponderomotive squeezing induced by the radiation-pressure rigidity was experimentally analyzed [37]. Recently, the ponderomotive squeezing, produced by the back-action of the ultracold atoms’ collective motion which is analogous to the motion of a mirror in a cavity optomechanics was detected [38]. The squeezing of the reflected light’s fluctuation spectrum at the few percent level around the mechanical resonance frequency in the MHz range was observed through homodyne detection in an optomechanical system comprising a micromechanical resonator coupled to a nanophotonic cavity [39], and the production of ponderomotive squeezing in the kilohertz range was demonstrated experimentally in an optomechanical system formed by a Fabry-Pérot cavity with a micromechanical mirror [40]. A larger squeezing of 1.7 dB below the shot-noise level from the optomechanical system composed of a semitransparent membrane placed inside an optical cavity was observed [41].

Besides the optomechanical squeezings discussed above, the reversible transfer of the quantum information between the mechanical oscillator and the electromagnetic field is another remarkable property of the optomechanical system, which attracts much attention due to its potential applications for enabling the distribution of quantum information among distant quantum systems. One kind of the coherent optomechanical transfer is the quantum state transfer between the mechanical oscillator and the electromagnetic field via optomechanical interface. The scheme based on the radiation pressure effects for transferring quantum states as well as entanglement from the propagating light fields to macroscopic, collective vibrational degree of freedom of a massive mirror was proposed [42]. Additionally, the coherent state swapping between two spatially and frequency separated resonators was demonstrated [43]. And, Tian has investigated the transfer of the quantum states between different cavity modes by adiabatically varying the effective optomechanical couplings [44]. The coherent transfer of optical or microwave photon states to mechanical modes was experimentally demonstrated [45–47].

Another kind of coherent optomechanical transfer is the coherent wavelength conversion between photons with vastly differing wavelengths, which is mediated by a mechanical mode coupling to two cavity fields by radiation pressures in a two-optical-mode optomechanical system [48–50]. To achieve efficient conversion fidelity, Wang et al proposed an effective mechanically dark mode, which is immune to mechanical dissipation, to perform high-fidelity quantum conversion between optical and microwave cavities [49]. And, the coherent wavelength conversion of optical photons using photon-phonon translation in a cavity-optomechanical system was theoretically proposed and experimentally demonstrated [50].

Although the coherent optomechanical transfer including quantum state and wavelength conversion has been extensively studied, it is necessary to consider the squeezing transfer between two cavity fields via optomechanical interface due to the fact that the optical squeezing is one of essential properties of light and has application in continuous-variable information processing. The squeezing of a mechanical oscillator within an optical cavity generated from a squeezed vacuum under the condition of resolved-sideband ground state cooling [51], and in the non-resolved-sideband regime [52] has been investigated. To our knowledge, the squeezing transfer between two electromagnetic cavities (e.g., optical and microwave) via optomechanical interface has not been considered yet. The squeezing transfer can facilitate the development of scalable quantum information processors with applications in conversion of information between optical and microwave photons.

In the present paper, we shall investigate the squeezing transfer from a squeezed vacuum injected in one cavity (left-hand cavity) to the output spectrum from the other cavity (right-hand cavity) in a two-mode optomechanical system, in which the two cavity are coupled to a common mechanical oscillator by radiation pressure. It is shown that the squeezing of the output spectrum in two symmetrical region is partially contributed by the destructive interference between the noise fluctuation of the input field and its optomechanically modified one, and mainly by the injected squeezed vacuum. This paper is organized as follows. In Sec.2, we present the model and give the quantum Langevin equations of the system. In Sec.3, we study the squeezing of the output field in the two-mode optomechanical system by calculating the quadrature spectrum of the output field from the right-hand cavity with a shot noise injected into the left-hand cavity. We investigate the quadrature squeezing of the output field in the case where a squeezed vacuum is injected into the left-hand cavity in Sec.4. A brief conclusion is presented in Sec.5.

2. Model and dynamical equation

A two-mode optomechanical system under our consideration, illustrated in Fig. 1, consists of two optical cavities which are coupled to a common mechanical oscillator by the radiation pressures. To enhance the optomechanical interactions of the two cavities with the oscillator, two coupling fields with power P_k (k = 1, 2) and amplitude $E_{k} = \sqrt{2 P_{k} κ_{k} / ℏ ω_{k}}$ are, respectively, used to drive the two cavities, in which ω_k and κ_k denote the frequency of coupling fields and the loss rate of the cavity fields. Here, â_k( ${\hat{a}}_{k}^{†}$ ) is the annihilation (creation) operator of the kth cavity field with resonance frequency ω_{c_k}. And, b̂(b̂^†) denotes the annihilation (creation) operator of the movable mechanical mode with its resonance frequency, effective mass and quality factor denoted by ω_m, m and Q_m, respectively. The Hamiltonian of the system in the rotating frame at the frequency ω_k (k = 1, 2) of the coupling fields reads (ħ = 1) [33,34,48–50]

H = \sum_{k = 1, 2} Δ_{c_{k}} a_{k}^{†} {\hat{a}}_{k} + ω_{m} {\hat{b}}^{†} \hat{b} - (g_{1} {\hat{a}}_{1}^{†} {\hat{a}}_{1} - g_{2} {\hat{a}}_{2}^{†} {\hat{a}}_{2}) (\hat{b} + {\hat{b}}^{†}) + \sum_{k = 1, 2} i E_{k} ({\hat{a}}_{k}^{†} - {\hat{a}}_{k}) .

Here, Δ_{c_k} = ω_{c_k} − ω_k(k = 1, 2) is the detuning of the kth cavity field from the corresponding coupling field. The coupling between the kth cavity field and the movable mechanical oscillator via radiation pressure is described by the term with the coupling parameter g_k, which is given by g_k = ω_{c_k} x_zpt /L_k with x_zpt being the zero-point fluctuation amplitude.

Fig. 1 Schematic of the two-cavity optomechanical system. The system consists of two optical cavities which are coupled to a common mechanical oscillator by the radiation pressures. Two coupling fields with power P_k (k = 1, 2) and frequency ω_k are used to drive the two cavities, respectively.

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The dynamical evolution of the system can be described by the quantum Heisenberg-Langevin equations of the cavity and oscillator variables, which are given by

{\dot{\hat{a}}}_{1} = - (i Δ_{c_{1}} + κ_{1}) {\hat{a}}_{1} + i g_{1} {\hat{a}}_{1} (\hat{b} + {\hat{b}}^{†}) + E_{1} + \sqrt{2 κ_{1}} {\hat{a}}_{in, 1},

{\dot{\hat{a}}}_{2} = - (i Δ_{c_{2}} + κ_{2}) {\hat{a}}_{2} - i g_{2} {\hat{a}}_{2} (\hat{b} + {\hat{b}}^{†}) + E_{2} + \sqrt{2 κ_{2}} {\hat{a}}_{in, 2},

\dot{\hat{b}} = - (i ω_{m} + γ_{m}) \hat{b} + i (g_{1} {\hat{a}}_{1}^{†} {\hat{a}}_{1} - g_{2} {\hat{a}}_{2}^{†} {\hat{a}}_{2}) + {\hat{b}}_{in}

where the operator â_in,k(k = 1, 2) describes the input field in the kth cavity with zero mean value.

γ_{m} = \frac{ω_{m}}{Q_{m}}

is the damping rate and b̂_in describes the Brownian force of the mechanical resonator. The equations of the cavity and oscillator variables are nonlinear ones because of the terms with g_k contributed by the optomechanical interactions. Due to the driving of the strong coupling fields on the cavities, we can divide the cavity and mechanical operators in Eqs. (2)–(4) into the steady-state parts and fluctuation ones, i.e. ô = o_s + δô with ô = â_k, b̂. The dynamical behavior of the system can be exhibited by solving the equations of motion for the fluctuations (δâ_k, δb̂) around their steady-state parts (a_ks, b_s). By setting the derivatives of the dynamical equations to be zero, the steady-state solutions of the system can be obtained as

a_{k s} = \frac{E_{k}}{κ_{k} + i Δ_{k}}, (k = 1, 2)

b_{s} = \frac{i g_{1} a_{1 s} a_{1 s}^{*} - i g_{2} a_{2 s} a_{2 s}^{*}}{γ_{m} + i ω_{m}}

where Δ_k = Δ_{c_k} + (−1)^k

g_{k} (b_{s} + b_{s}^{*})

, (k = 1, 2) is an effective detuning. By neglecting the higher-order terms of the fluctuation operators, the linearized quantum Langevin equations of the fluctuation parts are given by

δ {\dot{\hat{a}}}_{k} = - (i Δ_{k} + κ_{k}) δ {\hat{a}}_{k} - {(- 1)}^{k} i g_{k} a_{k s} (δ \hat{b} + δ {\hat{b}}^{†}) + \sqrt{2 κ_{k}} δ {\hat{a}}_{in, k}, (k = 1, 2)

δ \dot{\hat{b}} = - (i ω_{m} + γ_{m}) δ \hat{b} + i [g_{1} (a_{1 s}^{*} δ {\hat{a}}_{1} + a_{1 s} δ {\hat{a}}_{1}^{†}) - g_{2} (a_{2 s}^{*} δ {\hat{a}}_{2} + a_{2 s} δ {\hat{a}}_{2}^{†})] + δ {\hat{b}}_{in} .

The input squeezed vacuum noise operator δâ_in,k and the mechanical input operator δb̂_in in Eqs. (7) and (8) satisfy the following correlation relations: 〈δâ_in,j(ω)δâ_in,k(Ω)〉 = 2πMδ(ω+Ω−2ω_m)δ_j,k,

〈 δ {\hat{a}}_{in, j} (ω) δ {\hat{a}}_{in, k}^{†} (- Ω) 〉 = 2 π (N + 1) δ (ω + Ω) δ_{j, k}

, and

〈 δ {\hat{b}}_{in} (ω) δ {\hat{b}}_{in}^{†} (- Ω) 〉 = 2 π γ_{m} (n_{m} + 1) δ (ω + Ω)

. Here, N = sinh[r]² and M = sinh[r] cosh[r]e^iφ in which r is the squeezing parameter and φ the phase of the squeezed vacuum light. And,

n_{m} = (e^{\frac{ℏ ω_{m}}{κ_{B} T}} - 1)

is the mean thermal photon occupation number of energy ħω_m at environment temperature T with κ_B being Boltzmans constant. For simplicity, we choose φ = 0. Here, we mainly consider the squeezing transfer from the squeezed field injected into the left-hand cavity to the output spectrum in the right-hand cavity, then we assume that the input field into the right-hand cavity is in a vacuum (white) noise, i.e. 〈δâ_in,2(ω)δâ_in,2(Ω)〉 = 0 and

〈 δ {\hat{a}}_{in, 2} (ω) δ {\hat{a}}_{in, 2}^{†} (- Ω) 〉 = 2 π δ (ω + Ω)

.

When transforming the Langevin equations of the fluctuation parts shown in Eqs. (7) and (8) into the frequency domain by using the Fourier transform and substituting the fluctuations into the input-output formula $δ {\hat{a}}_{out, 2} (ω) = \sqrt{2 κ_{2}} δ {\hat{a}}_{2} (ω) - δ {\hat{a}}_{in, 2} (ω)$ , the fluctuation of the output field from the second cavity is obtained as

\begin{array}{l} δ {\hat{a}}_{out, 2} (ω) & = & C_{1} (ω) δ {\hat{a}}_{in, 1} + C_{2} (ω) δ {\hat{a}}_{in, 1}^{†} (- ω) + V_{1} (ω) δ {\hat{a}}_{in, 2} + V_{2} (ω) δ {\hat{a}}_{in, 2}^{†} (- ω) \\ + W_{1} (ω) δ {\hat{b}}_{in} (ω) + W_{2} (ω) δ {\hat{b}}_{in}^{†} (- ω) \end{array}

where

C_{1} (ω) = \frac{2 i g_{1} g_{2} a_{1 s}^{*} a_{2 s} \sqrt{κ_{1} κ_{2}} Γ_{m m}}{[κ_{1} + i (Δ_{1} - ω)] [κ_{2} + i (Δ_{2} - ω)]},

C_{2} (ω) = \frac{2 i g_{1} g_{2} a_{1 s} a_{2 s} \sqrt{κ_{1} κ_{2}} Γ_{m m}}{[κ_{1} + i (Δ_{1} - ω)] [κ_{2} + i (Δ_{2} - ω)]},

V_{1} (ω) = \frac{2 i g_{2}^{2} {| a_{2 s} |}^{2} κ_{2} Γ_{m m}}{[κ_{2} + i (Δ_{2} - ω)] [κ_{2} + i (Δ_{2} - ω)]} + \frac{2 κ_{2}}{κ_{2} + i (Δ_{2} - ω)} - 1,

V_{2} (ω) = \frac{2 i g_{2}^{2} a_{2 s}^{2} κ_{2} Γ_{m m}}{[κ_{2} + i (Δ_{2} - ω)] [κ_{2} + i (Δ_{2} + ω)]},

W_{1} (ω) = - \frac{i g_{2} a_{2 s} \sqrt{2 κ_{2}} Γ_{m b}}{κ_{2} + i (Δ_{2} - ω)},

W_{2} (ω) = - \frac{i g_{2} a_{2 s} \sqrt{2 κ_{2}} Γ_{m a}}{κ_{2} + i (Δ_{2} - ω)},

Γ_{m a} = \frac{Γ_{+}}{Γ_{+} Γ_{-} - 2 i ω_{m} (Λ_{m} + i δ_{m})},

Γ_{m b} = \frac{Γ_{-}}{Γ_{+} Γ_{-} - 2 i ω_{m} (Λ_{m} + i δ_{m})},

Γ_{m m} = \frac{2 ω_{m}}{Γ_{+} Γ_{-} - 2 i ω_{m} (Λ_{m} + i δ_{m})}

with Γ₊ = γ_m + i(ω_m − ω), Γ₋ = γ_m − i(ω_m + ω),

δ_{m} = \sum_{j = 1, 2} Im [\frac{1}{κ_{j} + i (Δ_{j} - ω)} - \frac{1}{κ_{j} - i (Δ_{j} + ω)}]

, and

Λ_{m} = \sum_{j = 1, 2} Re [\frac{1}{κ_{j} + i (Δ_{j} - ω)} - \frac{1}{κ_{j} - i (Δ_{j} + ω)}]

. The output spectrum of the optical field from the right-hand cavity is defined by

S_{θ} (ω) = \frac{1}{2 π} \int d Ω 〈 δ {\hat{X}}_{θ}^{out} (ω) δ {\hat{X}}_{θ}^{out} (Ω) 〉,

in which the quadrature of the output field is given by

δ {\hat{X}}_{θ}^{out} (ω) = e^{- i θ} δ {\hat{a}}_{out, 2} (ω) + e^{i θ} δ {\hat{a}}_{out, 2}^{+} (- ω)

. From the Eq. (9), we get the noise spectrum of the output field S_θ(ω), which is given by

S_{θ} (ω) = M F_{1} + (N + 1) F_{2} + N F_{3} + M^{*} F_{4} + G_{1} + γ_{m} (n_{b} + 1) H_{1} + γ_{m} n_{b} H_{2},

where

F_{1} = C_{1} (ω) C_{1} (Ω_{1}) e^{- 2 i θ} + C_{1} (ω) C_{2}^{*} (- Ω_{1}) + C_{2}^{*} (- ω) C_{1} (Ω_{1}) + C_{2}^{*} (- ω) C_{2}^{*} (- Ω_{1}) e^{2 i θ},

F_{2} = C_{1} (ω) C_{2} (Ω_{1}) e^{- 2 i θ} + C_{1} (ω) C_{1}^{*} (- Ω_{2}) + C_{2}^{*} (- ω) C_{2} (Ω_{2}) + C_{2}^{*} (- ω) C_{1}^{*} (- Ω_{2}) e^{2 i θ},

F_{3} = C_{2} (ω) C_{1} (Ω_{2}) e^{- 2 i θ} + C_{2} (ω) C_{2}^{*} (- Ω_{2}) + C_{1}^{*} (- ω) C_{1} (Ω_{2}) + C_{1}^{*} (- ω) C_{2}^{*} (- Ω_{2}) e^{2 i θ},

F_{4} = C_{2} (ω) C_{2} (Ω_{3}) e^{- 2 i θ} + C_{2} (ω) C_{1}^{*} (- Ω_{3}) + C_{1}^{*} (- ω) C_{2} (Ω_{3}) + C_{1}^{*} (- ω) C_{1}^{*} (- Ω_{3}) e^{2 i θ},

G_{1} = V_{1} (ω) V_{2} (Ω_{2}) e^{- 2 i θ} + V_{1} (ω) V_{1}^{*} (- Ω_{2}) + V_{2}^{*} (- ω) V_{2} (Ω_{2}) + V_{2}^{*} (- ω) V_{1}^{*} (- Ω_{2}) e^{2 i θ},

H_{1} = W_{1} (ω) W_{2} (Ω_{2}) e^{- 2 i θ} + W_{1} (ω) W_{1}^{*} (- Ω_{2}) + W_{2}^{*} (- ω) W_{2} (Ω_{2}) + W_{2}^{*} (- ω) W_{1}^{*} (- Ω_{2}) e^{2 i θ},

H_{2} = W_{2} (ω) W_{1} (Ω_{2}) e^{- 2 i θ} + W_{2} (ω) W_{2}^{*} (- Ω_{2}) + W_{1}^{*} (- ω) W_{1} (Ω_{2}) + W_{1}^{*} (- ω) W_{2}^{*} (- Ω_{2}) e^{2 i θ}

with Ω₁ = −ω + 2ω_m, Ω₂ = −ω and Ω₃ = −ω − 2ω_m. The output field from the right-hand cavity is squeezed if the quadrature spectrum meets S_θ(ω) < 1.

3. Quadrature squeezing generated by optomechanical interactions

We investigate the squeezing transfer between the cavities via optomechanical interface by calculating the noise spectrum of the output field from the right-hand optomechanical cavity. To distinguish the squeezing of the output field transferred from the input squeezed field in the left-hand cavity field from that originated by the optomechanical interactions in the system, we firstly investigate the variation of the noise spectrum with the two coupling fields in the case where the left-hand cavity is in a quantum vacuum noise, i.e. 〈δâ_in,2(ω)δâ_in,2(Ω)〉 = 0 and $〈 δ {\hat{a}}_{in, 2} (ω) δ {\hat{a}}_{in, 2}^{†} (- Ω) 〉 = 2 π δ (ω + Ω)$ . To make the following results within experimental realizations, we use the parameters in a recent experiment for demonstrating coherent wavelength conversion of photons using photon-phonon translation in a cavity-optomechanical system [50]: ω_m = 2π × 3.993 × 10⁹Hz, Q_m = 87 × 10³, g₁ = 2π × 960 × 10³, g₂ = 2π × 430 × 10³, κ₁ = 2π ×520×10⁶Hz, κ₂ = 1.73×10⁹Hz, ω₁ = 2π ×205.3×10¹²Hz, ω₂ = 2π ×194.1×10¹²Hz. The following discussions are explored under the resonance condition of Δ₁ = Δ₂ = ω_m. The resonance relation is the key condition for transferring the squeezing between the cavity fields in the present optomechanical system, which is addressed in the following section.

First, we consider the effects of the left coupling field (driving the left-hand cavity) on the noise spectrum S_θ(ω) when the left-hand cavity is injected by the quantum vacuum noise. In Fig. 2(a), the power of the left coupling field is set by different values as 1mW, 5mW, 10mW and that of the right coupling field (driving the right-hand cavity) is fixed at 1mW. It is shown that there appears a symmetric noise spectrum, in which the spectra around ω = ±ω_m display dispersion-like behavior and the other parts is in shot-noise. The dips with S_θ(ω) < 1 near ω = ±ω_m indicate the production of the squeezing of the output field, and the peaks with S_θ(ω) > 1 describe the optomechanically induced heating. Additionally, the squeezing degree is decreased with the left coupling filed. Second, the effects of the right coupling field on the noise spectrum are displayed in Fig. 2(b). At this time, the two symmetric dips become deeper and the squeezing of the output field is increased with the right coupling field. Correspondingly, the optomechanically induced heating becomes stronger and makes the noise peaks higher.

Fig. 2 The squeezing spectra S_θ (ω) as a function of the normalized frequency ω/ω_m with P₂ = 1 mW and for the different powers of the left coupling field (a): P₁ = 1 mW (black, solid curve); 5 mW (red, dashed curve); 10 mW (blue, dotted curve). The squeezing spectra S_θ (ω) as a function of the normalized frequency ω/ω_m with P₁ = 1 mW and for the different powers of the right coupling field (b): P₂ = 1 mW (black, solid curve); 5 mW (red, dashed curve); 10 mW (blue, dotted curve). The values of the parameters are given by: ω_m = 2π × 3.993 × 10⁹Hz, g₁ = 2π × 960 × 10³, g₂ = 2π × 430 × 10³, T = 10K, r = 0, Q_m = 87 × 10³, κ₁ = 2π × 520 × 10⁶Hz, κ₂ = 1.73 × 10⁹Hz, ω₁ = 2π × 205.3 × 10¹²Hz, ω₂ = 2π × 194.1 × 10¹²Hz, θ = π/2.

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The optomechanically induced heating describes the thermal phonon excitation, which is induced by the photonic excitation in the optomechanical interaction and is related to the thermal phonons (or the environment temperature T). This can be verified by Fig. 6(a) in the following section, in which the peaks are dramatically increased with the environment temperature T. Specifically, the thermal phonon excitation is described by the last two terms in Eq. (20), which are related to the thermal phonon occupation number at environment temperature T. These two terms are respectively originated from the last two terms in Eq. (9) with the coefficients addressed by the linearized coupling g₂a_2s, which describe the thermal noise from the environment of the mechanical oscillator through the optomechanical interaction between the second cavity and the mechanical oscillator. Also, this can explain why the peaks in Fig. 2(b) are remarkably increased with the power P₂.

The squeezing of the output field induced by the optomechanical interactions can be understood by using the quadrature fluctuations of cavity and mechanical variables, which are defined as ${\hat{A}}_{k} = δ {\hat{a}}_{k} + δ {\hat{a}}_{k}^{+}$ (k = 1, 2), ${\hat{B}}_{k} = \frac{1}{i} (δ {\hat{a}}_{k} - δ {\hat{a}}_{k}^{+})$ , Q̂ = δb̂ + δb̂⁺ and $\hat{P} = \frac{1}{i} (δ \hat{b} - δ {\hat{b}}^{+})$ . In fact, the quadrature $δ {\hat{X}}_{θ}^{out} (ω)$ with θ = π/2 adopted in the present paper is the phase quadrature, i.e. $δ {\hat{X}}_{θ = π / 2}^{out} (ω) = {\hat{B}}_{2} (ω)$ . Without loss of generality, the steady-state variable a_ks is assumed to be real. Then the phase quadrature B̂₂ in the frequency domain is

{\hat{B}}_{2} (ω) = - \frac{2 (κ_{2} - i ω) g_{2} a_{2 s}}{{(κ_{2} - i ω)}^{2} + Δ_{2}^{2}} \hat{Q} (ω) - \frac{Δ_{2} \sqrt{2 κ_{2}}}{{(κ_{2} - i ω)}^{2} + Δ_{2}^{2}} {\hat{A}}_{in, 2} + \frac{(κ_{2} - i ω) \sqrt{2 κ_{2}}}{{(κ_{2} - i ω)}^{2} + Δ_{2}^{2}} {\hat{B}}_{in, 2} .

The first term comes from the optomechanical interaction between the right-hand cavity and the mechanical oscillator, and the next two terms are resulted from the noise of the input field in the right-hand cavity. Correspondingly, the position operator is obtained as

\begin{array}{l} \hat{Q} (ω) & = & \frac{1}{D} (\sqrt{2 κ_{1}} K_{1} Ω_{1} {\hat{A}}_{in, 1} + \sqrt{2 κ_{1}} Δ_{1} Ω_{1} {\hat{B}}_{in, 1} - \sqrt{2 κ_{2}} K_{2} Ω_{2} {\hat{A}}_{in, 2} - \sqrt{2 κ_{2}} Δ_{2} Ω_{2} {\hat{B}}_{in, 2} \\ + Γ_{m} R_{1} R_{2} {\hat{Q}}_{in} + ω_{m} R_{1} R_{2} {\hat{P}}_{in}) \end{array}

where D = R_mR₁R₂ − 2(G₁Δ₁Ω₁ + G₂Δ₂Ω₂) with Ω_i = 2ω_mG_iR_j, (i, j = 1, 2, i ≠ j),

R_{m} = Γ_{m}^{2} + ω_{m}^{2}

,

R_{i} = K_{i}^{2} + Δ_{i}^{2}

, G_i = g_ia_is, Γ_m = γ_m −iω, K_i = k_i −iω. The first (next) two terms in Eq. (29) are contributed from the input vacuum in left(right)-hand cavity via their optomechanical interactions. The last two terms describe the contributions from the mechanical damping and the quantum Brownian noise. Substitution of Eq. (29) into Eq. (28) leads to the phase quadrature as

{\hat{B}}_{2} (ω) = a_{11} {\hat{A}}_{in, 1} + b_{11} {\hat{B}}_{in, 1} + a_{22} {\hat{A}}_{in, 2} + b_{22} {\hat{B}}_{in, 2} + q_{11} {\hat{Q}}_{in} + p_{11} {\hat{P}}_{in}

where

a_{11} = - \frac{1}{R_{2} D} \times 2 K_{1} K_{2} G_{2} Ω_{1} \sqrt{2 κ_{1}},

b_{11} = - \frac{1}{R_{2} D} \times 2 Δ_{1} K_{2} G_{2} Ω_{1} \sqrt{2 κ_{1}},

a_{22} = - \frac{1}{R_{2} D} \times \sqrt{2 κ_{2}} (Δ_{2} D - 2 K_{2}^{2} G_{2} Ω_{2}),

b_{22} = - \frac{1}{R_{2} D} \times \sqrt{2 κ_{2}} K_{2} (D + 2 Δ_{2}^{2} G_{2} Ω_{2}),

q_{11} = - \frac{1}{R_{2} D} \times 2 K_{2} G_{2} R_{1} R_{2} Γ_{m},

p_{11} = - \frac{1}{R_{2} D} \times 2 K_{2} G_{2} ω_{m} R_{1} R_{2} .

In above discussions, the squeezing of the output field from the right-hand cavity with a quantum vacuum injected in the left-hand cavity is increased with the right coupling field, while it is decreased with the left coupling field. This can be explained by using quadrature spectrum shown in Eq. (30). In the present system, there exist the optical noises from the cavities and the mechanical thermal noise affecting the squeezing of the output spectrum. The optical noise associated with the second cavity is fed into the output spectrum through two ways: (i) it directly enters into the second cavity; and (ii) it is indirectly transferred into the second cavity as a modified noise through the optomechanical interaction between the mechanical oscillator and the second cavity field. This can be verified by the third and fourth terms in Eq. (30), in each coefficient of which there are two terms: the first part in Eqs. (33) and (34) is directly contributed from the noise and the second one comes from the optomechanically modified noise dressed by the linearized coupling G₂. It is the destructive interference between the two paths that leads to the squeezing in the output spectrum. Consequently, the squeezing is enhanced by the right coupling field through the optomechanically modified noise, which is shown by the Figs. 2(b) and 3(b).

Fig. 3 The same settings are used as those in Fig. 2 except for the presence of squeezed vacuum injected in the left cavity with squeezing parameter r = 1.

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On the hand, the optical noise from the left cavity is transferred to the output spectrum only through the optomechanical interactions and enhances the fluctuation in the noise spectrum. Specifically, the modified noise from the left cavity by the optomechanical interactions is described by the first two terms in Eq. (30), in each of which there is only one term of the modified fluctuation by the optomechanical interactions. When the left coupling field is increased, the modified noise is stronger and the squeezing in the output field is suppressed. This is why the squeezing is decreased with the left coupling field, which are shown in Figs. 2(a) and 3(a).

4. Squeezing transfer of light between two cavities via the optomechanical interactions

In the following, we shall discuss the squeezing of the output field from the right-hand cavity when the left-hand cavity is injected by a squeezed vacuum. To effectively contrast the squeezing properties between the cases of the squeezed field and the quantum vacuum injected in the left-hand cavity, we use same settings in Fig. 3 as those in Fig. 2 except that a squeezed vacuum with squeezing parameter r = 1 is injected in the left-hand cavity. In Fig. 3(a), the power of the left coupling field is set by different values as 1mW, 5mW, 10mW and that of the right coupling field is fixed at 1mW. It is shown that the maximum squeezing around ω = ±ω_m becomes smaller when the left coupling field becomes stronger. This is similar to that shown in Fig. 2(a). However, the squeezing dips become deeper and wider than the corresponding curves in Fig. 2(a) due to the effects from the injected squeezed field. The effects of the right coupling field on the squeezing of the output spectrum in the case of the squeezed field injected into the left-hand cavity are exhibited in Fig. 3(b). It is shown that the maximal squeezing is increased with the power of the right coupling. By comparing with Fig. 2(b), the squeezing amount in Fig. 3(b) is much larger than that in the corresponding curve in the case of injected vacuum, which is shown in Fig. 2(b). To see clearly the enhancement of the squeezing of the output field due to the squeezed vacuum injected in the left-hand cavity, we plot the noise spectra with the quantum vacuum and the squeezed vacuum injected in the left-hand cavity, which are shown by the solid and dashed curves in Fig. 4, respectively. It is seen that in the case of the squeezed vacuum with r = 1, the amount of the maximal squeezing reaches larger than 60% for P₁ = 1mW and P₂ = 10mW, while the squeezing shown by the solid curve is 8% for the vacuum injected in the left-hand cavity.

Fig. 4 The squeezing spectra S_θ(ω) as a function of the normalized frequency ω/ω_m with P₁ = 1 mW, P₂ = 5 mW for the injected vacuum field r = 0 (black, solid curve), and for the injected squeezed field r = 1 (red, dashed curve). The others are set with the same values as in Fig. 2.

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By comparing the noise spectra in Fig. 3 with those in Fig. 2, it is shown that the enhanced squeezing in the Fig. 3 mainly comes from the squeezed field injected in the left cavity. To provide a specific demonstration of this squeezing transfer, we shall display the dependence of the squeezing of the output field from the right-hand cavity on the squeezing parameter r of the injected squeezed field. In Fig. 5, the squeezing parameter r is set by different values: r = 0.3, 0.6, 1, which correspond to solid, dashed and dotted curves, respectively. It is shown that the squeezing of the output field from the right-hand cavity is increased with the squeezing parameter r of the injected squeezed field, which maps the essential property of the variation of the injected squeezed field with the squeezing parameter on the output spectrum.

Fig. 5 The squeezing spectra S_θ(ω) as a function of the normalized frequency ω/ω_m with P₁ = 1 mW, P₂ = 5 mW and for different squeezing parameter r of the squeezed field injected in the left cavity: r = 0.3(black, solid curve); r = 0.6(red, dashed curve); r = 1(blue, dotted curve). The others are set with the same values as in Fig. 2.

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On the other hand, the injected squeezed field can mitigate the adverse effects of the temperature on the squeezing of the output field. By comparing with Fig. 2 for a vacuum injected in the left-hand cavity, the peaks in Fig. 3 are dramatically decreased in the case of the injected squeezed field. Now we see the effects of the temperature of the environment on the noise spectra in the cases of a vacuum and a squeezed field injected in the left-hand cavity, which are shown by Figs. 6(a) and 6(b), respectively. Although the noise spectrum in Fig. 6(a) behaves like that in Fig. 3(b), there are different physics mechanisms: the right coupling field in Fig. 3(b) leads to optomechanically induced heating and makes the noise peaks higher, meanwhile it induces the destructive interference of quantum noise and makes the squeezing dips deeper. While the temperature of the environment surrounding the mechanical oscillator in Fig. 6(a) provides the thermal phonons of the mechanical oscillator to increase the noise peaks and suppresses the squeezing induced by the optomechanical interactions. When a squeezed field is injected in the left-hand cavity, shown in Fig. 6(b), the noise spectra are almost invariant with the temperature. This is different from the case of a injected vacuum in the left-hand cavity shown in Fig. 6(a), in which the noise spectrum varies drastically with the temperature. It implies that the adverse effects of the environment temperature on the output spectra can be strongly mitigated by the injected squeezed field.

Fig. 6 The spectrum S_θ(ω) (a) as a function of the normalized frequency ω/ω_m for the injected vacuum r = 0 (a) and the injected squeezed vacuum r = 1 (b) in the left-hand cavity. The temperature for the mechanical environment are given by different values: T₁ = 10K (black, solid curve); 50 K (red, dashed curve); 100 K (blue, dotted curve). The others are set with the same values as in Fig. 2.

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In the squeezing transfer between the two cavities via the optomechanical interactions, the key condition is the resonance conditions of Δ₁ = Δ₂ = ω_m, which leads the two-mode optomechanical system to be a double-beam-splitter form. In above discussions, we use the linearization approach of quantum optics and get the linearized quantum Heisenberg-Langevin equations for the fluctuations in Eqs. (7) and (8), which corresponds to the linearized Hamiltonian: Ĥ_E ∝ Ĥ₁ + Ĥ₂ with ${\hat{H}}_{1} = ℏ \sum_{k = 1, 2} {(- 1)}^{k} G_{k} (δ {\hat{a}}_{k} δ \hat{b} + δ {\hat{a}}_{k}^{†} δ {\hat{b}}^{†})$ and ${\hat{H}}_{2} = ℏ \sum_{k = 1, 2} {(- 1)}^{k} G_{k} (δ {\hat{a}}_{k} δ \hat{b} + δ {\hat{a}}_{k}^{†} δ {\hat{b}}^{†})$ . The optomechanical coupling $G_{k} = g_{k} \sqrt{n_{k}}$ is enhanced from the single-photon optomechanical coupling g_k by the intracavity photon number n_k = |a_ks|² (k = 1, 2). The first part Ĥ₁ of the linearized Hamiltonian describes the two-mode squeezing interaction form, which has been employed for generating the entanglement between optical and mechanical modes and two-mode squeezing [4, 27, 53, 54]. The second part Ĥ₂, taking the beam-splitter-like form, can enable quantum transfer between the optical and the mechanical systems, such as state swapping [43,45–47] and optical wavelength conversion [44,48–50].

When the two intense red-detuned coupling fields are applied on the two cavities according to the conditions of Δ₁ = Δ₂ = ω_m, this makes the beam-splitter-like interactions dominant over the two-mode squeezing ones. Thus the optomechanical interactions in Eq. (1) can be effectively characterized by the beam-splitter Hamiltonian shown in Ĥ₂. As is proposed that a reversible transfer of light squeezing conveyed to the membrane or mirror via optomechanical interface can be realized under this conditions [51,52]. Then, the squeezing of the squeezed field injected into the left-hand cavity is transferred to the squeezing of the mechanical oscillator, which is in turn conveyed to that of the right-hand cavity. Additionally, the beam-splitter-like interaction, for red-detuned driving at the resonance conditions of Δ₁ = Δ₂ = ω_m, makes energy conversion between the intracavity field and the movable mirror and results in ground-state cooling of the mechanical resonator by using the optical vacuum bath to successively extract the energy. This can suppress the damages of the mechanical heating excitation on the squeezing of the output spectrum.

To further demonstrate the key role played by the beam-splitter-like interaction in the squeezing transfer, we numerically investigate three cases: noise spectra in the single optomechanical cavity under the conditions of Δ₂ = ω_m and Δ₂ = −ω_m, and in the two-mode optomechanical system under the condition of Δ₂ = −ω_m, which are shown in Figs. 7(a)–7(c), respectively. In Fig. 7(a), we investigate the squeezing properties in the single-mode optomechanical system under the condition of Δ₂ = ω_m by setting g₁ = 0. There appear two noise peaks lie in the inner sides of ω = ±ω_m and two squeezing dips on their outsides, which become deeper with the right coupling field. The noise spectrum behaves like the case of two-mode optomechanical system, which is shown in Fig. 2(b). When we adjust the right coupling to detune at Δ₂ = −ω_m in Fig. 7(b), the spectrum properties are different from the case of Δ₂ = ω_m, shown in Fig. 7(a). Here, the positions of the squeezing dips are changed in the inner sides of ω = ±ω_m, and those of the noise peaks are located on their outsides. Additionally, the noise peaks become shorter and the squeezing dips turn slightly deeper with the right coupling field. Thus, the squeezing of the output spectrum with a vacuum injected in the left-hand cavity is mainly resulted from the destructive interference between the noise and the modified one by the optomechanical interaction in the right-hand cavity, which can be described by the beam-splitter-like Hamiltonian $δ {\hat{a}}_{2}^{†} δ \hat{b} + δ {\hat{a}}_{2} δ {\hat{b}}^{†}$ at Δ₂ = ω_m.

Fig. 7 The spectrum S_θ(ω) (a) as a function of the normalized frequency ω/ω_m for g₁ = 0, r = 0 at Δ₂ = ω_m (a) and Δ₂ = −ω_m (b), and for g₁ = 2π × 960 × 10³, r = 1 at Δ₂ = −ω_m (c). The power of the right coupling field is set as: P₂ = 1 mW (black, solid curve); 3 mW (red, dashed curve); 6 mW (blue, dotted curve). The others are set with the same values as in Fig. 2.

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In the following, we consider the squeezing transfer in the two-mode optomechanical system under the condition of Δ₂ = −ω_m when a squeezed vacuum is injected in the first cavity by turning on its optomechanical interaction. Now, the squeezing dips disappear, instead there appear two noise peaks symmetrically located at ω = ±ω_m, in which the magnitudes of the peaks are increased with the right coupling field. The comparison in these three cases can show that the squeezing transfer mainly benefits from the beam-splitter-like part of the linearized Hamiltonian $δ {\hat{a}}_{2}^{†} δ \hat{b} + δ {\hat{a}}_{2} δ {\hat{b}}^{†}$ by setting the detuning of the coupling field to the red side of the optomechanical cavity resonance at Δ₂ = ω_m, which is used to cool the mechanical oscillator as well as to generate a state swap between the mechanical and the optical mode. On the contrary, two-mode squeezing interaction (counter-rotating interaction term) $δ {\hat{a}}_{2}^{†} δ \hat{b} + δ {\hat{a}}_{2} δ {\hat{b}}^{†}$ , which can be dominated in the linearized Hamiltonian by tuning the coupling field to the blue side of the optomechanical cavity resonance at Δ₂ = −ω_m and leads to the optomechanical induced amplification [15], destroys the squeezing transfer in the present model.

5. Conclusions

We have investigated the squeezing of the output field from the right-hand cavity when the left cavity is injected by a squeezed field in a two-mode optomechanical system, and found that there appear two squeezing regions symmetrically located at ω = ±ω_m. By comparing the squeezing properties in the case of a squeezed field with that for a vacuum injected in the left-hand cavity, we find that the squeezing of the output field from the right-hand cavity partially comes from the optomechanical interactions, and it is mainly transferred from the squeezed field injected in the left-hand cavity. Additionally, the squeezing amount is increased with the right coupling field but it is decreased with left coupling field. Finally, we find that the injected squeezed field can mitigate the effects of the temperature. Combination of the quantum transfer and the continuous-variable information processing related to the squeezed state can be useful in quantum communications in the optomechanical interface.

Funding

National Natural Science Foundation of China (10647007, 11874273); Science Foundation of Sichuan Province of China (2018JY0180).

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Squeezing transfer of light in a two-mode optomechanical system

Abstract

1. Introduction

2. Model and dynamical equation

3. Quadrature squeezing generated by optomechanical interactions

4. Squeezing transfer of light between two cavities via the optomechanical interactions

5. Conclusions

Funding

References

Cited By

Figures (7)

Equations (36)

Optics Express