Optomechanically induced sum sideband generation

Hao Xiong; Liu-Gang Si; Xin-You Lü; Ying Wu

doi:10.1364/OE.24.005773

1. Introduction

Cavity optomechanics [1], which combines a mechanical degree of freedom to an optical cavity via resonantly enhanced feedback backaction mechanism [2], has attracted great interest and seen remarkable progress in many fields of physics, including precision force sensing [3–6], manipulation of mechanical motion at the quantum limit [7–9], generation of dark states of a moving mirror [10], slowing and storage of light pulses [11, 12], and optomechanically induced transparency [14–20]. Recently, effects arise from the nonlinear optomechanical interaction emerging as a new frontier in cavity optomechanics due to breakthroughs of intrinsic characteristics of the linearized optomechanical interaction, and have enabled some interesting topics in both classical (or semiclassical) and quantum mechanism [21–23]. In the classical (or semiclassical) mechanism, nonlinear interactions in cavity optomechanics lead to second- and higher-order sideband generation [24], frequency comb [25], and chaos [26]. In the single-photon strong-coupling regime g₀ > κ, viz. the single photon optomechanical coupling rate exceeding the cavity decay rate, large optical Kerr nonlinearity as well as photon blockade can be realized via nonlinear interactions in the optomechanical system [27, 28].

Previous studies of the nonlinear optomechanical interaction usually focus on the single-probe-field-driven case [21–23]. The nonlinear features of double- and multiple-probe-field-driven optomechanical systems, however, remain poorly studied. Studying the nonlinear optomechanical interactions in a double-probe-field-driven optomechanical systems is an interesting topic. It has been shown that signals at the second order sideband, which is of great importance in understanding the optomechanical nonlinearity, reveals the nonlinear quantum nature of the optomechanical interactions in an optomechanical system with single probe field driven [22,23]. Here we focus on the nonlinear process in a double-probe-field-driven optomechanical system, that is the optomechanical system is driven by a strong control field with the frequency ω_c and two relatively weak probe fields with frequencies ω₁ and ω₂, respectively [shown in Fig. 1(a)]. Based on the analytical treatment proposed in [24], we show that there are signals at the sum-sideband (with frequencies ±Ω₊ in a frame rotating at ω_c) generation in such an optomechanical system due to the nonlinear terms iλ₀δxδa/h̄ and λ₀δa^*δa with Ω₊ = δ₁ + δ₂, δ₁ = ω₁ − ω_c, and δ₂ = ω₂ − ω_c. The frequency spectrogram of sum sideband generation in a generic optomechanical system is shown in Fig. 1(b). The influences of the pump power of the control field as well as the frequencies of the probe fields on the sum sideband generation are analyzed in detail. The results clearly indicate that sum sideband generation can be greatly enhanced by achieving the matching conditions. The signals at the sum sideband may be importance in understanding the nonlinear optomechanical interactions. From the precision measurement perspective, this system may provide an potential method for determination of parameters of optomechanical systems.

Fig. 1 (a) Schematic diagram of a double probe fields driven optomechanical system. The optomechanical system is driven by a strong control field with the frequency ω_c and two relatively weak probe fields with frequencies ω₁ and ω₂, respectively. (b) Frequency spectrogram of sum sideband generation in the optomechanical system with double probe fields driven. The frequency of the control field is detuned by Δ̄ from the cavity resonance frequency. There are sum sideband generation (frequency components ±Ω₊ in a frame rotating at ω_c) in the optomechanical system due to the nonlinear terms iλ₀δxδa/h̄ and λ₀δa^*δa with Ω₊ = δ₁ + δ₂, δ₁ = ω₁ − ω_c, and δ₂ = ω₂ − ω_c. Other frequency components in the spectrogram, such as second-order sideband of each probe field, are not shown.

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This paper is organized as follows. In Sec. 2, by taking account of nonlinear terms analytically, we derive the amplitude of the optomechanically induced sum sideband generation, including both upper and lower sidebands. In Sec. 3, the features of sum sideband generation are discussed. We analyze in detail the influences of the pump power of the control field as well as the frequencies of the probe fields on the sum sideband generation. Finally, we end our paper with a short summary in Sec. 4.

2. Derivation of optomechanically induced sum sideband generation

In this section, we will give a full description of the derivation of optomechanically induced sum sideband generation. We consider that the optical cavity is driven by three fields: a strong control field with the frequency ω_c and two relatively weak probe fields with frequencies ω₁ and ω₂. The resonance frequency of the cavity is ω₀ with the line width κ in the resolved-sideband regime. The Hamiltonian formulation of such system is [14]:

\begin{matrix} \hat{H} = {\hat{H}}_{mech} + {\hat{H}}_{cav} + {\hat{H}}_{int} + {\hat{H}}_{control} + {\hat{H}}_{probe} \\ {\hat{H}}_{mech} = \frac{{\hat{p}}^{2}}{2 m} + \frac{m Ω_{m}^{2} {\hat{x}}^{2}}{2}, {\hat{H}}_{cav} = \bar{h} ω_{0} {\hat{a}}^{†} \hat{a}, {\hat{H}}_{int} = - λ_{0} \hat{x} {\hat{a}}^{†} \hat{a}, \\ {\hat{H}}_{control} = i \bar{h} \sqrt{η κ} ε_{c} ({\hat{a}}^{†} e^{- i ω_{c} t} - \hat{a} e^{- i ω_{c} t}), \\ {\hat{H}}_{probe} = i \bar{h} \sqrt{η κ} ({\hat{a}}^{†} ε_{1} e^{- i ω_{1} t} + {\hat{a}}^{†} ε_{2} e^{- i ω_{2} t} - H . c .), \end{matrix}

where H.c. is the Hermitian conjugate, p̂ and x̂ are the momentum and position operators of the movable mirror with effective mass m and angular frequency Ω_m. â (â^†) is the annihilation (creation) operator of the cavity fields. The term −λ₀x̂â^†â, where λ₀ = −h̄G with G the optomechanical coupling constant, denotes the interaction between the cavity field and the movable mirror [29]. The total loss rate of the cavity fields κ consists of an intrinsic loss rate κ₀ and an external loss rate κ_ex, and the coupling parameter η = κ_ex/κ is chosen to be the critical coupling value 1/2 [14] throughout the work.

ε_{c} = \sqrt{P_{c} / \bar{h} ω_{c}}

and

ε_{i} = \sqrt{P_{i} / \bar{h} ω_{i}}

(i=1, 2) are the amplitudes of the control field and the probe fields inside the cavity, respectively, with P_c the pump power of the control field, and P₁ and P₂ are the powers of the two probe field with frequencies ω₁ and ω₂, respectively.

In a frame rotating at ω_c, the Heisenberg-Langevin Eqs. can be obtained as follows:

\dot{\hat{a}} = [i (Δ + λ_{0} \hat{x} / \bar{h}) - κ] \hat{a} + \sqrt{η κ} (ε_{c} + s_{in}) + {\hat{a}}_{in},

\dot{\hat{x}} = \hat{p} / m,

\dot{\hat{p}} = - m Ω_{m}^{2} \hat{x} + λ_{0} {\hat{a}}^{†} \hat{a} - Γ_{m} \hat{p} + {\hat{F}}_{th},

where Δ = ω_c − ω₀, s_in = ε₁e^−iδ₁t + ε₂e^−iδ₂t with δ₁ = ω₁ − ω_c and δ₂ = ω₂ − ω_c. The decay rate of the mechanical oscillator is introduced classically. Here, we are interested in the mean response of the system, so the operators can be reduced to their expectation values [14], viz. a(t) ≡ 〈â(t)〉, a^*(t) ≡ 〈â^†(t)〉, x(t) ≡ 〈x̂(t)〉, and p(t) ≡ 〈p̂(t)〉. The Heisenberg-Langevin Eqs. then become:

\dot{a} = (i Δ_{x} - κ) a + \sqrt{η κ} (ε_{c} + ε_{1} e^{- i δ_{1} t} + ε_{2} e^{- i δ_{2} t}),

(m \frac{d^{2}}{d t^{2}} + m Γ_{m} \frac{d}{d t} + m Ω_{m}^{2}) x = λ_{0} a^{*} a,

where Δ_x = Δ + λ₀x/h̄, the mean-field approximation by factorizing averages is used, viz. 〈Qc〉 = 〈Q〉〈c〉, and the quantum noise terms are dropped [14, 24]. Equations (5) and (6) are coupled nonlinear Eqs., and the solution of the intracavity field and the mechanical displacement can be written as a = ā + δa and x = x̄+ δx, where

\bar{a} = \frac{\sqrt{η κ} ε_{c}}{- i (Δ + λ_{0} \bar{x} / \bar{h}) + κ}, \bar{x} = \frac{λ_{0} {| \bar{a} |}^{2}}{m Ω_{m}^{2}},

and δa and δx are described by the following Eqs.:

\begin{matrix} \frac{d}{d t} δ a = (i \bar{Δ} - κ) δ a + i λ_{0} (\bar{a} δ x + δ x δ a) / \bar{h} + \sqrt{η κ} s_{in}, \\ (m \frac{d^{2}}{d t^{2}} + m Γ_{m} \frac{d}{d t} + m Ω_{m}^{2}) δ x = λ_{0} (\bar{a} δ a^{*} + \bar{a^{*}} δ a + δ a^{*} δ a), \end{matrix}

with Δ̄ = Δ + λ₀x̄/h̄. This system has bistability if the control field is strong enough [30]. In the present work the power of the control field is about a few mW, which is too low to achieve the bistable regime. Equations (8) can be linearized for the case |ε_i| ≪ |ε_c| (i=1, 2):

\begin{matrix} \frac{d}{d t} δ a = (i \bar{Δ} - κ) δ a + i \frac{λ_{0} \bar{a} δ x}{\bar{h}} + \sqrt{η κ} s_{in}, \\ (m \frac{d^{2}}{d t^{2}} + m Γ_{m} \frac{d}{d t} + m Ω_{m}^{2}) δ x = λ_{0} (\bar{a} δ a^{*} + \bar{a^{*}} δ a), \end{matrix}

which can be solved analytically by using the ansatz:

\begin{array}{l} δ a^{L} = a_{δ_{1}}^{+} e^{- i δ_{1} t} + a_{δ_{1}}^{-} e^{- i δ_{1} t} + a_{δ_{2}}^{+} e^{- i δ_{2} t} + a_{δ_{2}}^{-} e^{- i δ_{2} t}, \\ δ x^{L} = x_{δ_{1}} e^{- i δ_{1} t} + x_{δ_{1}}^{*} e^{i δ_{1} t} + x_{δ_{2}} e^{- i δ_{2} t} + x_{δ_{2}}^{*} e^{i δ_{2} t}, \end{array}

Many phenomena arise in cavity optomechanics, such as optomechanical ground-state cooling [7] and optomechanically induced transparency [14, 31, 32], can be understood through the linearized Eqs.. However, such linearized Eqs. are insufficient to discuss the process of sum sideband generation, and the nonlinearity must be taken account. These nonlinear terms leads to generation of cavity fields with new frequencies.

A full treatment of sum sideband generation in the perturbative regime can be performed by introducing the following ansatz:

\begin{array}{l} δ a = δ a_{1}^{(1)} + δ a_{2}^{(1)} + δ a_{1}^{(2)} + δ a_{2}^{(2)} + δ a_{sum}^{(2)} + \dots, \\ δ x = δ x_{1}^{(1)} + δ x_{2}^{(1)} + δ x_{1}^{(2)} + δ x_{2}^{(2)} + δ x_{sum}^{(2)} + \dots, \end{array}

where

\begin{matrix} δ a_{1}^{(1)} = a_{δ_{1}}^{+} e^{- i δ_{1} t} + a_{δ_{1}}^{-} e^{- i δ_{1} t}, δ a_{2}^{(1)} = a_{δ_{2}}^{+} e^{- i δ_{2} t} + a_{δ_{1}}^{-} e^{- i δ_{2} t}, \\ δ a_{1}^{(2)} = a_{2 δ_{1}}^{+} e^{- 2 i δ_{1} t} + a_{2 δ_{1}}^{+} e^{2 i δ_{1} t}, δ a_{2}^{(2)} = a_{2 δ_{2}}^{+} e^{- 2 i δ_{2} t} + a_{2 δ_{2}}^{-} e^{2 i δ_{2} t}, \\ δ x_{1}^{(1)} = x_{δ_{1}} e^{- i δ_{1} t} + x_{δ_{1}}^{*} e^{i δ_{1} t}, δ x_{2}^{(1)} = x_{δ_{2}} e^{- i δ_{2} t} + x_{δ_{2}}^{*} e^{i δ_{2} t}, \\ δ x_{1}^{(2)} = x_{2 δ_{1}} e^{- 2 i δ_{1} t} + x_{2 δ_{1}}^{*} e^{2 i δ_{1} t}, δ x_{2}^{(2)} = x_{2 δ_{2}} e^{- 2 i δ_{2} t} + x_{2 δ_{2}}^{*} e^{2 i δ_{2} t}, \\ δ a_{sum}^{(2)} = a_{s}^{+} e^{- i Ω_{+} t} + a_{s}^{-} e^{i Ω_{+} t}, δ x_{sum}^{(2)} = x_{s} e^{- i Ω_{+} t} + x_{s}^{*} e^{i Ω_{+} t}, \end{matrix}

with Ω₊ = δ₁ + δ₂. The physical picture of such ansatz is that there are output fields with frequencies ±Ω₊ generation due to the nonlinear terms iλ₀δxδa/h̄ and λ₀δa^*δa, which is very similar to sum frequency generation in a nonlinear medium. These components is the so-called sum sideband generation. There are also other components, such as difference sideband (frequencies δ₁ − δ₂ and δ₂ − δ₁ which similar to difference frequency generation in a nonlinear medium [33,34]) and the well discussed second- and higher-order sidebands [24]. It can be easily verified that these components contribute little to the amplitude of sum sideband generation. Here, we focus on the sum sideband generation, so the second- and higher-order sidebands can be ignored, and the ansatz is simplified as follows:

\begin{matrix} δ a = a_{1}^{+} e^{- i δ_{1} t} + a_{1}^{-} e^{i δ_{1} t} + a_{2}^{+} e^{- i δ_{2} t} + a_{2}^{-} e^{i δ_{2} t} + a_{s}^{+} e^{- i Ω_{+} t} + a_{s}^{-} e^{i Ω_{+} t} + \dots, \\ δ x = x_{1} e^{- i δ_{1} t} + x_{1}^{*} e^{i δ_{1} t} + x_{2} e^{- i δ_{2} t} + x_{2}^{*} e^{i δ_{2} t} + x_{s} e^{- i Ω_{+} t} + x_{s}^{*} e^{i Ω_{+} t} + \dots, \end{matrix}

Substitution of Eqs. (12) into Eqs. (8) leads to twelve Eqs.:

\begin{matrix} s a_{1}^{+} - i δ_{1} a_{1}^{+} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{1} - i \frac{λ_{0}}{\bar{h}} (a_{2}^{-} x_{s} + a_{s}^{+} x_{2}^{*}) - \sqrt{η κ} ε_{1} = 0, \\ s a_{1}^{-} + i δ_{1} a_{1}^{-} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{1}^{*} - i \frac{λ_{0}}{\bar{h}} (a_{2}^{+} x_{s}^{*} + a_{s}^{-} x_{2}) = 0, \\ s a_{2}^{+} - i δ_{2} a_{2}^{+} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{2} - i \frac{λ_{0}}{\bar{h}} (a_{1}^{-} x_{s} + a_{s}^{+} x_{1}^{*}) - \sqrt{η κ} ε_{2} = 0, \\ s a_{2}^{-} + i δ_{2} a_{2}^{-} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{2}^{*} - i \frac{λ_{0}}{\bar{h}} (a_{1}^{+} x_{s}^{*} + a_{s}^{-} x_{1}) = 0, \\ s a_{s}^{+} - i Ω_{+} a_{s}^{+} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{s} - i \frac{λ_{0}}{\bar{h}} (a_{1}^{+} x_{2} + a_{2}^{+} x_{1}) = 0, \\ s a_{s}^{-} + i Ω_{+} a_{s}^{-} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{s}^{*} - i \frac{λ_{0}}{\bar{h}} (a_{1}^{-} x_{2}^{*} + a_{2}^{-} x_{1}^{*}) = 0, \\ m Ω_{m}^{2} x_{1} - m {δ_{1}}^{2} x_{1} - i m Γ_{m} δ_{1} x_{1} - λ_{0} [a_{2}^{-} {(a_{s}^{-})}^{*} + a_{s}^{+} {(a_{2}^{+})}^{*} + \bar{a^{*}} a_{1}^{+} + \bar{a} {(a_{1}^{-})}^{*}] = 0, \\ m Ω_{m}^{2} x_{1}^{*} - m {δ_{1}}^{2} x_{1}^{*} + i m Γ_{m} δ_{1} x_{1}^{*} - λ_{0} [a_{2}^{+} {(a_{s}^{+})}^{*} + a_{s}^{-} {(a_{2}^{-})}^{*} + \bar{a^{*}} a_{1}^{-} + \bar{a} {(a_{1}^{+})}^{*}] = 0, \\ m Ω_{m}^{2} x_{2} - m {δ_{2}}^{2} x_{2} - i m Γ_{m} δ_{2} x_{2} - λ_{0} [a_{1}^{-} {(a_{s}^{-})}^{*} + a_{s}^{+} {(a_{1}^{+})}^{*} + \bar{a^{*}} a_{2}^{+} + \bar{a} {(a_{2}^{-})}^{*}] = 0, \\ m Ω_{m}^{2} x_{2}^{*} - m {δ_{2}}^{2} x_{2}^{*} + i m Γ_{m} δ_{2} x_{2}^{*} - λ_{0} [a_{1}^{+} {(a_{s}^{+})}^{*} + a_{s}^{-} {(a_{1}^{-})}^{*} + \bar{a^{*}} a_{2}^{-} + \bar{a} {(a_{2}^{+})}^{*}] = 0, \\ m Ω_{m}^{2} x_{s} - m {Ω_{+}}^{2} x_{s} - i m Γ_{m} Ω_{+} x_{s} - λ_{0} [a_{1}^{+} {(a_{2}^{-})}^{*} + a_{2}^{+} {(a_{1}^{-})}^{*} + \bar{a^{*}} a_{s}^{+} + \bar{a} {(a_{s}^{-})}^{*}] = 0, \\ m Ω_{m}^{2} x_{s}^{*} - m {Ω_{+}}^{2} x_{s}^{*} + i m Γ_{m} Ω_{+} x_{s}^{*} - λ_{0} [a_{1}^{-} {(a_{2}^{+})}^{*} + a_{2}^{-} {(a_{1}^{+})}^{*} + \bar{a^{*}} a_{s}^{-} + \bar{a} {(a_{s}^{+})}^{*}] = 0, \end{matrix}

with s = κ − iΔ̄. These twelve Eqs. are not independent and can be reduced into nine independent Eqs., which finally can be simplified into three groups by considering that sum sideband generation are second order processes:

The first group describes the linear process of the probe field with frequency ω₁ $\begin{matrix} (s - i δ_{1}) a_{1}^{+} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{1} - \sqrt{η κ} ε_{1} = 0 \\ (s + i δ_{1}) a_{1}^{-} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{1}^{*} = 0 \\ (m Ω_{m}^{2} - m {δ_{1}}^{2} - i m Γ_{m} δ_{1}) x_{1} - λ_{0} [\bar{a^{*}} a_{1}^{+} + \bar{a} {(a_{1}^{-})}^{*}] = 0 . \end{matrix}$
The second group describes the linear process of the probe field with frequency ω₂ $\begin{matrix} (s - i δ_{2}) a_{2}^{+} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{2} - \sqrt{η κ} ε_{2} = 0 \\ (s + i δ_{2}) a_{2}^{-} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{2}^{*} = 0 \\ (m Ω_{m}^{2} - m {δ_{2}}^{2} - i m Γ_{m} δ_{2}) x_{2} - λ_{0} [\bar{a^{*}} a_{2}^{+} + \bar{a} {(a_{2}^{-})}^{*}] = 0 . \end{matrix}$
The third group describes the process of sum-sideband generation $\begin{matrix} (s - i Ω_{+}) a_{s}^{+} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{s} - i \frac{λ_{0}}{\bar{h}} (a_{1}^{+} x_{2} + a_{2}^{+} x_{1}) = 0 \\ (s + i Ω_{+}) a_{s}^{-} - i \frac{λ_{0}}{\bar{h}} \bar{a} x_{s}^{*} - i \frac{λ_{0}}{\bar{h}} (a_{1}^{-} x_{2}^{*} + a_{2}^{-} x_{1}^{*}) = 0 \\ (m Ω_{m}^{2} - m {Ω_{+}}^{2} - i m Γ_{m} Ω_{+}) x_{s} \\ - λ_{0} [a_{1}^{+} {(a_{2}^{-})}^{*} + a_{2}^{+} {(a_{1}^{-})}^{*} + \bar{a^{*}} a_{s}^{+} + \bar{a} {(a_{s}^{-})}^{*}] = 0 . \end{matrix}$ The Eqs. (14) and (15), which ignore the second order quantities, are exactly the linear results obtained in some previous works [14], and are used to study the effect of optomechanically induced transparency. The solution to these Eqs. can be obtained as follows: $\begin{matrix} a_{1}^{+} = \frac{\sqrt{η κ} ε_{1} τ (δ_{1})}{θ (- δ_{1}) τ (δ_{1}) - τ}, x_{1} = \frac{λ_{0} \bar{a^{*}} a_{1}^{+}}{τ (δ_{1})}, a_{1}^{-} = \frac{i λ_{0} \bar{a}}{\bar{h} θ (δ_{1})} x_{1}^{*}, \\ a_{2}^{+} = \frac{\sqrt{η κ} ε_{2} τ (δ_{2})}{θ (- δ_{2}) τ (δ_{2}) - τ}, x_{2} = \frac{λ_{0} \bar{a^{*}} a_{2}^{+}}{τ (δ_{2})}, a_{2}^{-} = \frac{i λ_{0} \bar{a}}{\bar{h} θ (δ_{2})} x_{2}^{*}, \\ a_{s}^{+} = i \frac{λ_{0}}{\bar{h}} \frac{λ_{0} \bar{a} ξ_{s} + (a_{1}^{+} x_{2} + a_{2}^{+} x_{1}) τ (Ω_{+})}{τ (Ω_{+}) θ (- Ω_{+}) - α}, \\ a_{s}^{-} = \frac{i λ_{0} (\bar{a} x_{s}^{*} + a_{1}^{-} x_{2}^{*} + a_{2}^{-} x_{1}^{*})}{\bar{h} θ (Ω_{+})}, x_{s} = \frac{λ_{0} (ξ_{s} + \bar{a^{*}} a_{s}^{+})}{τ (Ω_{+})}, \end{matrix}$ where α = iλ₀² |ā|²/h̄, θ(x) = s + ix, $σ (x) = m Ω_{m}^{2} - m x^{2} - i m Γ_{m} x$ , τ(x) = σ(x) + α/θ(x)^*, and $ξ_{s} = a_{1}^{+} {(a_{2}^{-})}^{*} + a_{2}^{+} {(a_{1}^{-})}^{*} - i λ_{0} \bar{a} [{(a_{1}^{-})}^{*} x_{2} + {(a_{2}^{-})}^{*} x_{1}] / [\bar{h} θ {(Ω_{+})}^{*}]$ .

Using the input-output relation $s_{out} = s_{in} - \sqrt{η κ} a$ , the output fields (in a frame rotating at ω_c) of the optomechanical system can be obtained as follows:

\begin{array}{l} s_{out} = ε_{c} - \sqrt{η κ} \bar{a} + (ε_{1} - \sqrt{η κ} a_{1}^{+}) e^{- i δ_{1} t} + (ε_{2} - \sqrt{η κ} a_{2}^{+}) e^{- i δ_{2} t} \\ - \sqrt{η κ} a_{1}^{-} e^{i δ_{1} t} - \sqrt{η κ} a_{2}^{-} e^{i δ_{2} t} - \sqrt{η κ} a_{s}^{+} e^{- i Ω_{+} t} - \sqrt{η κ} a_{s}^{-} e^{i Ω_{+} t} . \end{array}

The transmission of the i-th probe field, which is defined as

T_{i} = (ε_{i} - \sqrt{η κ} a_{i}^{+}) / ε_{i}

, can be obtain as follows

T_{i} = 1 - η κ \frac{τ (δ_{i})}{θ (- δ_{i}) τ (δ_{i}) - α},

which has been used to discuss optomechanically induced transparency in previous works [14]. The terms of

- \sqrt{η κ} a_{s}^{+} e^{- i Ω_{+} t}

and

- \sqrt{η κ} a_{s}^{-} e^{i Ω_{+} t}

describe the upper and lower sum-sideband process, respectively.

For the degenerate case δ₁ = δ₂, the given results (17) does not go back to the single probe case where second-order sideband generation occur [24]. The physical interpretation is that the two probe fields are coherent in the degenerate case and there is interference between two scattering paths in the second-order sideband field [24], while the two probe fields are incoherent (frequency is different) in the nondegenerate case and the sum sideband fields result from the incoherent superposition of the two scattering paths.

3. Features and enhancement of sum sideband generation

We define $η_{s}^{+} = | - \sqrt{η κ} a_{s}^{+} / ε_{1} |$ , which is the ratio between amplitudes of the upper sum-sideband and the first probe field and thus dimensionless, as the efficiency of the upper sum sideband generation process. Similarly, $η_{s}^{-} = | - \sqrt{η κ} a_{s}^{-} / ε_{1} |$ is the efficiency of the lower sum sideband generation process.

The efficiencies (in logarithmic form) of upper and lower sum sideband generation as a function of the pump power of the control field P_c and the frequency of the first probe field δ₁ is shown in Fig. 2(a) and Fig. 2(b). The parameters used in the calculation are chosen from the recent experiment [14]. The strengths of sum sideband generation are often quite small due to the weak nonlinearity. Very different from the second-order sideband generation [24], efficiencies of sum sideband generation (including both upper and lower case) increase monotonically with the power of the control field P_c. For second-order sideband generation, there is a threshold value which determines the effect of the power of the control field: If the power of the control field is larger than the threshold value, there is a suppressive window for the second-order sideband field [24] which results from the interference between two scattering paths. For sum sideband generation, interference conditions are not met due to the nonzero frequency difference between two scattering paths. Consequently, with the increases of the power of the control field P_c, the efficiencies of sum sideband generation increase rapidly until P_c ≈ 0.8 mW and increase gradually for P_c > 0.8 mW. The enhancement of sum sideband generation by improving the power of the control field is often not obvious for P_c > 1 mW.

Fig. 2 Dependencies of the efficiencies (in logarithmic form) of (a) the upper sum sideband generation ${log}_{10} η_{s}^{+}$ , (b) lower sum sideband generation ${log}_{10} η_{s}^{-}$ on the pump power of the control field P_c and the frequency of the first probe field δ₁ for δ₂ = 0.1Ω_m. (c) Calculation results of ${log}_{10} η_{s}^{+}$ and ${log}_{10} η_{s}^{-}$ vary with δ₁ for δ₂ = −0.05Ω_m, P_c = 20 μW and P₁ = P₂ = 1 μW. (d) The amplitude of the mechanical oscillation (in unit of femtometer) at the sum-sideband vary with δ₁ and δ₂. The parameters used in the calculation are m=20 ng, G/2π=−12 GHz/nm, Γ_m/2π=41.0 kHz, κ/2π=15.0 MHz, Ω_m/2π=51.8MHz, and Δ=−Ω_m. The wavelength of the control field is chosen to be 532 nm here.

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Another control parameter for the improvement of sum sideband generation is the detuning between the control field and the probe fields. As shown in Fig. 2(a) and Fig. 2(b), the efficiencies of sum sideband generation exhibit peak structure for some specific values of δ₁. Thus, the processes of sum sideband generation can be enhanced significantly through the suitable selection of δ₁. In Fig. 2(a), ${log}_{10} η_{s}^{+}$ has a peak in the parameter range δ₁ ∈ [0.8Ω_m, 1.2Ω_m]. In Fig. 2(b), ${log}_{10} η_{s}^{-}$ has two peaks in the parameter range δ₁ ∈ [0.8Ω_m, 1.2Ω_m]. We call the specific values of δ₁ (δ₂) corresponding to these peaks as the matching conditions. The efficiencies (in logarithmic form) of upper and lower sum sideband generation vary with the frequency of the first probe field is shown in Fig. 2(c), and we find that the efficiency of upper sum sideband generation is much larger than the lower one. From Fig. 2(c) we can identify the matching conditions of sum sideband generation achieving the maximum value. ${log}_{10} η_{s}^{+}$ achieves the maximum at δ₁ = Ω_m while ${log}_{10} η_{s}^{-}$ achieves the maximum at both δ₁ = Ω_m and δ₁ = 1.05Ω_m. Here the condition δ₁ = 1.05Ω_m comes down to δ₁ + δ₂ = Ω_m. We confirm the results after examining the matching conditions carefully with various values of δ₂.

These matching conditions can be understood through the features of the mechanical oscillation at the sum sideband. Dependencies of the amplitude of the mechanical oscillation at the sum sideband on the variables δ₁ and δ₂ are illustrated in Fig. 2(d). It is shown that x_s becomes remarkable on the line δ₁ + δ₂ = Ω_m and at four points (δ₁, δ₂)=(0, ±Ω_m) and (±Ω_m, 0), which are exactly corresponding to the matching conditions for sum sideband generation. The amplitude of the mechanical oscillation x_s near the two points (0, Ω_m) and (Ω_m, 0) is about 5 fm, while at the other two points (0, −Ω_m) and (−Ω_m, 0) is about 2 fm.

To describe sum sideband generation in a systematic way, calculation results of sum sideband generation as a function of both δ₁ and δ₂ are shown in Fig. 3, from which one can identify the general matching conditions for sum sideband generation. Figure 3 shows the calculation results of upper and lower sum-sideband generation. For the case of upper sum-sideband generation in Fig. 3(a), as expected, upper sum-sideband generation is enhanced when δ₁ → Ω_m, which is the matching condition shown in Fig. 2(c). There is another local maximum at δ₁ = −Ω_m. Figure 3(a) is exactly symmetrical for δ₁ and δ₂ since the parameters δ₁ and δ₂ are on equal footing in the sum-sideband generation, so the upper sum-sideband generation also reaches local maximum at δ₂ = ±Ω_m. We also note that upper sum-sideband generation become quite remarkable near the point of (δ₁, δ₂)=(Ω_m, Ω_m). In contrast, upper sum-sideband generation is far less obvious near the point of (δ₁, δ₂)=(−Ω_m, −Ω_m). The physical interpretation of the different behavior near the two points is that the former one is near the resonance condition of the cavity Ω = −Δ̄. For the case of lower sum-sideband generation in Fig. 3(b), the general matching conditions for lower sum-sideband generation achieving the maximum are δ₁ = ±Ω_m, δ₂ = ±Ω_m, and δ₁ + δ₂ = ±Ω_m. The efficiency of lower sum-sideband generation ${log}_{10} η_{s}^{-}$ become remarkable at the intersections of these matching conditions.

Fig. 3 Efficiencies (in logarithmic form) of (a) upper sum-sideband generation and (b) lower sum-sideband generation as a function of δ₁ and δ₂. The parameters are the same as Fig. 2.

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Comparison between Fig. 3(a) and Fig. 3(b) reveals a difference in the matching conditions between upper and lower sum-sideband generation. The upper sum-sideband generation achieving the maximum when δ₁ = ±Ω_m or δ₂ = ±Ω_m, while the lower sum-sideband generation has an additional matching condition δ₁ + δ₂ = ±Ω_m, which is related to the two oblique lines in Fig. 3(b), except the conditions δ₁, δ₂ = ±Ω_m. Such additional matching condition δ₁ + δ₂ = ±Ω_m gives rise to additional peaks for the lower sum sideband generation. In Fig. 2(a), there appears a single peak for the upper sum sideband generation, while there exist two peaks in the lower sum sideband shown in Fig. 2(b).

A robust optical sum sideband generation, which deepens the understanding of the nonlinear interaction between cavity fields and mechanical motion, may also be useful for information processing. Unlike previously reported second- and higher-order sideband generation, sum sideband generation in the optomechanical system arise from the nonlinear interactions between the mechanical motion and multiple driven fields. In addition, sum sideband generation works under low operating power. These attributes make it attractive as a potential component for photonic information processing. The present mechanism of sum sideband generation may also be applied to other similar systems, such as quantum dot and quantum well system [35,36], metasurfaces [37], artificial molecule [38].

4. Conclusion

By analyzing a generic optomechanical system driven by a strong control field with the frequency ω_c and two relatively weak probe fields with frequencies ω₁ and ω₂, we show that such double-probe-field driven optomechanical system can lead to the generation of sum sideband signals (with frequencies ±Ω₊ in a frame rotating at ω_c) by taking account nonlinear terms. We give a full description of the derivation of optomechanically induced sum sideband generation beyond the linearized optomechanical interaction. The analytical expressions describe the amplitude of the upper and lower sum sideband are obtained. Meanwhile, we also show from numerical simulations that the generated sum sideband can be observed and controlled in the experimentally available parameter range. Especially, sum sideband generation can be greatly enhanced even at low power through the satisfaction of matching conditions. This investigation may provide further insight into the understanding of optomechanical system and find applications in precision measurement and optical communications.

Acknowledgments

The work is supported in part by the National Fundamental Research Program of China (Grant No. 2012CB922103) and the National Natural Science Foundation (NNSF) of China (Grant Nos. 11375067, 11275074, 11374116, 11204096, 11405061, and 11574104).

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Optomechanically induced sum sideband generation

Abstract

1. Introduction

2. Derivation of optomechanically induced sum sideband generation

3. Features and enhancement of sum sideband generation

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (20)

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