Generation of mechanical squeezing and entanglement via mechanical modulations

Wen-Ju Gu; Zhen Yi; Li-Hui Sun; Yan Yan

doi:10.1364/OE.26.030773

1. Introduction

With the rapid advancement of micro-fabrication and manipulation techniques, cavity optomechanics has achieved significant experimental breakthroughs, including ground-state cooling of mechanical oscillators [1–3], strong optomechanical couplings [4–6], and optomechanical interfaces for hybrid networks [7–9]. These achievements motivate further studies on quantum effects in macroscopic mechanical systems, which would provide a pathway to new quantum information tools [10], as well as fundamental research of the classical-to-quantum boundary [11–13]. In this framework, mechanical squeezing also possesses the potential to promote the ultra-high sensitive detections in fields such as atomic force microscopy [14]. Squeezing of mechanical fluctuations was originally demonstrated by the degenerate parametric amplification through parametrically modulating the mechanical spring constant [15]. However, the stationary squeezing is limited to 3dB due to the stability condition, and thus the thermal occupation of mechanical mode must be well below one phonon to achieve squeezing below the zero-point fluctuations. To surpass 3dB limit, there are many theoretical proposals and experimental realizations via combining with quantum control techniques [16–18]. Moreover, if the oscillator is embedded in an optical cavity, optomechanical interaction can be utilized to generate squeezing while simultaneously cooling, resulting in a squeezed-vacuum state [19, 20].

In addition, a different class of macroscopic, strongly non-classical states involves more than one mechanical modes, such as mechanical entanglement, Greenberger-Horne-Zeilinger (GHZ) and cluster states [21–23]. In particular, experimental realization of hybridizing two mechanical modes provides a basic ingredient of mechanical entanglement [24]. The EPR-type entanglement can be characterized by two-mode squeezing (TMS), where the variance of combination of quadratures reduces below the standard quantum limit that exhibits correlated fluctuations between quadratures of two modes. Generation of optical twin beams in parametric down-conversion offers a route towards a mechanical entanglement, which has been realized by implementing a modulation of coupling constant at the sum of the resonance frequencies of two oscillators [25–27]. However, the stationary squeezing is also limited to 3dB by the stability condition in parametric down-conversion. Differently, if a periodic modulation is employed on resonance frequencies, the system will also appear a rich phase diagram [28–31]. Recently, there exist some experimental platforms to investigate the effects induced by the parametrical modulation of mechanical frequency. One of such platforms is based on the cantilever optomechanical system, where a harmonically oscillating optical trap (optical force) is applied to modulating the effective frequency of cantilever [32]. Another platform is levitated optomechanics where a nanoparticle is confined within a hybrid electro-optical trap formed by a Paul trap within a single-mode optical cavity [33, 34]. In this system periodic modulations are generic and occur in optically trapped setups where the equilibrium point of the oscillator is varied cyclically.

In this work, distinct from the generation of mechanical entanglement via parametric down-conversion, we study the generation of strong mechanical squeezing and entanglement following the reservoir engineering procedure [20, 35]. Through modulating the mechanical oscillating frequency in two-and three-mode optomechanics, we can simultaneously realize the beam-splitter and parametric interactions between the cavity and mechanical modes, which is critical to the generation of squeezing and entanglement. In the two-mode modulated optomechanics, the degree of squeezing is controllable by the relative ratio of parametric and beam-splitter coupling strengths, and the associated optically-induced cooling process ensures the squeezing robust to thermal noise. In the three-mode optomechanics, strong EPR-type entanglement surpassing 3dB limit is also attainable. However, the ideal entanglement is unachievable here since only one of mechanical Bogoliubov modes can be cooled down to ground state by the cavity mode while the other Bogoliubov mode decouples from the interaction, which is in a thermal state and also makes the entanglement fragile to the mechanical noise.

The paper is organized as follows. In section 2 the generation of strong mechanical squeezing is discussed in modulated two-mode optomechanics, and the generation of EPR-type mechanical entanglement in the modulated three-mode optomechanics is studied in section 3. At last the conclusion is drawn in section 4.

2. Generation of mechanical squeezing in modulated two-mode optomechanics

We consider a two-mode optomechanical system in which a mechanical oscillator couples to an optical cavity and its frequency is periodically modulated, as schematically shown in Fig. 1. Cavity mode with frequency ω_c is described by the annihilation operator $\hat{a}$ . The mechanical oscillator is described by the annihilation operator $\hat{b}$ , and the oscillating frequency is modulated in the form of ω_m + ϵ sin(νt + φ), where ω_m is the static frequency, and ϵ, ν, φ are the modulation amplitude, frequency and phase. The cavity mode couples to mechanical oscillator via the radiation pressure with the single-photon strength g₀. In addition, to enhance the optomechanical coupling rate a laser field with the driving amplitude E and frequency ω_L is employed. The Hamiltonian of the system can be written as (ħ=1)

\hat{H} = ω_{c} {\hat{a}}^{†} \hat{a} + [ω_{m} + ϵ \sin (v t + φ)] {\hat{b}}^{†} \hat{b} - g_{0} {\hat{a}}^{†} \hat{a} ({\hat{b}}^{†} + \hat{b}) + i (E {\hat{a}}^{†} e^{- i ω_{L} t} - E^{*} \hat{a} e^{i ω_{L} t}),

Fig. 1 Schematic of the two-mode optomechanical system consisted of cavity mode and the frequency-modulated mechanical oscillator. The cavity mode $\hat{a}$ is driven by an external laser field with amplitude E and frequency ω_L , and couples to the modulated oscillator via radiation pressure with the strength g.

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In the rotating frame of laser frequency ω_L, the Hamiltonian becomes

\hat{H} = Δ {\hat{a}}^{†} \hat{a} + [ω_{m} + ϵ \sin (v t + φ)] {\hat{b}}^{†} \hat{b} - g_{0} {\hat{a}}^{†} \hat{a} ({\hat{b}}^{†} + \hat{b}) + i (E {\hat{a}}^{†} - E^{*} \hat{a}),

where Δ = ω_c − ω_L is the detuning between cavity frequency ω_c and laser frequency ω_L.

Then we introduce an unitary operator

\hat{U} (t) = \exp [i η \cos (v t + φ) {\hat{b}}^{†} \hat{b}]

with η = ϵ/ν. The transformed Hamiltonian follows the expression

\tilde{\hat{H}} = {\hat{U}}^{†} (t) \hat{H} \hat{U} (t) + i \frac{d {\hat{U}}^{†} (t)}{d t} \hat{U} (t),

and becomes

\tilde{\hat{H}} = Δ {\hat{a}}^{†} \hat{a} + ω_{m} {\hat{b}}^{†} \hat{b} - g_{0} {\hat{a}}^{†} \hat{a} [{\hat{b}}^{†} e^{- i η \cos (v t + φ)} + \hat{b} e^{i η \cos (v t + φ)}] + i (E {\hat{a}}^{†} - E^{*} \hat{a}) .

In order to achieve the enhanced optomechanical coupling strength, a strong driving laser is typically applied, and we could treat the system following the standard linearization procedure, where cavity and mechanical modes are written as the sum of steady-state values and fluctuations, i.e., $\hat{a} = α + \hat{c}$ , $\hat{b} = β + \hat{d}$ with $α = 〈 \hat{a} 〉$ , $β = 〈 \hat{b} 〉$ . For the weak single-photon coupling strength g₀, the steady-state values are approximately equal to α ≈ E/(iΔ + κ/2) and β ≈ 0, where κ is cavity damping rate. Via appropriately choosing the phase of E, we assume α real-valued. Then with use of the Jacobi-Anger identity

\exp [i η \cos (v t + φ)] = \sum_{n = - \infty}^{\infty} i^{n} J_{n} (η) \exp [i n (v t + φ)],

with J_n(η) being the nth Bessel function of first kind and fulfilling J_n(−η) = (−1)ⁿJ_n(η), the Hamiltonian for fluctuation operators becomes

\tilde{\hat{H}} = Δ {\hat{c}}^{†} \hat{c} + ω_{m} {\hat{d}}^{†} \hat{d} - g ({\hat{c}}^{†} + \hat{c}) \sum_{n = - \infty}^{\infty} i^{n} e^{i n (v t + φ)} J_{n} (η) [\hat{d} + {(- 1)}^{n} {\hat{d}}^{†}],

where g = g₀α is the effectively laser enlarged optomechanical coupling constant. In the rotating frame of cavity and mechanical frequencies, the Hamiltonian turns to the form

\tilde{\hat{H}} = - \sum_{n = - \infty}^{\infty} i^{n} e^{i n (v t + φ)} {\tilde{g}}_{n} [{\hat{c}}^{†} \hat{d} e^{- i (ω_{m} - Δ) t} + {(- 1)}^{n} {\hat{c}}^{†} {\hat{d}}^{†} e^{i (ω_{m} + Δ) t}] + H .c,

where

{\tilde{g}}_{n} = g J_{n} (η)

relates to the nth Bessel function.

When the laser field is tuned to the red sideband of cavity mode, i.e., Δ = ω_m, the resonant beam-splitter interaction between mechanical mode and cavity field is achievable. In addition, to simultaneously obtain the resonant parametric interaction, the modulation frequency should fulfill nν = 2ω_m. A sinusoidal modulation of mechanical frequency has been employed to investigate the formation of squeezing with a perturbation method in Ref. [36], where numerical results reveal that optimal choice of modulation frequency of maximal squeezing is ν = 2ω_m and at ν = ω_m the level of squeezing is also enhanced, which is consistent with nν = 2ω_m. Here we also consider ν = 2ω_m and phase φ = −π/2. By neglecting the non-resonant terms the effective Hamiltonian is achieved in the form

{\tilde{\hat{H}}}_{eff} = - {\tilde{g}}_{0} ({\hat{c}}^{†} \hat{d} + {\hat{d}}^{†} \hat{c}) - {\tilde{g}}_{1} (\hat{c} \hat{d} + {\hat{c}}^{†} {\hat{d}}^{†}) .

The beam-splitter and parametric interaction between the cavity and mechanical modes are simultaneously realized, which would benefit for the generation of squeezing by reservoir engineering [19, 20, 35].

From the Hamiltonian (9) and taking into account of cavity and mechanical dissipations, the Langevin equation of $\hat{c}$ and $\hat{d}$ can be obtained as

\begin{array}{l} \frac{d}{d t} \hat{c} = - \frac{κ}{2} \hat{c} + i {\tilde{g}}_{0} \hat{d} + i {\tilde{g}}_{1} {\hat{d}}^{†} + \sqrt{κ} {\hat{c}}_{in}, \\ \frac{d}{d t} \hat{d} = - \frac{γ_{m}}{2} \hat{d} + i {\tilde{g}}_{0} \hat{c} + i {\tilde{g}}_{1} {\hat{c}}^{†} + \sqrt{γ_{m}} {\hat{d}}_{in}, \end{array}

where γ_m is the mechanical dissipation rate,

{\hat{c}}_{in}

and

{\hat{d}}_{in}

are the cavity and mechanical input noises which fulfill the correlations

\begin{array}{l} 〈 {\hat{c}}_{in} (t) {\hat{c}}_{in}^{†} (t^{'}) 〉 = δ (t - t^{'}), \\ 〈 {\hat{d}}_{in} (t) {\hat{d}}_{in}^{†} (t^{'}) 〉 = (n_{th} + 1) δ (t - t^{'}) . \end{array}

The average thermal phonon occupation number n_th = [exp(ħω_m/k_BT)− 1]⁻¹, where k_B is the Boltzmann constant, and T is the temperature of the environment surrounding the mechanical oscillator.

In order to achieve mechanical squeezing, following reservoir engineering procedure in the regime of $κ ≫ {{\tilde{g}}_{0}, {\tilde{g}}_{1}}$ , we can adiabatically eliminate the cavity mode

\hat{c} \approx \frac{i {\tilde{g}}_{0}}{κ / 2} \hat{d} + \frac{i {\tilde{g}}_{1}}{κ / 2} {\hat{d}}^{†} + \frac{2}{\sqrt{κ}} {\hat{c}}_{in} .

By substituting $\hat{c}$ into the motion equations of mechanical mode, we have

\begin{array}{l} \frac{d}{d t} \hat{d} = (\frac{G_{1}}{2} - \frac{G_{0}}{2} - \frac{γ_{m}}{2}) \hat{d} + i \sqrt{G_{0}} {\hat{c}}_{in} + i \sqrt{G_{1}} {\hat{c}}_{in}^{†} + \sqrt{γ_{m}} {\hat{d}}_{in}, \\ \frac{d}{d t} {\hat{d}}^{†} = (\frac{G_{1}}{2} - \frac{G_{0}}{2} - \frac{γ_{m}}{2}) {\hat{d}}^{†} - i \sqrt{G_{0}} {\hat{c}}_{in}^{†} - i \sqrt{G_{1}} {\hat{c}}_{in} + \sqrt{γ_{m}} {\hat{d}}_{in}^{†}, \end{array}

where

G_{0} = 4 {\tilde{g}}_{0}^{2} / κ

and

G_{1} = 4 {\tilde{g}}_{1}^{2} / κ

are the optically induced mechanical damping and gain rates. Generally, the mechanical gain will induce instability, but here we consider the stationary mechanical squeezing in the region of G₁ < G₀ + γ_m. It is convenient to solve the equations in frequency domain via performing the Fourier transformation

\begin{array}{l} \hat{d} (ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} \hat{d} (t) e^{i ω t} d t, \\ {\hat{d}}^{†} (ω) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} {\hat{d}}^{†} (t) e^{i ω t} d t, \end{array}

where the noise correlations in frequency domain become

〈 {\hat{c}}_{in} (ω) {\hat{c}}_{in}^{†} (ω^{'}) 〉 = δ (ω + ω^{'})

,

〈 {\hat{d}}_{in} (ω) {\hat{d}}_{in}^{†} (ω^{'}) 〉 = (n_{th} + 1) δ (ω + ω^{'})

.

In frequency domain, the solution of Eq. (13) can be written as

\begin{array}{l} \hat{d} (ω) = - \frac{i \sqrt{G_{0}} {\hat{c}}_{in} (ω) + i \sqrt{G_{1}} {\hat{c}}_{in}^{†} (ω) + \sqrt{γ_{m}} {\hat{d}}_{in} (ω)}{i ω + (G_{1} - G_{0} - γ_{m}) / 2}, \\ {\hat{d}}^{†} (ω) = - \frac{i \sqrt{G_{0}} {\hat{c}}_{in}^{†} (ω) + i \sqrt{G_{1}} {\hat{c}}_{in} (ω) - \sqrt{γ_{m}} {\hat{d}}_{in}^{†} (ω)}{i ω + (G_{1} - G_{0} - γ_{m}) / 2} . \end{array}

The squeezing of mechanical motion is characterized by the variance of generalized quadrature operator, which is defined in the frequency domain as

{\hat{X}}_{θ} (ω) = \frac{1}{\sqrt{2}} [e^{i θ} \hat{d} (ω) + e^{- i θ} {\hat{d}}^{†} (ω)] .

The phase angle θ is experimentally controllable in a homodyne detection scheme. The squeezing spectrum of quadrature operator is

\begin{matrix} S_{X_{θ}} (ω) = 〈 {\hat{X}}_{θ} (ω) {\hat{X}}_{θ} (ω^{'}) 〉 δ (ω + ω^{'}) \\ = \frac{1}{2} \frac{G_{0} + G_{1} - 2 \sqrt{G_{0} G_{1}} \cos 2 θ + γ_{m} (2 n_{th} + 1)}{ω^{2} + {(G_{1} - G_{0} - γ_{m})}^{2} / 4}, \end{matrix}

and the stationary variance of quadrature fluctuation in the time domain is determined by

〈 {\hat{X}}_{θ}^{2} 〉 = \frac{1}{2 π} \int_{- \infty}^{\infty} S_{X_{θ}} (ω) d ω .

After the integration, the variance becomes

〈 {\hat{X}}_{θ}^{2} 〉 = \frac{G_{1} + γ_{m} n_{th} - \sqrt{G_{0} G_{1}} \cos 2 θ}{G_{0} - G_{1} + γ_{m}} + \frac{1}{2},

and the optimal squeezing (minimum of variance) is achieved when θ = 0,

〈 {\hat{X}}_{θ}^{2} 〉 = \frac{G_{1} + γ_{m} n_{th} - \sqrt{G_{0} G_{1}}}{G_{0} - G_{1} + γ_{m}} + \frac{1}{2} .

To achieve the mechanical squeezing, we should have $〈 {\hat{X}}_{0}^{2} 〉 < 1 / 2$ , which is determined by the ratio between optically induced mechanical gain and damping rates, and we define λ = G₁/G₀ for convenience. For the recent cavity-optomechanical experiments which have reached the quantum regime (mechanical phonon number occupancy ≤ 1) within the resolved sideband laser cooling regime [1, 2], optically induced damping rate G₀ ≫ γ_m, and then the variance becomes

〈 {\hat{X}}_{0}^{2} 〉 \approx - \frac{\sqrt{λ}}{1 + \sqrt{λ}} + \frac{γ_{m} n_{th}}{G_{0} (1 - λ)} + \frac{1}{2},

which corresponds to squeezing induced by parametric interaction, optically induced cooling of mechanical thermal noise, and vacuum fluctuation. In the limit of G₁ → 0, the system becomes a typical sideband cooling model and the mechanical motion approaches the vacuum state with

〈 {\hat{X}}_{0}^{2} 〉 = γ_{m} n_{th} / G_{0} + \frac{1}{2}

, where mechanical squeezing is unreachable. However, the increase of G₁ has two conflicting effects: it can increase the degree of squeezing but also decrease the efficiency of cooling the initial phonon thermal noise. The optimal balance is achieved at

\sqrt{λ} = 1 + \frac{γ_{m} n_{th}}{G_{0}} - \sqrt{\frac{γ_{m}^{2} n_{th}^{2}}{G_{0}^{2}} + 2 \frac{γ_{m} n_{th}}{G_{0}}} .

In the experimental setup of laser cooling of micromechanical membrane [37], mechanical dissipation rate γ_m = 2π × 0.18Hz, optically induced damping rate G₀ < 2π × 30kHz, and at the cryostat temperature initial phonon number n_th ∼ 10³. In Fig. 2 we have shown that in the certain region of λ, strong mechanical squeezing (> 3dB) is achievable.

Fig. 2 Mechanical squeezing (dB) versus controllable parameter λ (G₁/G₀) with different initial thermal occupations. The other parameters are: γ_m = 2π ×0.18Hz, G₀ = 2π ×20kHz.

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The mechanical noise will decrease the squeezing. With the increase of initial phonon occupations, such as n_th = 2 × 10⁴, the squeezing decreases to even below the 3dB limit. Moreover, there exists a critical value of γ_mn_th/G₀ = 1/4. If the initial noise is below the critical value, squeezing can appear for certain values of λ, and otherwise, squeezing cannot appear.

3. Generation of mechanical entanglement in modulated three-mode optomechanics

In this section, we consider a three-mode optomechanical system consisted of two frequency-modulated mechanical oscillators coupled to a common cavity mode, as shown in Fig. 3. A laser field with amplitude E and frequency ω_L is employed to drive the cavity mode, and the static frequency of each mechanical oscillator is $ω_{m}^{(j)} (j = 1, 2)$ and modulated with amplitude ϵj , frequency ν_j and phase φ_j . The Hamiltonian of the system can be written as

\begin{matrix} \hat{H} = ω_{c} {\hat{a}}^{†} \hat{a} + \sum_{j = 1}^{2} {[ω_{m}^{(j)} + ϵ_{j} \sin (v_{j} t + φ_{j})] {\hat{b}}_{j}^{†} {\hat{b}}_{j} \\ - g_{0}^{(j)} {\hat{a}}^{†} \hat{a} ({\hat{b}}_{j}^{†} + {\hat{b}}_{j})} + i (E {\hat{a}}^{†} e^{- i ω_{L} t} - E^{*} \hat{a} e^{i ω_{L} t}), \end{matrix}

where

\hat{a}

and

{\hat{b}}_{j}

are the annihilation operators of cavity and mechanical modes, ω_c is the frequency of cavity mode, and

g_{0}^{(j)}

is the jth single-photon optomechanical coupling strength. In the rotating frame of laser frequency and introducing the unitary operator

\hat{U} (t) = \prod_{j = 1}^{2} \exp [i η_{j} \cos (v_{j} t + φ_{j}) {\hat{b}}_{j}^{†} {\hat{b}}_{j}]

with η_j = ϵ_j /ν_j , following Eq. (4) the transformed Hamiltonian becomes

\begin{matrix} \tilde{\hat{H}} = Δ {\hat{a}}^{†} \hat{a} + \sum_{j = 1}^{2} {ω_{m}^{(j)} {\hat{b}}_{j}^{†} {\hat{b}}_{j} - g_{0}^{(j)} {\hat{a}}^{†} \hat{a} [{\hat{b}}_{j}^{†} e^{- i η_{j} \cos (v_{j} t + φ_{j})} \\ + {\hat{b}}_{j} e^{i η_{j} \cos (v_{j} t + φ_{j})}]} + i (E {\hat{a}}^{†} - E^{*} \hat{a}), \end{matrix}

with the detuning Δ = ω_c − ω_L .

Fig. 3 Schematic of the three-mode optomechanical system consisted of a cavity mode and two frequency-modulated mechanical oscillators. The cavity mode $\hat{a}$ is driven by an external laser field with amplitude E and frequency ω_L , and couples to two modulated oscillators via radiation pressure with the strengths g₁ and g₂.

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Similarly, we investigate the optomechanical interaction following the linearization procedure with cavity and two mechanical modes considered as a sum of coherent and quantum fluctuation parts, i.e., $\hat{a} = α + \hat{c}$ , ${\hat{b}}_{j} = β_{j} + {\hat{d}}_{j}$ , where $α = 〈 \hat{a} 〉$ and $β_{j} = 〈 {\hat{b}}_{j} 〉$ are the average amplitudes and $\hat{c}$ and ${\hat{d}}_{j}$ are the quantum fluctuation operators. For the weak single-photon coupling strength $g_{0}^{(j)}$ , the steady-state values are approximately α ≈ E/(iΔ + κ/2) and β_j ≈ 0, where κ is cavity damping rate. Via appropriately choosing the phase of E, we assume α real-valued. Then with use of the Jacobi-Anger identity in Eq. (6), the form of Hamiltonian becomes

\begin{matrix} \tilde{\hat{H}} = Δ {\hat{c}}^{†} \hat{c} + \sum_{j = 1}^{2} {ω_{m}^{(j)} {\hat{d}}_{j}^{†} {\hat{d}}_{j} - g_{j} ({\hat{c}}^{†} + \hat{c}) \\ \times \sum_{n = - \infty}^{\infty} i^{n} e^{i n (v_{j} t + φ_{j})} J_{n} (η_{j}) [{\hat{d}}_{j} + {(- 1)}^{n} {\hat{d}}_{j}^{†}]} . \end{matrix}

where

g_{j} = g_{0}^{(j)} α

are the effective optomechanical coupling constants.

In the rotating frame of cavity and mechanical frequencies, the Hamiltonian becomes

\begin{matrix} {\tilde{\hat{H}}}_{int} = - \sum_{j = 1}^{2} \sum_{n = - \infty}^{\infty} g_{j} i^{n} J_{n} (η_{j}) e^{i n (v_{j} t + φ_{j})} [{\hat{c}}^{†} {\hat{d}}_{j} e^{i (Δ - ω_{m}^{(j)}) t} \\ + {(- 1)}^{n} {\hat{c}}^{†} {\hat{d}}_{j}^{†} e^{i (Δ + ω_{m}^{(j)}) t}] + H . c . \end{matrix}

In order to entangle ${\hat{d}}_{1}$ and ${\hat{d}}_{2}$ , we should first create the parametric interaction to entangle ${\hat{d}}_{2}$ and $\hat{c}$ , then the beam-splitter interaction to swap $\hat{c}$ and ${\hat{d}}_{1}$ , and finally achieve the desired entanglement [38, 39]. To obtain the desired interactions, the laser field is red-detuned with $Δ = ω_{m}^{(1)}$ to generate the resonant beam-splitter interaction between $\hat{c}$ and ${\hat{d}}_{1}$ , and meanwhile the modulation frequency ν₁ should satisfy $n v_{1} \neq 2 ω_{m}^{(1)}$ to avoid the resonant parametric interaction. In addition, modulation frequency ν₂ satisfies $n v_{2} = ω_{m}^{(1)} + ω_{m}^{(2)}$ to generate the resonant parametric interaction between $\hat{c}$ and ${\hat{d}}_{2}$ , and $n v_{2} \neq ω_{m}^{(1)} - ω_{m}^{(2)}$ to avoid the resonant beam-splitter interaction. For example, in unit of $ω_{m}^{(1)}$ , $ω_{m}^{(2)} = 0.8 ω_{m}^{(1)}$ , $v_{1} = v_{2} = 0.9 ω_{m}^{(1)}$ can fulfill the requirement. Then via neglecting the non-resonant terms, the Hamiltonian of resonant interactions becomes

{\tilde{\hat{H}}}_{int} = - {\tilde{g}}_{1} ({\hat{c}}^{†} {\hat{d}}_{1} + {\hat{d}}_{1}^{†} \hat{c}) - {\tilde{g}}_{2} ({\hat{c}}^{†} {\hat{d}}_{2}^{†} + \hat{c} {\hat{d}}_{2}),

where

{\tilde{g}}_{1} = g_{1} J_{0} (η_{1})

,

{\tilde{g}}_{2} = g_{2} J_{2} (η_{2})

with the phase angle

φ_{2} = - \frac{π}{2}

.

From the Hamiltonian (28) and taking into account of cavity and mechanical dissipations, the Langevin equation becomes

\begin{array}{l} \frac{d}{d t} \hat{c} = - \frac{κ}{2} \hat{c} + i {\tilde{g}}_{1} {\hat{d}}_{1} + i {\tilde{g}}_{2} {\hat{d}}_{2}^{†} + \sqrt{κ} {\hat{c}}_{in}, \\ \frac{d}{d t} {\hat{d}}_{1} = - \frac{γ_{m}^{(1)}}{2} {\hat{d}}_{1} + i {\tilde{g}}_{1} \hat{c} + \sqrt{γ_{m}^{(1)}} {\hat{d}}_{1,in}, \\ \frac{d}{d t} {\hat{d}}_{2}^{†} = - \frac{γ_{m}^{(2)}}{2} {\hat{d}}_{2}^{†} - i {\tilde{g}}_{2} \hat{c} + \sqrt{γ_{m}^{(2)}} {\hat{d}}_{2,in}^{†}, \end{array}

where κ is the decay rate of optical cavity, and

γ_{m}^{(1)}

and

γ_{m}^{(2)}

are the dissipation rates of two mechanical oscillators. The noise operators fulfill the relations

〈 {\hat{c}}_{in} (t) {\hat{c}}_{in}^{†} (t^{'}) 〉 = δ (t - t^{'})

,

〈 {\hat{d}}_{1, in}^{†} (t) {\hat{d}}_{1,in} (t^{'}) 〉 = n_{th}^{(1)} δ (t - t^{'})

,

〈 {\hat{d}}_{2, in}^{†} (t) {\hat{d}}_{2,in} (t^{'}) 〉 = n_{th}^{(2)} δ (t - t^{'})

with the initial phonon occupations

n_{th}^{(1)}

and

n_{th}^{(2)}

. We consider the regime of bad cavity limit, i.e.,

κ ≫ {{\tilde{g}}_{1}, {\tilde{g}}_{2}}

, and the cavity mode can be adiabatically eliminated,

\hat{c} \approx \frac{i {\tilde{g}}_{1}}{κ / 2} {\hat{d}}_{1} + \frac{i {\tilde{g}}_{2}}{κ / 2} {\hat{d}}_{2}^{†} + \frac{2}{\sqrt{κ}} {\hat{c}}_{in} .

Via substituting the cavity mode into the equations of mechanical motion, we have

\begin{array}{l} \frac{d}{d t} {\hat{d}}_{1} = - (\frac{G_{1}}{2} + \frac{γ_{m}^{(1)}}{2}) {\hat{d}}_{1} - \frac{\sqrt{G_{1} G_{2}}}{2} {\hat{d}}_{2}^{†} + i \sqrt{G_{1}} {\hat{c}}_{in} + \sqrt{γ_{m}^{(1)}} {\hat{d}}_{1, in}, \\ \frac{d}{d t} {\hat{d}}_{2}^{†} = (\frac{G_{2}}{2} - \frac{γ_{m}^{(2)}}{2}) {\hat{d}}_{2}^{†} + \frac{\sqrt{G_{1} G_{2}}}{2} {\hat{d}}_{1} - i \sqrt{G_{2}} {\hat{c}}_{in} + \sqrt{γ_{m}^{(2)}} {\hat{d}}_{2, in}^{†}, \end{array}

with optically induced mechanical damping rate

G_{1} = 4 {\tilde{g}}_{1}^{2} / κ

and optically induced mechanical gain rate

G_{2} = 4 {\tilde{g}}_{2}^{2} / κ

for modes

{\hat{d}}_{1}

and

{\hat{d}}_{2}

. The mechanical gain rate will induce instability, and to analyze the stationary entanglement of the system, the parameters should fulfill the Routh-Hurwitz criteria [40]

\begin{array}{l} G_{1} > G_{2} - γ_{m}^{(2)} - γ_{m}^{(1)}, \\ G_{1} > G_{2} γ_{m}^{(1)} / γ_{m}^{(2)} - γ_{m}^{(1)} . \end{array}

We employ the Fourier transformation (14) to solve the motion equations, and have the solutions in the form

\begin{array}{l} {\hat{d}}_{1} (ω) = χ_{1 c} (ω) {\hat{c}}_{in} (ω) + χ_{11} (ω) {\hat{d}}_{1,in} (ω) + χ_{12} (ω) {\hat{d}}_{2, in}^{†} (ω), \\ {\hat{d}}_{2}^{†} (ω) = χ_{2 c} (ω) {\hat{c}}_{in} (ω) + χ_{21} (ω) {\hat{d}}_{1,in} (ω) + χ_{22} (ω) {\hat{d}}_{2, in}^{†} (ω), \end{array}

with the susceptibility coefficients

\begin{array}{l} χ_{1 c} (ω) = i \sqrt{G_{1}} (- i ω + γ_{m}^{(2)} / 2) / D (ω), \\ χ_{11} (ω) = \sqrt{γ_{m}^{(1)}} [- i ω - (G_{2} - γ_{m}^{(2)}) / 2] / D (ω), \\ χ_{12} (ω) = - \sqrt{γ_{m}^{(2)} G_{1} G_{2}} / 2 D (ω), \\ χ_{2 c} (ω) = - \sqrt{G_{2}} (i ω - γ_{m}^{(1)} / 2) / D (ω), \\ χ_{21} (ω) = \sqrt{γ_{m}^{(1)} G_{1} G_{2}} / 2 D (ω), \\ χ_{22} (ω) = \sqrt{γ_{m}^{(2)}} [- i ω + (G_{1} + γ_{m}^{(1)}) / 2] / D (ω), \\ D (ω) = [i ω - \frac{G_{1} + γ_{m}^{(1)}}{2}] [i ω + \frac{G_{2} - γ_{m}^{(2)}}{2}] + \frac{G_{1} G_{2}}{4} . \end{array}

Through utilizing the correlations of noise operators, we can achieve the phonon numbers and correlation in frequency domain, which are

\begin{array}{l} 〈 {\hat{d}}_{1}^{†} {\hat{d}}_{1} 〉 (ω) = {| χ_{11} (- ω) |}^{2} n_{th}^{(1)} + {| χ_{12} (- ω) |}^{2} (n_{th}^{(2)} + 1), \\ 〈 {\hat{d}}_{2}^{†} {\hat{d}}_{2} 〉 (ω) = {| χ_{21} (ω) |}^{2} (n_{th}^{(1)} + 1) + {| χ_{22} (ω) |}^{2} n_{th}^{(2)} + {| χ_{2 c} (ω) |}^{2}, \\ 〈 {\hat{d}}_{1} {\hat{d}}_{2} 〉 (ω) = χ_{11} (ω) χ_{21}^{*} (ω) (n_{th}^{(1)} + 1) + χ_{12} (ω) χ_{22}^{*} (ω) n_{th}^{(2)} + χ_{1 c} (ω) χ_{2 c}^{*} (ω) . \end{array}

The steady-state mean values of phonon numbers and correlations in the time domain is obtainable by integrating the frequency as Eq. (18). For simplification, we consider $γ_{m}^{(1)} = γ_{m}^{(2)} = γ m$ , and then after integration we have

\begin{array}{l} 〈 {\hat{d}}_{1}^{†} {\hat{d}}_{1} 〉 = \frac{[(G_{2} - γ_{m}) (G_{2} - 2 γ_{m}) + G_{1} γ_{m}] n_{th}^{(1)} + G_{1} G_{2} (n_{th}^{(2)} + 1)}{(G_{1} - G_{2} + γ_{m}) (G_{1} - G_{2} + 2 γ_{m})}, \\ 〈 {\hat{d}}_{2}^{†} {\hat{d}}_{2} 〉 = \frac{G_{1} G_{2} (n_{th}^{(1)} + 1) + [(G_{1} + γ_{m}) (G_{1} + 2 γ_{m}) - G_{2} γ_{m}] n_{th}^{(2)}}{(G_{1} + G_{2} + γ_{m}) (G_{1} - G_{2} + 2 γ_{m})} + \frac{G_{2}}{G_{1} - G_{2} + γ_{m}}, \\ 〈 {\hat{d}}_{1} {\hat{d}}_{2} 〉 = - \frac{\sqrt{G_{1} G_{2}} [(G_{2} - γ_{m}) n_{th}^{(1)} + (G_{1} + γ_{m}) (n_{th}^{(2)} + 1)]}{(G_{1} - G_{2} + γ_{m}) (G_{1} - G_{2} + 2 γ_{m})} . \end{array}

In the limit of G₂ → 0, the system becomes a typical cavity-assisted sideband cooling model for mode ${\hat{d}}_{1}$ [41, 42], while ${\hat{d}}_{2}$ is decoupled from the cavity field within a thermal state. Then the phonon numbers and correlation become

〈 {\hat{d}}_{1}^{†} {\hat{d}}_{1} 〉 = \frac{γ_{m} n_{th}^{(1)}}{G_{1} + γ_{m}}, 〈 {\hat{d}}_{2}^{†} {\hat{d}}_{2} 〉 = n_{th}^{(2)}, 〈 {\hat{d}}_{1} {\hat{d}}_{2} 〉 = 0.

The EPR-type entanglement of ${\hat{d}}_{1}$ and ${\hat{d}}_{2}$ is measurable by the degree of TMS [43], which is characterized by the variance

V = \frac{1}{2} [〈 {({\hat{X}}_{1} + {\hat{X}}_{2})}^{2} 〉 + 〈 {({\hat{P}}_{1} - {\hat{P}}_{2})}^{2} 〉],

with the amplitude and phase quadratures

{\hat{X}}_{j} = \frac{1}{\sqrt{2}} ({\hat{d}}_{j}^{†} + {\hat{d}}_{j}), {\hat{P}}_{j} = \frac{1}{\sqrt{2}} ({\hat{d}}_{j}^{†} - {\hat{d}}_{j}) .

By substituting the quadratures into Eq. (38) we have

V = 〈 {\hat{d}}_{1}^{†} {\hat{d}}_{1} 〉 + 〈 {\hat{d}}_{2}^{†} {\hat{d}}_{2} 〉 + 2 〈 {\hat{d}}_{1} {\hat{d}}_{2} 〉 + 1.

In Fig. 4 we plot the degree of TMS V with the change of relative ratio of G₂/G₁ with different initial phonon occupations. If V < 1, the variance is below the vacuum level, and mechanical modes ${\hat{d}}_{1}$ and ${\hat{d}}_{2}$ are entangled. Moreover, for stationary TMS generated by parametric down-conversion process, the minimum variance equals to 0.5, which is limited by stability requirement. In Fig. 4, the minimum value can approach 0.25, where 0.5 limit is surpassed and strong entanglement can be achieved. However, the variance still cannot reach the value of an ideal TMS state, in which the variance should approach zero and the mechanical modes become the perfect EPR-type entanglement. In the following we will discuss the cause of the limit value of variance obtained here.

Fig. 4 Entanglement indicated by degree of two-mode squeezing V versus controllable parameter G₂/G₁ with different initial thermal occupations. The other parameters are: γ_m = 2π × 0.18Hz, G₁ = 2π × 20kHz.

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The expression of V can be simplified for high-Q mechanical oscillators, where mechanical intrinsic damping rate is much smaller than the optically induced damping and gain rates, i.e., γ_m ≪ {G₁, G₂}, and we can obtain

V = \frac{G_{2} n_{th}^{(1)} + G_{1} n_{th}^{(2)} - \sqrt{G_{2}} (2 \sqrt{G_{1}} + \sqrt{G_{2}})}{{(\sqrt{G_{1}} + \sqrt{G_{2}})}^{2}} + 1.

First we suppose that mechanical oscillators are precooled to ground state, i.e., $n_{th}^{(1)} = 0$ , $n_{th}^{(2)} = 0$ , privileging a clearer focus on the main mechanism of entanglement generation via optomechanical interactions. The optically mediated coupling between two mechanical oscillators shown in Eq. (31) forms two Bogoliubov modes

\begin{array}{l} \hat{w} = \sqrt{\frac{G_{1}}{G_{1} - G_{2}}} {\hat{d}}_{1} + \sqrt{\frac{G_{2}}{G_{1} - G_{2}}} {\hat{d}}_{2}^{†}, \\ \hat{u} = \sqrt{\frac{G_{2}}{G_{1} - G_{2}}} {\hat{d}}_{1} + \sqrt{\frac{G_{1}}{G_{1} - G_{2}}} {\hat{d}}_{2}^{†}, \end{array}

where ŵ can be dissipatively cooled to ground state while û is a constant of motion (neglecting the mechanical dissipation) [44]. An ideal dissipatively generated entanglement lies on the cooling of both Bogoliubov modes [45,46]. But in our proposal, only one of Bogoliubov modes can be cooled down, and thus the minimum value of TMS can only reach 0.25 in the limit of G₂ → G₁. It is insufficient to create a TMS vacuum, in which V can approach zero, and actually the final state is a two-mode squeezed thermal state [38].

Moreover, Eq. (41) shows that the initial mechanical noise brings significant influence on the entanglement. It is because that even the mechanical noise of ŵ is cooled down which can suppress its influence on the entanglement, but the mechanical noise of û is unaffected which would still decrease the entanglement. Therefore, mechanical entanglement is sensitive to the initial thermal noise, and to achieve a better entanglement, mechanical precooling is necessary.

It should be noted that different from the experimentally demonstrated steady-state squeezing of a micrometer-scale mechanical resonator via implementing reservoir-engineering scheme with two-tone pumps [19], the frequency-modulation method is employed here that may possess the potential in manipulating multiple resonators (more than two). Since the coupling of each mechanical resonator to a common cavity field is independently tunable via parametrically modulating the mechanical frequency and easy to extend, and the method in superconducting system has experimentally realized a perfect quantum state transfer in a chain of four coupled qubits [47].

4. Conclusion

To conclude, we have proposed an protocol for generating mechanical squeezing and entanglement in the modulated two- and three-mode optomechanics. Via appropriately choosing the laser detuning and modulation frequency on mechanical motion, both the beam-splitter and parametric interactions between the cavity and mechanical modes can be simultaneously achieved, which is crucial to squeezing and entanglement based on reservoir engineering. In the two-mode modulated optomechanics, the degree of squeezing is determined by the ratio of parametric and beam-splitter interaction strengths and also robust to thermal noise due to the simultaneously optically induced cooling. Strong squeezing beyond 3dB limit is attainable. In the three-mode optomechanics, strong EPR-type entanglement is also attainable, while the ideal entanglement is impossible since only one of mechanical Bogoliubov modes is cooled by the cavity mode, which also makes the entanglement sensitive to the mechanical noise. Moreover, our results can be extended to levitated nanoparticle systems.

Funding

National Natural Science Foundation of China (11504031, 61505014, 11704045); Yangtze Youth Talents Fund; Yangtze Funds for Youth Teams of Science and Technology Innovation (2015cqt03).

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Generation of mechanical squeezing and entanglement via mechanical modulations

Abstract

1. Introduction

2. Generation of mechanical squeezing in modulated two-mode optomechanics

3. Generation of mechanical entanglement in modulated three-mode optomechanics

4. Conclusion

Funding

References

Cited By

Figures (4)

Equations (42)

Optics Express