Ground-state cooling of mechanical oscillator via quadratic optomechanical coupling with two coupled optical cavities

Jun-Ya Yang; Dong-Yang Wang; Cheng-Hua Bai; Si-Yu Guan; Xiao-Yuan Gao; Ai-Dong Zhu; Hong-Fu Wang

doi:10.1364/OE.27.022855

1. Introduction

Cavity optomechanics as a crucial branch of physics which focuses on the interaction of controllable radiation pressure between light and mechanical motion has aroused wide interest in recent years [1, 2]. Cooling the mechanical resonators to the quantum ground-state provides a crucial application prospect in cavity optomechanics [3, 4], and has been widely used to quantum information processing [5–10], high-precision control and measurement [11–14], and quantum-classical boundary probe of quantum behaviour of macroscopic system [15–17], etc. So far, various methods have been used for cooling a mechanical resonator to its ground-state, such as pure cryogenic cooling [18], feedback cooling [19, 20],and sideband cooling [21–26]. Among them, the sideband cooling is the most common method which require the condition of resolved sideband limit to be fulfilled, i.e. the decay rate of the cavity mode is much less than the mechanical frequency. This is a strict restriction since a high quality factor is needed for the cavity and it is hard to implement in most physical systems. Once the condition of sideband cooling cannot be satisfied by the cavity mode, the Stokes heating cannot be well suppressed, which will result in a failure in cooling. In contrast, the restriction for implementing ground-state cooling is relaxed in the unresolved sideband regime which can be achievable with a bad cavity. Recently some approaches have been proposed such as the dissipative coupling mechanism [27–31], optomechanically induced transparency [32, 33], atom-optomechanical systems [34–36], and coupled-cavity configurations [37–39], etc. These schemes are promising to be realized in unresolved sideband conditions where the decay rate of cavity mode would be larger than the mechanical frequency.

In most optomechanical theoretical and experimental studies, researchers mainly focused on the linear coupling optomechanical system, where the optical cavity mode is parametrically coupled to the position of a mechanical oscillator. However, some studieshave started to focus on the nonlinear mechanical cooling process, for instance, the cooling in high optomechanical coupling regime of the single-photon [40] and the secondary sideband laser cooling of trapped ions [41]. This leads to nonthermal steady state, even nonclassical sub-Poissonian state. Quantum systems far from thermal equilibrium have great prospects for fundamental physics research and the realization of practical quantum devices [42, 43]. At present, a neotype of optomechanical system, quadratic optomechanics (also known as “quadratic nonlinearity”) has been proposed, where the coupling term is proportional to the square of mechanical displacement x². It’s realizable in the setups of membrane-in-the-middle geometry [44, 45], which is placed at a node or antinode of the cavity field as well as the ultracold atom cloud loaded into the optical cavity, the centroid coordinate of the cloud is regarded as the mechanical degree of freedom [46, 47]. Up to now, theoretical literature has focused on the detection of phonon Fock states by using quadratic coupling, as well as the cooling and squeezing of mechanical oscillators [48–50]. At the same time, electromagnetically induced transparency (EIT) or optomechanically induced transparency (OMIT) have also been investigated in quadratic coupled optomechanical systems, which means two phonon processes [51, 52]. The development of experimental technology has made it become more and more attractive to investigate the nonlinear cooling process caused by intrinsic quadratic coupling. However, the main limitation in experiment comes from the weak quadratic coupling rate when quadratic coupling is used. Hence, many efforts have been devotedto improve the nonlinear coupling strength in the optomechanical system [53]. Enlightened by theses works, here we propose an EIT-like ground-state cooling scheme for the mechanical oscillator in a coupled-cavity optomechanical system with weak quadratic coupling. In our scheme, the first cavity is coupled parametrically to the square of the position of a mechanical oscillator via the radiation pressure force. We demonstrate that the cooling of mechanical oscillator can be achieved even if the optomechanical cavity decay is in unresolved sideband regime due to the coherent coupling to the auxiliary optical cavity mode. By analyzing the optical force spectrum and deriving the optimal cooling conditions, the EIT-like spectrum splits from the single Lorentzian peak of the standard optomechanical cavity into two narrower peaks with a dip emerging between them, which leads to more efficient cooling for a weak optomechanical coupling strength. Compared with the previous works involving single-cavity quadratic optomechanical coupling [54, 55], an auxiliary cavity is included in our present system which brings out the quantum interference effect [56], thus is more efficient and the mechanical mode can be cooled to a lower mean phonon number even when the resolved sideband condition is not fulfilled. In addition, different from the linear optomechanical coupling scheme with double cavities [37], our present scheme with quadratic coupling can cool the oscillator to its ground state with a faster velocity due to the two-phonon absorption. The cooling is achievable even with a relatively weak quadratic coupling strength by tunning the coupling between two cavities to reach the optimum cooling conditions, thus provides an useful way to overcome the limitations of weak quadratic coupling rate in experiments.

This paper is organized as follows. In Sec. 2, the quadratic optomechanical system is introduced and the effective Hamiltonian is given. In Sec. 3, the rate equation for the phonon of the mechanical oscillator is given and the properties of optical force spectrum are derived. In Sec. 4, the optimum cooling conditions are obtained by parameters modulation, and the cooling limits in weak quadratic coupling condition are discussed. In Sec. 5, cooling dynamics of the system is investigated by using master equation and numerical simulations. In Sec. 6, we give some discussions on experimental feasibility. Finally, a brief summary of our work is presented in Sec. 7.

Fig. 1 Schematic of the optomechanical system. A membrane oscillator is placed in the middle of cavity 1 which is driven by a continuous-wave input laser, the coupling between them is proportional to the square of position of the oscillator. The cavity2 as an auxiliary cavity only directly couples to the optomechanical cavity 1 with coupling strength J.

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2. Model and Hamiltonian

We consider a optomechanical system including double coupled cavities as shown in Fig. 1, where the optomechanical cavity is coupled parametrically to the square of the position of a mechanical oscillator and driven by an continuous-wave input laser, the second cavity as an auxiliary cavity couples directly to the first one with the coupling strength $J$ . The total system Hamiltonian can be written as ( $ℏ = 1$ )

H = ω_{1} a_{1}^{†} a_{1} + ω_{2} a_{2}^{†} a_{2} + ω_{m} b^{†} b + g a_{1}^{†} a_{1} {(b^{†} + b)}^{2} + Ω (a_{1}^{†} e^{- i ω_{L} t} + a_{1} e^{i ω_{L} t}) + J (a_{1}^{†} a_{2} + a_{1} a_{2}^{†}),

in which a₁, a₂ and b denote the annihilation operators for the two optical cavity modes and the mechanical oscillator, respectively, with the frequencies ω₁, ω₂ and ω_m; g is the quadratic optomechanical coupling coefficient; ω_L is the frequency of the input laser and Ω is the pumping strength. In the frame rotating at the input laser frequency ω_L, the system Hamiltonian becomes

H^{'} = Δ_{1} a_{1}^{†} a_{1} + Δ_{2} a_{2}^{†} a_{2} + ω_{m} b^{†} b + g a_{1}^{†} a_{1} {(b^{†} + b)}^{2} + Ω (a_{1}^{†} + a_{1}) + J (a_{1}^{†} a_{2} + a_{1} a_{2}^{†}),

where

Δ_{i} = ω_{i} - ω_{L}, (i = 1, 2)

are detunnings between the laser and two cavities. To obtain the linearized Hamiltonian the three modes are split into an average amplitude and a fluctuation term, i.e.,

a_{1} \to α_{1} + a_{1}

,

a_{2} \to α_{2} + a_{2}

and

b \to β + b

, where α₁, α₂ and β denote the steady-state displacements of the optical and mechanical modes which can be calculated as

\begin{array}{l} α_{1} = \frac{i Ω}{(- i Δ_{1} - \frac{κ_{1}}{2}) + \frac{J^{2}}{- i Δ_{2} - \frac{κ_{2}}{2}}}, \\ α_{2} = \frac{i J α_{1}}{- i Δ_{2} - \frac{κ_{2}}{2}}, \\ β = 0. \end{array}

After the standard linearization procedure, the effective linearized Hamiltonian can be expressed as

H_{eff} = Δ_{1} a_{1}^{†} a_{1} + Δ_{2} a_{2}^{†} a_{2} + ω_{m} b^{†} b + (G a_{1}^{†} + G^{*} a_{1}) {(b^{†} + b)}^{2} + J (a_{1}^{†} a_{2} + a_{1} a_{2}^{†}),

where

G = α_{1} g

is the cavity-enhanced optomechanical coupling strength.

The energy levels of the system are presented in Fig. 2. Due to the coupling between two cavities, quantum interference may take place among different excitation pathways. For example, quantum interference occurs between the excitation paths $| 1 〉 \to | 2^{'} 〉$ and $| 1 〉 \to | 2^{'} 〉 \to | 3^{'} 〉 \to | 3^{'} 〉$ which is corresponding to the cooling process for the mechanical mode. On the other side, that occurs between the paths $| 1 〉 \to | 2 〉$ and $| 1 〉 \to | 2 〉 \to | 3 〉 \to | 3 〉$ which is corresponding to the heating process for the mechanical mode and should be suppressed in order to get a efficient cooling. Next we are going to harness the quantum interference to suppress the heating process and enhance the cooling process, which can be achievable effectively in the current scheme even in the unresolved sideband regime $κ_{1} > ω_{m}$ .

Fig. 2 Energy level diagram of the system in the displaced frame. $| n_{1}, n_{2}, m 〉$ donates the state of n₁ photons in mode a₁, n₂ photons in mode a₂, and m phonons in mode b. The orange double arrow denotes the couplingbetween two cavities with coupling strength J.

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3. Optical force spectrum and quantum rate equation

In order to show the cooling regime, we calculate the optical force spectrum. In the weak-coupling regime, the optical force spectrum can be analyzed by using perturbation theory. Regarding the optomechanical coupling as perturbation to the optical field, the optical force spectrum $S_{FF} (ω) = \int_{- \infty}^{\infty} 〈 F^{†} (ω) F (ω^{'}) 〉 d ω^{'}$ can be calculated from the optical part of the linearized Hamiltonian Eq. (4) without coupling to the mechanical oscillator

H_{op} = Δ_{1} a_{1}^{†} a_{1} + Δ_{2} a_{2}^{†} a_{2} + J (a_{1}^{†} a_{2} + a_{1} a_{2}^{†}),

and the optical force is described by

F = (G^{*} a_{1} + G a_{1}^{†}) / x_{ZPF}^{2}

from the Eq. (4) with x_ZPF denoting the zero-point fluctuation of the oscillator. Then

S_{FF} (ω)

can be obtained from the corresponding quantum Langevin equations in the frequency domain

\begin{array}{l} - i ω a_{1} (ω) = (- i Δ_{1} - \frac{κ_{1}}{2}) a_{1} (ω) - i J a_{2} (ω) - \sqrt{κ_{1}} a_{in, 1} (ω), \\ - i ω a_{2} (ω) = (- i Δ_{2} - \frac{κ_{2}}{2}) a_{2} (ω) - i J a_{1} (ω) - \sqrt{κ_{2}} a_{in, 2} (ω), \end{array}

where the corresponding noise operators

a_{in, 1}

,

a_{in, 2}

satisfy the correlation in time domain

〈 a_{in, 1} (t) a_{in, 1}^{†} (t^{'}) 〉 = 〈 a_{in, 2} (t) a_{in, 2}^{†} (t^{'}) 〉 = δ (t - t^{'})

. Calculating from the Eq. (6) we obtain

a_{1} (ω) = \frac{\sqrt{κ_{1}} a_{in, 1} (ω) - i J χ_{2} (ω) κ_{2} a_{in, 2} (ω)}{χ (ω)},

with

\begin{array}{l} χ_{1} (ω) = \frac{1}{κ_{1} / 2 - i (ω - Δ_{1})}, \\ χ_{2} (ω) = \frac{1}{κ_{2} / 2 - i (ω - Δ_{2})}, \\ χ (ω) = \frac{1}{χ_{1} (ω)} + J^{2} χ_{2} (ω) . \end{array}

As a result, we have

S_{FF} (ω) = \frac{| G |^{2}}{x_{ZPF}^{4}} [\frac{1}{χ (ω)} + \frac{1}{χ^{*} (ω)}] .

The two-phonon cooling and amplification rates can be obtained from Fermi’s Golden rule by using the methods given in Ref. [57]. Concentrating on the diagonal terms of the density matrix $ρ_{nn} = P_{n}$ , we write down a set of rate equations

\begin{matrix} \dot{P_{n}} = - γ_{m} [n_{th} (n + 1) + (n_{th} + 1) n] P_{n} \\ + γ_{m} n_{th} n P_{n - 1} + γ_{m} (n_{th} + 1) (n + 1) P_{n + 1} \\ - [A_{-} n (n - 1) + A_{+} (n + 2) (n + 1)] P_{n} \\ + A_{-} (n + 2) (n + 1) P_{n + 2} + A_{+} n (n - 1) P_{n - 2}, \end{matrix}

where P_n is the phonon number distribution in the n phonon state. The terms in the first two lines are due to the coupling of the mechanical oscillator to its thermal bath with rate γ_m and thermal phonon number n_th. The last two lines describe the transition effects induced by the effective optomechanical coupling, where

A_{\mp} = x_{ZPF}^{4} S_{FF} (\pm 2 ω_{m})

stand for the cooling rate and heating rate for the mechanical oscillator. Then the net cooling rate is given as

Γ_{opt} = A_{-} - A_{+}

.

4. Cooling manipulation and cooling limits

4.1. Cooling improvement by parameter modulation

In order to obtain the optimal cooling effect, it is necessary to get the optimum optical coupling strength between the two optical cavities as well as the effective optical detunings.

For the case of single cavity (i.e. J = 0), the optical force spectrum $S_{FF} (ω)$ has a Lorentzian noise spectrum with single peak localed at $ω = Δ_{1}$ . According to the sideband cooling regime, it is necessary to meet the condition of $κ_{1} < ω_{m}$ to realize ground-state cooling in the resolved sideband. In the unresolved sideband regime ( $κ_{1} > ω_{m}$ ), the peaks of the cooling spectrum $S_{FF} (2 ω_{m})$ and heating spectrum $S_{FF} (- 2 o m e g a_{m})$ are comparable, thus leads to the ground-state cooling for the mechanical oscillator unachievable. However, in our present scheme, the optical force spectrum $S_{FF} (ω)$ becomes a complex line shape due to the coupling between the two optical modes. We focus on the the double-cavity optomechanical system in the unresolved sideband regime ( $κ_{1} > ω_{m}$ ).

Fig. 3 (a) The optical force spectrum S_FF (ω) (in arbitrary units) for different optical coupling strengths J, with Δ₁ = 2ω_m, κ₂ = 0.01ω_m. (b) The optical force spectrum S_FF (ω) (in arbitrary units) for different detunings of the second cavity κ₂ in the optimum conditions with Δ₁ = −4ω_m, $J = 2 \sqrt{6} ω_{m}$ . The other parameters are taken as Δ₂ = −2ω_m, κ₁ = 10ω_m, G = 0.08ω_m.

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In Fig. 3(a), the optical force spectrum $S_{FF} (ω)$ is plotted versus the frequency ω with different optical coupling strength J. It can be seen that the single Lorentzian peak splits into two peaks with a dip between them due to the coupling between two optical cavities. The dip appears similar to the two-photon resonance in the electromagnetically induced transparency (EIT) of a three-level system [58]. Obviously, the minimal value of the optical force spectrum $S_{FF} (ω)$ locates at $ω = Δ_{2} = - 2 ω_{m}$ , corresponding to the two-photon resonance condition in EIT. In this case, the phenomenon of quantum destructive interference happens, thereby the heating process of mechanical oscillator is suppressed effectively. Therefore in order to locate the minimum value of the optical force spectrum $S_{FF} (ω)$ at $ω = - 2 ω_{m}$ , the corresponding optimal condition is taken as $Δ_{2} = - 2 ω_{m}$ . In the meantime, it can be seen from the Fig. 3(a) that the two peaks of the optical force spectrum move with different optical coupling strength. In order to achieve the maximum cooling efficiency, it is necessary to locate the right-hand peak at $ω = 2 ω_{m}$ corresponding to the cooling process.

Next, we investigate the relation between the detuning Δ₁ and the optical coupling strength J to implement the optimal cooling effect. In fact, the two split peaks of the optical force spectrum origin from the normal mode splitting owing to the coupling between the two cavities. So the optimal coupling strength can be ascertained by the eigen-energies of the dressed states of Hamiltonian H_op in Eq. (5),

\begin{array}{l} E_{+} = \frac{1}{2} (Δ_{1} + Δ_{2} + \sqrt{{(Δ_{1} + Δ_{2})}^{2} + 4 J^{2}}), \\ E_{-} = \frac{1}{2} (Δ_{1} + Δ_{2} - \sqrt{{(Δ_{1} + Δ_{2})}^{2} + 4 J^{2}}) . \end{array}

Here, $E_{+}$ is corresponding to the right-hand peak of the optical force spectrum. In order to obtain the optimal cooling effect, the relation $E_{+} = 2 ω_{m}$ should be satisfied. Then the optimal coupling strength can be obtained as

J = \sqrt{4 ω_{m} (2 ω_{m} - Δ_{1})} .

With this optimal coupling strength in the Eq. (12) and $Δ_{2} = - 2 ω_{m}$ , in Fig. 3(b), the optical force spectrum $S_{FF} (ω)$ is illustrated under different decay rate κ₂ of the auxiliary cavity. Here the effective detuning of the first cavity has been selected as $Δ_{1} = - 4 ω_{m}$ , which means the corresponding optimal coupling strength $J = 2 \sqrt{6} ω_{m}$ .

Fig. 4 The net cooling rate Γ_opt = A₋A₊ versus the dacay rate of the second cavity κ₂ and the coupling strength J. Here the detuning and the decay rate of the first cavity are taken as Δ₁ = −4ω_m, κ₁ = 10ω_m, the detuning of the second cavity is taken as Δ₂ = −2ω_m, and the optomechanical coupling strength is taken as G = 0.08ω_m.

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It’s worth to be noticed that even if the above optimal conditions are satisfied, i.e., $Δ_{2} = - 2 ω_{m}$ corresponding to a minimal heating effect and $E_{+} = 2 ω_{m}$ corresponds to a maximal cooling effect, we still have to control the decay rate κ₂ appropriately which directly determines the depth of the dip, in order to limit the minimal value of the optical force spectrum to be as close as possible to zero. Then the optimal ground cooling can be implemented. For this purpose the net cooling rate $Γ_{opt} = A_{-} - A_{+}$ versus the decay rate of the auxiliary cavity κ₂ and the coupling strength J is plotted in Fig. 4. As can be seen from that, the net cooling rate increases with decreasing of the decay rate κ₂ of the auxiliary cavity. That is because a smaller decay rate κ₂ leads to a lower minimum of the dip in the optical force spectrum, even close to zero. In this way, the heating process can be significantly suppressed. According to the above conditions, when $Δ_{1} = - 4 ω_{m}$ , the position of the maximum cooling rate in the Fig. 4 corresponds to the proximity of $J = 2 \sqrt{6} ω_{m}$ , which conforms to the optimum conditions we deduced. Compared with the case of single cavity, i.e. J = 0, the net cooling rate is obviously much lower than that of our scheme.

Fig. 5 A comparison of the net cooling rates $Γ_{opt} = A_{-} - A_{+}$ between the current coupled-cavity optomechanical system and the single-cavity optomechanical system. The former is plotted as the green circle curve under the optimum coupling conditions in Eq. (12) versus the detuning of the auxiliary cavity Δ₂ for $Δ_{1} = - 4 ω_{m}$ , $κ_{1} = 30 ω_{m}$ , $κ_{2} = 0.01 ω_{m}$ , and the latter is plotted as the blue triangle curve versus Δ₁ for $κ = 30 ω_{m}$ . The effective optomechanical coupled strength $G = 0.08 ω_{m}$ is taken.

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In order to illustrate the effect the auxiliary cavity, a comparison of the net cooling rates between the current system and the single optomechanical system is shown in the Fig. 5. The net cooling rate $Γ_{opt}$ of versus the detuning of the auxiliary cavity Δ₂ in our double-cavity system is plotted with the green circle line in the optimum conditions with $Δ_{1} = - 4 ω_{m}$ selected. In this case, the maximum cooling rate can be achieved by modulating the detunning of the auxiliary cavity. Compared with the net cooling rate for the single-cavity optomechanical system (plotted with the blue triangle line), it can be clearly seen that the optimal net cooling rate in the coupled cavity system is much higher than that of single cavity system. In our coupled cavity system, the minimum and maximum net cooling rate is located at the point $Δ_{2} = \pm 2 ω_{m}$ , while in the case of single cavity, the peak of net cooling rate moves and becomes smooth under the unresolved sideband regime, thus the mechanical oscillator can hardly be cooled.

4.2. Cooling in weak quadratical coupling limits

Here we’re going to discuss the cooling limits of the quadratic optomechanical coupling. The infinite set of rate equations Eq. (10) can be replaced by a single differential equation for the generating function $F (z, t) = \sum_{n = 0}^{\infty} P_{n} (t) z^{n}$ [59]. By solving the rate equations in the limit of strong optical damping and small thermal temperature, i.e. $γ_{m} n_{th} ≪ A_{\pm}$ , only pure two-phonon transitions be obtained, we have

P_{2 n + j}^{(2)} = (1 - m) m^{n} (γ + j - 1) {(- 1)}^{j - 1}, j = 0, 1,

where the additional parameter γ is determined by the initial conditions, characterizing a relative weight of the odd oscillator phonon states,

m = A_{+} / A_{-}

. When the above optimum conditions are satisfied, which will lead to

A_{-} ≫ A_{+}

, i.e.

m ≪ 1

. This means that the distribution of phonon number is only concentrated in zero-phonon state

P_{0} = 1 - γ

and one-phonon state

P_{1} = γ

. This indicates that the cooling process of two-phonon maintains the parity of phonon number, making the initial odd phonon state cool to one-phonon state and even phonon state cool to zero-phonon state.

By using the optimal conditions Eq. (12), in the strong two-phonon absorption condition $A_{-} ≫ γ_{m} n_{th} ≫ A_{+}$ , we obtain [59]

\begin{array}{l} P_{0} = \frac{1 + 2 ξ}{1 + 3 ξ} + O (\frac{γ_{m} n_{t h}}{A_{-}}), \\ P_{1} = \frac{ξ}{1 + 3 ξ} + O (\frac{γ_{m} n_{t h}}{A_{-}}), ξ = \frac{n_{t h}}{n_{t h} + 1}, \end{array}

with all other phonon number distribution P_n = 0 and the minimal mean phonon number

〈 \hat{n} 〉 = 1 / (4 + 1 / n_{th})

.

5. Quantum master equation and numerical simulation

The quantum master equation of the system reads

\begin{matrix} \dot{ρ} = - i [H_{eff}, ρ] + \frac{κ_{1}}{2} (2 a_{1} ρ a_{1}^{†} - a_{1}^{†} a_{1} ρ - ρ a_{1}^{†} a_{1}) \\ + \frac{κ_{2}}{2} (2 a_{2} ρ a_{2}^{†} - a_{2}^{†} a_{2} ρ - ρ a_{2}^{†} a_{2}) \\ + \frac{γ_{m}}{2} (n_{th} + 1) (2 b ρ b^{†} - b^{†} b ρ - ρ b^{†} b) \\ + \frac{γ_{m}}{2} n_{th} (2 b^{†} ρ b - b b^{†} ρ - ρ b b^{†}), \end{matrix}

here, H_eff is the linearized effective system Hamiltonian given by Eq. (4).

n_{th} = {(exp (ℏ ω_{m} / k_{B} T) - 1)}^{- 1}

corresponds to the bath thermal phonon number at the environmental temperature T.

Fig. 6 (a) The average phonon number $〈 b^{†} b 〉$ of the steady state versus the detuning of the second cavity Δ₂ and the optomechanical coupling strength G. The detuning of first cavity Δ₁ and the coupling strength between two cavities J meet the optimum condition in Eq. (12). (b) The average phonon number $〈 b^{†} b 〉$ versus the effective detuning of the first cavity Δ₁ and the coupling strength J. Here the effective detuning of the second cavity $Δ_{2} = - 2 ω_{m}$ is taken. The white dotted curve satisfies the Eq. (12). The other parameters are taken as $κ_{1} = 10 ω_{m}$ , $κ_{2} = 0.01 ω_{m}$ , $γ_{m} = 10^{- 6} ω_{m}$ , $G = 0.08 ω_{m}$ and $n_{th} = 10$ .

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By employing the master equation, the numerical simulation for the average phonon number n_f of the steady state versus the detunings of second optical cavities Δ₂ and the optomechanical coupling strength G are shown in Fig. 6(a),where Δ₁ satisfies the relation of optimal coupling strength in the Eq. (12). It can be found that the average phonon number $〈 b^{†} b 〉$ is far less than unity for $Δ_{2} = - 2 ω_{m}$ , i.e. we can obtain the optimal cooling effect. In addition, it can be seen from the figure that the optomechanical coupling strength satisfies the weak coupling mechanism, i.e. $G ≪ κ_{1}$ . When Δ₂ satisfies the optimum condition, the optomechanical coupling strength also has an optimum range, that is, $G / ω_{m} < 0.1$ . To further verify the above optimum condition, in the density plot Fig. 6(b), the average phonon number $〈 b^{†} b 〉$ is shown as a function of effective detuning of the first cavity Δ₁ and the coupling strength J. The white dotted line in this figure represents the derived optimum condition in the Eq. (12). Obviously, the optimal cooling effect obtained by numerical simulation always meet with this optimum cooling line.

Fig. 7 A comparison of the cooling dynamics between the current coupled-cavity optomechanical system and the single-cavity optomechanical system. The numerical results of the average phonon number $〈 b^{†} b 〉$ for the coupled-cavity system are plotted with the magenta diamond and green square curves, corresponding to the initial even and odd phonon numbers $n_{th} = 10$ and 11, respectively, with $Δ_{1} = - 4 ω_{m}$ , $Δ_{2} = - 2 ω_{m}$ , $J = 2 \sqrt{6} ω_{m}$ . The numerical results for single-cavity optomechanical system are plotted with orange circle and blue triangle curves corresponding to initial even and odd phonon numbers $n_{th} = 10$ and 11, respectively, with $Δ_{1} = 2 ω_{m}$ . The shadow area denotes the optimal cooling region for $〈 b^{†} b 〉 < 1$ .

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For the high-Q mechanical oscillator and lower initial temperature, we can ignore the effects of thermal damping, and the pure two-phonon transitions occur. Therefore, based on the characteristics of the two-phonon transitions, the quadratic coupling scheme proposed here has a faster cooling rate than the linear coupled double-cavity system [37], which comes from the nature of the quadratic coupling. In Fig. 7, the time evolutions of the final average phonon number $〈 b^{†} b 〉$ in the cases of double-cavity and of single-cavity are numerically simulated, respectively. As mentioned above, the final mean phonon number distribution is concentrated in one-phonon state or zero-phonon state. As shown in Fig. 7, the two lines below represent our coupled-cavity system, in which the magenta diamond line represents the case of initial even phonon number which finally can be cooled to near 0; similarly, the green square line represents the case of initial odd phonon number which finally can be cooled to near 1. The upper two lines represent the situation of single-cavity optomechanical system [54, 55], from which we can see that the ground-state cooling can not be achieved in unresolved sideband regime. Obviously, the cooling effect in the current coupled-cavity system is significantly improved compared to that in single-cavity system due to the interference effect caused by the participation of the auxiliary cavity, which suppresses the Stokes heating processes in unresolved sideband regime and weak coupling mechanism. This will be benefit to the practical experiments in weak coupling limit even beyond the resolved sideband regime.

Fig. 8 Exact numerical result of the average phonon number $〈 b^{†} b 〉$ in quadratic optomechanical system for different initial thermal phonon number $n_{th} = 10$ (green pentagrams), $n_{th} = 20$ (magenta circles), $n_{th} = 30$ (blue triangles). The other parameters are taken as $Δ_{1} = - 4 ω_{m}$ , $Δ_{2} = - 2 ω_{m}$ , $κ_{1} = 10 ω_{m}$ , $κ_{2} = 0.01 ω_{m}$ , $γ_{m} = 10^{- 6} ω_{m}$ , $J = 2 \sqrt{6} ω_{m}$ and $G = 0.08 ω_{m}$ . The shadow area denotes the optimal cooling region for $〈 b^{†} b 〉 < 1$ .

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In Fig. 8, the cooling dynamics under different temperatures are plotted with blue triangles, magenta circles and green stars standing for different thermal phonon number $n_{th} = 10, 20, 30$ . It can be seen from the figure that our current quadratic cooling scheme is robust to the bath noise under the appropriate temperature. We also notice in most cases, that the number of phonons is actually arbitrary in practical experiments. It is very difficult for people to really control phonons to integers. Assuming the initial thermal phonon number is $n_{th} = 2 k + k_{0}$ or $n_{th} = (2 k + 1) + k_{0}$ with k being an integer and $0 < k_{0} < 1$ , the corresponding final average phonon number will be cooled to k₀ or $k_{0} + 1$ , respectively. So in order to enhance the cooling efficiency as much as possible, cryogenic precooling is still required to reduce the environmental temperature.

6. Experimental feasibility

The present cooling scheme can be implemented with different kinds of optomechanical cavities. For instance, a membrane-in-the middle setup which formed by a high-finesse Fabry-Perot cavity with a thin semi-transparent SiN membrane inside can be used like described in the Ref. [60]. They demonstrated that the position and orientation of the membrane in the cavity can be used to fine-tune the frequency of the cavity modes, and make these modes couple with the vibration modes of the membrane. In most membrane positions, the frequency shifts of the cavity mode is linearly related to the membrane deformation, so there is a traditional radiation pressure coupling between the optical and mechanical modes. However, at the nodes and antinodes of the cavity field, the linear term disappears, and one has a dispersive interaction, which is quadratic in the position operator of the mechanical mode.

In existing experiments with the membrane-in-the-middle geometry work in the resolved sideband limit at large thermal phonon number, the optomechanical coupling is small compared to the cavity linewidth $G / κ = 10^{- 5}$ [61] for the parameters $κ = 105 Hz$ , $ω_{m} = 106 Hz$ , $γ = 0.1 Hz$ , and $n_{th} = 10^{7}$ at $T = 300 K$ . Therefore, many experiments are devoted to improve the optomechanical coupling strength. Experiments with ultracold atoms in optical resonators have already realized quadratic optomechanical coupling $G / κ \approx 1$ at small thermal phonon number n_th [62]. In single cavity optomechanical system, ground-state cooling is achieved for the coupling strength within the range of $0.5 \leq G / κ \leq 0.8$ [54]. In our current system, the cavity decay rate and the optomechanical coupling strength are taken as $κ_{1} = 30 ω_{m}$ (or $10 ω_{m})$ and $G = 0.08 ω_{m}$ for numerical simulations, i.e., $G / κ_{1} \approx 10^{- 3}$ , which is achievable based on the above experimental parameters. Meanwhile, it is noticeable that the ground-state cooling can be realized in our current scheme without the request for a large quadratic coupling strength due to the participation of the second cavity.

7. Conclusions

In conclusions, we have proposed a EIT-like nonlinear ground-state cooling scheme for the mechanical resonator with weak quadratic optomechanical coupling in unresolved sideband regime. The optical force spectrum splits into a heating peak and a cooling peak from original single Lorentz peak due to the auxiliary cavity introduced. Considering the normal mode splitting, the optimum cooling condition can be obtained. Based on the destructive interference between different transitions and the optimum cooling conditions, the heating process is well suppressed, and the mechanical resonator can be cooled to near its ground state even in unresolved sideband regime. It is demonstrated by the numerical simulations that in our current scheme the mechanical oscillator can be cooled to its ground state with a faster rate than in the linear-coupling system due to the two-phonon absorption. Moreover, given the condition of quadratic optomechanical coupling, the cooling effect is greatly improved compared to the single-cavity system due to contribution of the second auxiliary cavity. By tunning the coupling between two cavities to fulfill the optimum cooling conditions, the ground-state cooling can be achieved even with a weak quadratic coupling strength, thus providesan solution for overcoming the limitations of weak quadratic coupling rate in experiments. The scheme is feasible and promising to be used in quantum manipulation of macroscopic mechanical devices beyond the resolved sideband limit and linear coupling regime.

Funding

National Natural Science Foundation of China under Grant 11564041, 61822114.

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Ground-state cooling of mechanical oscillator via quadratic optomechanical coupling with two coupled optical cavities

Abstract

1. Introduction

2. Model and Hamiltonian

3. Optical force spectrum and quantum rate equation

4. Cooling manipulation and cooling limits

4.1. Cooling improvement by parameter modulation

4.2. Cooling in weak quadratical coupling limits

5. Quantum master equation and numerical simulation

6. Experimental feasibility

7. Conclusions

Funding

References

Cited By

Figures (8)

Equations (15)

Optics Express