Squeezing Light via Levitated Cavity Optomechanics

Abstract

Squeezing light is a critical resource in both fundamental physics and precision measurement. Squeezing light has been generated through optical-parametric amplification inside an optical resonator. However, preparing the squeezing light in an optomechanical system is still a challenge for the thermal noise inevitably coupled to the system. We consider an optically levitated nano-particle in a bichromatic cavity, in which two cavity modes could be excited by the scattering photons of the dual tweezers, respectively. Based on the coherent scattering mechanism, the ultra-strong coupling between the cavity field and the torsional motion of nano-particle could be achieved for the current experimental conditions. With the back-action of the optically levitated nano-particle, the broad single-mode squeezing light can be realized in the bad cavity regime. Even at room temperature, the single-mode light can be squeezed for more than 17 dB, which is far beyond the 3 dB limit. The two-mode squeezing light can also be generated, if the optical tweezers contain two frequencies, one is on the red sideband of the cavity mode, the other is on the blue sideband. The two-mode squeezing can be maximized near the boundary of the system stable regime and is sensitive to both the cavity decay rate and the power of the optical tweezers.

1. Introduction

Squeezing light, in which the quantum fluctuation is modulated below the shot noise level, has been regarded as a powerful resource in fundamental physics, e.g., improving the sensitivity of gravitational wave detection [1], cooling the motion of a macroscopic mechanical object below the quantum backaction limit [2], engineering matter interactions [3], and inducing the topological phase transitions [4], among many others. Squeezing light can be generated via the nonlinear optics, such as parametric down-conversion process [5,6,7,8]. The motion of the mechanical oscillator in cavity optomechanical systems could be regarded as an effective nonlinear optical medium, generating the pondermotive squeezing state by the back-action interaction [8,9,10,11]. With various optomechanical coupling mechanisms, cavity optomechanical systems produce abundant squeezing light sources [11,12,13,14,15], even in pulse driving regime [11,15]. However, due to the noise inevitably induced from the thermal bath, as well as the limitation of the nonlinear interaction, generating the substantial pondermotive squeezing light is still a challenge for the optomechanical system [9].

The levitated optomechanical system has raised widespread interest in macroscopic quantum superposition [16,17,18], quantum time crystals [19,20], and quantum information processing [21]. The squeezing has been studied in the levitated optomechanical system, in which the nano-particle is optically levitated and coupled with a cavity field [22]. Under ultra-high vacuum (gas pressure $p={10}^{-10}$ Torr), the squeezing light could be estimated over 15 dB below the vacuum noise level [22], which is much higher than the squeezing light source based on a membrane mechanical resonator embedded in an optical cavity [9]. In the levitated optomechanical system, the motional state of the optically levitated nano-particle has been squeezed $2.7$ dB below the thermal noise [23]. Recently, the coherent scattering mechanism has been theoretically investigated to cool both the axial and the in-plane motion of the levitated optomechanical system via a cavity mode [24,25]. The axial motion of the optically levitated nano-particle has been cooled to the ground state, either through the coherent scattering induced cavity cooling [26,27], or through the feedback cooling [28,29]. Moreover, the strong coherent scattering coupling between the cavity field and the motion of the nano-particle has also been observed in experiment [30]. With the attainable ultra-high quality factor (beyond ${10}^9$) of the optically nano-particle [31,32,33], it is expected that the stronger and robust squeezing light can be realized via the coherent scattering [22,24,32,34,35].

In this paper, we aim to generate both single-mode and two-mode squeezing light via a vacuum levitated optomechanical system, in which the nano-particle is optically levitated by the single and dual tweezers, respectively [22,24]. Both schemes are based on the coherent scattering mechanism [24]. We find that the ultra-strong coupling between the cavity mode and the torsion mode of the nano-particle is available by the current experimental parameters [36,37,38,39]. In order to obtain the squeezed single-mode light in steady state, the optical tweezers are detuned to the red sideband of the cavity mode. By the back-action of the optically levitated nano-particle [11], the single-mode light can be squeezed over 17 dB. The strong and broad single-mode squeezing light is observed in a bad cavity. Furthermore, in order to generate the two-mode squeezing lightly, we consider the optical tweezers with two frequencies, one is on the red sideband of the cavity mode, the other is on the blue sideband. We find that the two-mode squeezing can be maximized when the dynamics approach system instability [40]. The squeezing is sensitive to both the cavity decay rate and optical tweezers’ power. Unlike the single-mode squeezing light, the generated two-mode squeezing light is also an entangling source, which is very sensitive to thermal noise. We also find that the two-mode squeezing degree is smaller than the single-mode squeezing.

The paper is organized as follows. The model and the system dynamics are depicted in Section 2. Then the results of one-mode squeezing light and two-mode squeezing light are discussed in Section 3 and Section 4. At last, a brief conclusion is given in Section 5.

2. Model and Dynamics

We consider a uniform isotropic non-dispersive nano-ellipsoid, which is optically levitated by linearly polarized dual tweezers (A and B) in a bichromatic cavity as shown in Figure 1a. The optical tweezers A (B) with frequency ${\omega }_{A\left(B\right)}$ are on the red (blue) sideband of cavity mode ${\widehat{a}}_A$ (${\widehat{a}}_B$). Under this condition, the nano-ellipsoid will be cooled and heated by the optical tweezers A and B, respectively. If the cooling rate is larger than the heating rate, the motional state can be cooled down to a low temperature and the system remains dynamically stable.

The motion of the levitated nano-ellipsoid is characterized by the five degrees of freedom where $\{x,y,x\}$ for the center-of-mass motion in position $\widehat{R}$ and $\{\theta ,\varphi \}$ for the torsional motion in orientation $\widehat{\mathsf{\Omega }}$, as shown in Figure 1b. By locating the nano-ellipsoid to the node (anti-node) of the cavity modes, the center-of-mass motion and torsional motion can be decoupled from each other [37]. Besides, intrinsic optomechanical coupling between the cavity mode and torsional motion of the nano-ellipsoid is typically much weaker than the coherent scattering coupling strength [24,26,37]. Therefore, we only consider the coherent scattering coupling between the cavity mode and the motion of the nano-ellipsoid. Without loss of generality, we take the torsional motion as an example in the following discussion [38,39].

With respect to the interaction picture defined by the Hamiltonian $H_0=\hslash {\displaystyle \sum _{j=A,B}}{\omega }_j{a}_j^{†}{a}_j$, the interaction Hamiltonian for the system can be written as [24]

$$\widehat{H}=\hslash \sum _{j=A,B}{\Delta }_j{\widehat{a}}_j^{†}{\widehat{a}}_j+\hslash {\omega }_m{\widehat{b}}^{†}\widehat{b}-\hslash \sum _{j=A,B}{g}_j\left({\widehat{a}}_j^{†}+{\widehat{a}}_j\right)\left({\widehat{b}}^{†}+\widehat{b}\right),$$

(1)

where ${\widehat{a}}_j^{†}$ and ${\widehat{a}}_j$ (${\widehat{b}}^{†}$ and $\widehat{b}$) are the bosonic creation and annihilation operators with the commutation relations $\left[{\widehat{a}}_j,{\widehat{a}}_j^{†}\right]=1$$(\left[\widehat{b},{\widehat{b}}^{†}\right]=1)$ [41], ${\Delta }_j={\omega }_c^j-{\omega }_j$ is the detuning between the cavity frequency ${\omega }_c^j$ and the optical tweezers frequency ${\omega }_j$. ${\omega }_m$ is the torsional mode frequency and $g_j$ is the coherent scattering coupling. They can be calculated by (see Appendix B)

$$g_j=\left({\alpha }_a-{\alpha }_b\right)E_0^j{\xi }_{0}cos\left(\phi \right)\sqrt{\frac{{\omega }_c^j}{8\hslash {\epsilon }_0{V}_c^j}}$$

(2)

and

$${\omega }_m=\sqrt{\sum _{j=A,B}{E}_0^{j2}\left({\alpha }_a-{\alpha }_b\right)/\phantom{\left(E_0^A{E}_0^A+E_0^B{E}_0^B\right)\left({\alpha }_a-{\alpha }_b\right)2I}2I}$$

(3)

respectively. For the current experimental parameters, the torsional frequency can be in the order of MHz as shown in Figure 2a. Interestingly, the ratio ($g_j/{\omega }_m$) between the coherent scattering coupling and the torsional frequency is larger than $0.1$ as shown in Figure 2b, in which the ultra-strong optomechanical coupling regime has been reached [36]. In this regime, both the rotating and anti-rotating wave terms have to be considered. As ${\widehat{a}}_j$ (${\widehat{a}}_j^{†}$) depends on $\widehat{b}+{\widehat{b}}^{†}$, one field quadrature of the cavity mode will be squeezed, while another will be anti-squeezed [5,6].

Derived from the system Hamiltonian (1), the dynamics of the system is characterized by the following quantum Langevin equations:

$$\frac{d\widehat{b}}{dt}=-\left(\frac{{\gamma }_m}{2}+i{\omega }_m\right)\widehat{b}+i\sum _{j=A,B}{g}_j\left({\widehat{a}}_j^{†}+{\widehat{a}}_j\right)+\sqrt{{\gamma }_m}{\widehat{b}}^{in},$$

(4)

$$\frac{d{\widehat{a}}_j}{dt}=-\left(\frac{{\kappa }_j}{2}+i{\Delta }_j\right){\widehat{a}}_j+i{g}_j\left({\widehat{b}}^{†}+\widehat{b}\right)+\sqrt{{\kappa }_j}{\widehat{a}}_j^{in},$$

(5)

where ${\gamma }_m$ is the damping rate of the torsional mode (see Appendix C). Note that the operators are already replaced by $\delta \widehat{O}\to \widehat{O}$, where $\delta \widehat{O}$ denotes the fluctuation of $\widehat{O}=\{{\widehat{a}}_j,\widehat{b}$}. ${\widehat{a}}_j^{in}$ and ${\widehat{b}}^{in}$ are the zero-mean Gaussian noises, satisfying the correlation relations $〈{\widehat{a}}_j^{in}\left(t\right){\widehat{a}}_j^{in†}\left(t^{\prime }\right)〉=\delta \left(t-t^{\prime }\right)$ and $〈{\widehat{b}}^{in}\left(t\right){\widehat{b}}^{in†}\left(t^{\prime }\right)〉=\left(\overline{n}+1\right)\delta \left(t-t^{\prime }\right)$ respectively [41]. Here $\overline{n}=\{exp(\hslash {\omega }_m/k_{B}T)$${-1\}}^{-1}$ is the mean thermal excitation numbers at bath temperature T, and $k_B$ is Boltzmann constant. The linearized Langevin Equations (4) and (5) can be rewritten in the matrix form

$$\dot{\widehat{f}}\left(t\right)=A\widehat{f}\left(t\right)+\widehat{D}\left(t\right)$$

(6)

where ${\widehat{f}}^T\left(t\right)=\{\widehat{b}\left(t\right),{\widehat{b}}^{†}\left(t\right),{\widehat{a}}_A\left(t\right),{\widehat{a}}_A^{†}\left(t\right),{\widehat{a}}_B\left(t\right),{\widehat{a}}_B^{†}\left(t\right)\}$ and ${\widehat{D}}^T\left(t\right)=\{\sqrt{{\gamma }_m}{\widehat{b}}^{in}\left(t\right),\sqrt{{\gamma }_m}{\widehat{b}}^{in†}\left(t\right),\sqrt{{\kappa }_A}{\widehat{a}}_A^{in}\left(t\right),\sqrt{{\kappa }_A}{\widehat{a}}_A^{in†}\left(t\right),\sqrt{{\kappa }_B}{\widehat{a}}_B^{in}\left(t\right),\sqrt{{\kappa }_B}{\widehat{a}}_B^{in†}\left(t\right)\}$. The drift matrix A is given by

$$A=\left(\begin{array}{c}\begin{array}{cccccc}-\frac{\gamma }{2}-i{\omega }_m& 0& i{g}_A& i{g}_A& i{g}_B& i{g}_B\\ 0& -\frac{\gamma }{2}+i{\omega }_m& -i{g}_A& -i{g}_A& -i{g}_B& -i{g}_B\\ i{g}_A& i{g}_A& -\frac{{\kappa }_A}{2}-i{\Delta }_A& 0& 0& 0\\ -i{g}_A& -i{g}_A& 0& -\frac{{\kappa }_A}{2}+i{\Delta }_A& 0& 0\\ i{g}_B& i{g}_B& 0& 0& -\frac{{\kappa }_B}{2}-i{\Delta }_B& 0\\ -i{g}_B& -i{g}_B& 0& 0& 0& -\frac{{\kappa }_B}{2}+i{\Delta }_B\end{array}\end{array}\right).$$

(7)

After the Fourier transform $\widehat{O}\left(t\right)={\int }_{-\infty }^{+\infty }\widehat{O}\left(\omega \right)e^{-i\omega t}d\omega$, the steady state solutions of the Langevin Equations (4) and (5) in the frequency domain are

$$\widehat{b}\left(\omega \right)=\frac{i{\displaystyle \sum _{j=A,B}}{g}_j\left[{\widehat{a}}_j^{†}\left(-\omega \right)+{\widehat{a}}_j\left(\omega \right)\right]+\sqrt{{\gamma }_m}{\widehat{b}}^{in}}{\frac{{\gamma }_m}{2}+i\left({\omega }_m-\omega \right)},$$

(8)

$${\widehat{a}}_j\left(\omega \right)=\frac{i{g}_j\left({\widehat{b}}^{†}\left(-\omega \right)+\widehat{b}\left(\omega \right)\right)+\sqrt{{\kappa }_j}{\widehat{a}}_j^{in}}{\frac{{\kappa }_j}{2}+i\left({\Delta }_j-\omega \right)},$$

(9)

where the corresponding correlation noises are $〈{\widehat{b}}^{in}\left(\omega \right){\widehat{b}}^{in†}\left(-{\omega }^{\prime }\right)〉=2\pi \left(2\overline{n}+1\right)\delta \left(\omega +{\omega }^{\prime }\right)$ and $〈{\widehat{a}}_j^{in}\left(\omega \right){\widehat{a}}_{j^{\prime }}^{in†}\left(-{\omega }^{\prime }\right)〉=2\pi {\delta }_{j{j}^{\prime }}\left(\omega +{\omega }^{\prime }\right)$, respectively. Furthermore, Equations (8) and (9) can be rewritten as $\widehat{A}\left(\omega \right)=J^{-1}\left(\omega \right)\widehat{B}\left(\omega \right)$ where $\widehat{A}\left(\omega \right)={\{\widehat{b}\left(\omega \right),{\widehat{b}}^{†}(-\omega ),{\widehat{a}}_A\left(\omega \right),{\widehat{a}}_A^{†}(-\omega ),{\widehat{a}}_B\left(\omega \right),{\widehat{a}}_B^{†}(-\omega )\}}^T$, $B\left(\omega \right)={\{\sqrt{{\gamma }_m}{\widehat{b}}^{in}\left(\omega \right),\sqrt{{\gamma }_m}{\widehat{b}}^{in†}(-\omega ),\sqrt{{\kappa }_A}{\widehat{a}}_A^{in}\left(\omega \right),\sqrt{{\kappa }_A}{\widehat{a}}_A^{in†}(-\omega ),\sqrt{{\kappa }_B}{\widehat{a}}_B^{in}\left(\omega \right),\sqrt{{\kappa }_B}{\widehat{a}}_B^{in†}(-\omega )\}}^T$,

$$J\left(\omega \right)=\left[\begin{array}{cccccc}{u}_1& 0& -i{g}_A& -i{g}_A& -i{g}_B& -i{g}_B\\ 0& u_2& i{g}_A& i{g}_A& i{g}_B& i{g}_B\\ -i{g}_A& -i{g}_A& v_1& 0& 0& 0\\ i{g}_A& i{g}_A& 0& v_2& 0& 0\\ -i{g}_B& -i{g}_B& 0& 0& w_1& 0\\ i{g}_B& i{g}_B& 0& 0& 0& w_2\end{array}\right],$$

(10)

$u_1=\frac{{\gamma }_m}{2}+i\left({\omega }_m-\omega \right)$, $u_2=\frac{{\gamma }_m}{2}-i\left({\omega }_m+\omega \right)$, $v_1=\frac{{\kappa }_A}{2}+i\left({\Delta }_A-\omega \right)$, $v_2=\frac{{\kappa }_A}{2}-i\left({\Delta }_A+\omega \right)$, $w_1=\frac{{\kappa }_B}{2}+i\left({\Delta }_B-\omega \right)$, and $w_2=\frac{{\kappa }_B}{2}+i\left({\Delta }_B-\omega \right)$. Based on the standard input–output relation

$${\widehat{a}}_j^{out}\left(\omega \right)=\sqrt{{\kappa }_j}{\widehat{a}}_j\left(\omega \right)-{\widehat{a}}_j^{in}\left(\omega \right),$$

(11)

the output cavity mode $a_j^{out}\left(\omega \right)$ can be attainable.

3. Single-Mode Squeezing

We consider the case that the nano-ellipsoid is levitated by only one optical tweezer A, while the other optical tweezer B is turned off. Only one cavity mode ${\widehat{a}}_A$ is excited by the scattering photons of the optical tweezers A. From Equations (8) and (9), the cavity mode ${\widehat{a}}_A\left(\omega \right)$ can be solved as

$${\widehat{a}}_A\left(\omega \right)=m_1^A\left(\omega \right){\widehat{b}}^{in}\left(\omega \right)+m_2^A\left(\omega \right){\widehat{b}}^{in†}(-\omega )+m_3^A\left(\omega \right){\widehat{a}}_A^{in}\left(\omega \right)+m_4^A\left(\omega \right){\widehat{a}}_A^{in†}(-\omega )$$

(12)

where

$$\begin{array}{c}{m}_1^A\left(\omega \right)=i\sqrt{{\gamma }_m}{g}_A{u}_2{v}_2/z,\hfill \\ \\ m_2^A\left(\omega \right)=i\sqrt{{\gamma }_m}{g}_A{u}_1{v}_2/z,\hfill \\ \\ m_3^A\left(\omega \right)=\sqrt{{\kappa }_A}\{g_A^2(u_1-u_2)+u_1{u}_2{v}_2\}/z,\hfill \\ \\ m_4^A\left(\omega \right)=\sqrt{{\kappa }_A}{g}_A^2(u_1-u_2)/z,\hfill \end{array}$$

(13)

with $z=g_A^2(u_1-u_2)(v_1-v_2)+u_1{u}_2{v}_1{v}_2$. Furthermore, substituting Equation (12) in Equation (11), one can obtain the stationary squeezing spectrum of the transmitted field [6]:

$$S_{\vartheta }\left(\omega \right)=〈\delta X_{\vartheta }^{out}\left(\omega \right)\delta X_{\vartheta }^{out}\left({\omega }^{\prime }\right)〉$$

(14)

where $\delta X_{\vartheta }^{out}\left(\omega \right)=e^{-i\vartheta }{a}_A^{out}\left(\omega \right)+e^{i\vartheta }{a}_A^{out†}\left(-\omega \right)$ with $\vartheta$ being the measurement phase angle in homodyne detection. Or more specifically, Equation (14) can be rewritten as the form of $S_{\vartheta }\left(\omega \right)=S_{\widehat{a}{\widehat{a}}^{†}}+S_{{\widehat{a}}^{†}\widehat{a}}+\left[e^{-2i\vartheta }{S}_{\widehat{a}\widehat{a}}+c.c\right]$ where

$$\begin{array}{cc}\hfill S_{\widehat{a}{\widehat{a}}^{†}}\left(\omega \right)=& 〈{\widehat{a}}_A^{out}\left(\omega \right){\widehat{a}}_A^{out†}\left(-{\omega }^{\prime }\right)〉\hfill \\ \hfill =& {\kappa }_A\left[m_3^A\left(\omega \right)-1/\sqrt{{\kappa }_A}\right]\left[m_3^{A*}\left(\omega \right)-1/\sqrt{{\kappa }_A}\right]\hfill \\ \hfill & +{\kappa }_A\left[(\overline{n}+1)m_1^A\left(\omega \right)m_1^{A*}\left(\omega \right)+\overline{n}{m}_2^A\left(\omega \right)m_2^{A*}\left(\omega \right)\right],\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill S_{{\widehat{a}}^{†}\widehat{a}}\left(\omega \right)& =〈{\widehat{a}}_A^{out†}\left(-\omega \right){\widehat{a}}_A^{out}\left({\omega }^{\prime }\right)〉\hfill \\ \hfill & ={\kappa }_A\left[{\left|m_4^A\left(-\omega \right)\right|}^2+\left(\overline{n}+1\right){\left|m_2^A\left(-\omega \right)\right|}^2+\overline{n}{\left|m_1^A\left(-\omega \right)\right|}^2\right],\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill S_{\widehat{a}\widehat{a}}\left(\omega \right)=& 〈{\widehat{a}}_A^{out}\left(\omega \right){\widehat{a}}_A^{out}\left({\omega }^{\prime }\right)〉\hfill \\ \hfill =& {\kappa }_A\left[m_3^A\left(\omega \right)-1/\sqrt{{\kappa }_A}\right]m_4^A(-\omega )\hfill \\ \hfill & +{\kappa }_A\left[(\overline{n}+1)m_1^A\left(\omega \right)m_2^A(-\omega )+\overline{n}{m}_2^A\left(\omega \right)m_1^A(-\omega )\right].\hfill \end{array}$$

(17)

It is noted that the maximum squeezing could reach when $d{S}_{\vartheta }\left(\omega \right)/\vartheta =0$. Then it is easy to find out that the maximum squeezing of single output cavity mode is [13,42]

$$S_1\left(\omega \right)=S_{\widehat{a}{\widehat{a}}^{†}}\left(\omega \right)+S_{{\widehat{a}}^{†}\widehat{a}}\left(\omega \right)-2\left|S_{\widehat{a}\widehat{a}}\left(\omega \right)\right|$$

(18)

as $e^{2i\vartheta }=-S_{\widehat{a}\widehat{a}}/\phantom{S_{\widehat{a}\widehat{a}}\left|S_{\widehat{a}\widehat{a}}\right|}\left|S_{\widehat{a}\widehat{a}}\right|$. The output cavity mode is squeezed if $S_1\left(\omega \right)$ decreases below the shot-noise level ($S_1\left(\omega \right)<1$).

In order to maintain the system stability, we assume that the frequency of the optical tweezer A is on the red sideband of the cavity mode, e.g. ${\Delta }_A={\omega }_m$. We find that the Routh–Hurwitz criterion could be fulfilled, even under ultra-strong coupling regime $g_A/{\omega }_m>0.1$ [41]. Besides, the cavity mode could be highly squeezed in this regime. As shown in Figure 3, we find that the output cavity mode can be squeezed even at room temperature. Obviously, the optimal squeezing of the output cavity field occurs around resonance regime $\omega =0$, where the dissipative part of the mechanical susceptibility has no response in dynamics [6]. In the good cavity regime ${\kappa }_A<{\omega }_m$, the pondermotive squeezing will be suppressed [15]. As the cavity decay rate increases from ${\kappa }_A=0.1 {\omega }_m$ to ${\kappa }_A=1 {\omega }_m$, the minimum of $S_1\left(\omega \right)$ decreases. The output cavity mode can be squeezed up to 17 dB when the lifetime of cavity photons approximately equals to a period of torsional motion (${\kappa }_A\approx {\omega }_m$). As the cavity decay rate continues to increase, the broadband and strong squeezing generate a bad cavity condition ${\kappa }_A\approx 2 {\omega }_m$. Then the broadband squeezing will split into two dips under the bad cavity regime, and the squeezing degree will reduce at $\omega =0$ for the larger dissipation [15].

Note that the model we proposed is an open system, in which the optically levitated nano-ellipsoid irreversibly couples to the environment by the collision of the surrounding gas [33]. With the back-action interaction, the thermal disturbance could act on cavity mode, leading to decoherence. Here we investigate the effects on the squeezing light from the perspective of the surrounding gas pressure and temperature of the torsional motion. As shown in Figure 4a, $S_1\left(\omega \right)$, this depends on the gas pressure p. The lower pressure of the surrounding gas leads to the higher squeezing light. However, limited by the stability in ultra-high vacuum and the photons recoiling [33], the squeezing threshold exists. On the other hand, the higher temperature for the torsional mode will extremely enlarge the number of thermal phonons, which would destroy the squeezing state of the output lights. Therefore, many schemes based on optomechanics require a low-temperature environment [9,11,23]. In order to realize the stronger squeezing light, the high-quality factor and low-temperature environment are necessary for the levitated optomechanical system [11]. For our scheme, the squeezing is robust to the thermal environment as shown in Figure 4b.

4. Two-Mode Squeezing

Our scheme can also be used for generating two-mode squeezing light, if we consider the nano-ellipsoid is optically levitated by dual tweezers, both A and B. Driven by the scattering photons, two cavity modes ${\widehat{a}}_A$ and ${\widehat{a}}_B$ are excited in the bichromatic cavity. Based on Equations (8), (9), and (11), the output cavity mode ${\widehat{a}}_j^{out}\left(\omega \right)$ can be solved by

$$\begin{array}{cc}\hfill {\widehat{a}}_j^{out}\left(\omega \right)=& m_1^j\left(\omega \right){\widehat{b}}^{in}\left(\omega \right)+m_2^j\left(\omega \right){\widehat{b}}^{in†}(-\omega )+m_3^j\left(\omega \right){\widehat{a}}_A^{in}\left(\omega \right)\hfill \\ & +m_4^j\left(\omega \right){\widehat{a}}_A^{in†}(-\omega )+m_5^j\left(\omega \right){\widehat{a}}_B^{in}\left(\omega \right)+m_6^j\left(\omega \right){\widehat{a}}_B^{in†}(-\omega )\end{array}$$

(19)

where

$$\begin{array}{c}{m}_1^A=i\sqrt{{\kappa }_A{\gamma }_m}{g}_A{u}_2{v}_2{w}_1{w}_2/z,\hfill \\ \\ m_2^A=i\sqrt{{\kappa }_A{\gamma }_m}{g}_A{u}_1{v}_2{w}_1{w}_2/z,\hfill \\ \\ m_3^A={\kappa }_A\left(\left(u_1-u_2\right)\left(g_B^2{v}_2\left(w_1-w_2\right)+g_A^2{w}_1{w}_2\right)+u_1{u}_2{v}_2{w}_1{w}_2\right)/z-1,\hfill \\ \\ m_4^A={\kappa }_A\left(g_A^2\left(u_1-u_2\right)w_1{w}_2\right)/z,\hfill \\ \\ m_5^A=\sqrt{{\kappa }_A{\kappa }_B}\left(g_A{g}_B\left(u_1-u_2\right)v_2{w}_2\right)/z,\hfill \\ \\ m_6^A=\sqrt{{\kappa }_A{\kappa }_B}\left(g_A{g}_B\left(u_1-u_2\right)v_2{w}_1\right)/z,\hfill \end{array}$$

(20)

and

$$\begin{array}{c}{m}_1^B=i\sqrt{{\kappa }_B{\gamma }_m}{g}_B{u}_2{v}_1{v}_2{w}_2/z,\hfill \\ \\ m_2^B=i\sqrt{{\kappa }_B{\gamma }_m}{g}_B{u}_1{v}_1{v}_2{w}_2/z,\hfill \\ \\ m_3^B=\sqrt{{\kappa }_A{\kappa }_B}\left(g_A{g}_B\left(u_1-u_2\right)v_2{w}_2\right)/z,\hfill \\ \\ m_4^B=\sqrt{{\kappa }_A{\kappa }_B}\left(g_A{g}_B\left(u_1-u_2\right)v_1{w}_2\right)/z,\hfill \\ \\ m_5^B={\kappa }_B\left(\left(u_1-u_2\right)\left(g_B^2{v}_1{v}_2+g_A^2\left(v_1-v_2\right)w_2\right)+u_1{u}_2{v}_1{v}_2{w}_2\right)/z-1,\hfill \\ \\ m_6^B={\kappa }_B\left(g_B^2\left(u_1-u_2\right)v_1{v}_2\right)/z,\hfill \end{array}$$

(21)

with $z=\left(u_1-u_2\right)\left(g_B^2{v}_1{v}_2\left(w_1-w_2\right)+g_A^2\left(v_1-v_2\right)w_1{w}_2\right)+u_1{u}_2{v}_1{v}_2{w}_1{w}_2$. Furthermore, by introducing two phase-quadrature operators

$$X\left(\omega \right)=\frac{1}{2}\sum _{j=A,B}\left[a_j^{out}\left(\omega \right)+a_j^{out†}\left(-\omega \right)\right],$$

(22)

and

$$Y\left(\omega \right)=\frac{1}{2i}\sum _{j=A,B}\left[a_j^{out}\left(\omega \right)-a_j^{out†}\left(-\omega \right)\right],$$

(23)

the squeezing spectra $S_{XX}\left(\omega \right)$ and $S_{YY}\left(\omega \right)$ can be calculated by $〈X\left(\omega \right)X\left({\omega }^{\prime }\right)〉=2\pi S_{XX}\left(\omega \right)\delta \left(\omega +{\omega }^{\prime }\right)$ and $〈Y\left(\omega \right)Y\left({\omega }^{\prime }\right)〉=2\pi S_{YY}\left(\omega \right)\delta \left(\omega +{\omega }^{\prime }\right)$ [43]. The output two-mode cavity field is squeezed if $S_2\left(\omega \right)=S_{XX}\left(\omega \right)+S_{YY}\left(\omega \right)<1$.

In order to realize the two-mode squeezing light in steady state, two optical tweezers A and B are detuned to the red sideband (red-red), or the optical tweezers A is detuned to red sideband while the optical tweezers B is detuned to blue sideband (red-blue). In the red-blue case, the system will be stable in the blue sideband if the cavity photons are quickly lost to the vacuum. So, the cavity decay rates are assumed to ${\kappa }_B=10{\kappa }_A=3{\omega }_m$. As the torsional mode resonance in the blue sideband, the dynamics of the torsional motion will be gradually amplified. In that case, the enhanced back-action interaction on the cavity mode will induce deeper squeezing. As shown in Figure 5, the squeezing of the red-blue case is much larger than that of the red-red case as the mechanical susceptibility of the levitated nano-ellipsoid responds in the resonant region.

Next, we will analyze the two-mode squeezing from the view of system stability. As shown in Figure 6a, none of $S_2\left(\omega \right)$ is lower than 0.5 (3 dB). Obviously, there is a narrow and suddenly interrupted area, which splits the blue area into two parts. By comparing with Figure 6b, it corresponds to the critical boundary between the stable and unstable regions. The larger ${\kappa }_B$ and smaller ${\kappa }_A$ would make the system more stable. However, the squeezing will be reduced for the weak back-action. The maximum $S_2\left(\omega \right)$ reaches its maximum as the system approaches the dynamical instability [40].

In Figure 7, we assume that the power of optical tweezer A is fixed at $P_A=0.1$ W. Noted that the optical tweezer A is set to the red sideband of cavity mode, while optical tweezer B is detuned to the blue sideband. With the power $P_B$ from 0.05 W to 0.15 W, $S_2\left(\omega \right)$ decreases at $\omega \approx {\omega }_m$ for the enhancement of the back-action interaction. As the power $P_B$ up to 0.5 W, there is no squeezing on $\omega \approx {\omega }_m$. The amplified torsional motion is overlarge on resonance. It results in the instability of the system, which shows the destruction of squeezing.

5. Conclusions

We have proposed the squeezing light source based on an optically levitated nano-ellipsoid, which couples with a cavity. By the coherent scattering mechanism, the ultra-strong coupling between the cavity field and torsional mode of the nano-ellipsoid could be realized under the current experimental parameters. If the optical tweezers are on the red sideband of the cavity mode, the single-mode squeezed light source (17 dB squeezing) can be realized even at room temperature. It is noteworthy that both the strong and broadband squeezing light can be achieved under the bad cavity condition. In order to achieve the two-mode squeezing light two optical tweezers are applied to trap the nano-particle at the same time. One is on the red sideband, while another is on the blue sideband. The two-mode squeezing of the lights is sensitive to the system stability, which depends on both the cavity decay rates and the power of the optical tweezers. When the system dynamics are close to the boundary of the stable regime, the two-mode squeezing can be maximized. However, two-mode squeezing can not be over the 3 dB limit in our current scheme. In future, the 3 dB limit can be overcome either by feedback [44] or by the reservoir-engineering method [45].