# Squeezing Light via Levitated Cavity Optomechanics

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## Abstract

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## 1. Introduction

## 2. Model and Dynamics

## 3. Single-Mode Squeezing

## 4. Two-Mode Squeezing

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Polarization Tensor

## Appendix B. Hamiltonian for Coherent Scattering

## Appendix C. Damping for Torsional Motion

## References

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**Figure 1.**(

**a**) Schematic diagram of the levitated optomechanical system. The nano-ellipsoid is placed into a bichromatic cavity and optically levitated by the dual tweezers with two frequencies ${\omega}_{A}$ and ${\omega}_{B}$, and amplitudes ${E}_{A}$ and ${E}_{B}$, respectively. Two cavity modes ${\widehat{a}}_{A}^{\u2020}$ (${\widehat{a}}_{A}$) and ${\widehat{a}}_{B}^{\u2020}$ (${\widehat{a}}_{B}$) are excited by the scattering photons with the decay rates ${\kappa}_{A}$ and ${\kappa}_{B}$. (

**b**) The orientation of the nano-ellipsoid $\{{x}_{E},{y}_{E},{z}_{E}\}$ rotates under the tweezers coordinate $\{{x}_{T},{y}_{T},{z}_{T}\}$ with a small angel $\varphi $. $\beta $ is the angel between the cavity coordinate axis ${x}_{C}$ and tweezers coordinate axis ${x}_{T}$. Tweezers propagate along the direction of axis ${z}_{T}$.

**Figure 2.**The torsional frequency ${\omega}_{m}$ (

**a**) and the ratio between the coherent scattering coupling and torsional frequency (

**b**) as a function of the tweezers power ${P}_{t}$. The parameters are given as follows: the cavity length $L=1$ mm, the wavelength of the optical tweezers ${\lambda}_{A}=780$ nm (${\lambda}_{B}=980$ nm), the beam waist of the tweezers in focus ${w}_{0}^{j}=1$$\mu $m, the principle axes of the nano-ellipsoid $a=2b=2c=100$ nm, the relative permittivity of the nano-ellipsoid $\epsilon =2.1$, and the density of the nano-ellipsoid $\rho =2200$ kg/m${}^{3}$. For convenience, the power of two optical tweezers are assumed to be the same ${P}_{j}={P}_{t}$.

**Figure 3.**${S}_{1}\left(\omega \right)$ as a function of the cavity decay rate ${\kappa}_{A}$. Parameters are listed as follows: optical tweezers’ power in focus ${P}_{A}=0.05$ W, pressure of the residual gas $p={10}^{-4}$ Pa, temperature of the residual gas ${T}_{a}=300$ K, bath temperature for the torsional mode $T=300$ K, and the accommodation efficient ${\gamma}_{ac}=0.9$ (see Appendix C). Other parameters are the same as Figure 2.

**Figure 4.**${S}_{1}\left(\omega \right)$ as a function of the pressure of the surrounding gas p (

**a**) and the temperature of torsional mode T (

**b**). In pictures (

**a**), the temperature of the torsional mode is assumed $T=300$ K, while the pressure of residual gas is set to ${10}^{-2}$ Pa in picture (

**b**). The cavity decay rate is ${\kappa}_{A}=3{\omega}_{m}$ in both picture (

**a**) and (

**b**). Other parameters are the same as Figure 3.

**Figure 5.**Squeezing spectrum of two output modes. The legend red-red represents both two optical tweezers in red sideband of the cavity mode. The legend red-blue denotes the optical tweezers A in red sideband while another tweezers B in blue sideband. The decay rates of the cavity mode are ${\kappa}_{A}=0.3{\omega}_{m}$ and ${\kappa}_{B}=3{\omega}_{m}$. Two tweezers are different in wavelength ${\lambda}_{A}=780$ nm (${\lambda}_{B}=980$ nm). Other parameters are the same as Figure 3.

**Figure 6.**${S}_{2}\left(\omega \right)$ (

**a**) and the maximum eigenvalue of Equation (7) (

**b**) as a function of the cavity decay rate ${\kappa}_{j}$. The detunings are set to the red and blue sideband ${\Delta}_{A}=-{\Delta}_{B}={\omega}_{m}$, respectively. In picture (

**b**), the yellow area means the maximum eigenvalue of Equation (7) is non-negative. According to the Routh–Hurwitz stability criterion [41], the system becomes instable in the long time limit. Conversely, the blue area denotes the system is stable. Other parameters are the same as Figure 5.

**Figure 7.**${S}_{2}\left(\omega \right)$ as a function of ${P}_{B}$. The power of optical tweezer A in focus is set to ${P}_{A}=0.1$ W. Tweezer A is detuned to the red sideband while tweezer B is set to be the blue sideband. Other parameters are the same as Figure 5.

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**MDPI and ACS Style**

Li, G.; Yin, Z.-Q. Squeezing Light via Levitated Cavity Optomechanics. *Photonics* **2022**, *9*, 57.
https://doi.org/10.3390/photonics9020057

**AMA Style**

Li G, Yin Z-Q. Squeezing Light via Levitated Cavity Optomechanics. *Photonics*. 2022; 9(2):57.
https://doi.org/10.3390/photonics9020057

**Chicago/Turabian Style**

Li, Guoyao, and Zhang-Qi Yin. 2022. "Squeezing Light via Levitated Cavity Optomechanics" *Photonics* 9, no. 2: 57.
https://doi.org/10.3390/photonics9020057