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Normal mode splitting and optical squeezing in a linear and quadratic optomechanical system with optical parametric amplifier

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Abstract

We theoretically investigate the normal mode splitting (NMS) in the displacement spectrum of a moving membrane and the output cavity field squeezing spectrum in an optomechanical system with a degenerate optical parametric amplifier (OPA) placed inside the cavity. In our proposed system, the cavity mode is coupled to this moving membrane (which acts as mechanical mode) through both the linear optomechanical coupling and the quadratic optomechanical coupling. We have shown that the nonlinear OPA gain G and the phase angle \(\theta \) can effectively alter this NMS behavior in the displacement spectrum of the mechanical mode as well as output cavity field spectrum qualitatively as well as quantitatively. Furthermore, we can also enhance the squeezing bandwidth of the output cavity quadrature through both the parameters of OPA even at higher environment temperature T or in other words a significant mechanical thermal noise. Our study provides an efficient method to control the NMS behavior and the squeezing properties in such kind of generalized optomechanical systems.

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Authors and Affiliations

  1. Department of Physics, Muffakham Jah College of Engineering and Technology, Hyderabad, Telangana, 500034, India

    S. K. Singh

  2. Department of Basic Science, Hamedan University of Technology, Mardom, Hamedan, 65169-1-3733, Iran

    M. Mazaheri

  3. State Key Laboratory of Precision Spectroscopy, Quantum Institute for Light and Atoms, Department of Physics, East China Normal University, Shanghai, 200062, China

    Jia-Xin Peng

  4. Department of Physics, Government College University (GCUF), Faisalabad, 38000, Pakistan

    Amjad Sohail

  5. The Key Laboratory of Weak-Light Nonlinear Photonics, Ministry of Education, School of Physics and TEDA Applied Physics Institute, Nankai University, Tianjin, 300071, China

    Zhidong Gu

  6. Advanced Communication Engineering (ACE) Centre of Excellence, Universiti Malaysia Perlis, Allama Iqbal, Kangar Perlis, 01000, Kangar Perlis, Malaysia

    M. Asjad

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Appendix A

Appendix A

We can use Eq. 19 to find the amplitude and phase quadratures, \(\delta {\hat{x}_{\textrm{out}}}\) and \(\delta {\hat{y}_{\textrm{out}}}\) given as

$$\begin{aligned} \delta {\hat{x}_{\textrm{out}}}= & {} \delta {\hat{c}_{\textrm{out}}}+\delta {\hat{c}_{\textrm{out}}^{\dagger }}\nonumber ,\\ \delta {\hat{y}_{\textrm{out}}}= & {} i[\delta {\hat{c}_{\textrm{out}}^{\dagger }}-\delta {\hat{c}_{\textrm{out}}}]. \end{aligned}$$
(A-1)

The spectral density of the quadratures of the output field in the frequency domain, \(S_{x,\textrm{out}}(\omega )\) and \(S_{y,\textrm{out}}(\omega )\) can be found using the following equations:

$$\begin{aligned}{} & {} \left\langle \delta {\hat{x}_{\textrm{out}}^{\dagger }}(\omega ')\delta {\hat{x}_{\textrm{out}}}(\omega )\right\rangle =2 \pi S_{x,\textrm{out}}(\omega )\delta (\omega '+\omega ),\nonumber \\{} & {} \left\langle \delta {\hat{y}_{\textrm{out}}^{\dagger }}(\omega ')\delta {\hat{y}_{\textrm{out}}}(\omega )\right\rangle =2 \pi S_{y,\textrm{out}}(\omega )\delta (\omega '+\omega ). \end{aligned}$$
(A-2)

Using Eqs. 17–20, we get

(A-3)

Here we have taken \(D_{g}(\omega )=[D(\omega )]^{*}\), \(D_{t}(\omega )=D(-\omega )\), \(D_{tg}(\omega )=[D_{t}(\omega )]^{*}\), and \(E_{jg}(\omega )=[E_{j}(\omega )]^{*}\), \(E_{jt}(\omega )=E_{j}(-\omega )\), \(E_{jtg}(\omega )=[E_{jt}(\omega )]^{*}\), (\(j=\hat{c},\hat{c}^{\dagger },\xi \)).

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Singh, S.K., Mazaheri, M., Peng, JX. et al. Normal mode splitting and optical squeezing in a linear and quadratic optomechanical system with optical parametric amplifier. Quantum Inf Process 22, 198 (2023). https://doi.org/10.1007/s11128-023-03947-w

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