Non-linear effects of quadratic coupling and Kerr medium in a hybrid optomechanical cavity system

Abstract

We investigate theoretically, the optical response for a cavity optomechanical system containing a Kerr non-linear substrate. The optomechanical cavity’s moving membrane interacts linearly and quadratically with the radiation pressure of the photons. We found that the quadratic non-linear coupling and Kerr non-linearity strongly influence the system’s optical multistability. Further, the transmission spectrum is observed under the influence of both non-linearities. Our theoretical results show that the width of the spectrum alters significantly by tuning the non-linearities present in the system. In addition, our results show that a peculiar positive value of quadratic optomechanical coupling is instrumental in generating Normal Mode splitting while the absence and negative value of quadratic optomechanical coupling inhibit the Normal-Mode splitting.

Appendices

Appendix A

In our present system, we found the stability conditions using Routh-Hurwitz Criterion (Gradshteyn and Ryzhik 1964) as,

$$\begin{aligned}&S_{1}=R_{0}>0, \end{aligned}$$

(37)

$$\begin{aligned}&S_{2}=(R_{3}R_{2}R_{1}-R_{4}R_{1}^2-R_{0}R_{3}^2)>0. \end{aligned}$$

(38)

Here,

$$\begin{aligned} R_{0}= & {} -2ia_{1}a_{2}\omega _{m}\Gamma _{1}-(\Delta _{1}+\delta _{1})a_{1}^{2}\omega _{m} \nonumber \\&-(\delta _{1}-\Delta _{1})a_{2}^2\omega _{m}+\omega _{t}\omega _{m}(\Delta _{1}^{2}-\delta _ {1}^{2}-\Gamma _{1}^2+\kappa ^2), \end{aligned}$$

(39)

$$\begin{aligned}&R_{1}=\gamma _{m}(\Delta _{1}^{2}-\delta _{1}^{2}-\Gamma _{1}^{2}+\kappa ^{2})+2\gamma _{m}\kappa , \end{aligned}$$

(40)

$$\begin{aligned}&R_{2}=\Delta _{1}^{2}-\delta _{1}^{2}-\Gamma _{1}^{2}+\kappa ^{2}+2\gamma _{m}\kappa , \end{aligned}$$

(41)

$$\begin{aligned}&R_{3}=2\Gamma _{1}+\gamma _{m}, \end{aligned}$$

(42)

$$\begin{aligned}&R_{4}=-1. \end{aligned}$$

(43)

Appendix B

$$\begin{aligned}&C_{1}(\omega )=\left( \kappa -i\omega \right) ^{2}-\Gamma _{1}^{2}+\Delta _{1}^{2}-\delta _{1}^{2}, \end{aligned}$$

(44)

$$\begin{aligned}&C_{2}(\omega )=\kappa +\Gamma _{1}-i\omega , \end{aligned}$$

(45)

$$\begin{aligned}&C_{3}(\omega )=a_{1}C_{2}(\omega )-ia_{2}(\delta _{1}-\Delta _{1}), \end{aligned}$$

(46)

$$\begin{aligned}&C_{4}(\omega )=\omega _{m}\left\{ -C_{3}(\omega )\left[ ia_{2}C_{2}(\omega )+(\Delta _{1}+\delta _{1})a_{1}\right] +\left[ ia_{2}a_{1}+\omega _{t}C_{2}(\omega )\right] C_{1}(\omega )\right\} \nonumber \\&-i\omega \left( \gamma _{m}-i\omega \right) C_{2}(\omega )C_{1}(\omega ). \end{aligned}$$

(47)