# The Dynamical Casimir Effect in a Dissipative Optomechanical Cavity Interacting with Photonic Crystal

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Floquet–Liouvillian

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Floquet–Liouvillian Complex Eigenvalue Problem and Effective Operator

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**Figure 1.**(

**a**) optomechanical cavity interacting with a photonic crystal; (

**b**) the frequency of a single cavity mode is periodically changed by mechanical pumping, and the cavity mode photon decays into a one-dimensional photonic band; (

**c**) dispersion relation of a one-dimensional photonic band with a bandwidth of $2B$ and the central frequency ${\omega}_{B}$.

**Figure 3.**(

**a**) imaginary part of the complex eigenvalues of ${\mathcal{L}}_{\mathrm{eff}}$ as a function of ${\omega}_{0}$ for the coupled optomechanical cavity with the photonic band, where the parameters are taken $\Omega =2{\omega}_{B}$, $B=1$, ${f}_{0}=0.2$, and $g=1/\pi $. The horizontal axis represents ${\omega}_{0}-\Omega /2$. The bifurcation points are indicated by the red filled circles, and the stationary points are indicated by the green filled circles; (

**b**) Floquet–Liouvillian level scheme of the $|{a}^{*},1)\phantom{\rule{-0.166667em}{0ex}}\rangle ,|a,0)\phantom{\rule{-0.166667em}{0ex}}\rangle ,|{b}_{k}^{*},1)\phantom{\rule{-0.166667em}{0ex}}\rangle $, and $|{b}_{k},0)\phantom{\rule{-0.166667em}{0ex}}\rangle $ states, where the vertical axis denotes the frequencies of the modes in the Floquet–Liouvillian. The dotted line is drawn at $-\Omega /2$ as a guide.

**Figure 4.**Imaginary part of the complex eigenvalues of the phenomenological model as a function of ${\omega}_{0}$ for the values of ${f}_{0}=0.2$ and $\gamma =0.2$, where the horizontal axis represents ${\omega}_{0}-\Omega /2$. (

**a**) $\theta =\pi /2$, and (

**b**) $\theta =\pi $.

**Figure 5.**(

**a**) overall picture of the imaginary part of the complex eigenvalues of ${\mathcal{L}}_{\mathrm{eff}}$ as a function of ${\omega}_{0}$ for the coupled optomechanical cavity with the photonic band, where the parameters are taken $\Omega =2{\omega}_{B}-\Delta $, $\Delta =3B/2$, $B=1$, ${f}_{0}=0.2$, and $g=1/\pi $. The horizontal axis represents ${\omega}_{0}-\Omega /2$. The bifurcation points are indicated by the red filled circles, and the stationary points are indicated by the green filled circle; (

**b**) expanded picture of the dotted box area in (a). Non-local multimode stationary point is indicated by the yellow filled circle; (

**c**) Floquet–Liouvillian level scheme of the $|{a}^{*},1)\phantom{\rule{-0.166667em}{0ex}}\rangle ,|a,0)\phantom{\rule{-0.166667em}{0ex}}\rangle ,|{b}_{k}^{*},1)\phantom{\rule{-0.166667em}{0ex}}\rangle $, and $|{b}_{k},0)\phantom{\rule{-0.166667em}{0ex}}\rangle $ states, where the vertical axis denotes the frequencies of the modes in the Floquet–Liouvillian. The dotted line is drawn at $-\Omega /2$ as a guide.

**Figure 6.**Floquet–Liouvillian level scheme in a situation direct dissipation to the photonic band is suppressed and multimode parametric amlification is induced with the external driving frequency $\Omega $ much smaller than $2{\omega}_{0}$.

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**MDPI and ACS Style**

Tanaka, S.; Kanki, K.
The Dynamical Casimir Effect in a Dissipative Optomechanical Cavity Interacting with Photonic Crystal. *Physics* **2020**, *2*, 34-48.
https://doi.org/10.3390/physics2010005

**AMA Style**

Tanaka S, Kanki K.
The Dynamical Casimir Effect in a Dissipative Optomechanical Cavity Interacting with Photonic Crystal. *Physics*. 2020; 2(1):34-48.
https://doi.org/10.3390/physics2010005

**Chicago/Turabian Style**

Tanaka, Satoshi, and Kazuki Kanki.
2020. "The Dynamical Casimir Effect in a Dissipative Optomechanical Cavity Interacting with Photonic Crystal" *Physics* 2, no. 1: 34-48.
https://doi.org/10.3390/physics2010005