Heat transfer mediated by the dynamical Casimir effect in an optomechanical system

Heat transfer in an optomechanical system consisting of one optical cavity and two movable mirrors with or without a laser field is studied by using the quantum master-equation method. The radiation-pressure interaction between the cavity and the mirrors contains the single-photon optomechanical coupling (SPOC) and the dynamical Casimir effect (DCE). When the laser is not applied to the cavity, the heat transfer is induced by a nonlocal interaction between mirrors that builds up only due to the presence of the DCE. Once the monochromatic field is turned on, both the SPOC and the DCE can be used to modify the energy transfer process. In both cases, the heat transfer can be drastically enhanced at resonant conditions manifested by the DCE. Our results can motivate further studies to actively control heat transfer mediated by the DCE in optomechanical devices.

Figure 1

Setup for the heat transfer in an optomechanical system consisting of one cavity and two mechanically moving mirrors. The cavity mode ($a_c$) with frequency ${\omega }_c$ is coupled to the mechanical modes ($b_1$ and $b_2$) with frequencies ${\omega }_1$ and ${\omega }_2$ via the radiation pressure. The radiation pressure drives the SPOC and the DCE, which are marked by $g_s$ and $g_{\mathrm{DCE}}$. The cavity and mechanical oscillators are coupled to external baths with coupling strengths ${\gamma }_1, {\gamma }_2$, and ${\kappa }_c$. Assuming that the baths are in thermal equilibrium with temperatures $T_L, T_R$, and $T_C$. The cavity is driven by an external weak laser field with frequency ${\omega }_L$ and amplitude $E$.

Figure 2

(a) Equal-time second-order correlation functions $g_1^{\left(2\right)}\left(0\right), g_2^{\left(2\right)}\left(0\right)$, and $g_c^{\left(2\right)}\left(0\right)$ as a function of the Casimir interaction $g_{\mathrm{DCE}}$. (b) Effective temperature of the mechanical modes $T_1^{\mathrm{eff}}$ and $T_2^{\mathrm{eff}}$ versus the Casimir interaction $g_{\mathrm{DCE}}$. (c) Heat fluxes $J_L, J_R$, and $J_C$ versus the Casimir interaction $g_{\mathrm{DCE}}$ calculated by Eqs. (8) and (9). (d) Cavity mean photon number $\langle a_c^{†}{a}_c\rangle$ and correlation function $\mathrm{Re}\left[\langle a_c^{†}{a}_c^{†}\rangle \right]$ versus the Casimir interaction $g_{\mathrm{DCE}}$. The temperatures of the three baths are taken as $T_L=T_0+\delta T, T_R=T_0$, and $T_C=T_0$. The other parameters are ${\omega }_1/2\pi ={\omega }_2/2\pi ={\omega }_m/2\pi =5$ GHz, ${\omega }_c=2{\omega }_m, g_s=0.04{\omega }_m, {\kappa }_c={10}^{-6}{\omega }_m, {\gamma }_1={10}^{-4}{\omega }_m, {\gamma }_2={10}^{-4}{\omega }_m, T_0=50$ mK, $\delta T=25$ mK, ${\omega }_L=0$, and $E=0$.

Figure 3

Frequency dependence of the mechanical power spectrum ${\mathcal{S}}_1\left(\omega \right)$ (black solid line) and ${\mathcal{S}}_2\left(\omega \right)$ (red dashed line) calculated for different $g_{\mathrm{DCE}}$. The other parameters are the same as in Fig. 2.

Figure 4

(a) Effective temperature of the mechanical modes $T_1^{\mathrm{eff}}$ and $T_2^{\mathrm{eff}}$ as a function of the SPOC $g_s$ for the indicated values of the Casimir interaction $g_{\mathrm{DCE}}$. The temperature of bath L (R) is marked by upper (lower) shading. (b) Similar to (a), but for heat fluxes $J_L$ and $J_R$. The other parameters are the same as in Fig. 2.

Figure 5

(a) Heat fluxes $J_L, J_R$, and $J_C$ as a function of the cavity frequency ${\omega }_c$. Frequency dependence of the power spectrum calculated for (b) ${\omega }_c=0.49{\omega }_m$, (c) ${\omega }_c=0.5{\omega }_m$, and (d) ${\omega }_c=0.51{\omega }_m$. Both the SPOC and the DCE are considered here, and we take $g_s=g_{\mathrm{DCE}}=0.0004{\omega }_m$. The other parameters are the same as in Fig. 2.

Figure 6

(a) Heat fluxes $J_L$ and $J_R$ as a function of the optomechanical-coupling strength $g_s=g_{\mathrm{DCE}}$ for different cavity frequencies ${\omega }_c$. (b) Enlarged image of the region $(0,0.001{\omega }_m)$ in (a). The other parameters are the same as in Fig. 5.

Figure 7

Heat fluxes $J_L$ and $J_R$ as a function of the cavity-bath coupling rate ${\kappa }_c$ for two cavity frequencies, ${\omega }_c=0.49{\omega }_m$ and ${\omega }_c=0.5{\omega }_m$. The other parameters are the same as in Fig. 5.

Figure 8

Heat fluxes $J_L, J_R$, and $J_C$ as a function of the mechanical frequency ${\omega }_2$ with ${\omega }_1={\omega }_m$ and ${\omega }_c=0.5{\omega }_m$. The inset shows an enlarged image of the region $(0.993{\omega }_2/{\omega }_1,0.995{\omega }_2/{\omega }_1)$. The other parameters are the same as in Fig. 5.

Figure 9

Heat flux $J_L$ as a function of the optomechanical-coupling strength ($g_s,g_{\mathrm{DCE}}$). For the laser, we take ${\omega }_L={\omega }_c-{\omega }_m$ and $E=0.05{\omega }_m$. The other parameters are the same as in Fig. 2.

Figure 10

Laser field dependence of heat fluxes $J_L$ and $J_R$ for $g_s=g_{\mathrm{DCE}}=0.0004{\omega }_m$. (b) The same as in (a) calculated for the cavity equal-time second-order correlation function $g_c^{\left(2\right)}\left(0\right)$ and mean photon number $\langle a_c^{†}{a}_c\rangle$. The other parameters are the same as in Fig. 9.