Phonon Cooling by an Optomechanical Heat Pump

We propose and analyze theoretically a cavity optomechanical analog of a heat pump that uses a polariton fluid to cool mechanical modes coupled to a single precooled phonon mode via external modulation of the substrate of the mechanical resonator. This approach permits us to cool phonon modes of arbitrary frequencies not limited by the cavity-optical field detuning deep into the quantum regime from room temperature.

Figure 1

Dynamics of the mean polariton populations $⟨{\widehat{N}}_A⟩$ (green dashed line) and $⟨{\widehat{N}}_B⟩$ (dotted red line) and of the population $⟨{\widehat{N}}_c⟩$ of the phonon mode $c$ (solid blue line) for a few cooling cycles deep in the quantum regime. Here ${\omega }_b=\delta =2\times 10^3$, $\kappa =40$, ${\mathrm{\Delta }}_i=-6\times 10^3$, ${\mathrm{\Delta }}_f=-6\times 10^2$, $g={\mathrm{\Omega }}_0=2\times 10^2$, ${\tau }_1={\tau }_3=4\times 10^{-2}$, ${\tau }_2=8\times 10^{-3}$, and ${\tau }_4=0.1$, with frequencies in units of the phonon decay rate $\gamma$ and times in units of ${\gamma }^{-1}$. The initial average populations are $⟨{\widehat{N}}_A⟩(0)=0.5$, $⟨{\widehat{N}}_B⟩(0)=2$ and $⟨{\widehat{N}}_c⟩(0)=12$. The four dots on the horizontal axis mark the end of the first expansion step and beginning of heat exchange, the end of heat exchange and beginning of the compression step, the beginning of thermalization, and the end of the first cooling cycle.

Figure 2

Example of two-mode cooling, showing the dynamics of the mean polariton population $⟨{\widehat{N}}_A⟩$ (green dotted line) and of the populations $⟨{\widehat{N}}_c⟩$ (solid red line) and $⟨{\widehat{N}}_d⟩$ (dashed blue line) of the phonon modes $c$ and $d$ for two cooling cycles deep in the quantum regime. The parametric coupling between the modes $A$ and $c$ occurs at times $\gamma t=0.004$ and 0.0416, and the parametric coupling between $A$ and $d$ at times $\gamma t=0.02228$ and 0.0604. Same parameters as in Fig. 1, except that $\gamma =\kappa /400$. The initial average populations are $⟨{\widehat{N}}_A⟩(0)=0$, $⟨{\widehat{N}}_c⟩(0)=10$, and $⟨{\widehat{N}}_d⟩(0)=7.4$. The frequencies of modes $c$ and $d$ are ${\omega }_c=1.5{\omega }_b$ and ${\omega }_d=2{\omega }_b$.