Theory of an optomechanical quantum heat engine

Coherent interconversion between optical and mechanical excitations in an optomechanical cavity can be used to engineer a quantum heat engine. This heat engine is based on an Otto cycle between a cold photonic reservoir and a hot phononic reservoir [K. Zhang, F. Bariani, and P. Meystre, Phys. Rev. Lett. 112, 150602 (2014)]. Building on our previous work, we (i) develop a detailed theoretical analysis of the work and the efficiency of the engine and (ii) perform an investigation of the quantum thermodynamics underlying this scheme. In particular, we analyze the thermodynamic performance in both the dressed polariton picture and the original bare photon and phonon picture. Finally, (iii) a numerical simulation is performed to derive the full evolution of the quantum optomechanical system during the Otto cycle, by taking into account all relevant sources of noise.

Figure 1

Frequencies of the two polaritons (normal modes) of the optomechanical system for $G/{\omega }_m=0.05$ in the red-detuned case ${\Delta }_p<0$. The dashed curves correspond to the frequencies of the bare photon and phonon modes.

Figure 2

Schematics of the Otto cycles associated with the polariton branches $A$ and $B$. See text for details.

Figure 3

Contour maps of the thermal efficiency and the total work (absolute value) of the $B$-branch Otto cycle for ${\delta }_i=-3$ with value legends aside. The thermal mean population of the normal mode $B$ is calculated through a numerical Bogoliubov transformation with the thermal mean photon and phonon numbers, ${\overline{n}}_a=0$ and ${\overline{n}}_b=10$, respectively. The white region is mechanically unstable.

Figure 4

Hierarchical structure of the quantum engine where the coupled photon and phonon modes constitute a polariton mode which exchanges heat and work with the external control field.

Figure 5

Intuitive physical picture of the Otto cycle for the optomechanical heat engine. The numbered arrows correspond to the engine strokes in Fig. 2. Initially (first figure) the mechanics undergoes relatively large thermal fluctuations due to its coupling to a hot thermal reservoir. After the adiabatic step 1, the polariton becomes photonlike, with its still-unchanged mean occupation becoming photonlike, resulting in added radiation pressure force on the mechanics. Thermalization at the low radiation field temperature significantly reduces the polariton occupation number in step 2. After the adiabatic step 3 the polariton has regained its phononic nature with a small adjustment of the mirror position. Finally the polariton is in contact with the hot thermal bath in step 4, regaining the initial thermal occupation number at rate $\gamma$.

Figure 6

The time evolution of the number of the bare photon mode $a$ (blue dashed line), the bare phonon mode $b$ (red solid line), the polariton mode $A$ (black dotted line), and the polariton mode $B$ (pink dot-dashed line) in a loop of the Otto cycle. All parameters are normalized by the mechanical resonator frequency ${\omega }_m=2\pi \times 200$ MHz. The optomechanical coupling strength is $G=0.2{\omega }_m$, the cavity decay is $\kappa =0.03{\omega }_m$, and the mechanical damping is $\gamma ={10}^{-3}{\omega }_m$. The stroke times are ${\tau }_1={\tau }_3=25{\omega }_m^{-1}$, ${\tau }_2=50{\omega }_m^{-1}$, and ${\tau }_4={10}^4{\omega }_m^{-1}$. All the parameters are chosen according to the hierarchy relationship (87).

Figure 7

The number state probability distribution $P_n$ of the quantum states of the photon mode $a$ (red solid line and circles) and the phonon mode $b$ (blue dashed line and squares). (a–d) correspond to the nodes of the Otto cycle (see text). ${\overline{n}}_{a\left(b\right)}$ are the corresponding mean particle numbers.