Efficiency at maximum power of a quantum heat engine based on two coupled oscillators

We propose and theoretically investigate a system of two coupled harmonic oscillators as a heat engine. We show how these two coupled oscillators within undamped regime can be controlled to realize an Otto cycle that consists of two adiabatic and two isochoric processes. During the two isochores the harmonic system is embedded in two heat reservoirs at constant temperatures $T_h$ and $T_c(<T_h)$, respectively, and it is tuned slowly along a protocol to realize an adiabatic process. To illustrate the performance in finite time of the quantum heat engine, we adopt the semigroup approach to model the thermal relaxation dynamics along the two isochoric processes, and we find the upper bound of efficiency at maximum power (EMP) ${\eta }^{*}$ to be a function of the Carnot efficiency ${\eta }_C(=1-T_c/T_h)$: ${\eta }^{*}\le {\eta }_{+}\equiv {\eta }_C^2/[{\eta }_C-(1-{\eta }_C)ln(1-{\eta }_C)]$, identical to those previously derived from ideal (noninteracting) microscopic, mesoscopic, and macroscopic systems.

Figure 1

Sketch of the two coupled harmonic oscillators.

Figure 2

Eigenvalues of the coupled harmonic system, as a function of the ratio of the bare modes ${\omega }_b/{\omega }_a$ for coupling strength $\lambda =0.3{\omega }_a$ (solid lines) and $\lambda =0$ (dashed lines). The frequency splitting $\delta ={\omega }_B-{\omega }_A$ is proportional to the dimensionless coupling strength $g=\lambda /{\omega }_a$.

Figure 3

Schematic diagram of a quantum Otto cycle along $\alpha$ branch in the $({\omega }_{\alpha },n_{\alpha })$ plane, with $\alpha =A,B$. The nodes of the cycle $i$ (with $i=1,2,3,4$) connecting two adiabats $1\to 2$ and $3\to 4$ and two isochores $2\to 3$ and $4\to 1$ are indicated by circles. $n_{\alpha }^{eqh}$ and $n_{\alpha }^{eqc}$ are thermal mean populations of the system approaching thermal equilibrium with the hot and cold reservoirs at inverse temperatures ${\beta }_h$ and ${\beta }_c$, respectively.