Finite-time performance of a quantum heat engine with a squeezed thermal bath

We consider the finite-time performance of a quantum Otto engine working between a hot squeezed and a cold thermal bath at inverse temperatures ${\beta }_h$ and ${\beta }_c(>{\beta }_h)$ with $(k_B\equiv 1)\beta =1/T$. We derive the analytical expressions for work, efficiency, power, and power fluctuations, in which the squeezing parameter is involved. By optimizing the power output with respect to two frequencies, we derive the efficiency at maximum power as ${\eta }_{mp}={\left({\eta }_C^{\mathrm{gen}}\right)}^2/[{\eta }_C^{\mathrm{gen}}-(1-{\eta }_C^{\mathrm{gen}})ln(1-{\eta }_C^{\mathrm{gen}})]$, where the generalized Carnot efficiency ${\eta }_C^{\mathrm{gen}}$ in the high-temperature or small squeezing limit simplifies to an analytic function of squeezing parameter $\gamma$: ${\eta }_C^{\mathrm{gen}}=1-{\beta }_h/[{\beta }_{c}cosh\left(2\gamma \right)]$. Within the context of irreversible thermodynamics, we demonstrate that the expression of efficiency at maximum power satisfies a general form derived from nonlinear steady state heat engines. We show that, the power fluctuations are considerably increased, although the engine efficiency is enhanced by squeezing.

Figure 1

Schematic diagram of a quantum Otto cycle working with a harmonic system in the $(\omega ,n)$ plane. Here ${\langle n_h\rangle }^{sq,eq}\equiv {[exp\left({\omega }_h{\beta }_h^{\mathrm{eff}}\right)-1]}^{-1}$ (with the effective temperature ${\beta }_h^{\mathrm{eff}}$ of the hot squeezed bath) is excitation number of the system after an infinite long time interaction with the bath, ${\langle n_h\rangle }^{eq}$ (${\langle n_c\rangle }^{eq}$) is excitation number of the system at thermal equilibrium with the hot (cold) bath of inverse temperature ${\beta }_h({\beta }_c$).

Figure 2

The efficiency at maximum power ${\eta }_{mp}$ as a function of the Carnot efficiency for fixed squeezing parameter $\gamma =0.2$. The analytical expression ${\eta }_{mp}^{*}$ given by Eq. (41) and the generalized CA efficiency ${\eta }_{CA}^{\mathrm{gen}}$ are denoted by a red solid line and a blue dashed one, respectively.

Figure 3

The efficiency $\eta$ as a function of the squeezing parameter $\gamma$ for given Carnot value (dotted black line). The solid red line and dashed blue one show ${\eta }_{mp}^{*}$ and ${\eta }_{CA}^{\mathrm{gen}}$, both of which can surpasses the standard Carnot efficiency for finite squeezing parameters, but obey the generalized Carnot value (green dot-dashed line).