Power-efficiency trade-off for the finite-time quantum harmonic Otto heat engine via a phase-space approach

Thermodynamic constraints impose a trade-off between power and efficiency in heat engines, preventing the simultaneous achievement of high power and high efficiency. For classical microscopic engines, explicit inequalities have been discovered, demonstrating the inherent inevitability of this power-efficiency trade-off. However, extensions of these results to quantum engines have so far been limited to cases of slow operation. In this study, we derive a power-efficiency trade-off relation for a paradigmatic quantum engine operating within a finite time, specifically the Otto cycle of a quantum harmonic oscillator. By utilizing a phase-space approach based on quasiprobability representations, we establish a universal trade-off relation applicable to arbitrary time-dependent protocols during the adiabatic processes. Our results reveal that the power of the quantum engine vanishes as the efficiency approaches the quantum-mechanical efficiency bound, which is stricter than the Carnot bound. Furthermore, we identify the conditions under which the upper bound is attained, which indicate maximum power is achieved when the generation of quantum coherence is reduced, and the difference in time durations of the isochoric processes increases. These findings are validated through numerical calculations, which confirm their applicability across various types of protocols for heat engine cycles.