Quantum evolution of mixed states and performance of quantum heat engines

TL;DR

This paper introduces a new technique for analyzing the quantum evolution of coupled harmonic oscillators and applies it to understand the performance of quantum heat engines, revealing optimal conditions and fundamental efficiency limits.

Contribution

The authors develop an exact solution method for quantum dynamics of coupled oscillators and analyze their role in quantum heat engines, including efficiency and state transformations.

Findings

Maximum work occurs when oscillators swap excitations at parametric resonance.

Carnot efficiency remains the upper limit for quantum heat engines.

Density operators of Gaussian states can be decomposed into thermal states with different temperatures.

Abstract

We introduce a technique for calculating the density operator time evolution along the lines of Heisenberg representation of quantum mechanics. Using this technique, we find the exact solution for the quantum evolution of two and three coupled harmonic oscillators initially prepared in thermal states at different temperatures. We show that such systems exhibit interesting quantum dynamics in which oscillators swap their thermal states due to correlation induced in the process of energy exchange and yield noise induced coherence. A photonic quantum heat engine (QHE) composed of two optical cavities can be modeled as coupled harmonic oscillators with time-dependent frequencies. Photons in the cavities become correlated during the engine operation. We show that the work done by such an engine is maximum if at the end of the cycle the oscillators swap numbers of excitations which can be achieved when the engine operates under the condition of parametric resonance. We also show that Carnot formula yields limiting efficiency for QHEs under general assumptions. Moreover, we show that, by making a canonical transformation, density operator of arbitrary n-mode Gaussian state can be written as a product of n thermal density operators describing independent collective excitations with different temperatures. Thus, operation of QHEs based on the correlated Gaussian states is equivalent to that based on uncorrelated thermal reservoirs. Our results deepen understanding of quantum evolution of mixed states which could be useful to design quantum machines with better performance.