Minimum-Time Transitions Between Thermal Equilibrium States of the Quantum Parametric Oscillator

In this note, we use geometric optimal control to completely solve the problem of minimum-time transitions between thermal equilibrium states of the quantum parametric oscillator, which finds applications in various physical contexts. We discover a new kind of optimal solutions, absent from all the previous treatments of the problem.

I. Introduction

Constantin Carathéodory, the famous mathematician with seminal contributions in the calculus of variations, which paved the way to optimal control theory, pioneered the axiomatic formulation of thermodynamics along a purely geometric approach [1], at the dawn of the 20th century. Based on these foundations, the geometry of thermodynamics [2] was developed many decades later. In this context, thermal equilibrium states are represented as points on a manifold and tools from differential geometry are used to quantify the distance between them and to express the laws of thermodynamics. This approach is closely related to the subject of finite-time thermodynamics [3], which aims to optimize the performance of a thermodynamic system under restrictions on the available time, for example, to maximize the extracted power. Optimal control theory [4] is the mathematical tool used to tackle this kind of problems, thus the connection between thermodynamics and control is not restricted to the emblematic figure of Carathéodory but is actually deeper.