Jarzynski equation for a simple quantum system: Comparing two definitions of work

TL;DR

This paper examines the validity of the Jarzynski equation in a simple quantum system, comparing two different definitions of work and their implications for accurately determining free energy differences.

Contribution

It analyzes two definitions of work in quantum systems and demonstrates how the Jarzynski equation's validity depends on the chosen definition, especially at low temperatures.

Findings

The Jarzynski equation holds for the measurement-based work definition.

The Jarzynski equation fails at low temperatures for the operator-based work definition.

Different work definitions impact the applicability of fluctuation theorems in quantum systems.

Abstract

The validity of the Jarzynski equation for a very simple, exactly solvable quantum system is analyzed. The implications of two different definitions of work proposed in the literature are investigated. The first one derives from measurements of the system energy at the beginning and at the end of the process under consideration making the work a classical stochastic variable with transition probabilities derived from quantum mechanics. In the second definition an operator of work is introduced and the average in the Jarzynski equation is a quantum expectation value. For the first definition a general quantum mechanical version of the Jarzynski equation is known to hold. For the second one the Jarzynski equation fails to yield the free energy difference at low temperature.