Lindbladian operators, von Neumann entropy and energy conservation in time-dependent quantum open systems

TL;DR

This paper investigates the evolution of von Neumann entropy in time-dependent quantum open systems governed by Lindblad equations, and derives conditions for energy conservation, exemplified by a harmonic oscillator.

Contribution

It provides a simple formula for entropy evolution under Lindblad dynamics and derives a Lindblad operator that conserves energy in time-dependent systems.

Findings

Von Neumann entropy non-decreases over time in these systems.

A unique Lindblad operator is constructed for energy conservation.

The harmonic oscillator example illustrates the theoretical results.

Abstract

The Lindblad equation is widely employed in studies of Markovian quantum open systems. Here, firstly, a simple result is presented on the time evolution of the non Neumann entropy under the Lindblad equation, which enables one to examine if the entropy increases/decreases. Then, secondly, the following question is posed: In a quantum open system with a time-dependent Hamiltonian, what is the corresponding Lindblad equation for the quantum state that keeps the internal energy of the system constant in time? This issue is of importance in realizing quasi-stationary states of open systems such as quantum circuits and batteries. As an illustrative example, the time-dependent harmonic oscillator is analyzed. The Lindbladian operator is uniquely determined with the help of a Lie-algebraic structure, and the time derivative of the von Neumann entropy is shown to be nonnegative depending on the time-dependence of the Hamiltonian.