Weak invariants of time-dependent quantum dissipative systems

TL;DR

This paper introduces the concept of weak invariants for time-dependent quantum dissipative systems, explicitly constructing such invariants for a damped harmonic oscillator, extending the Lewis-Riesenfeld invariant to nonunitary dynamics.

Contribution

It generalizes the Lewis-Riesenfeld invariant to nonunitary, dissipative quantum systems using Lie algebraic methods, specifically for Lindblad-type master equations.

Findings

Constructed explicit weak invariant for damped harmonic oscillator

Extended invariant concept to nonunitary, dissipative systems

Utilized su(1,1) Lie algebraic structure for derivation

Abstract

The concept of weak invariant is introduced. Then, the weak invariants associated with time-dependent quantum dissipative systems are discussed in the context of master equations of the Lindblad type. In particular, with the help of the su(1,1) Lie-algebraic structure, the weak invariant is explicitly constructed for the quantum damped harmonic oscillator with the time-dependent frequency and friction coefficient. This generalizes the Lewis-Riesenfeld invariant to the case of nonunitary dynamics in the Markovian approximation.