On dynamical semigroup for damped driven Jaynes-Cummings equations

TL;DR

This paper constructs a contraction dynamical semigroup for the damped driven Jaynes-Cummings model, providing a rigorous mathematical framework for its evolution under broad damping and pumping conditions.

Contribution

It introduces a novel construction of a contraction semigroup for the model's equations, extending the mathematical understanding of the system's dynamics.

Findings

Proves nonpositivity of the basic dissipation operator in Quantum Optics.

Constructs a semigroup whose trajectories are solutions to the Jaynes-Cummings equations.

Provides a rigorous mathematical foundation for the system's evolution.

Abstract

The article addresses the damped driven Jaynes-Cummings for quantised one-mode Maxwell field coupled to a two-level molecule. We consider a broad class of damping and pumping which are polynomial in the creation and annihilation operators. Our main result is the construction of a contraction dynamical semigroup in the Hilbert space of Hermitian Hilbert-Schmidt operators in the case of a nonpositive dissipation operator and time-independent pumping. All trajectories of the semigroup are generalised solutions to the Jaynes-Cummings equations. As a key example, we prove nonpositivity for the basic dissipation operator of Quantum Optics.