On the relaxation to equilibrium of a quantum oscillator interacting with a radiation field

TL;DR

The paper analyzes the relaxation to thermodynamic equilibrium of a one-dimensional quantum harmonic oscillator in contact with a heat bath by performing a spectral analysis of an infinite system of Pauli master equations. It constructs a self-adjoint generator on a weighted sequence space with a compact resolvent, establishing a purely discrete spectrum and exponential convergence to the Gibbs distribution, and it links the dynamics to a holomorphic contracting semigroup. A related linear pencil is introduced to connect the operator spectrum with a family of shifted operators, revealing the full complex spectrum while identifying the physical relaxation modes as its point spectrum. The work introduces a novel compact-embedding framework and provides a rigorous spectral perspective on relaxation in open quantum systems with potential relevance to quantum optics and black-body radiation theory.

Abstract

In this article we investigate from the point of view of spectral theory the problem of relaxation to thermodynamical equilibrium of a quantum harmonic oscillator interacting with a radiation field. Our starting point is a system of infinitely many Pauli master equations governing the time evolution of the occupation probabilities of the available quantum states. The system we consider is derived from the evolution equation for the reduced density operator obtained after the initial interaction of the oscillator with the radiation field, the latter acting as a heat bath. We provide a complete spectral analysis of the infinitesimal generator of the equations, showing thereby that it generates an infinite-dimensional system of hyperbolic type. This implies that every global solution to the equations converge exponentially rapidly toward the corresponding Gibbs equilibrium state. We also provide a complete spectral analysis of a linear pencil naturally associated with the Pauli equations. All of our considerations revolve around the notion of compactness, more specifically around the existence of a new compact embedding result involving a space of sequences of Sobolev type into a weighted space of square-integrable summable sequences.