Inverse spectral theory for semiclassical Jaynes-Cummings systems

TL;DR

This paper establishes that, under certain conditions, the joint spectrum of semiclassical Jaynes-Cummings systems uniquely determines the system's structure, advancing inverse spectral theory in quantum integrable models.

Contribution

It proves that the joint spectrum of two semiclassical Jaynes-Cummings type systems uniquely identifies the system up to isomorphism, assuming Bohr-Sommerfeld rules.

Findings

Joint spectrum determines the system up to isomorphism.

Spectral coincidence up to a7(a7)2 implies system equivalence.

Results apply to quantum integrable systems with circular symmetry.

Abstract

Quantum semitoric systems form a large class of quantum Hamiltonian integrable systems with circular symmetry which has received great attention in the past decade. They include systems of high interest to physicists and mathematicians such as the Jaynes\--Cummings model (1963), which describes a two-level atom interacting with a quantized mode of an optical cavity, and more generally the so-called systems of Jaynes\--Cummings type. In this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of Jaynes\--Cummings type. We prove, assuming the Bohr\--Sommerfeld rules hold, that if the joint spectrum of two of these systems coincide up to $\mathcal{O}(\hbar^2)$, then the systems are isomorphic.