On the spectrum of a class of quantum models

TL;DR

This paper presents a method to determine the spectrum of certain quantum models with three-term recurrence eigenvalue equations using a transcendental function, aiding analysis of complex quantum systems.

Contribution

It introduces an analytical approach to find spectra of quantum models like the Rabi and Jaynes-Cummings models via a transcendental function derived from recurrence coefficients.

Findings

Spectrum can be obtained as zeros of a transcendental function.

Method applicable to models beyond exactly solvable cases.

Computer code for spectrum calculation is publicly available.

Abstract

The spectrum of any quantum model which eigenvalue equation reduces to a three-term recurrence, such as a displaced harmonic oscillator, the Jaynes-Cummings (JC) model, the Rabi model, and a generalized Rabi model, can be determined as zeros of a corresponding transcendental function F(x). The latter can be analytically determined as an infinite series defined solely in terms of the recurrence coefficients. The ease in obtaining the spectrum is of importance regarding recent experimental advances in preparing ultrastrongly interacting quantum systems, which can no longer be reliably described by the exactly solvable JC model. The relevant computer code has been made freely available online.