Continued Fractions and the Rabi Model

TL;DR

This paper reviews two different continued fraction techniques for numerically computing the spectrum of the quantum Rabi model, highlighting their methodologies, advantages, and limitations.

Contribution

It compares and analyzes two approaches using continued fractions for the quantum Rabi model, clarifying their theoretical foundations and practical implications.

Findings

The first method uses Bargmann space representation and approximates the spectrum with high-order truncation.

The second method considers finite-dimensional models, but its results are ambiguous without analyticity justification.

Both methods provide insights into the spectral properties of the quantum Rabi model.

Abstract

Techniques based on continued fractions to compute numerically the spectrum of the quantum Rabi model are reviewed. They are of two essentially different types. In the first case, the spectral condition is implemented using a representation in the infinite-dimensional Bargmann space of analytic functions. This approach is shown to approximate the correct spectrum of the full model if the continued fraction is truncated at sufficiently high order. In the second case, one considers the limit of a sequence of models defined in finite-dimensional state spaces. Contrary to the first, the second approach is ambiguous and can be justified only through recourse to the analyticity argument from the first method.