On the 2-mode and $k$-photon quantum Rabi models

TL;DR

This paper investigates the mathematical properties of the two-mode and k-photon quantum Rabi models, revealing conditions for normalizable wavefunctions and showing that for k≥3, these models are not diagonalizable, contrasting with simpler models.

Contribution

It provides a rigorous analysis of the existence of entire wavefunctions in the Bargmann-Hilbert space for these models and establishes their non-diagonalizability for k≥3.

Findings

Two-mode and 2-photon Rabi models have normalizable wavefunctions only under specific parameter constraints.

For k≥3, the k-photon Rabi model lacks wavefunctions in the Bargmann-Hilbert space for all non-trivial parameters.

The k≥3 models are not diagonalizable, unlike the k-photon Jaynes-Cummings models.

Abstract

By mapping the Hamiltonians of the two-mode and 2-photon Rabi models to differential operators in suitable Hilbert spaces of entire functions, we prove that the two models possess entire and normalizable wavefunctions in the Bargmann-Hilbert spaces only if the frequency $\omega$ and coupling strength $g$ satisfy certain constraints. This is in sharp contrast to the quantum Rabi model for which entire wavefunctions always exist. For model parameters fulfilling the aforesaid constraints we determine transcendental equations whose roots give the regular energy eigenvalues of the models. Furthermore, we show that for $k\geq 3$ the $k$-photon Rabi model does not possess wavefunctions which are elements of the Bargmann-Hilbert space for all non-trivial model parameters. This implies that the $k\geq 3$ case is not diagonalizable, unlike its RWA cousin, the $k$-photon Jaynes-Cummings model which can be completely diagonalized for all $k$.