The $k$-photon quantum Rabi model

TL;DR

The paper investigates a nonlinear generalization of the quantum Rabi model in which each spin flip is coupled to $k$ photons. Using Bargmann space representations and asymptotic analysis of the coupled eigenvalue problem, it shows that for all $k\ge 3$ the Hamiltonian $H_{kp}$ is not essentially self-adjoint, yielding a continuous spectrum filling $\mathbb{C}$ and normalizable eigenfunctions that do not belong to the natural domain, thus precluding unitary time evolution. Consequently, the model is unphysical despite formal symmetry, a feature not detectable by truncating to finite-dimensional spaces. The results have implications for $k$-photon interactions in extended models such as the Dicke model, where some finite-photon-number states may exist but do not cure the fundamental non-self-adjointness of the generic Hamiltonian.

Abstract

A generalization of the quantum Rabi model is obtained by replacing the linear (dipole) coupling between the two-level system and the radiation mode by a non-linear expression in the creation and annihilation operators, corresponding to multi-photon excitations. If each spin flip involves $k$ photons, it is called the "$k$-photon" quantum Rabi model. While the formally symmetric Hamilton operator is self-adjoint in the case $k=2$, it is demonstrated here that the Hamiltonian is not self-adjoint for $k\ge 3$. Therefore it does not generate a unitary time evolution and is unphysical. This result cannot be obtained by numerical calculations in finite-dimensional spaces which attempt to approximate an unbounded operator by a finite-rank operator.