Exact Solvability of the two-photon Rabi Hamiltonian

TL;DR

This paper derives the exact spectrum of the two-photon Rabi Hamiltonian by extending methods used for the one-photon case, revealing symmetries and a complete set of solutions in Bargmann space.

Contribution

It provides an exact analytical solution for the two-photon Rabi Hamiltonian, including symmetry analysis and a recurrence scheme for spectrum calculation.

Findings

Complete spectrum obtained through roots of analytic functions

Hilbert space splits into four symmetry-based subspaces

Eigenstates constructed in Bargmann space

Abstract

Exact spectrum of the two-photon Rabi Hamiltonian is found, proceeding in full analogy with the solution of standard (one-photon) Rabi Hamiltonian, published by Braak in Phys. Rev. Lett. 107, 100401 (2011). The Hamiltonian is rewritten as a set of two differential equations. Symmetries that get hidden after further treatment are found. One can plainly see, how the Hilbert space splits into four disjunct subspaces, categorized by four values of the symmetry parameter $c=\pm1,\pm i$. There were only two values $\pm1$ for the standard Rabi model. Four analytic functions are introduced by a recurrence scheme for the coefficients of their series expansion. All their roots yield the complete spectrum of the Hamiltonian. Eigenstates in Bargmann space are also at disposal.