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

Understand

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.