Cat states in one- and two-mode Z₃ Rabi models

We investigate one- and two-mode variants of the Z₃-symmetric quantum Rabi model, which describe the interaction of a qutrit with one or two bosonic modes and are directly relevant for circuit-QED and spin-qudit platforms. We find a canonical transformation that allows one to obtain the spectrum of the Z₃ Rabi model using perturbation theory in a magnetic field. We show that in a certain parameter regime (deep-strong coupling and a small magnetic field), the three lowest eigenstates become Z₃ qutrit-boson cat states. In order to characterize these states, we introduce a joint qutrit-boson Wigner function and derive its closed-form expression for the qutrit-boson cat states. Numerical calculations across a wide range of coupling strengths show that the proposed Wigner function is a useful tool that allows one to unambiguously identify the states of the Z₃ qutrit-boson cat states.