Semi-analytical RWA formalism to solve Schr\"odinger equations for multi-qubit systems with resonator couplings

In this study, we develop a semi-analytical framework to solve generalized Jaynes-Tavis-Cummings Hamiltonians describing multi-qudit systems coupled via EM resonators. Besides the multi-level generalization we allow for an arbitrary number of resonators and/or modes, with nonidentical couplings to the qudits. Our method is based on generic excitation-number operators which commute with the respective Hamiltonians in the rotating wave approximation (RWA). The validity of the RWA is assessed explicitly. The formalism enables the study of eigenstates, eigenenergies and corresponding time evolutions of such coupled multi-qudit systems. The technique can be applied in cavity quantum electrodynamics and circuit quantum electrodynamics. It is also applicable to atomic physics, describing the coupling of a single-mode photon to an atom. As an example, we solve the Schr\"odinger equation for a two-qubit-one-resonator system, in principle to arbitrary high excitations. We also solve the Tavis-Cummings Hamiltonian in the one-excitation subspace for an arbitrary number of identical qubits resonantly coupled to one resonator. As a final example, we calculate of the low-excitation spectrum of a coupled two-transmon system.