  We study the two-qubit Rabi model in the most general case where the qubits
are different from each other. The spectrum of the system in the
ultrastrong-coupling regime is shown to converge to two forced oscillator
chains by perturbation theory. An even and odd decomposition of the Hilbert
space allows us to calculate the spectra in any given parameter regime; the
cases studied confirm our perturbation theory prediction in the
ultrastrong-coupling regime and point to crossings in the spectra within each
parity subspace in the moderate-coupling regime. The normal modes of the system
are calculated by two different methods, the first a linear algebra approach
via the parity bases that delivers a four-term recurrence relation for the
amplitudes of the proper states and, the second, via Bargmann representation
for the field that delivers five-term recurrence relations. Finally, we show
some examples of the time evolution of the mean photon number, population
inversion, von Neuman entropy and Wootters concurrence under the two-qubit
quantum Rabi Hamiltonian by taking advantage of the parity decomposition.
