We have studied the two-qubit quantum Rabi model in the asymmetric case and its generalizations with dipole and Heisenberg-type qubit–qubit interactions. The solutions are obtained analytically with eigenstates given in terms of the extended coherent states or photon number states. For identical qubit–photon couplings, a novel type of quasi-exact solution which exists for all coupling values with constant eigenenergy is found, leading to level crossings within the same parity subspace even for non-identical qubits. In contrast to the quasi-exact eigenstates of the single-qubit model, these exceptional eigenstates are formed by just a few Fock states (photon number bounded from above at one or three), and the condition for them depends only on a fine-tuning of the qubit level splittings with respect to the photon energy, but not on the coupling to the photon field if the qubit–qubit interaction is not taken into consideration. This makes them excellent candidates for direct experimental observation within circuit quantum electrodynamics (QED) and application in single photon experiments. Besides, exceptional solutions with finite photon numbers N are also found.

The exact solutions of the Duffin–Kemmer–Petiau (DKP) oscillator for spin-0 particles have been studied in noncommutative phase space. The results show that due to the noncommutative effect, the energy spectrum of the DKP oscillator for spin-0 particles is no longer degenerate. In addition, we obtain the nonrelativistic limit of the energy spectrum.

The simplified two-qubit Rabi model is proposed and its analytical solution is presented. There are no level crossings in the spectral graph of the model, which indicates that it is not integrable. The criterion of integrability for the Rabi model proposed by Braak (2011 Phys. Rev. Lett. 107 100401) is also used for the simplified two-qubit Rabi model and the same conclusion, consistent with what the spectral graph shows, can be drawn, which indicates that the criterion remains valid when applied to the two-qubit case. The simplified two-qubit Rabi model is another example of a non-integrable but exactly solvable system except for the generalized Rabi model.

We study the Dirac oscillators in a noncommutative phase space. The results show that the energy gap of Dirac oscillator was changed by noncommutative effect. In addition, we obtain the non-relativistic limit of the energy spectrum.

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