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Exact solution for the quantum Rabi model with the $\boldsymbol{\mathsf{A}}^{2}$ term

Quantum Rabi model (QRM) is widely used for the analysis of the radiation-matter interaction at the fundamental level in cavity quantum electrodynamics. Typically the QRM Hamiltonian includes only $\boldsymbol{\mathsf{p}} \cdot \boldsymbol{\mathsf{A}}$ term, however, the complete nonrelativistic Hamiltonian of quantum electrodynamics includes $\boldsymbol{\mathsf{A}}^{2}$ term as well. Here we find an exact solution and demonstrate with the help of the exact canonical transformations that the QRM Hamiltonian with the $\boldsymbol{\mathsf{A}}^{2}$ term (QRMA) is reduced to the standard QRM model Hamiltonian with the renormalized frequency and the coupling constant and the eigenstates are expressed through the squeezed states of the field. As a result, the $\boldsymbol{\mathsf{A}}^{2}$ term qualitatively changes the behavior of the QRM with purely electromagnetic interaction in the strong coupling regime: the value of the ground state energy of an atom inside the cavity is higher than in vacuum and the number of crossing of energy levels with different quantum numbers decreases.