  The scientific interest in the analytical solution of the quantum Rabi model
is due to the widespread use of this simple model in quantum optics, quantum
computing, cavity QED, and nanoelectromechanical systems. This interest is
related to the need for the theoretical description of the interaction of a
two-level system with a quantum oscillator in the case when the rotating wave
approximation fails. In this Letter, we present an approach to the exact
diagonalization of the quantum Rabi Hamiltonian. This approach is based on the
properties of the Pauli operators and allows us to readily solve the stationary
Schrodinger equation for a two-level system. First, we demonstrate the
applicability of the approach to the Jaynes-Cummings Hamiltonian to get the
well-known solution. Then, we obtain the eigenvalues and eigenstates for the
quantum Rabi Hamiltonian using the proposed approach. It is shown that the
obtained eigenstates can be represented in the basis of the eigenstates of the
Jaynes-Cummings Hamiltonian.
