The Quantization of the Rabi Hamiltonian

The Bargmann-Fock representation of the Rabi Hamiltonian is expressed by a system of two coupled first-order differential equations in the complex field, which may be rewritten in a canonical form under the Birkhoff transformation. The transformation gives rise to leapfrog recurrence relations, from which the eigenvalues and eigenvectors could be obtained. The interesting feature of this approach is that it generates integer quantum numbers, which relate the solutions to the Juddian baselines. The relationship with Braak's integrability claim [PRL 107, 100401 (2011)] is discussed