A hidden analytic structure of the Rabi model Original Research Article

The Rabi model describes the simplest interaction between a cavity mode with a frequency image and a two-level system with a resonance frequency image. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to image, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy image, which are orthogonal on an equidistant lattice. A non-zero value of image leads to non-classical discrete orthogonal polynomials image and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. 1350 calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak’s solution. Any first image eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of image of at least the degree image. The value of image, which is slowly increasing with image, depends on the required precision. For instance, image for image and dimensionless interaction constant image, if double precision is required. Given that the sequence of the imageth zeros image’s of image’s defines a monotonically decreasing discrete flow with increasing image, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for dimensionless interaction constant image, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for image.