Incipience of quantum chaos in the spin-boson model

The peculiar spectral properties of the spinboson model make it suitable for an investigation of quantum nonintegrability effects and level statistics from a new perspective. For fixed spin quantum numbers, its energy spectrum consists of 2s+1 sequences of levels with no upper bound. These sequences are identified and labelled consecutively by means of a quantum invariant calculated from the time average of a non-stationary operator. For integrable cases, level repulsion (on the energy axis) is limited to states within each sequence. From the observed spectral properties, we infer a series ofs-dependent level-spacing distributions. They converge towards a Poisson distribution fors. For nonintegrable cases, level repulsion becomes a universal phenomenon, but the amount of repulsion between two states decreases with increasing separation (in label) of the two sequences to which they belong. For smalls, the quantum nonintegrability effects are compelling but not at all chaotic. Nevertheless, they contain all the ingredients necessary to produce the symptoms commonly described as indicators of quantum chaos. In this model, we can observe quantum chaos in the making under very controllable conditions.