Incipience of quantum chaos in the spin-boson model

Abstract

The peculiar spectral properties of the spin-boson model make it suitable for an investigation of quantum nonintegrability effects and level statistics from a new perspective. For fixed spin quantum number s, 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 of s-dependent level-spacing distributions. They converge towards a Poisson distribution for s → ∞. 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 small s, 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.

Small denominators, frequency operators, and Lie transforms for nearly integrable quantum spin systems

“…Related phenomena have been observed in particle systems by Bohigas, Tomsovic, and Ullmo [15]. In a study of the spin-boson model the observed "resonance phenomena" can be attributed to the presence of small denominators becoming more or less effective in representations of different dimensions [16]. An analytic study of these numerically observed effects in the vein of the present paper is under way [17].…”

Quantum integrability and action operators in spin dynamics

“…This type of resonances and their impact on the level statistics have been investigated more systematically in a recent study of the spin-boson model. [17] The numerical evidence for the nonintegrability of a classical two-spin system as derived, for example, from a Poincaré map or a strange invariant-surface, is virtually unmistakable if carried out with sufficient circumspection, even though it is not rigorous. Formal proofs of nonintegrability are scarce, essentially limited to systems that are as non-generic as integrable ones are.…”

The quantum Rabi model: solution and dynamics

“…Under these conditions, the coefficients h n vanish for n > N. The Heun function thus truncates to N terms. Applying the conditions (84) and (85) in this way leads to Judd's isolated exact solutions, with a finite set of recurrence relations following from (62). The coefficients are given by…”