Quantum integrability and nonintegrability in the spin-boson model

Abstract

We study the spectral properties of a spin-boson Hamiltonian that depends on two continuous parameters $0\le \Lambda <\infty$ (interaction strength) and $0\le \alpha \le \pi /2$ (integrability switch). In the classical limit this system has two distinct integrable regimes, $\alpha =0$ and $\alpha =\pi /2$. For each integrable regime we can express the quantum Hamiltonian as a function of two action operators. Their eigenvalues (multiples of $\hslash$) are the natural quantum numbers for the complete level spectrum. This functional dependence cannot be extended into the nonintegrable regime $(0<\alpha <\pi /2)$. Here level crossings are prohibited and the level spectrum is naturally described by a single (energy sorting) quantum number. In consequence, the tracking of individual eigenstates along closed paths through both regimes leads to conflicting assignments of quantum numbers. This effect is a useful and reliable indicator of quantum chaos -- a diagnostic tool that is independent of any level-statistical analysis.