Abstract
We study the spectral properties of a spin-boson Hamiltonian that depends on two continuous parameters 0 < or = Lambda < infinity (interaction strength) and 0 < or = alpha < or = 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 variant Planck's over 2pi ) 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.
Highlights
Classical integrability of a system with two degrees of freedom guarantees that the Hamiltonian can be expressed as a piecewise smooth function of two action coordinates Hp1, q1 ; p2, q2͒ = HCJ1, J2͒
Level crossings are prohibited and the level spectrum is naturally described by a singleenergy sortingquantum number
Whereas each surviving torus can still be characterized by two local action coordinates J1, J2 via line integralspidqi along pairs of topologically independent closed paths, the functional relation HCJ1, J2͒ breaks down at the edge of the integrable regime
Summary
Classical integrability of a system with two degrees of freedom guarantees that the Hamiltonian can be expressed as a piecewise smooth function of two action coordinates Hp1 , q1 ; p2 , q2͒ = HCJ1 , J2͒ No such functional relation exists if the system is nonintegrable1–4͔. Studies in one or the other classical limit of the spin-boson model revealed chaotic phase space flow turning regular in the rotating wave approximation6,7,11,12͔. In the two-dimensional parameter space spanned by thepolarcoordinates⌳ , ␣͒, the two integrable regimes are located on two perpendicular straight lines that intersect each other at the point of zero coupling strength. Each quadrant of this parameter plane represents a nonintegrable regime. VI for the identification of the two regimes in purely quantum mechanical terms
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