Quantum chaos entails the investigation of the quantization of classical Hamiltonian systems with chaotic dynamics.1,2 It is particularly interesting to consider the quantization of systems with mixed regular ͑integrable͒ and chaotic dynamics, as typical classical Hamiltonian systems possess mixed dynamics, with islands of integrability situated in a chaotic sea.One particularly appealing example consists of a particle on a ring coupled linearly to a harmonic oscillator on one section of the ring. 3 The classical system possesses large, well-divided regions of integrable and chaotic dynamics. In the present short paper, we study the quantum signatures of these dynamics. 4 The eigenvalues of the Hamiltonian describing the quantum system can approach each other very closely, exhibiting so-called "avoided crossings," as the coupling strength ␣ is varied. The top center figure displays what is known as a "sharp" avoided crossing. The figure's inset provides a magnification of the closest approach of the levels. Adjusting the length L of the uncoupled region to increase the fraction of the classical system's phase space that is chaotic yields broader avoided crossings in which some eigenvalues are nearly degenerate over a longer range of coupling strengths. The bottom center figure illustrates an interacting cluster of such "broad" avoided crossings.To compare the quantum dynamics to corresponding classical surfaces of section ͑SOS͒, we show Husimi distributions 5 on the left ͑right͒ for the top ͑bottom͒ center plot. We depict these distributions as contour plots that range from zero probability density in black to higher probability densities in progressively brighter shades of red. For each Husimi distribution, we overlay ͑in blue͒ the corresponding SOS, whose empty spaces indicate regions of integrability. The left figure shows a continuous and complete exchange of structure between the participating levels in a narrow window around their closest approach; this exchange occurs between a state clearly localized in an integrable region and one localized around the edge of the chaotic sea. As ␣ is varied through the avoided crossing, probability density continuously flows through bridges between the integrable and chaotic regions. The right figure shows a more complicated exchange of structure in the Husimi distributions near the cluster of avoided level crossings. Clusters that include broad avoided crossings often involve complex level interactions, leading to a "mixing" of initial Husimi distributions as well as delocalization of the probability density in one or more of the participating levels.The study of avoided level crossings in the spectra of quantum Hamiltonians whose classical counterparts exhibit mixed regular-chaotic dynamics reveals important information about the quantum signatures of such dynamics. Among other topics, one can study delocalization phenomena and exchange of structure in Husimi distributions and compare the quantum dynamics one observes to the dynamics of corresponding classical sys...

We investigate the quantization of a free particle coupled linearly to a harmonic oscillator. This system, whose classical counterpart has clearly separated regular and chaotic regions, provides an ideal framework for studying the quantization of mixed systems. We identify key signatures of the classically chaotic and regular portions in the quantum system by constructing Husimi distributions and investigating avoided level crossings of eigenvalues as functions of the strength and range of the interaction between the system's two components. We show, in particular, that the Husimi structure becomes mixed and delocalized as the classical dynamics becomes more chaotic. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2819060͔ Typical classical Hamiltonians systems are neither fully integrable nor fully chaotic, but instead possess mixed dynamics, with islands of stability situated in a chaotic sea. In this paper, we investigate the quantization of a recently-studied system with mixed dynamics.1 This example consists of a free particle that moves around a ring that is divided into two regions. At the boundaries between these regions, the particle is kicked impulsively by a harmonic oscillator (in a manner that conserves the system's total energy), but the particle and oscillator otherwise evolve freely. Although the system is not generic, its separation into regular and chaotic components also allows more precise investigations (both classically and quantum-mechanically) than is typically possible, making this an ideal example to achieve a better understanding of the quantization of mixed systems. By examining avoided level crossings and Husimi distributions in the quantum system, we investigate the quantum signatures of mixed dynamics, demonstrating that the Husimi structures of nearby states become mixed and delocalized as chaos becomes a more prominent feature in the classical phase space.