Onset of quantum chaos in molecular systems and the zeros of the Husimi function

Localization measures of parity adapted U(D)-spin coherent states applied to the phase space analysis of the D-level Lipkin-Meshkov-Glick model

We study phase space properties of critical, parity symmetric, N-qudit systems undergoing a quantum phase transition (QPT) in the thermodynamic N→∞ limit. The D=3 level (qutrit) Lipkin-Meshkov-Glick model is eventually examined as a particular example. For this purpose, we consider U(D)-spin coherent states (DSCS), generalizing the standard D=2 atomic coherent states, to define the coherent state representation Q_{ψ} (Husimi function) of a symmetric N-qudit state |ψ〉 in the phase space CP^{D-1} (complex projective manifold). DSCS are good variational approximations to the ground state of an N-qudit system, especially in the N→∞ limit, where the discrete parity symmetry Z_{2}^{D-1} is spontaneously broken. For finite N, parity can be restored by projecting DSCS onto 2^{D-1} different parity invariant subspaces, which define generalized "Schrödinger cat states" reproducing quite faithfully low-lying Hamiltonian eigenstates obtained by numerical diagonalization. Precursors of the QPT are then visualized for finite N by plotting the Husimi function of these parity projected DSCS in phase space, together with their Husimi moments and Wehrl entropy, in the neighborhood of the critical points. These are good localization measures and markers of the QPT.

Geometrodynamic approach to conjugate points and the Maslov index

Maslov indices are an essential ingredient in the semiclassical approaches to quantum mechanics, as they are also related to the conjugate points of the corresponding trajectory, which reflects the dynamics in its neighborhood. In this paper, we show how these important topological parameters can be computed using the geometrodynamic approach to dynamics. Illustrations in two- and three-dimensional systems are presented and discussed.

Shannon entropy at avoided crossings in the quantum transition from order to chaos

Shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics. In order to be able to study jointly this entropy for discrete and continuous probability, calculations have been performed on a quantized map, the kicked Harper map, resulting in a different behavior, as order-chaos transition takes place, for the discrete (position representation) and continuous (coherent state representation) cases. This different behavior is analyzed in terms of the distribution of zeros of the Husimi function.

Identifying the order of a quantum phase transition by means of Wehrl entropy in phase space

We propose a method to identify the order of a quantum phase transition by using area measures of the ground state in phase space. We illustrate our proposal by analyzing the well known example of the quantum cusp and four different paradigmatic boson models: Dicke, Lipkin-Meshkov-Glick, interacting boson model, and vibron model.

Fermi resonance in dynamical tunneling in a chaotic billiard

We elucidate that Fermi resonance ever plays a decisive role in dynamical tunneling in a chaotic billiard. Interacting with each other through an avoided crossing, a pair of eigenfunctions are coupled through tunneling channels for dynamical tunneling. In this case, the tunneling channels are an islands chain and its pair unstable periodic orbit, which equals the quantum number difference of the eigenfunctions. This phenomenon of dynamical tunneling is confirmed in a quadrupole billiard in relation with Fermi resonance.

Distribution of zeros of the Husimi function in systems with degeneracy

The distribution of zeros of the Husimi function of energy eigenstates has proven to be a very useful tool for the characterization of the quantum transition from order to chaos in systems without degeneracy. In this paper, we show, by means of calculations on the Hénon-Heiles Hamiltonian system, that the quantum order-chaos transition in systems with degeneracy can also be characterized through the distribution of the Husimi function zeros, providing that the appropriate linear combination of degenerate eigenstates is used. If using an arbitrary linear combination, spurious results can be achieved, suggesting a wrong premature quantum transition to chaos.