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Physical Review A

Structure of eigenfunctions in terms of classical trajectories in an SU(3) schematic shell model

P. Lebœuf, Marcos Saraceno

Published May 1, 1990• 31 citations
Publisher

Abstract

The Husimi distribution for the eigenstates of a classically nonintegrable two-dimensional system with mixed phase space was calculated in order to analyze its structure in terms of classical trajectories. Besides eigenfunctions concentrated on invariant tori or with support on chaotic regions of phase space, we also found others that combine different classical invariant sets (like a stable and an unstable periodic orbit) and families of eigenstates scarred by classically unstable periodic orbits.

DOI
10.1103/physreva.41.4614

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The Husimi distribution for the eigenstates of a classically nonintegrable two-dimensional system with mixed phase space was calculated in order to analyze its structure in terms of classical trajectories. Besides eigenfunctions concentrated on invariant tori or with support on chaotic regions of phase space, we also found others that combine different classical invariant sets (like a stable and an unstable periodic orbit) and families of eigenstates scarred by classically unstable periodic orbits.
The authors use the generalised coherent state distribution to study in phase space the eigenfunctions of non-integrable Hamiltonians. The methods available in the Weyl group (the Husimi distribution) are extended to the more complicated phase spaces that arise from other Lie groups. They demonstrate the feasibility of numerical calculations in two models with W 2 and SU(3) symmetries, emphasising the similarities and showing the emergence of classical invariant structures in the semiclassical limit.
To find out how quantum bifurcation phenomena manifest themselves in the space of states, various periodic trajectories accompanied with several sets of bifurcated trajectories are studied for a simple quantum model Hamiltonian under a time-dependent mean-field approximation. Making use of the quantum-classical correspondence, a method of classifying the eigenstates and of visualizing the global structures of the whole space of eigenstates is proposed. By means of the profile functions of the eigenstates, appropriate sets of eigenstates representing the quantum correspondents of the bifurcations are found. The idea of quantum bifurcations is supported by similar behavior of the energy vs period time ( E-T ) plots of the periodic trajectories and the eigenstates.
We study the quantum spectra and eigenfunctions of the three-junction superconducting quantum interference device designed for the Josephson flux qubit at high energies. We analyze the spectral statistics of the parameter region where the system has a mixed classical phase space where regular and chaotic orbits can be found at the same classical energy. We perform a numerical calculation of eigenvalues and eigenstates for different values of the ratio of the Josephson and charging energies, EJ∕EC{E}_{J}∕{E}_{C}EJ​∕EC​, which is directly related to an effective $\ensuremath{\hbar}$ parameter. We find that the nearest-neighbor distributions P(s)P(s)P(s) of the energy-level spacings are well fitted by the Berry-Robnik theory employing as free parameters the pure classical measures of the chaotic and regular regions of phase space in the different energy regions in the semiclassical case. The phase-space representation of the wave functions is obtained via the Husimi distributions, and the localization of the states of classical structures is analyzed. We discuss for which values of EJ∕EC{E}_{J}∕{E}_{C}EJ​∕EC​ it can be possible to perform experiments that could be sensitive to the structure of a mixed classical phase space.
The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.
The wave functions of the Dicke Hamiltonian, describing a spin coupled to a bosonic mode, are studied via Husimi distributions. A classical analogue of this system is also obtained. For several energy ranges studied, the Husimi distribution of the wave functions show the scar of simple periodic orbits when projected into the boson phase space. Surprisingly, these same distributions, when projected into the spin phase space, are spread through large regions. An explanation of this fact is given in terms of semiclassical theory and border effects associated with nonsemiclassical behaviour.