Quantum chaos of SU3 observables

Hamiltonians built of SU 3 generators are presented which allow for qualitatively different classical limits. The generators of the group SU 3 give eight independent observables. In the classical limit two invariant functions of these observables, the so-called Casimir functions c 2 and c 3 , take fixed values in the ranges ½ c 2 (2/3) and -(2/9) c 3 (2/9) (an SU 2 system, in contrast, has just one such Casimir function, the squared total angular momentum). Generically, this leads to a six-dimensional phase space. However, for two special (`degenerate') pairs of values c 2 = (2/3),c 3 = -(2/9) and c 2 = (2/3),c 3 = (2/9) the classical dynamics is only four-dimensional. One and the same Hamilton function of the generators may be integrable in the four-dimensional phase space but generate chaos in the six-dimensional case. We give two examples of Hamiltonians for which that alternative does arise; one of these is closely related to the Lipkin Hamiltonian familiar from nuclear shell models. The transition from integrable to chaotic classical behaviour is accompanied by the usual quantum transition from level clustering to level repulsion.

[1]  J. Eberly,et al.  Nonlinear constants of motion for three-level quantum systems , 1982 .

[2]  G. Casati,et al.  STRUCTURE OF EIGENSTATES AND LOCAL SPECTRAL DENSITY OF STATES : A THREE-ORBITAL SCHEMATIC SHELL MODEL , 1997, chao-dyn/9708020.

[3]  L. Biedenharn,et al.  On the Representations of the Semisimple Lie Groups. II , 1963 .

[4]  D. Kusnezov,et al.  Classical limit for Lie algebras , 1990 .

[5]  M. Berry,et al.  Level clustering in the regular spectrum , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[6]  O. Bohigas,et al.  Characterization of chaotic quantum spectra and universality of level fluctuation laws , 1984 .

[7]  F. Berezin.  General concept of quantization , 1975 .

[8]  Laurence G. Yaffe,et al.  Large N limits as classical mechanics , 1982 .

[9]  F. Haake.  Quantum signatures of chaos , 1991 .

[10]  A. Barut,et al.  Theory of group representations and applications , 1977 .

[11]  K. Johnson.  Functional integrals for spin , 1989 .

[12]  W. Lisiecki.  Coherent state representations. A survey , 1995 .

[13]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[14]  Koonin,et al.  Quantum chaos in a schematic shell model. , 1988, Physical review. A, General physics.

[15]  P. Leboeuf,et al.  Eigenfunctions of non-integrable systems in generalised phase spaces , 1990 .

[16]  Leboeuf,et al.  Semiclassical approximations in the coherent-state representation. , 1989, Physical review. A, General physics.

[17]  L. Biedenharn.  On the Representations of the Semisimple Lie Groups. I. The Explicit Construction of Invariants for the Unimodular Unitary Group in N Dimensions , 1963 .

[18]  Diercksen,et al.  Intramanifold chaos in Rydberg atoms in external fields. , 1994, Physical review letters.

[19]  A. Perelomov.  Generalized Coherent States and Their Applications , 1986 .

[20]  Sven Gnutzmann,et al.  Coherent states and the classical limit on irreducible representations , 1998 .