Clones and other interference effects in the evolution of angular-momentum coherent states

Abstract

The aim of this paper is to present the interference effects that occur during the time evolution of simple angular wave packets (WP's) which can be associated with a diatomic rigid molecule (heteronuclear) or with a quantum rigid body with axial symmetry like a molecule or a nucleus. The time evolution is understood entirely within the framework of fractional revivals discovered by Averbukh and Perelman (Phys. Lett. A 39, 449 (1989); Usp. Fiz. Nauk 161, 41 (1991) [Sov. Phys. Usp. 37, 572 (1991)]), since the energy spectrum is exactly quadratic. Our objectives are to study how these interference effects differ when there is a change of the initial WP. For this purpose we introduce a two-parameter set of angular-momentum coherent states. On the one hand, this set emerges quite naturally from the three-dimensional coherent states of the harmonic oscillator; on the other hand, this set is shown to be built from intelligent spin states. By varying one parameter $(\ensuremath{\eta}),$ a scenario of interferences occurs on the sphere at fractional parts of the revival time that strongly depend on $\ensuremath{\eta}.$ For $\ensuremath{\eta}=\ifmmode\pm\else\textpm\fi{}1$ the WP, which coincides with a WP found by Mostowski [Phys. Lett. A 56, 369 (1976)], is a superposition of Bloch [Phys. Rev. 70, 460 (1946)] or Radcliffe [J. Phys. A 4, 313 (1971)] states, and clone exactly in time according to a scenario found for the infinite square well in one dimension, and also for a two dimensional rotor. In the context of intelligent spin states it is also natural to study the evolution by changing $\ensuremath{\eta}.$ For $\ensuremath{\eta}=0$ the WP is called linear, and in time produces a set of rings with axial symmetry over the sphere. The WP's for other values of $\ensuremath{\eta}$ are called elliptic, and sets of fractional waves are generated which make a transition between two symmetries. We call these fractional waves ``mutants.'' For specific times a clone is produced that stands among the mutants. Therefore the change in $\ensuremath{\eta}$ produces a change in the quantum spread on the sphere. We have also constructed simple coherent states for a symmetric rotor which are applicable to molecules and nuclei. Their time evolution also shows a cloning mechanism for the rational ratio of moments of inertia. For irrational values of this ratio, the scenario of partial revivals completed by Bluhm, Kostelecky, and Tudose [Phys. Lett. A 222, 220 (1996)] is valid.