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arXiv:quant-ph/9810024 · Journal of Physics A Mathematical and General

Geometrical properties of intelligent spin states and time evolution of coherent states

Robert Arvieu, P Rozmej

Published Jan 1, 1999• 7 citations
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Abstract

We recall the properties of the intelligent (and quasi-intelligent) spin states introduced by Aragone et al (1974 J. Phys. A: Math. Gen. 7 L149, 1976 J. Math. Phys. 17 1963). We use these states to construct families of coherent wavepackets (WP) on the sphere and we sketch the time evolution of these WP for a rigid-body molecule.
DOI
10.1088/0305-4470/32/14/005

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In this paper we give a more compact representation of the intelligent spin states defined by Aragone, Guerri, Salamó, and Tani. Using this new representation, we discuss the differences between minimum uncertainty states, coherent Bloch spin states and intelligent states. The evolution of these states under a particular time dependent Hamiltonian is studied, showing the relevance of the noncompact subgroup K of the Lorentz group. Finally we analyze the radiative properties connected with the intelligent states for a pointlike medium. The main results are: (I) they have a nonvanishing dipole moment (as the Bloch states) and (II) the proper intelligent states give a spontaneous emission intensity which is different from the one provided by the Bloch states.
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.