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arXiv:math-ph/0102023

About the Overcompleteness of Coherent State Systems with a Line Bundles Viewpoint

A. I. Shimabukuro

Published Feb 21, 2001• 0 citations
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Abstract

Standart Coherent State Systems have an analysis based on lattices (von Neumann's lattices) in terms of wich they are classified, looking at the size of the minimun cell, by: complete, overcomplete and not complete. In this work we analize overcomplete systems with a geometrical viewpoint (holomorphic line-budles). We apply the method to evaluate the degeneracy of the lowest Landau level.

DOI
10.48550/arxiv.math-ph/0102023

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Standart Coherent State Systems have an analysis based on lattices (von Neumann's lattices) in terms of wich they are classified, looking at the size of the minimun cell, by: complete, overcomplete and not complete. In this work we analize overcomplete systems with a geometrical viewpoint (holomorphic line-budles). We apply the method to evaluate the degeneracy of the lowest Landau level.
The growth of Bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states. Using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of Bargmann functions of an order not exceeding (and of a type not exceeding if of order ). These results generalize known results on the completeness of von Neumann lattices. The practical significance of this formalism in the context of quantum optics is also discussed.
A new general subset of coherent states, which is complete and overcomplete by just one element, is introduced. It is the union of NNN arbitrary nonintersecting von Neumann lattices whose unit-cell areas have a harmonic mean of Nh\mathrm{Nh}Nh. First applications are given in the problems of crystal electrons in rational magnetic fields, incommensurate structures in a magnetic field, and quantum dynamics.
We investigate the completeness of coherent states subsystems with respect to several classes of states on rational lattices. Our main results show that the phase-space shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L^2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for windows that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a discrete set of coherent states.
Arbitrary quantum states can be expanded with a very good accuracy in terms of coherent states on a truncated von Neumann lattice. Optimization techniques are used to further reduce the size of the basis, whilst still preserving the same degree of accuracy. Various p -norms are adopted as accuracy measures, which reflect different practical needs. Numerical examples demonstrate that a significant reduction in the size of the basis can be achieved with optimization techniques, without substantial loss of accuracy.