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arXiv:quant-ph/9809020 · Journal of Physics A Mathematical and General

Coherent states on the circle

José A González, M. A. del Olmo

Published Nov 6, 1998• 73 citations
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Abstract

A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González (in 1992 Semiclassical behaviour of the Weyl correspondence on the circle Group Theoretical Methods in Physics vol I (Madrid: Ciemat)), is carried out. They were obtained from the Weyl-Heisenberg coherent states in by means of the Weil-Brezin-Zak transformation, they are labelled by the points of the cylinder , and they provide a realization of by entire functions (similar to the well known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.

DOI
10.1088/0305-4470/31/44/012

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A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González (in 1992 Semiclassical behaviour of the Weyl correspondence on the circle Group Theoretical Methods in Physics vol I (Madrid: Ciemat)), is carried out. They were obtained from the Weyl-Heisenberg coherent states in by means of the Weil-Brezin-Zak transformation, they are labelled by the points of the cylinder , and they provide a realization of by entire functions (similar to the well known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.
A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González in [DG 92], is carried out. They were obtained from the Weyl-Heisenberg coherent states in L2(R)L^2(\\R)L2(R) by means of the Weil-Brezin-Zak transformation, they are labeled by the points of the cylinder S1timesRS^1 \\times \\RS1timesR, and they provide a realization of L2(S1)L^2(S^1)L2(S1) by entire functions (similar to the well-known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.
Abstract We extend former results for coherent states on the circle in the literature in two ways. On the one hand, we show that expectation values of fractional powers of momentum operators can be computed exactly analytically by means of Kummer’s confluent hypergeometric functions. Earlier, these expectation values have only been obtained by using suitable estimates. On the other hand, we consider the Zak transformation not only to map harmonic oscillator coherent states to coherent states on the circle as it has been discussed before, but we also use the properties of the Zak transformation to derive a relation between matrix elements with respect to coherent states in <?CDATA L2(R){L}_{2}({\mathbb{R}})L2​(R)?> L 2 ( R ) and L 2(S 1). This provides an alternative way for computing semiclassical matrix elements for coherent states on the circle. In certain aspects, this method simplifies the semiclassical computations in particular if one is only interested in the classical limit, that is the zeroth order term in the semiclassical expansion.
Covariant integral quantisation using coherent states for semi-direct product groups is implemented for the motion of a particle on the circle. In this case, the phase space is the cylinder, which is viewed as a left coset of the Euclidean group E(2). Coherent states issued from fiducial vectors are labeled by points in the cylinder and depend also on extra parameters. We carry out the corresponding quantisations of the basic classical observables, particularly the angular momentum and the 2π-periodic discontinuous angle function. We compute their corresponding lower symbols. The quantum localisation on the circle is examined through the properties of the angle operator yielded by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Heisenberg inequality. Comparison with other approaches to the long-standing question of the quantum angle is discussed.
We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S 1 including the Aharonov–Bohm-type quantum description. Coherent states are constructed by Perelomov's method as group-related coherent states generated by Weyl operators on the quantum phase space . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction, we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was . The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, some of their further properties are also studied demonstrating similarities as well as substantial differences. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
The Heisenberg-Weyl group HW(d)HW(d)HW(d) related to a ddd-dimensional Hilbert space H(d)H(d)H(d), is enlarged into the Heisenberg-Weyl-parity group HWP(d)HWP(d)HWP(d) that incorporates parity transformations. It consists of 2d32d^32d3 elements, of which d3d^3d3 elements belong to the HW(d)HW(d)HW(d) subgroup, and extra d3d^3d3 elements which are related through a Fourier transform with the former ones. It is shown that HWP(d)HWP(d)HWP(d) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. HWP(d)HWP(d)HWP(d) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.2d22d^22d2 coherent states related to the HWP(d)HWP(d)HWP(d) group are introduced, which consist of d2d^2d2 coherent states related to the HW(d)HW(d)HW(d) subgroup, and extra d2d^2d2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2d22d^22d2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d2d^2d2 coherent states related to HW(d)HW(d)HW(d). One of the consequences of the HWP(d)HWP(d)HWP(d) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.