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Physical Review A

Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems

John R. Klauder, K. A. Penson, J. M. Sixdeniers

Published Jun 13, 2001• 155 citations
Publisher

Abstract

We construct a large number of coherent states satisfying the resolution of unity with a positive weight function, obtained through analytic solutions of the Stieltjes moment problem (coherent states on a plane) and the Hausdorff moment problem (coherent states on a disk). These solutions are obtained through the method of inverse Mellin transform. In addition, these coherent states induce a deformation of the metric that has been calculated analytically.

DOI
10.1103/physreva.64.013817

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We construct a large number of coherent states satisfying the resolution of unity with a positive weight function, obtained through analytic solutions of the Stieltjes moment problem (coherent states on a plane) and the Hausdorff moment problem (coherent states on a disk). These solutions are obtained through the method of inverse Mellin transform. In addition, these coherent states induce a deformation of the metric that has been calculated analytically.
We construct explicit families of coherent states whose parametrization permits an arbitrarily small deviation from conventional coherent states. They possess the resolution of unity with a positive weight function given as the solution of a Stieltjes power-moment problem, with moments analytically expressible through Tricomi's confluent hypergeometric function. These states are in general squeezed, can be either sub- or super-Poissonian in nature, and induce a deformation of the metric of the complex plane.
We construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation dε(q,x)/dx=ε(q,qx) with 0⩽q⩽1 and ε(q,0)=1. We use this function in place of the usual exponential to generate new coherent states |q,z〉 from the vacuum, which are normalized and continuous in their label z. These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally-ordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is sub-Poissonian. For any q<1 the states |q,z〉 are squeezed; we obtain and discuss their signal to quantum noise ratio. The function ε(q,x) allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For q=1 all the above results reduce to those for conventional coherent states. Finally, we establish a link with q-deformed bosons.
Abstract. The canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ(m). The requirement that these states realize a resolution of the identity typically ends up in a moment problem, where the moments form the positive sequence of real numbers {ρ(m)} ∞ m=0. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ(m) of the canonical coherent states by n × n matrices. A matrix moment problem is proposed and it is solved by introducing a matrix weight. Associated oscillator algebras are discussed. Finally, as a physical application, a generalized version of the Jaynes-Cummings model is discussed. 1.
Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on natural numbers. We follow in this work the same path by considering sequences of non-negative numbers and their associated “non-linear” coherent states. We illustrate our approach with the 2-d motion of a charged particle in a uniform magnetic field. By solving the involved Stieltjes moment problem we construct a family of coherent states for this model. We then proceed with the corresponding coherent state quantization and we show that this procedure takes into account the circle topology of the classical motion.