Coherent State Quantization and Moment Problem

J P Gazeau,M C Baldiotti,D M Gitman

doi:10.14311/1185

Abstract

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.

Highlights

One of the most interesting properties of standard or Glauber coherent states |z [1, 2, 3, 4] is the Bayesian duality [5, 6] that they encode between the discrete Poisson probability distribution, n → e−|z|2|z|2/n!, of obtaining n quantum excitations (“photons” or “quanta”) in a measurement through some counting device, and the continuous Gamma probability distribution measure |z|2 → e−|z|2|z|2/n! on the classical phase space
For this latter distribution, |z|2 is itself a random variable, denoting the average number of photons, given that n photons have been counted. Such a duality underlies the construction of all types of coherent state families, provided they satisfy a resolution of the unity condition
We carry out the corresponding coherent state quantization and we examine the consequences in terms of its probabilistic, functional, and localization aspects

Summary

Introduction

One of the most interesting properties of standard or Glauber coherent states |z [1, 2, 3, 4] is the Bayesian duality [5, 6] that they encode between the discrete Poisson probability distribution, n → e−|z|2|z|2/n!, of obtaining n quantum excitations (“photons” or “quanta”) in a measurement through some counting device, and the continuous Gamma probability distribution measure |z|2 → e−|z|2|z|2/n! on the classical phase space. It turns out that this condition is equivalent to setting up a “positive operator valued measure” (POVM) [7, 4] on the phase space Such a measure, in turn, leads to the quantization of the classical phase√space, which associates to each point z ≡ (q + ip)/ 2 the one dimensional projection operator Pz, projecting onto to the subspace generated by the coherent state vector, and for z = z , PzPz = Pz Pz). By introducing a kind of squeezing parameter q = eλ > 1 we extend the definition of the latter and solve the corresponding Stieltjes moment problem This allows us to proceed with the quantization of the physical quantities and illustrate our study with numerical investigation

Quantization with standard coherent states and

Coherent states for generic sequences

CS quantization with sequence X

An example: charged particle in a magnetic field