Coherent States of su(1,1): Correlations, Fluctuations, and the Pseudoharmonic Oscillator

TL;DR

This paper extends the analysis of coherent states to su(1,1), deriving semiclassical expansions and examining their application to the pseudoharmonic oscillator, highlighting differences in classical limit proximity between two types of states.

Contribution

It introduces systematic semiclassical expansions for su(1,1) coherent states and compares different state types in the context of the pseudoharmonic oscillator.

Findings

Perelomov-Gilmore states approach classical limit with constant energy variance.

Barut-Girardello states have energy variance decreasing with inverse square root of energy.

Leading order energy uncertainty depends on classical variable dynamics.

Abstract

We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov-Gilmore coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the Perelomov-Gilmore states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.