$SU(1,1)$ and $SU(2)$ Perelomov number coherent states: algebraic approach for general systems

TL;DR

This paper explores the properties and time evolution of $SU(1,1)$ and $SU(2)$ Perelomov number coherent states, including uncertainty relations and specific examples like the pseudoharmonic oscillator.

Contribution

It provides an algebraic approach to analyze Perelomov number coherent states for $SU(1,1)$ and $SU(2)$, including their uncertainty properties and dynamics.

Findings

Uncertainty is minimized for standard coherent states.

Derived time evolution for states with Hamiltonian proportional to $K_0$.

Explicit examples for pseudoharmonic and isotropic harmonic oscillators.

Abstract

We study some properties of the $SU(1,1)$ Perelomov number coherent states. The Schr\"odinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator $K_0$ of the $su(1,1)$ Lie algebra. Analogous results for the $SU(2)$ Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.