Phase operators, phase states and vector phase states for SU(3) and SU(2,1)

TL;DR

This paper develops a unified algebraic framework for phase operators and states in SU(3) and SU(2,1) symmetries, introducing a generalized oscillator algebra that encompasses various Lie algebra representations and their phase properties.

Contribution

It introduces a one-parameter generalized oscillator algebra A(k,2) that unifies the treatment of SU(3), SU(2,1), and related symmetries, and constructs associated phase states.

Findings

Defined phase operators for A(k,2) algebra.

Constructed temporally stable phase and vector phase states.

Linked quantized phase states to quadratic discrete Fourier transform.

Abstract

This paper focuses on phase operators, phase states and vector phase states for the sl(3) Lie algebra. We introduce a one-parameter generalized oscillator algebra A(k,2) which provides a unified scheme for dealing with su(3) (for k < 0), su(2,1) (for k > 0) and h(4) x h(4) (for k = 0) symmetries. Finite- and infinite-dimensional representations of A(k,2) are constructed for k < 0 and k > 0 or = 0, respectively. Phase operators associated with A(k,2) are defined and temporally stable phase states (as well as vector phase states) are constructed as eigenstates of these operators. Finally, we discuss a relation between quantized phase states and a quadratic discrete Fourier transform and show how to use these states for constructing mutually unbiased bases.