Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems

TL;DR

This paper introduces a generalized oscillator algebra and constructs phase operators and stable states for finite and infinite-dimensional quantum systems, with applications to quantum systems and information theory.

Contribution

It develops a generalized algebra A(k), constructs phase operators and stable states, and introduces a truncation procedure for finite-dimensional cases, extending prior methods.

Findings

Constructed phase operators for generalized oscillator algebra.

Developed a truncation method for infinite-dimensional cases.

Applied to quantum systems with finite and infinite spectra and to mutually unbiased bases.

Abstract

We introduce a one-parameter generalized oscillator algebra A(k) (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter k. We define an (Hamiltonian) operator associated with A(k) and examine the degeneracies of its spectrum. For the finite (when k < 0) and the infinite (when k > 0 or = 0) representations of A(k), we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated generalized oscillator algebra A(k,s), where s denotes the truncation order. We construct two types of temporally stable states for A(k,s) (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of A(k,s)). Two applications are considered in this article. The first concerns physical realizations of A(k) and A(k,s) in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Poeschl-Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.