Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras

TL;DR

This paper demonstrates that generalized oscillator algebras can produce nonlinear coherent states with classical-like P-representations, yet these states exhibit nonclassical properties such as anti-bunching, challenging their classical interpretation.

Contribution

It introduces a unified construction of nonlinear coherent states using Meijer G-functions and clarifies their nonclassical nature despite delta-function P-representations.

Findings

Nonlinear coherent states satisfy a Meijer G-function closure relation.

These states exhibit anti-bunching, indicating nonclassical behavior.

They lack second-order coherence despite classical P-representations.

Abstract

The purposes of this work are (1) to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form; and (2) to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are P-represented by a delta function. In (1) we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer G-function. This property automatically defines the delta distribution as the P-representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the su(1,1) Lie algebra that is realized as a deformation of the oscillator algebra. In (2), we use a beam splitter to show that the nonlinear coherent states exhibit properties like anti-bunching that prohibit a classical description for them. We also show that these states lack second order coherence. That is, although the P-representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered `classical' in the context of the quantum theory of optical coherence.