Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras

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, theremust 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 beamsplitter to showthat the nonlinear coherent states exhibit properties like antibunching 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.