This paper has been motivated by a recent paper by Dey [Phys. Rev. D 91, 044024 (2015)] on the known Arik-Coon
q oscillator. We construct
q coherent, even and odd
q-cat states in Fock representation for the Biedenharn-Macfarlane
q oscillator with
q>1 and study their nonclassical properties. The
q-coherent states minimize the Heisenberg uncertainty relation between the generalized position and momentum operators as well as the
x and
y components of a
q-deformed
su(1,1) algebra in the Schwinger boson representation. The latter is also minimized by the even and odd
q-cat states. We show that, contrary to the undeformed harmonic oscillator, the squeezing effect in both position and momentum operators can be exhibited by odd
q-cat states. It is also violated by even
q-cat states. Furthermore, it is shown that the antibunching effect and sub-Poissonian or super-Poissonian statistics can simultaneously appear by each of the even or odd
q-cat states. Finally, a unitary Fock representation of the
q-deformed
su(1,1) algebra is obtained by the
q-deformed Bargmann-Fock realization.
The purposes of this work are 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 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 -represented by a delta function. In 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 -function. This property automatically defines the delta distribution as the -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 Lie algebra that is realized as a deformation of the oscillator algebra. In , we use a beam splitter to show that 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 -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.