Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators

The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.

Completeness and Nonclassicality of Coherent States for Generalized Oscillator Algebras

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 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.

From the Wigner–Eckart theorem to the Hilbert–Schmidt scalar product for an adjoint representation of the quantum algebra Up,q(su2)

Irreducible tensor operators as the irreducible submodules of an adjoint representation of the two-parametric quantum ∗-algebra [Formula: see text] are constructed by using its Jordan–Schwinger formulation on two independent [Formula: see text]-oscillator ∗-algebras. All [Formula: see text]-submodules are equipped with an appropriate Hilbert–Schmidt scalar product with the help of the Wigner–Eckart theorem. We show that with respect to this scalar product, not only the bases of all irreducible submodules of the adjoint representation are orthonormal, but also the adjoint representation is a ∗-representation.

New version of pseudo-hermiticity in the two-sided deformation of Heisenberg algebra

The recently introduced two- and three-parameter [Formula: see text]- and [Formula: see text]-deformed extensions of the Heisenberg algebra (HA) were explored under the condition of their connectedness with the respective nonstandard (other than known ones) deformed quantum oscillator algebras. In this paper, we show that such connection dictates certain new [Formula: see text]-pseudo-Hermitian conjugation rule between the creation and annihilation operators with [Formula: see text] depending on the particle number operator [Formula: see text]. In turn, that leads to the related [Formula: see text]-pseudo-hermiticity of the position–momentum operators, though the involved Hamiltonian is Hermitian. Different possible cases are studied, and some interesting features implied by the use of such [Formula: see text]-based conjugation rule are emphasized.