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arXiv:quant-ph/0411094 · Journal of Mathematical Physics

Construction of the dual family of Gazeau–Klauder coherent states via temporally stable nonlinear coherent states

R. Roknizadeh, Mohammad Kazem Tavassoly

Published Mar 28, 2005• 49 citations
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Abstract

Using the analytic representation of the so-called Gazeau–Klauder coherent states (CSs), we shall demonstrate that how a new class of generalized CSs, namely the family of dual states associated with theses states, can be constructed through viewing these states as temporally stable nonlinear CSs. Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the supersymmetric quantum mechanics (SUSYQM) techniques. Then, we have applied this method to some physical systems with known spectrum, such as Pöschl–Teller, infinite well, Morse potential and hydrogenlike spectrum as some quantum mechanical systems. Finally, we propose the generalized form of the Gazeau–Klauder CS and the corresponding dual family.

DOI
10.1063/1.1861276

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Using the analytic representation of the so-called Gazeau–Klauder coherent states (CSs), we shall demonstrate that how a new class of generalized CSs, namely the family of dual states associated with theses states, can be constructed through viewing these states as temporally stable nonlinear CSs. Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the supersymmetric quantum mechanics (SUSYQM) techniques. Then, we have applied this method to some physical systems with known spectrum, such as Pöschl–Teller, infinite well, Morse potential and hydrogenlike spectrum as some quantum mechanical systems. Finally, we propose the generalized form of the Gazeau–Klauder CS and the corresponding dual family.
In this tutorial I intend to present some of the results I obtained through my PhD work in the "quantum optics group of the University of Isfahan" under consideration Dr. R. Roknizadeh and Prof. S. Twareque Ali as my supervisor and advisor, respectively. I will revisit some of the pioneering proposals recently developed the concept of generalized CSs. As it can be observed the customary three generalization methods {\it (symmetry, algebraic and dynamical)} have never been considered in neither of them. Our intention in this work is at first to investigate the lost ring between the customary three methods and the recently developed ones, as possible. For this purpose it has been devised general analytic descriptions, which successfully demonstrate how different varieties of CSs (which are nonlinear in nature) can be obtained by two processes, first the {\it "nonlinear CSs"} method and second by {\it "basis transformations on an underlying Hilbert spaces"}. As a result, I will systematize the recently introduced generalized CSs in a clear and concise way. It will be clear also, that how our results can be considered as a first step in the generation process of the mathematical physics CSs in the context of quantum optics. Besides this, some new results emerge from our studies. I introduce a large classes of generalized CSs, namely the {\it "dual family"} associated with each set of early known CSs. But, in this relation, the previous processes for constructing the dual pair of Gazeau-Klauder CSs fail to work well, so I outlined a rather different method based on the {\it "temporal stability"} requirement of generalized nonlinear CSs.
This paper deals with an extension of our previous work [J. Phys. A: Math. Theor. {\bf 40} F817] by considering an alternative construction of canonical and deformed vector coherent states (VCSs) of the Gazeau-Klauder type associated with generalized spin-orbit Hamiltonians. We define an annihilation operator which takes into account the finite dimensional space of states induced by the kkk-photon transition processes of the two-level atom interacting with the single-mode radiation field. The class of nonlinear VCSs (NVCSs) corresponding to the action of the annihilation operator is deduced and expressed in terms of generalized displacement operators. Various NVCSs including their "dual" counterparts are also discussed. Still by using the Hilbert space structure, a new family of NVCSs parameterized by unit vectors of the S3S^3S3 sphere has been identified without making use of the annihilation operator.
Using the nonlinear coherent states method , a formalism for the construction of the coherent states associated with ‘ inverse bosonic operators ’ and their dual family has been proposed. Generalizing the approach, the ‘inverse of f -deformed ladder operators’ corresponding to the nonlinear coherent states in the context of quantum optics and the associated coherent states have been introduced. Finally, after applying the proposal to a few known physical systems, particular nonclassical features as sub-Poissonian statistics and the squeezing of the quadratures of the radiation field corresponding to the introduced states have been investigated.
The so-called Gazeau–Klauder and Perelomov coherent states are introduced for an arbitrary quantum system. We give also the general framework to construct the generalized intelligent states which minimize the Robertson–Schrödinger uncertainty relation. As illustration, the Pöschl–Teller potentials of trigonometric type will be chosen. We show the advantage of the analytical representations of Gazeau–Klauder and Perelomov coherent states in obtaining the generalized intelligent states in analytical way.