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Maths-type q-deformed coherent states for q > 1

Quesne, C, K. A. Penson, Tkachuk, V M

Published Mar 19, 2003• 37 citations
OpenAlex

Abstract

Maths-type q-deformed coherent states with $q > 1$ allow a resolution of unity in the form of an ordinary integral. They are sub-Poissonian and squeezed. They may be associated with a harmonic oscillator with minimal uncertainties in both position and momentum and are intelligent coherent states for the corresponding deformed Heisenberg algebra.

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Maths-type q-deformed coherent states with q>1q > 1q>1 allow a resolution of unity in the form of an ordinary integral. They are sub-Poissonian and squeezed. They may be associated with a harmonic oscillator with minimal uncertainties in both position and momentum and are intelligent coherent states for the corresponding deformed Heisenberg algebra.
We construct a new family of q -deformed coherent states | z ⟩ q , where 0 < q < 1. These states are normalizable on the whole complex plane and continuous in their label z . They allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated Stieltjes power-moment problem and expressed in terms of one of the two Jacksons's q -exponentials. They also permit exact evaluation of matrix elements of physically-relevant operators. We use this to show that the photon number statistics for the states is sub-Poissonian and that they exhibit quadrature squeezing as well as an enhanced signal-to-quantum noise ratio over the conventional coherent state value. Finally, we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics.
We create a family of boson coherent states using the functions of Mittag-Leffler (ML) E ( z ), ( >0) and their generalizations E , ( z ), ( , >0) instead of exponentials. These states are shown to satisfy the usual requirements of normalizability, continuity in the label and the resolution of unity with a positive weight function. This last quantity is found for arbitrary , >0 as a solution of an associated Stieltjes moment problem. In addition, for = m = 1,2,3 ... and = 1 (corresponding to E m ( z )) we propose and analyse special q -deformations (0< q 1) of the functions E m ( z ) which serve as a tool to define q -deformed coherent states of ML type. We provide the expressions for expectation values of physical quantities for all the above states. We discuss physical properties of these states, noting that they are squeezed. The ML coherent states are sub-Poissonian in nature, whereas the q -deformed ML states can be sub- and super-Poissonian depending on q . All these states are shown to be eigenstates of deformed boson operators whose commutation relations are given.
We construct a family of deformed boson coherent states associated with deformed Bogoliubov (p, q)-transformations in an infinite dimensional subspace of the harmonic oscillator Hilbert space without first finite Fock states. We investigate their over-completeness and show that they allow the resolution of unity in the form of an ordinary integral (for Q = pq < 1) or a generalized Q-deformed one (for Q = pq > 1). We study in detail analytically and numerically some of the geometrical and physical properties of these deformed coherent states in the context of deformed quantum optics. In particular, we show that for Q > 1 they exhibit sub-Poissonian statistics and no quadrature squeezing occurs while for Q < 1 their photon number statistics is super-Poissonian and there is a simultaneous squeezing in both field quadratures (double squeezing). Additionally, by a natural extension, we construct the corresponding multi-photon deformed coherent states and investigate their properties.
We construct a new family of deformed states. These states are normalizable on the whole complex plane and continuous in their label z . Under fixed values of parameters, they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated Stieltjes power-moment problem. In addition to the mathematical characteristics, the quantum statistical properties of these states are analytically and numerically discussed in detail in the context of conventional as well as deformed quantum optics. We find that, for non-deformed photons, the states exhibit quadrature squeezing and their photon number statistics is sub-Poissonian. On the other hand, for deformed photons, the states are super-Poissonian and no quadrature squeezing occurs.
Adopting the framework of the generalized q -deformed Heisenberg–Weyl algebra , we present a mathematical procedure which leads us to obtain analytical expressions for a general class of q -deformed coherent states associated with the different patterns of the energy spectrum exhibited by the nonlinear f -oscillator. In particular, we establish the properties of a small group of q -deformed coherent states for α > γ > 0 with emphasis on the resolution of unity. As an application of these properties, we investigate the Robertson–Schrödinger uncertainty relation and the squeezing effect for the deformed coordinate and momentum operators, which are defined in terms of the abstract elements of this algebra. Furthermore, we also obtain the Wigner function and the correct quantum-mechanical marginal distributions in phase space.