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International Journal of Modern Physics B

TRICOMI COHERENT STATES

J. M. Sixdeniers, K. A. Penson, John R. Klauder

Published Dec 30, 2001• 6 citations
Publisher

Abstract

We construct explicit families of coherent states whose parametrization permits an arbitrarily small deviation from conventional coherent states. They possess the resolution of unity with a positive weight function given as the solution of a Stieltjes power-moment problem, with moments analytically expressible through Tricomi's confluent hypergeometric function. These states are in general squeezed, can be either sub- or super-Poissonian in nature, and induce a deformation of the metric of the complex plane.

DOI
10.1142/s0217979201007890

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We construct explicit families of coherent states whose parametrization permits an arbitrarily small deviation from conventional coherent states. They possess the resolution of unity with a positive weight function given as the solution of a Stieltjes power-moment problem, with moments analytically expressible through Tricomi's confluent hypergeometric function. These states are in general squeezed, can be either sub- or super-Poissonian in nature, and induce a deformation of the metric of the complex plane.
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 new family of boson coherent states using a specially designed function which is a solution of a functional equation dε(q,x)/dx=ε(q,qx) with 0⩽q⩽1 and ε(q,0)=1. We use this function in place of the usual exponential to generate new coherent states |q,z〉 from the vacuum, which are normalized and continuous in their label z. These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally-ordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is sub-Poissonian. For any q<1 the states |q,z〉 are squeezed; we obtain and discuss their signal to quantum noise ratio. The function ε(q,x) allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For q=1 all the above results reduce to those for conventional coherent states. Finally, we establish a link with q-deformed bosons.
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.
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.