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Journal of Physics Conference Series

First-Order Polynomial Heisenberg Algebras and Coherent States

Miguel Castillo-Celeita, David J. Fernández C.

Published Mar 1, 2016• 2 citations
Publisher

Abstract

The polynomial Heisenberg algebras (PHA) are deformations of the Heisenberg- Weyl algebra characterizing the underlying symmetry of the supersymmetric partners of the Harmonic oscillator. When looking for the simplest system ruled by PHA, however, we end up with the harmonic oscillator. In this paper we are going to realize the first-order PHA through the harmonic oscillator. The associated coherent states will be also constructed, which turn out to be the well known even and odd coherent states.

DOI
10.1088/1742-6596/698/1/012007

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The polynomial Heisenberg algebras (PHA) are deformations of the Heisenberg- Weyl algebra characterizing the underlying symmetry of the supersymmetric partners of the Harmonic oscillator. When looking for the simplest system ruled by PHA, however, we end up with the harmonic oscillator. In this paper we are going to realize the first-order PHA through the harmonic oscillator. The associated coherent states will be also constructed, which turn out to be the well known even and odd coherent states.
Abstract In this paper we will realize the polynomial Heisenberg algebras through the harmonic oscillator. We are going to construct then the Barut–Girardello coherent states, which coincide with the so-called multiphoton coherent states, and we will analyze the corresponding Heisenberg uncertainty relation and Wigner distribution function for some particular cases. We will show that these states are intrinsically quantum and cyclic, with a period being a fraction of the oscillator period. The associated geometric phases will be as well evaluated.
We review the higher‐order supersymmetric quantum mechanics (H‐SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first‐order SUSY transformations are used to derive in a simple way the higher‐order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H‐SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.
At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.
Abstract The multiphoton algebras for one-dimensional Hamiltonians with infinite discrete spectrum, and for their associated kth-order SUSY partners are studied. In both cases, such an algebra is generated by the multiphoton annihilation and creation operators, as well as by Hamiltonians which are functions of an appropriate number operator. The algebras obtained turn out to be polynomial deformations of the corresponding single-photon algebra previously studied in literature. The Barut-Girardello coherent states, which are eigenstates of the annihilation operator, are obtained and their uncertainty relations are explored by means of the associated quadratures.
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl [Formula: see text]-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.