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arXiv:2009.13425

Constructing Coherent States for the Rational Extensions of the Harmonic Oscillator Potential

Z. M. McIntyre, Robert Milson

Published Sep 28, 2020• 0 citations
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Abstract

Using the formalism of Maya diagrams and ladder operators, we describe the algebra of annihilating operators for the class of rational extensions of the harmonic oscillators. This allows us to construct the corresponding coherent state in the sense of Barut and Girardello. The resulting time-dependent function is an exact solution of the time-dependent Schrodinger equation and a joint eigenfunction of the algebra of annihilators.

DOI
10.48550/arxiv.2009.13425

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Using the formalism of Maya diagrams and ladder operators, we describe the algebra of annihilating operators for the class of rational extensions of the harmonic oscillators. This allows us to construct the corresponding coherent state in the sense of Barut and Girardello. The resulting time-dependent function is an exact solution of the time-dependent Schrodinger equation and a joint eigenfunction of the algebra of annihilators.
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We construct ladder operators, C̃ and C̃†, for a multistep rational extension of the harmonic oscillator on the half plane, x ≥ 0. These ladder operators connect all states of the spectrum in only infinite-dimensional representations of their polynomial Heisenberg algebra. For comparison, we also construct two different classes of ladder operator acting on this system that form finite-dimensional as well as infinite-dimensional representations of their respective polynomial Heisenberg algebras. For the rational extension, we construct the position wavefunctions in terms of exceptional orthogonal polynomials. For a particular choice of parameters and for the three lowest weights μ = −5, −3, and 5, we construct the coherent states, eigenvectors of C̃ with generally complex eigenvalues, z, as superposition of subsets of the energy eigenvectors. Then, we calculate the properties of these coherent states, looking for classical or nonclassical behavior. We calculate the energy expectations as functions of |z|. We plot position probability densities for the coherent states and for the even and odd cat states formed from these coherent states. We plot the Wigner functions for a particular choice of z. For these coherent states on one arm of a beamsplitter, we calculate the two excitation number distributions and the linear entropies of the output states. We plot the standard deviations in x and find squeezing in the regime considered in one of the cases. By plotting the Mandel Q parameters for the coherent states as functions of |z|, we find that the number statistics is sub-Poissonian in all cases.
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In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.
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