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arXiv:2310.08799 · Physica Scripta

Optical ladder operators in the Glauber-Fock oscillator array

Ivan A. Bocanegra-Garay, Leonardi Hernández-Sánchez, I. Ramos-Prieto, Francisco Soto, H. M. Moya-Cessa

Published Jan 19, 2024• 5 citations
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Abstract

Abstract In this study, we investigate the stationary states of the Glauber-Fock oscillator waveguide array. We begin by transforming the associated Hamiltonian into the form of a quantum harmonic oscillator Hamiltonian, allowing the implementation of a supersymmetric (SUSY) approach. By considering the simplest case for the intertwining operator, the optical ladder operators are straightforwardly constructed and shown to map eigensolutions into eigensolutions of the corresponding Hamiltonian operator, in pretty much the same manner as it is done for the quantum harmonic oscillator case. The ladder of the corresponding (eigen) supermodes is then easily established.

DOI
10.1088/1402-4896/ad20bd

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Abstract In this study, we investigate the stationary states of the Glauber-Fock oscillator waveguide array. We begin by transforming the associated Hamiltonian into the form of a quantum harmonic oscillator Hamiltonian, allowing the implementation of a supersymmetric (SUSY) approach. By considering the simplest case for the intertwining operator, the optical ladder operators are straightforwardly constructed and shown to map eigensolutions into eigensolutions of the corresponding Hamiltonian operator, in pretty much the same manner as it is done for the quantum harmonic oscillator case. The ladder of the corresponding (eigen) supermodes is then easily established.
A simple way to construct exactly solvable Hamiltonians whose spectra contain two equidistant ladders, one finite and another infinite, appears when applying supersymmetric quantum mechanics to the harmonic oscillator. Some of those supersymmetric partners have third order differential ladder operators, although the order of the transformation is higher than one. In this work the linearized coherent states for these specific Hamiltonians are studied. To each SUSY partner Hamiltonian corresponds two families of linearized coherent states: one inside the subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states inserted through the SUSY technique.
The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator (QMHO) carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general – except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalise the Heisenberg uncertainty principle. Coupled SUSY is explicitly realised by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalised harmonic oscillators.
In this work, we derive two equivalent non-rational extensions of the quantum harmonic oscillator using two different supersymmetric transformations. For these extensions, we built ladder operators as the product of the intertwining operators related with these equivalent supersymmetric transformations, which results in two-step ladder operators. We linearised these operators to obtain operators of the same nature that follow a linear commutation relation. After the linearisation, we derive coherent states as eigenstates of the annigilation operator and analyse some relevant mathematical and physical properties, such as the completeness relation, mean-energy values, temporal stability, time evolution of the probability densities, and Wigner distributions. From these properties, we conclude that these coherent states present both classical and quantum behaviour.
We present, for the isospectral family of oscillator Hamiltonians, a systematic procedure for constructing raising and lowering operators satisfying any prescribed “distorted” Heisenberg algebra (including the q-generalization). This is done by means of an operator transformation implemented by a shift operator. The latter is obtained by solving an appropriate partial isometry condition in the Hilbert space. Formal representations of the nonlocal operators concerned are given in terms of pseudo-differential operators. Using the new annihilation operators, new classes of coherent states are constructed for isospectral oscillator Hamiltonians. The corresponding Fock–Bargmann representations are also considered, with specific reference to the order of the entire function family in each case.
The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.