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Acta Polytechnica

Linearised coherent states for non-rational SUSY extensions of the harmonic oscillator

Alonso Contreras-Astorga, David J. Fernández C., César Muro-Cabral

Published Feb 28, 2022• 2 citations
Publisher

Abstract

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.

DOI
10.14311/ap.2022.62.0030

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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.
Abstract We construct the coherent states and Schrödinger cat states associated with new types of ladder operators for a particular case of a rationally extended harmonic oscillator involving type III Hermite exceptional orthogonal polynomials. In addition to the coherent states of the annihilation operator, c , we form the linearised version, c ~ , and obtain its coherent states. We find that while the coherent states defined as eigenvectors of the annihilation operator c display only quantum behaviour, those of the linearised version, c ~ , have position probability densities displaying distinct wavepackets oscillating and colliding in the potential. The collisions are certainly quantum, as interference fringes are produced, but the remaining evolution indicates a classical analogue.
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.
Abstract The systems we consider are rational extensions of the harmonic oscillator, the truncated oscillator and the radial oscillator. The wavefunctions for the extended states involve exceptional Hermite polynomials for the oscillator and truncated oscillator and exceptional Laguerre polynomials for the radial oscillator. In all cases it is possible to construct ladder operators that have infinite-dimensional representations of their polynomial Heisenberg algebras and couple all levels of the systems. We construct Barut-Girardello coherent states in all cases, eigenvectors of the respective annihilation operators with complex eigenvalues. Then we calculate their physical properties to look for classical or non-classical behaviour.
Abstract We construct the coherent states of general order, m , for the ladder operators and , which act on rational deformations of the harmonic oscillator. The position wavefunctions of the eigenvectors involve type III Hermite exceptional orthogonal polynomials. We plot energy expectations, time-dependent position probability densities for the coherent states and for the even and odd cat states, Wigner functions, and Heisenberg uncertainty relations. We find generally non-classical behaviour, with one exception: there is a regime of large magnitude of the coherent state parameter, z , where the otherwise indistinct position probability density separates into m + 1 distinct wavepackets oscillating and colliding in the potential, forming interference fringes when they collide. The Mandel Q parameter is calculated to find sub-Poissonian statistics, another indicator of non-classical behaviour. We plot the position standard deviation and find squeezing in many of the cases. We calculate the two ‘photon’ number probability density for the output state when the m = 4, coherent states (where μ labels the lowest weight in the superposition) are placed on one arm of a beamsplitter. We find that it does not factorize, again indicating non-classical behaviour. Calculation of the linear entropy for this beamsplitter output state shows significant entanglement, another non-classical feature. We also construct linearized versions, , of the annihilation operators and their coherent states and calculate the same properties that we investigate for the coherent states. For these we find similar behaviour to the coherent states, at much smaller magnitudes of z , but comparable average energies.
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.
Abstract The 2:1 two-dimensional anisotropic quantum harmonic oscillator is considered and new sets of states are defined by means of normal-ordering non-linear operators through the use of non-commutative binomial theorems as well as solving recurrence relations. The states generated are good candidates for the natural generalisation of the s u ( 2 ) coherent states of the two-dimensional isotropic oscillator. The two-dimensional non-linear generalised ladder operators lead to several chains of states which are connected in a non trivial way. The uncertainty relations of the defining chain of states are calculated and it is found that they admit a resolution of the identity and the spatial distribution of the wavefunction produces Lissajous figures in correspondence with the classical 2:1 oscillator.