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arXiv:1709.08835 · Journal of Physics A Mathematical and Theoretical

Non-classical behaviour of coherent states for systems constructed using exceptional orthogonal polynomials

Scott E. Hoffmann, Véronique Hussin, Ian Marquette, Yao-Zhong Zhang

Published Jan 5, 2018• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aaa553

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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.
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.
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.
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 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.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.