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Physica Scripta

X m –EOP/PDEM connection for Scarf I potentials and their coherent states revivals

Sid-Ahmed Yahiaoui, Othmane Cherroud

Published Dec 18, 2024• 1 citation
Publisher

Abstract

Abstract The revival structures for X m -exceptional orthogonal polynomials (EOP) of Scarf I potentials endowed with position-dependent effective mass (PDEM) are studied in the context of the generalized Gazeau-Klauder coherent states. It is shown that the deduced coherent states mimic full and fractional revivals phenomena in the case of the constant-mass. However, in the case of position-dependent effective mass, although full revivals take place during their time evolution, there is no trace of fractional revivals as defined in the common sense. These properties are illustrated numerically by means of some specific profiles of the mass-function, with and without singularities. We have also observed a close connection between the coherence time τ coh [ m ] and the mass-parameter λ . We suspect the effect that a larger mass-parameter λ contributes significantly to a longer coherence time τ coh [ m ] .

DOI
10.1088/1402-4896/ada11a

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Abstract The revival structures for X m -exceptional orthogonal polynomials (EOP) of Scarf I potentials endowed with position-dependent effective mass (PDEM) are studied in the context of the generalized Gazeau-Klauder coherent states. It is shown that the deduced coherent states mimic full and fractional revivals phenomena in the case of the constant-mass. However, in the case of position-dependent effective mass, although full revivals take place during their time evolution, there is no trace of fractional revivals as defined in the common sense. These properties are illustrated numerically by means of some specific profiles of the mass-function, with and without singularities. We have also observed a close connection between the coherence time τ coh [ m ] and the mass-parameter λ . We suspect the effect that a larger mass-parameter λ contributes significantly to a longer coherence time τ coh [ m ] .
The revival structures for the X_m exceptional orthogonal polynomials of the Scarf I potential endowed with position-dependent effective mass is studied in the context of the generalized Gazeau-Klauder coherent states. It is shown that in the case of the constant mass, the deduced coherent states mimic full and fractional revivals phenomena. However in the case of position-dependent effective mass, although full revivals take place during their time evolution, there is no fractional revivals as defined in the common sense. These properties are illustrated numerically by means of some specific profile mass functions, with and without singularities. We have also observed a close connection between the coherence time τ_coh^m? and the mass parameter ?.
Gazeau-Klauder coherent states of the extended trigonometric Scarf potential, underlying the quadratic energy spectrum and associated with Jacobi-type Xm exceptional orthogonal polynomials P(a,b,m) n (x), are constructed. The temporal evolution of wave-packet coherent states are performed by means of an autocorrelation function and the full revival properties are investigated in the usual time-domain analysis. This latter seems to be less useful for describing the fractional revivals due to the complicated nature of coherent wave-packet. Fortunately the autocorrelation function revels a little signature of fractional revivals at the vicinity of quarters of the revival time Trev due to the quadratic energy spectrum and the use of the wavelet-based time-frequency analysis of the autocorrelation function provides an analytical and numerical observation of the fractional revivals at different orders of the system.
Abstract We develop generalized coherent states for a class of nonlinear oscillators with position-dependent effective mass in the context of the Gazeau–Klauder formalism and discuss some of their properties. In order to investigate the temporal evolution we first explore the statistical properties by means of weighting distribution and the Mandel parameter. It is found that the temporal evolution of the coherent states may exhibit the phenomena of quantum revivals and fractional revivals for a particular choice of position-dependent mass oscillator.
We develop generalized coherent states based on the Gazeau–Klauder formalism for a particle with position-dependent mass trapped in an infinite square well. We study the quantum statistical properties of these states by means of the Mandel parameter and the second-order correlation function. Our analysis reveals that the constructed coherent states exhibit sub-Poissonian statistics. Moreover, theoretical investigations of wave packet revivals and fractional revivals for the pertaining system have been performed by means of the autocorrelation function and temporal evolution of probability density.
We show that the Scarf I potential problem in a position-dependent mass background of the type m(α;x)=(1+αsin⁡x)−2m(α;x) = (1 + α\sin x)^{-2}m(α;x)=(1+αsinx)−2, 0<α<10<α<10<α<1, can be solved by using a point canonical transformation mapping the corresponding Schr\" odinger equation onto that of the Scarf I potential with constant mass. The inverse point canonical transformation then provides some exactly-solvable rational extensions of the Scarf I potential with positive-dependent mass associated with XmX_mXm​-Jacobi exceptional orthogonal polynomials of type I, II, or III. The Scarf I potential problem with position-dependent mass is shown to exhibit a deformed shape invariance property in a deformed supersymmetric framework. Such a property is also valid for extended potentials of type I and II. The results are illustrated with a simple example.