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arXiv:1308.4470

Quantum Revivals of Morse Oscillators and Farey-Ford Geometry

Alvason Zhenhua Li, William G. Harter

Published Aug 21, 2013• 0 citations
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Abstract

Analytical eigensolutions for Morse oscillators are used to investigate quantum resonance and revivals and show how Morse anharmonicity affects revival times. A minimum semi-classical Morse revival time Tmin-rev found by Heller is related to a complete quantum revival time Trev using a quantum deviation parameter that in turn relates Trev to the maximum quantum beat period Tmax-beat. Also, number theory of Farey and Thales-circle geometry of Ford is shown to elegantly analyze and display fractional revivals. Such quantum dynamical analysis may have applications for spectroscopy or quantum information processing and computing.

DOI
10.48550/arxiv.1308.4470

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Analytical eigensolutions for Morse oscillators are used to investigate quantum resonance and revivals and show how Morse anharmonicity affects revival times. A minimum semi-classical Morse revival time Tmin-rev found by Heller is related to a complete quantum revival time Trev using a quantum deviation parameter that in turn relates Trev to the maximum quantum beat period Tmax-beat. Also, number theory of Farey and Thales-circle geometry of Ford is shown to elegantly analyze and display fractional revivals. Such quantum dynamical analysis may have applications for spectroscopy or quantum information processing and computing.
Analytical eigenfunctions and eigenvalues for the Morse oscillator were applied to investigate the quantum resonant beats and revivals of wave packet propagation. A concise way for exact prediction of the complete revival period of the Morse oscillator was given for the first time. It was suggested that any complete period was made of integer numbers of the minimum or fundamental period. Within the fundamental period, the anharmonicity of this oscillator appeared to cause interesting space-time phenomena that include relatively simple Farey-sum revival structures. In addition, a simple sum of two Morse oscillators led to a double-Morse well whose geometric symmetry provided analytical eigenfunctions and eigenvalues for certain low-lying energy levels. The quantum tunneling between the double-Morse well significantly affected the resonant beats and revivals local to each well, and gave rise to interesting tsunami-like waves in the middle of the double well. Furthermore, quantum rotor wave functions based upon Wigner-D matrix were applied to investigate the quantum resonant beats and revivals that occur in experimentally accessible spin systems. Interesting physical effects in quantum rotors between half-integer spin and integer spin systems were observed to show effects of symmetry. Essentially, the quantum revivals in these quantum systems exhibited number-information aspects of surprisingly simple Farey-sum and Ford circles geometry. Such quantum dynamics will provide a physical insight to further develop matter wave packet technology, and might have applications for quantum information processing and quantum computing.
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We consider the revival properties of quantum systems with an eigenspectrum E n n 2 , and compare them with the simplest member of this class - the infinite square well. In addition to having perfect revivals at integer multiples of the revival time t R , these systems all enjoy perfect fractional revivals at quarterly intervals of t R . A closer examination of the quantum evolution is performed for the Pöschel-Teller and Rosen-Morse potentials, and comparison is made with the infinite square well using quantum carpets.
We study the temporal behavior of generic transient signals originating from multilevel quantum systems. Such signals typically arise in the physics of wave packets in atoms, molecules, cavity QED, and ion traps and consist of a sum of a large number of harmonics whose frequencies depend nonlinearly on the sequential number of the harmonic. In particular, we focus on the semiclassical limit. Here, quantum beats between individual terms in the underlying sum lead to characteristic features of the signal in different time regimes, such as collapse, fractional revivals, and full revivals. We present a universal recipe for describing analytically all of the details of these features. Our approach is based on a specific representation of the sum of harmonics, which is most convenient in each of these time regions of interest. This brings out in a most natural way the phenomenon of fractional revivals and full revivals and explains their fine structures observed in recent experiments. \textcopyright{} 1996 The American Physical Society.