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Spatio Temporal Wave Packet Dynamics In Fermi Ulam Accelerator

Tasawar Abbas

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The idea of wave packet, that clarified many conceptual difficulties, has more recently been revisited to address interferometric and dynamical issues.Present work focuses on the spatio-temporal dynamics of the quantum wave packets in the Fermi-Ulam Accelerator Model.The specific system considered here is a one-dimensional box under periodic oscillations at one boundary. Such a seemingly simple system can be mapped to non-trivial problems like BEC trapped in an oscillating cavity or dynamics of atomic wave packet in different potential in presence or absence of gravitational field.The system is shown to display rich dynamical features including transitions from regular to stochastic and further to chaotic domain in classical phase space. We have obtained the approximated eigen-value solutions of the problem through time dependent Schrodinger’s equation under rescaling transformations that effectively replace moving boundary with a static one. Consequently the running time becomes a nonlinear function of the continuously changing boundary.Based on these solutions, revivals or the resurrection of the original wave packet, in the driven and un-driven Fermi-Ulam model are presented. Such revivals correspond to the constructive interference whereas the wave packet’s collapse owes to destructive interference. Furthermore, interplay of constructive and destructive interferences also readily mimics the fractional revivals.It is further demonstrated that initial position and momentum play a highly crucial role in the emergent revival and fractional revival patterns.It is shown that the symmetries exhibited by such structures in un-driven systems are usually lost under driven conditions and the revival/fractional revival times can only be predicted through a transcendental equation whose solution mainly depends on the strength of the external driving force.In present thesis we have proposed a comprehensive mathematical technique to calculate these hitherto unpredictable times.In addition, a graphical method is also suggested to observe these asymmetric revival/fractional revival times.This apparent asymmetry, do display an overall periodicity over a comparatively long span covering many times of revivals.It is found that whenever the ratio between the period of wall oscillation and quantum revival time `α/β' is a rational number then the revivals time shows a periodicity after β number of revival times over α wall oscillation. Standard tools of autocorrelation and quantum carpets, a consequence of quantum interference, have been employed for visual manifestation of these revivals.These carpets amply demonstrate the explicit asymmetric shifting of revival times for the driven case.These micro changes exhibited in quantum carpet morphologies consequently yield amended phase velocities associated with the canals and ridges in the driven framework.Furthermore, to grasp the quantum phase space dynamics in its entirety, we have also utilized the more sophisticated techniques of the Wigner function and information entropy.It is demonstrated mathematically that revivals not apparent via autocorrelation become prominent in the Wigner phase space description along with the appearance of smooth mini wave packets in the fractional revivals.These interference patterns emerge due to the state vector formed through the linear combination of Gaussians with different positions and momenta having pure Gaussian as well as oscillatory terms. Moreover, the information entropy, like the Wigner function also predicts the revivals when and where they happen.Thus a rich pattern of fractional revivals, including some altogether new ones, are observed through this tool in both driven and un-driven situations.Finally we have elucidated various scenarios including cold atoms in magneto-optical traps where such a system can be experimentally realized along with the suggestions for future work in this specific domain

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The idea of wave packet, that clarified many conceptual difficulties, has more recently been revisited to address interferometric and dynamical issues.Present work focuses on the spatio-temporal dynamics of the quantum wave packets in the Fermi-Ulam Accelerator Model.The specific system considered here is a one-dimensional box under periodic oscillations at one boundary. Such a seemingly simple system can be mapped to non-trivial problems like BEC trapped in an oscillating cavity or dynamics of atomic wave packet in different potential in presence or absence of gravitational field.The system is shown to display rich dynamical features including transitions from regular to stochastic and further to chaotic domain in classical phase space. We have obtained the approximated eigen-value solutions of the problem through time dependent Schrodinger’s equation under rescaling transformations that effectively replace moving boundary with a static one. Consequently the running time becomes a nonlinear function of the continuously changing boundary.Based on these solutions, revivals or the resurrection of the original wave packet, in the driven and un-driven Fermi-Ulam model are presented. Such revivals correspond to the constructive interference whereas the wave packet’s collapse owes to destructive interference. Furthermore, interplay of constructive and destructive interferences also readily mimics the fractional revivals.It is further demonstrated that initial position and momentum play a highly crucial role in the emergent revival and fractional revival patterns.It is shown that the symmetries exhibited by such structures in un-driven systems are usually lost under driven conditions and the revival/fractional revival times can only be predicted through a transcendental equation whose solution mainly depends on the strength of the external driving force.In present thesis we have proposed a comprehensive mathematical technique to calculate these hitherto unpredictable times.In addition, a graphical method is also suggested to observe these asymmetric revival/fractional revival times.This apparent asymmetry, do display an overall periodicity over a comparatively long span covering many times of revivals.It is found that whenever the ratio between the period of wall oscillation and quantum revival time `α/β' is a rational number then the revivals time shows a periodicity after β number of revival times over α wall oscillation. Standard tools of autocorrelation and quantum carpets, a consequence of quantum interference, have been employed for visual manifestation of these revivals.These carpets amply demonstrate the explicit asymmetric shifting of revival times for the driven case.These micro changes exhibited in quantum carpet morphologies consequently yield amended phase velocities associated with the canals and ridges in the driven framework.Furthermore, to grasp the quantum phase space dynamics in its entirety, we have also utilized the more sophisticated techniques of the Wigner function and information entropy.It is demonstrated mathematically that revivals not apparent via autocorrelation become prominent in the Wigner phase space description along with the appearance of smooth mini wave packets in the fractional revivals.These interference patterns emerge due to the state vector formed through the linear combination of Gaussians with different positions and momenta having pure Gaussian as well as oscillatory terms. Moreover, the information entropy, like the Wigner function also predicts the revivals when and where they happen.Thus a rich pattern of fractional revivals, including some altogether new ones, are observed through this tool in both driven and un-driven situations.Finally we have elucidated various scenarios including cold atoms in magneto-optical traps where such a system can be experimentally realized along with the suggestions for future work in this specific domain
Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.
The use of wave packet to analyze the dynamics of quantum mechanical systems is an increasingly important method to the study of the classical-quantum correspondence.Using the quantum Gaussian wave packet analysis method,we calculate the autocorrelation function of the rectangular billiard,the peak positions of the autocorrelation function match well with the periods of the classical periodic orbits,which show that the period of the classical orbits can be produced by the time-dependent quantum wave packet method.We also discuss wave packet revivals and fractional revivals in the rectangular billiard,the results show that there are exact revival for all wave packet at each revival time.We find additional cases of exact revivals with short revival times for zero-momentum wave packets initially located at special symmetry point inside the billiard.
We apply wave-packet methods to study an ion-trap system imposing neither the rotating wave nor the Lamb-Dicke approximations. By this approach we show the existence of states with restricted phase-space evolution as a genuine consequence of quantum interference between wave-packet fractions. A particular instance of such a state oscillates between maximal entanglement and pure disentanglement between the constitute subsystems, where the characteristic crossover time is very rapid. Over longer time periods the dynamics of these states exhibits collapse-revival patterns with well-resolved fractional revivals in autocorrelation, inversion, and entanglement.
Wave packet revivals and fractional revivals are striking quantum interference phenomena that can occur under suitable conditions in a system with a nonlinear spectrum. In the framework of a specific model (the propagation of an initially coherent wave packet in a Kerr-like medium), it is shown that distinctive signatures of these revivals and fractional revivals are displayed by the time evolution of the expectation values of physical observables and their powers, i.e., by experimentally measurable quantities. Moreover, different fractional revivals can be selectively identified by examining appropriate higher moments.
This paper presents a robust computational study of the Time-Dependent Schrödinger Equation (TDSE) dynamics for a spatial Gaussian wave packet confined within a one-dimensional infinite potential well. Employing a discrete Finite Difference Method (FDM) framework, we construct a tridiagonal Hamiltonian matrix to compute the unitary real-time evolution of the system. We systematically analyze two critical quantum phenomena: (1) scattering and quantum tunneling against an engineered central rectangular potential barrier, and (2) temporal dispersion and subsequent fractional quantum revival of the localized state. The numerical solver successfully resolved a distinct fractional revival and constructive self-interference peak at t = 1.416 time units. These computational benchmarks provide a rigorous pedagogical and diagnostic baseline for grid-based matrix methods in low-dimensional quantum mechanics.