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Structure transformation of an electron quantum packet under a laser impulse action

Andrey L. Sanin, Andrey T. Bagmanov

Published Feb 16, 2005• 0 citations

Abstract

Spatial-temporal structure of quantum packet of an electron in an one-dimensional system with impenetrable walls under laser impulse action had been investigated. The coupled equations in partial derivatives for a real and imaginary components of a wave function are solved by the numerical methods. The initial quantum packet is presented in the special trigonometric form with the arbitrary velocity. The calculations of the dynamical variables, its expected values and Fourier-spectra, phase trajectories are performed in details. Influence of a laser impulse duration on the transformation of the initial quantum packet was established. The accurate analytical solutions to the equations in partial derivatives for the free packet at initial trigonometric packet are discussed. These solutions allow us to understand the numerical results of basic part of a paper.

DOI: 10.1117/12.610588

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Spatial-temporal structure of quantum packet of an electron in an one-dimensional system with impenetrable walls under laser impulse action had been investigated. The coupled equations in partial derivatives for a real and imaginary components of a wave function are solved by the numerical methods. The initial quantum packet is presented in the special trigonometric form with the arbitrary velocity. The calculations of the dynamical variables, its expected values and Fourier-spectra, phase trajectories are performed in details. Influence of a laser impulse duration on the transformation of the initial quantum packet was established. The accurate analytical solutions to the equations in partial derivatives for the free packet at initial trigonometric packet are discussed. These solutions allow us to understand the numerical results of basic part of a paper.

Oscillatory motion of quantum packet at the specific initial and zeroth boundary conditions with arbitrary initial velocity is investigated analytically and numerically. Influence of integer and half-integer initial velocities on the spatially temporal properties of solutions has been studied in detail. Hydrodynamical variables such as the probability density, mean values of coordinate and current, phase trajectories for coordinate and current, and Fourier spectra of these quantities were calculated. The behavior of quantum potential and kinetic energy near the nodal point was also analyzed.

Quantum hydrodynamic equations are used to describe motion of single electron. The initial and boundary conditions are formulated from the analytic formula for the electron packet in a free space. The numerical integration of equations for the free packet is performed. Then, we investigated the wave packet propagation through the back-ground positive charge, when the electric field is self-consistent. Moreover, we studied the influence of the external harmonic electric fields on the properties of the motion. The purpose of the present work consists in study of the spatial-temporal processes including packets when the quantum wave effects are essential.

The wave function temporal evolution on the one-dimensional (1D) lattice is considered in the tight-binding approximation.The lattice consists of N equal sites and one impurity site (donor).The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor.The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2.The wave packet width grows with time ∼ t 1/3 and its amplitude decreases ∼ t -1/3 .The wave packet reflects multiply from the lattice ends.Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.

The wave function temporal evolution on the one-dimensional(1D) lattice is considered in the tight-binding approximation. The lattice consists of N equal sites and one impurity site(donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time ～ t1/3and its amplitude decreases ～ t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.

Dynamics of wave packets in quantum systems subjected to a friction force and impulse feedback is investigated in context of Schrodinger-Langevin-Kostin (SchLK) equation. The quantum systems with a quadratic potential bounded by impenetrable walls of a well are presented as examples. Numerical simulations are carried out for initial Gaussian packet at varied values of a friction coefficient and amplitudes of the impulse feedback. Informative and structural properties of the wave packets are analyzed by using Fourier transform of dynamical variables and their means, of autocorrelation Nauenberg's function. Coherent oscillations of the wave packets and transition to chaotic dynamics are found. Influence of friction on oscillatory dynamics is discussed.