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Hu'nan Shifan Daxue xuebao. Ziran kexue ban

Adiabatic Evolution of Coherent States of a One-Dimensional Electron

Hui Meng

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

By using an analytical method,exact solutions of the equation for a single tapped electron adiabatically interacting with an external field is obtained.Such exact solutions describe a series of exact squeezed coherent states of this quantum system,whose configuration of probability density is breathing and oscillatory wave-packets.The time evolutions of the expectation of coordinate,momentum and energy are greatly affected by the adiabatic factor in a time interval.Therefore,manipulation of quantum state can be realized by suitably adjusting adiabatic parameters.

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By using an analytical method,exact solutions of the equation for a single tapped electron adiabatically interacting with an external field is obtained.Such exact solutions describe a series of exact squeezed coherent states of this quantum system,whose configuration of probability density is breathing and oscillatory wave-packets.The time evolutions of the expectation of coordinate,momentum and energy are greatly affected by the adiabatic factor in a time interval.Therefore,manipulation of quantum state can be realized by suitably adjusting adiabatic parameters.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.
In this work we investigate the quantum dynamics of a particle trapped by oscillating fields. With the help of quadratic invariants and of the invariant method we solve analytically and exactly the Schrödinger equation for this system and write the corresponding wave functions in terms of solutions of the Milne–Pinney equation. We also construct coherent and squeezed states for this system and evaluate the quantum fluctuations in coordinate and momentum as well as the uncertainty product between canonical coordinate and momentum.
A general framework for a time-dependent variational approach in terms of squeezed coherent states is given with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. By using a formula given by Balian and Brezin for canonical decompositions of generalized Bogoliubov transformations, we show that this approach is entirely equivalent to that given by Jackiw and Kerman in terms of the most general Gaussian functions, which is originally motivated for the variational definition of the effective action in the quantum field theory. The resultant equations of motion have structure of classical Hamiltonian systems consisting of displacement degree of freedom and squeezing degree of freedom in the configuration space. The initial conditions for solving these equations, particularly for the squeezing degree of freedom, are discussed in connection with the minimum-uncertainty relations of the squeezed coherent state. For a simple one-dimensional quantum-mechanical Hamiltonian with potentials of quadratic terms or less, the present formulation gives an exact result for the time-evolution of the Gaussian wave packet.
We consider a harmonically trapped and strongly driven electron in a one-dimensional (1D) quantum wire. A set of exact complete solutions of the time-dependent Schrödinger equation is constructed, which shows the properties of generalized squeezed coherent states. The corresponding probability densities behave like some breathing and oscillating wave-packet-trains. The resonance ladders of the expectation energy are found and the transition probabilities between two stationary states are exactly calculated. The results reveal that the quantum motions of the considered system can be controlled by adjusting the laser driving field.
Motivated by the recent proposal of Vysotstkii and Vysotskyy of the possibility of correlated coherent state evolution of an ion wave function in a low density magnetized plasma leading to low energy nuclear reactions, we have considered the construction of a dynamical squeezed state wave packet solution for a particle with a dynamic charge in a constant magnetic field. For a z- directed magnetic field, the dynamics simplify if the wave packet is constructed to have the same dynamics in x and y. From the solution developed, evolution equations are obtained for the first and second moments. In the case of the first moments, the dynamics that result are Newton’s laws for a particle in a magnetic field. The second moments evolve according to what would be expected for a squeezed state solution, except that the breathing frequency is reduced by a factor of two. The uncertainty product for the squeezed state is a Schrodinger-Robertson relation. The amount of squeezing required for the fluctuation energy to be sufficient to result in nuclear reactions is extremely large, and such a wave packet would be quite fragile in the sense that collisions with other particles would prevent organized “breathing” mode evolution except at very low plasma density. Following the proposal of Vysotskii and Vysotskyy, we constructed generalization of the squeezed state wave packet in which an excited state SHO eigenfunction is squeezed, and find that the evolution of the squeezing parameters is similar as for a squeezed state solution. The uncertainty principle is of the form of the Schrodinger-Robertson relation, but with a prefactor that substantially increases the uncertainty. An issue that was identified in the proposed scheme is localization associated with electron ionization and recombination that has the potential to spoil the delocalization and phase of the wave packet. It is noted that in a minor modification of the scheme non-classical energy exchange may be possible for charged particles traveling through a dynamic and spatially inhomogeneous magnetic field, as long as the changes in the field as seen by the particle are non-adiabatic relative to the squeezing dynamics.