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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum dynamics under the influence of external magnetic fields

S. Midgley, Jingbo B. Wang

Published Apr 11, 2003• 0 citations
Publisher

Abstract

In this paper, we describe a very versatile numerical model, which can be applied to study the quantum dynamics of an electron wave packet in the presence of external electric and magnetic fields. Detailed numerical analysis is carried out in the paper on the validity of such a model, including (1). comparison with analytical solution in the case of uniform magnetic field, and (2). stringent numerical tests in the case of both electric and magnetic fields. The algorithm presented here is found to be highly accurate, computationally efficient, and numerically stable. It also has the advantage of being able to accommodate arbitrarily complex magnetic field profiles as well as time-varying external fields.

DOI
10.1103/physreve.67.046702

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In this paper, we describe a very versatile numerical model, which can be applied to study the quantum dynamics of an electron wave packet in the presence of external electric and magnetic fields. Detailed numerical analysis is carried out in the paper on the validity of such a model, including (1). comparison with analytical solution in the case of uniform magnetic field, and (2). stringent numerical tests in the case of both electric and magnetic fields. The algorithm presented here is found to be highly accurate, computationally efficient, and numerically stable. It also has the advantage of being able to accommodate arbitrarily complex magnetic field profiles as well as time-varying external fields.
For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as 〈qi〉〈{q}_{i}〉〈qi​〉, 〈pi〉〈{p}_{i}〉〈pi​〉 and 〈qipj〉〈{q}_{i}{p}_{j}〉〈qi​pj​〉, 〈pipj〉〈{p}_{i}{p}_{j}〉〈pi​pj​〉, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.
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.
It has been shown that the irreversible dynamics of a system in a dissipative environment can be described by an effective one-particle Hamiltonian (in the classical case) or an effective one-particle Schrödinger equation (in the quantum mechanical case) without taking into account the environmental degrees of freedom explicitly. In the quantum mechanical case, the description can be achieved either by a nonlinear modification of the Schrödinger equation or an explicitly time-dependent Hamiltonian operator — where both forms are connected via a non-unitary transformation, corresponding to a non-canonical transformation on the classical level. Taking into account an external magnetic field, in both formalisms, an additional unphysical gauge-dependent term occurs in the equations of motion. This shortcoming can be eliminated if the Lagrangian and Hamiltonian that provide Maxwell’s equations of electrodynamics undergo, in the dissipative case, the same kind of transformation as the mechanical degrees of freedom of the system. As a result, a consistent description in terms of exact analytical wave packet-solutions for the dissipative motion in a magnetic field can be obtained that will be discussed in detail.
We study the dynamics of a two-qubit system coupled through time dependent anisotropic XYZXYZXYZ Heisenberg interaction in presence of a time varying non-uniform external magnetic field. Exact results are presented for the time evolution of the system under certain integrability conditions. Furthermore, the corresponding entanglement of the system is studied for different values of the involved parameters. We found that the time evolution and different properties of entanglement such as amplitude, frequency and profile can be finely tuned by the interplay of the characteristics of the time-dependent magnetic field and exchange coupling. Also we show how the discrete symmetries of the system Hamiltonian, which splits its Hilbert space into two distinct subspaces, can be utilized to deduce the dynamics in one of its two distinct subspaces from the other one. Moreover, approximate results for the time evolution are provided utilizing the rotating wave as well as the perturbation approximations for the special case of either static magnetic field or exchange coupling. We compare the range of validity of the two approximation methods and their effectiveness in treating the considered system and determine their critical parameters.
A fast and stable numerical method is formulated to compute the time evolution of a wave function in a magnetic field by solving the time-dependent Schroedinger equation. This computational method is based on the finite element method in real space to improved accuracy without any increase of computational cost. This method is also based on Suzuki's exponential product theory to afford an efficient way to manage the TD-Schroedinger equation with a vector potential. Applying this method to some simple electron dynamics, we have confirmed its efficiency and accuracy.