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Journal of Physics A Mathematical and General

Tunnelling through moving barriers

Ashok V. Pimpale, S. Holloway, R J Smith

Published Aug 7, 1991• 64 citations
Publisher

Abstract

The authors present a variety of theoretical methods to treat the general problem of a quantum particle interacting with a time-dependent barrier. Assuming a sinusoidal form for the time dependence, they discuss the classical, semiclassical and quantum solutions for two specific problems: a barrier whose amplitude is modulated, and one whose mean position changes. While the classical solutions to these problems are quite different, the time-independent quantum solutions appear almost identical. It is only when the particle is modelled as a wavepacket that the quantum solutions behave like their classical counterparts.

DOI
10.1088/0305-4470/24/15/021

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The authors present a variety of theoretical methods to treat the general problem of a quantum particle interacting with a time-dependent barrier. Assuming a sinusoidal form for the time dependence, they discuss the classical, semiclassical and quantum solutions for two specific problems: a barrier whose amplitude is modulated, and one whose mean position changes. While the classical solutions to these problems are quite different, the time-independent quantum solutions appear almost identical. It is only when the particle is modelled as a wavepacket that the quantum solutions behave like their classical counterparts.
There has been a surprising degree of uncertainty about how, or even whether, the standard semiclassical theory of Van Vleck and Gutzwiller (VVG) can be used to treat the time-dependent dynamics of tunneling through barriers. At issue is how semiclassical theory describes the time evolution of tunneling. This uncertainty has hindered development of practical techniques for semiclassical calculations of multidimensional systems. It is shown here that the VVG theory can indeed be applied to barrier tunneling, but generally requires consideration of classical trajectories with both complex initial conditions and complex time paths, even for systems with one degree of freedom. The present approach is verified by applying it to the classically forbidden propagation of a wave packet through a one-dimensional barrier---a very basic problem that cannot be treated by the VVG theory without the modifications described here.
There has recently been a growth in interest in electron tunnelling through barriers, and in particular in the question of whether it is possible to talk of a tunnelling time and, if so, what is it? A brief account of a study of the signal velocity of wavepackets travelling into a rectangular barrier was given recently, and the present paper is an extended version of this theory. In particular it is shown that the signal velocity of a free electron is the same as its group velocity, which in turn is the same as that which a classical particle particle of the same energy would have. It is also shown that when the wavepacket enters a classically forbidden region the signal velocity continues to have a meaning, and indeed a pulse-like wavefunction travels very much like a light pulse in a dispersive and attenuating medium. The expression for the signal velocity is symmetrical with respect to the barrier height, so that pulses well below the barrier travel faster, though with greater attenuation, than pulses near the top of the barrier. This result agrees with conclusions reached by some other workers. It is also shown that an electron responds to an accelerating field in the barrier as would be expected for a classical particle. It is argued that the not uncommon statement that the non nor eigenstates of a free particle describe a beam of particles is incorrect for electrons, for such a solution can at most describe two electrons. The theory indicates that by using orthogonal wavepackets it is possible to construct many-electron wavefunctions which do describe beams of electrons, and then free electrons travel with the group (and signal) velocity.
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.
A variety of different theoretical procedures have been used to determine tunneling times. These include (1) phase time; (2) the Stevens procedure; (3) Larmor times; (4) a complex “time”; (5) dwell time; (6) the Büttikker-Landauer time; (7) Feynman path-integrals; (8) scattering theory; (9) a stochastic formulation; (10) oscillatory perturbation of the barrier; (11) kinetic time; and (12) the semiclassical solution. Interest in tunneling times is not purely pedagogical, since correction for changes in image potential during transit is required in determining the conductance of several new high-speed semiconductor devices. A new procedure to evaluate transit time is presented, which assumes that energy fluctuations keep a tunneling particle above the barrier while transiting the classically forbidden region. For large rectangular barriers, the most probable fluctuations minimize the product of their magnitude and the transit time. This results in the semiclassical solution. For the case of very small barriers, there is a separate solution in which the transit time is indeterminate but bounded. The new solution is compared with 12 different results by others. Numerical values, limiting forms, and interpretations are presented for these various tunneling times. Difficulties in using time domain numerical solutions to determine the transit time for a wave packet will also be described. © 1992 John Wiley & Sons, Inc.
Solutions to the time-dependent Schr\"odinger equation for a particle in a spatially uniform time-dependent field and some potentials of arbitrary form are proved to be like the time-independent eigenfunctions for an identical static potential. The field phase modulates the wave function and translates its envelope. This work applies directly to constant, linear, and parabolic static potentials and is self-consistent. Results for the modulation by a time-dependent field of the current through a rectangular barrier are relevant to the traversal time for tunneling and associated device limitations.