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arXiv:quant-ph/0305061

Classical trajectories and quantum tunneling

Boris I. Ivlev

Published May 12, 2003• 0 citations
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Abstract

The problem of inter-band tunneling in a semiconductor (Zener breakdown) in a nonstationary and homogeneous electric field is solved exactly. Using the exact analytical solution, the approximation based on classical trajectories is studied. A new mechanism of enhanced tunneling through static non-one-dimensional barriers is proposed in addition to well known normal tunneling solely described by a trajectory in imaginary time. Under certain conditions on the barrier shape and the particle energy, the probability of enhanced tunneling is not exponentially small even for non-transparent barriers, in contrast to the case of normal tunneling.

DOI
10.48550/arxiv.quant-ph/0305061

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The problem of inter-band tunneling in a semiconductor (Zener breakdown) in a nonstationary and homogeneous electric field is solved exactly. Using the exact analytical solution, the approximation based on classical trajectories is studied. A new mechanism of enhanced tunneling through static non-one-dimensional barriers is proposed in addition to well known normal tunneling solely described by a trajectory in imaginary time. Under certain conditions on the barrier shape and the particle energy, the probability of enhanced tunneling is not exponentially small even for non-transparent barriers, in contrast to the case of normal tunneling.
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The process of quantum tunneling of particles in various physical systems can be effectively controlled even by a weak and slow varying in time electromagnetic signal, especially if to adapt its shape to a particular system. During an under-barrier motion of a particle, such a signal provides a ``coherent'' assistance of tunneling by the multiquanta absorption resulting in a strong enhancement of the tunneling probability. The semiclassical approach based on trajectories in the complex time is developed for tunneling in a nonstationary field. Enhancement of tunneling occurs when a singularity of the signal coincides in position at the complex time plane with a singularity of the classical Newtonian trajectory of the particle. The developed theory is also applicable to the over-barrier reflection of particles and to reflection of classical waves (electromagnetic, hydrodynamic, etc.) from a spatially-smooth medium.
The nonadiabatic quantum tunneling picture, which may be called the many-body Schwinger-Landau-Zener mechanism, for the dielectric breakdown of Mott insulators in strong electric fields is studied in the one-dimensional Hubbard model. The tunneling probability is calculated by a method due to Dykhne-Davis-Pechukas with an analytical continuation of the Bethe-ansatz solution for excited states to a non-Hermitian case. A remarkable agreement with the time-dependent density-matrix renormalization-group result is obtained.
The analytical continuation of classical equations of motion to complex times suggests that a tunneling particle spends in the barrier an imaginary duration i|T| . Does this mean that it takes a finite time to tunnel, or should tunneling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards |T| being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it very little of the field's past behavior, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill posed, and we explain why. Published by the American Physical Society 2024
The extremely small probability of tunneling through an almost classical potential barrier may become not small under the action of the specially adapted nonstationary signal which selects the certain particle energy ER.{E}_{R}.ER​. For particle energies close to this value, the tunneling rate is not small during a finite interval of time and has a very sharp peak at the energy ER.{E}_{R}.ER​. After entering inside the barrier, the particle emits electromagnetic quanta and exits the barrier with a lower energy. The signal amplitude can be much less compared to the field of the static barrier. This phenomenon can be called the Euclidean resonance since the under-barrier motion occurs in imaginary time. The resonance may stimulate chemical and biochemical reactions in a selective way by adapting the signal to a certain particular chemical bond. The resonance may be used in search of the soft-\ensuremath{\alpha}-particle decay for which a conventional observation is impossible due to an extremely small decay rate.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).