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Physical Review Letters

Electromagnetic Radiation by a Tunneling Charge

M. I. Dyakonov, I. V. Gornyi

Published May 6, 1996• 32 citations
Publisher

Abstract

Electromagnetic radiation during the quasiclassical tunneling motion of a charge through a smooth potential barrier is considered. A general formula for the radiation spectral density per tunneling particle is derived, which is essentially classical (i.e., does not contain the Planck constant), and is given by a simple modification of the convenient result for the classical overbarrier motion.

DOI
10.1103/physrevlett.76.3542

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Electromagnetic radiation during the quasiclassical tunneling motion of a charge through a smooth potential barrier is considered. A general formula for the radiation spectral density per tunneling particle is derived, which is essentially classical (i.e., does not contain the Planck constant), and is given by a simple modification of the convenient result for the classical overbarrier motion.
The classical equation of motion of a charged point particle, including its radiation reaction, is described by the Lorentz-Dirac equation. We found a new class of solutions that describe tunneling (in a completely classical context!). For nonrelativistic electrons and a square barrier, the solution is elementary and explicit. We show the persistance of the solution for smoother potentials. For a large range of initial velocities, initial conditions may leave a (discrete) ambiguity on the resulting motion.
We show that the classical equation of motion of a radiating charged point particle (the Lorentz-Dirac equation) has ``tunneling'' solutions. For a given initial position and velocity we find that, contrary to common belief, several different physically acceptable solutions exist for a range of initial data. Both features are demonstrated for a rectangular barrier. To check that these phenomena are not dependent on the discontinuities of the potential, we also study in detail the solutions for a smoothened (single) potential step.
We make a brief review of the Kramers escape rate theory for the probabilistic motion of a particle in a potential well U(x), and under the influence of classical fluctuation forces. The Kramers theory is extended in order to take into account the action of the thermal and zero-point random electromagnetic fields on a charged particle. The result is physically relevant because we get a non null escape rate over the potential barrier at low temperatures (T -> 0). It is found that, even if the mean energy is much smaller than the barrier height, the classical particle can escape from the potential well due to the action of the zero-point fluctuating fields. These stochastic effects can be used to give a classical interpretation to some quantum tunneling phenomena. Relevant experimental data are used to illustrate the theoretical results.
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.
A Schrödinger type equation for a mathematical probability amplitude ψ(x,t), is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M, moving in a potential V(x). The particle phase space probability density is denoted W(x,p,t) and the entire system is immersed in the "vacuum" zero-point electromagnetic radiation . We show that the generalized Liouville equation is reduced to a non-quantized Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. Our derivation will be based on a simple Fourier transform of the non-quantized phase space probability distribution W(x,p,t). For convenience, we introduce in this Fourier transform an auxiliary constant α, with dimension of action, and an auxiliary coordinate denoted by y. We shall prove that α is equal to the Planck's constant present in the momentum operator of the Schrödinger equation. Moreover, we shall show that this momentum operator is deeply related with the ubiquitous zero-point electromagnetic radiation. It is also important to say that we do not assume that the mathematical amplitude ψ(x,t) is a de Broglie matter-wave, in other words, the wave-particle duality hypothesis is not used within our work. The implications of our study for the standard interpretation of the photoelectric effect is discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunnelling phenomenon and in the Compton's effect.
Abstract The question of fluctuations in electromagnetic radiation played an important part during the first period of the development of quantum theory. After having introduced (Einstein 1905) the conception of light quanta or photons in order to explain the observed phenomena of the photoelectric effect, Einstein (1909) considered the consequences of this idea for other properties of the radiation. Planck’s formula for the energy density of radiation implies, by arguments of general thermodynamics and statistics, the following expression for the mean square fluctuation of the energy contained in a volume v in terms of the mean energy Ēv.