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Fractals

Fractal Dimensions in Tunneling Processes and Effects of Environmental Fluctuations

Vassilios I. Kovanis, Paolo Grigolini A, V. M. Kenkre, George P. Tsironis

Published Mar 1, 1998• 0 citations
Publisher

Abstract

We study electron tunneling between two infinte potential square wells connected via an opaque barrier and find that time evolution of the probability of the presence of a Gaussian wave packet, localized initially in one of the wells, shares the fractal behavior of tunneling in a quartic potential, discovered by Dekker (H. Dekker, Phys. Rev.A35, 1825 (1987). However, the fractal dimensions are found to be closer to those of a conventional Weierstrass function than those appropriate to the quasi-Weierstrass behavior of Dekker. It is argued that the usual exponential decay predicted by conventional relaxation processes can be recovered only as an effect of thermal fluctuations.

DOI
10.1142/s0218348x98000079

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We study electron tunneling between two infinte potential square wells connected via an opaque barrier and find that time evolution of the probability of the presence of a Gaussian wave packet, localized initially in one of the wells, shares the fractal behavior of tunneling in a quartic potential, discovered by Dekker (H. Dekker, Phys. Rev.A35, 1825 (1987). However, the fractal dimensions are found to be closer to those of a conventional Weierstrass function than those appropriate to the quasi-Weierstrass behavior of Dekker. It is argued that the usual exponential decay predicted by conventional relaxation processes can be recovered only as an effect of thermal fluctuations.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.
We examine a fast decay process that arises in the transition period between the Gaussian and exponential decay processes in quantum decay systems. It is usually expected that the decay is decelerated by a confinement potential barrier. However, we find a case where the decay in the transition period is accelerated by tunneling through a confinement potential barrier. We show that the acceleration gives rise to an appreciable effect on the time evolution of the nonescape probability of the decay system.
We discuss quantum relaxation and fluctuation phenomena in the light of the corresponding classical behaviour. Quantum exponential localization and tunneling effects lead to the universal algebraic decay of the survival probability P ( t ) ∝ 1/ t . This behaviour is observed in particular in the time dependence of the ionization probability of Rydberg atoms driven by a monochromatic field. At a fixed time the quantum survival probability strongly fluctuates as some parameter is varied. These fluctuations have a fractal structure which is quantitatively related to the time dependence of the survival probability.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.
In order to better understand the trapping and response time of resonant electron wave packets in a resonant tunneling structure, we examined the time-dependent transport of a spherically symmetric Gaussian wave packet through a model double-barrier potential. Within the effective mass picture, the wave packet is projected onto the complete eigenstates of the model potential. The wave packet is initially localized to the left of the barriers and allowed to propagate through the structure. At wave packet energies corresponding to the resonant energies, the probability transmitted to the right of the barriers exhibits a steplike behavior as a function of time. A computation is made of the quantum mechanical delay time, to~tclass, where to is the packet transit time and tclass is the transit time for a classical particle. An increase in this delay time is observed when the packet energy corresponds to the energy of the resonant levels, and a comparison is made to the decay time of a particle in the first resonant state of the well. The delay time increases with increasing barrier area and decreasing packet energy width.