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arXiv:1507.04842

Tunneling as a Source for Quantum Chaos

Ofir Flom, Asher Yahalom A, Haggai Zilberberg, Lawrence Paul Horwitz, Jacob Levitan

Published Jul 17, 2015• 0 citations
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Abstract

We use an one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of the spatial entropy function defined by S = -\int |Ψ(x,t)|^2 ln |Ψ(x,t)|^2 dx. There is no classical counterpart to tunneling, but a decrease in the tunneling in a short time interval may be interpreted as an approach of a quantum system to a classical system. We show that changing the square barrier by increasing the height/width do not only decrease the tunneling but also slows down the rapid rise of the entropy function, indicating that the entropy growth is an essentially quantum effect.

DOI
10.48550/arxiv.1507.04842

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We use an one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of the spatial entropy function defined by S = -\int |Ψ(x,t)|^2 ln |Ψ(x,t)|^2 dx. There is no classical counterpart to tunneling, but a decrease in the tunneling in a short time interval may be interpreted as an approach of a quantum system to a classical system. We show that changing the square barrier by increasing the height/width do not only decrease the tunneling but also slows down the rapid rise of the entropy function, indicating that the entropy growth is an essentially quantum effect.
We use a one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of a locally defined spatial entropy function defined by S=-\int |\Psi(x,t)|^2 \ln |\Psi(x,t)|^2 dx . We show that changing the square barrier by increasing the height or width of the barrier not only decreases the tunneling but also slows down the rapid rise of the entropy function, indicating that the locally defined entropy growth is an essentially quantum effect.
It is demonstrated in the context of the simple one-dimensional example of a barrier in an infinite well that highly complex behavior of the time evolution of a wave function is associated with the near degeneracy of levels in the process of tunneling. Degenerate conditions are obtained by shifting the position of the barrier. The complexity strength depends on the number of almost degenerate levels which depend on geometrical symmetry. The presence of complex behavior is studied to establish correlation with spectral degeneracy.
We describe a quantum many-body system undergoing multiple resonant tunneling which exhibits chaotic behavior in numerical simulations of a mean field approximation. This phenomenon, which has no counterpart in the classical limit, is due to effective nonlinearities in the tunneling process and can be observed in principle within a heterostructure. 03.65.-w, 05.45.+b, 73.40.Gk Typeset using REVT E X In current studies the problem of quantum chaos is usually posed in the following terms. We are given a system which exhibits chaotic behavior classically: does this property survive quantization? Typically one studies the spectrum of the quantum mechanical Hamiltonian operator. However spectral differences at the quantum level between classically integrable and chaotic systems do not seem to imply dramatic differences in the time evolution of the quantum mechanical observables. Relics of classical chaos exist in low dimensional systems in short wavelength or large quantum numbers limits ...
Acknowledgments For two-dimensional, symmetric double well potentials separated by a tunneling barrier, it has been shown theoretically that quantum mechanical tunneling rates are dependent on the shape of the well in the limit of small quantum wavelength. Shapes that support chaotic wave functions produce statistically smaller fluctuations in the tunneling rate than classical (integrable) wells. This effect is called “regularization” of tunneling rates and can be analyzed by examining the splitting in energy level between symmetric and antisymmetric wave functions. Utilizing the similarity in the wave natures of a quantum mechanical particle in a symmetric double well and transverse electromagnetic waves in a 2-D cavity, we investigate chaos regularization in coupled microwave cavities both numerically and experimentally. The difference between the antisymmetric and symmetric eigenstate frequencies squared and evanescent electromagnetic wave propagation in a cutoff tunnel is shown to be analogous to energy splitting and quantum mechanical tunneling, respectively. This analogy allows investigation of regularization in various microwave cavities and, by extension to a lab-scale model, the behavior of complex quantum mechanical systems.