Commun. Comput. Phys., 1 (2006), pp. 81-99.


Chaotic Driven Tunneling in Rectangular Double-Well

V. L. Golo 1*, Yu. S. Volkov 1

1 Department of Mechanics and Mathematics, Moscow State University, Moscow 119 899 GSP-2, Russia.

Received 20 July 2005; Accepted (in revised version) 9 August 2005
Communicated by Dietrich Stauffer

Abstract

We consider a charged particle confined in a one-dimensional rectangular double-well potential, driven by an external periodic excitation at frequency $\Omega$ and with amplitude $A$. We find that there is the regime of the parametric resonance due to the modulation of the amplitude $A$ at the frequency $\omega_{prm}$, which results in the change in the population dynamics of the energy levels. The analysis relies on the Dirac system of Hamiltonian equations that are equivalent to the Schr\"odinger equation. Considering a finite dimensional approximation to the Dirac system, we construct the foliation of its phase space by subsets ${\cal F}_{ab}$ given by constraints $a \le N_0 \le b$ on the occupation probabilities $N_0$ of the ground state, and describe the tunneling by frequencies $\nu_{ab}$ of the system's visiting subsets ${\cal F}_{ab}$. The frequencies $\nu_{ab}$ determine the probability density and thus the Shannon entropy, which has the maximum value at the resonant frequency $\omega = \omega_{prm}$. The reconstruction of the state-space of the system's dynamics with the help of the Shaw-Takens method indicates that the quasi-periodic motion breaks down at the resonant value $\omega_{prm}$.


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Key words: Tunneling; double-well; parametric resonance.


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Correspondence to: V. L. Golo , Department of Mechanics and Mathematics, Moscow State University, Moscow 119 899 GSP-2, Russia. Email: golo@mech.math.msu.su
 

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