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Dissipative tunneling through a parabolic potential in the Lindblad theory of open quantum systems

Аурелиан Исар, A. Sãndulescu, W. Scheid

Published Jan 18, 2007• 7 citations
OpenAlex

Abstract

By using the Lindblad theory for open quantum systems, an analytical expression of the tunneling probability through an inverted parabola is obtained. This penetration probability depends on the environment coefficients. It is shown that the tunneling probability increases with the dissipation and the temperature of the thermal bath.

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Dissipative tunneling through a parabolic potential in the Lindblad theory of open quantum systems

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For a tunneling system we consider the coupling to the dissipative environment of the coordinate q, the momentum p and the potential function U(q). In the frame of the quantum theory of open systems we derive a quantum master equation, fundamental constraints and uncertainty relations. We obtain an analytical expression of the tunneling spectrum and show that at thermal equilibrium its shape entirely depends on the barrier characteristics while its amplitude is proportional to the strength of the dissipative coupling.
We study the barrier penetrability in the frame of the Lindblad theory of open quantum systems. In addition to the diagonal elements of the density matrix, leading to the Gamow’s formula, new terms, describing energy dissipation and spectral line broadening effects are obtained. It is shown that the presence of a dissipative environment increase the barrier penetrability, in accordance with a very simple physical interpretation: for a system initially found in its ground state the dissipation can lead only to transitions to the reaction channels where lower-energy levels exist.
Using Lindblad's axiomatic approach for the description of dissipation in quantum mechanics, we describe the dynamics of an open quantum system with a two-dimensional inverse parabolic potential. The derivation of the equations of motion for the first two moments of the quantum system from the general Markovian master equation is shown and their solution is given. We analyse the scattering of a Gaussian wave package in the considered potential and present an analytical expression for the probability density in position space and numerical results for the probabilities. A possible application of the developed formalism is shown for the nuclear physics example of the collision of 48 Ca on 248 Cm.
Applying a technique developed recently [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994); L. H. Yu, Phys. Lett. 202, 167 (1995)] for a harmonic oscillator coupled to a bath of harmonic oscillators, we present an exact solution for the tunneling problem in an Ohmic dissipative system with inverted harmonic potential. The result shows that while the dissipation tends to suppress the tunneling, the Brownian motion tends to enhance the tunneling. Whether the tunneling rate increases or not would then depend on the initial conditions. We give a specific formula to calculate the tunneling probability determined by various parameters and the initial conditions. \textcopyright{} 1996 The American Physical Society.
Abstract The inverted parabola remains an exactly solvable tunneling barrier when imaginary parts are added to the potential. They represent the effect of particle absorption. In the limit of strong absorption (“opaque barrier”) convenient limiting forms for the tunneling probability are derived using asymptotic expansions of the exact scattering solution. An application to Zener tunneling is made and the result is compared to the case of tunneling under viscous dissipation. Significant differences are found between these two approaches to tunneling under dissipation.