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Physical Review A

Coherent and incoherent tunneling in asymmetric double-well potentials

Anedio Ranfagni, Ilaria Cacciari, Maria Angela Vitali, G. Viliani, Paolo Moretti, R. Ruggeri

Published Jul 13, 2006• 3 citations
Publisher

Abstract

The determination of the time scale for coherent and incoherent tunneling in asymmetric double-well potentials is reconsidered according to the instanton-bounce method. In particular, by making use of Feynman's transition elements, a different, relatively simpler approach to this problem, with respect to conventional quantum-mechanical treatments, is obtained.

DOI
10.1103/physreva.74.014102

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The determination of the time scale for coherent and incoherent tunneling in asymmetric double-well potentials is reconsidered according to the instanton-bounce method. In particular, by making use of Feynman's transition elements, a different, relatively simpler approach to this problem, with respect to conventional quantum-mechanical treatments, is obtained.
Double-well potentials are widely used to model phenomena in physics and chemistry. The system is assumed to be formed in a metastable state, its dynamical evolution providing the clues for the interpretation of the experimental data. Quantum mechanics predicts coherent oscillations of probability between wells if the double-well potential is nearly symmetric and irreversible exponential decay if the final well has an infinite width. For very asymmetric double-well potentials, these two extreme behaviors are expected to coexist. The purpose of the present paper is to investigate this coexistence and its evolution as a function of the width of (or density of states in) the second well. In this sense, increasing the density of states can be regarded as a mechanism for coherence breakdown. The dynamical evolution of the metastable state can be simulated by solving numerically the time-dependent Schroedinger equation (TDSE). This approach is general, intuitive, and gives access to time scales and dynamical effects. It also allows the inclusion of phenomenological dissipation.
We present an alternative way to compute and interpret quantum tunneling in a one-dimensional double-well potential. For large transition time we show that the quantum action functional gives an analytical expression for tunneling amplitudes. This has been confirmed by numerical simulations giving relative errors in the order of ${10}^{\ensuremath{-}5}$. In contrast to the classical potential, the quantum potential has a triple well if the classical wells are deep enough. Its minima are located at the positions of extrema of the ground state wave function. The striking feature is that a single trajectory with a double instanton reproduces the tunneling amplitude. This is in contrast to the standard instanton approach, where infinitely many instantons and anti-instantons have to be taken into account. The quantum action functional is valid in the deep quantum regime in contrast to the semiclassical regime where the standard instanton approach holds. We compare both approaches via numerical simulations. While the standard instanton picture describes only the transition between potential minima of equal depth, the quantum action may give rise to instantons also for asymmetric potential minima. Such a case is illustrated by an example.