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Physical Review A

Information measures of quantum tunneling

Aleksei M. Zheltikov

Published Aug 1, 2023• 3 citations
Publisher

Abstract

A wave packet localized within a binding potential feeds an outgoing probability current as it tunnels through a finite-width potential barrier, imprinting its localization-related information on the outgoing probability-current waveform. The product of the wave number and the gradient of the outgoing probability current provides a readout for this information, connecting to the wave-packet localization, Fisher information, and related information entropy production, thus revealing the information-flow underpinning of tunneling dynamics.

DOI
10.1103/physreva.108.022201

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The article intertwines the study of wave packet dynamics with information-theoretic measurements in one dimensional (D =1) system. A localized wave packet at time t=0 has been considered here and its change at later instant of time t is calculated for the position space wave function. The momentum space wave function is obtained by taking the Fourier transform of the position space wave function at time t= 0 . These wave functions are then employed to construct the wave packet's respective probability densities in position and momentum space, and later have been used to compute the corresponding position and momentum space Shannon (S) and Fisher information (I) entropies. It has been observed that although the Shannon and Fisher information entropies explicitly depend on the standard deviation, neither the Shannon entropy sum nor the product of Fisher information entropies is. Moreover, for D dimensional systems, the computed values of the Shannon and Fisher entropies are found to satisfy the lower bounds of the Bialynicki-Birula and Myceilski (BBM) inequality relation and the Stam-Cramer-Rao inequalities better known as the Fisher based uncertainty relation. Thus, our theoretical study explores the validation of information-theoretic measurements for wave packet dynamics using the basic formulations of information theory.
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