Atop‐the‐barrier localization in periodically driven double wells: A minimization of information entropic sums in conjugate spaces

Abstract

The spatio‐temporal localization of a system in the presence of an oscillating electric field for a symmetric double‐well potential is examined via numerical simulations of the time‐dependent Schrödinger equation. For an initial state with equal probability densities in both the wells, stabilized localization atop the barrier can be achieved on a periodic high‐frequency driving. The barrier localization is characterized using Shannon information entropies in position and momentum spaces, defined as Sρ = − ∫ |ψ|2 ln |ψ|2 dx and Sγ = − ∫ |ϕ|2 ln |ϕ|2 dp, where ψ and ϕ refer to position and momentum space wave functions, respectively. The information entropy sum, Sρ + Sγ, goes through a minimum indicating the formation of the barrier‐localized state, when the peak intensity of the oscillating field is reached. The generalized uncertainty via the Białynicki‐Birula‐Mycielski inequality ( Sρ + Sγ ≥ 1 + lnπ) is saturated upon this minimization. This serves as a signature of the formation of the barrier‐atop localized state, in terms of Shannon entropies of measurable densities.