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

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 S_ρ = − ∫ |ψ|² ln |ψ|² dx and S_γ = − ∫ |ϕ|² ln |ϕ|² dp , where ψ and ϕ refer to position and momentum space wave functions, respectively. The information entropy sum, S_ρ + S_γ, 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 ( S_ρ + S_γ ≥ 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.