Buildup and decay dynamics of exponentially modulated quantum waves in resonant structures

Abstract

We derive an exact analytic solution of time-dependent Schrödinger’s equation for cutoff quantum waves with exponential modulation to explore the dynamics along the internal region of a one-dimensional potential of arbitrary shape. The modulation of the wave packet is controlled by a single parameter given by its momentum distribution width, Δ, which allows to explore on the same footing the transient features of both cutoff spatially localized wave packets and extended plane waves. From the formal solution, we derive a simple and reliable formula that accurately describes both the buildup and decay of the probability density inside a resonant structure. The buildup-decay formula involves two exponential terms governed respectively by the resonance width Γ1 of the lowest resonance of the system and the width of a ‘virtual’ state induced by the incident wave packet. During the buildup process, we demonstrate that the contributions of both Γ1 and become essential. In the decay process, on the other hand, we find that there are two regimes characterized by a critical value Δc, such that for Δ > Δc the exponential decay is governed by Γ1, while for Δ < Δc the exponential decay is governed by . We also derive an analytic formula for a relevant timescale for the buildup-decay process, which depends in a simple fashion on the decay rates Γ1 and . Deviations from the exponential decay law at long times are also discussed.