Free temporal evolution of a superbandwidth wave packet: The quantum tale of the hare and the tortoise

TL;DR

This paper explores the quantum-classical analogy in wavepacket dynamics, revealing local momenta with super- and sub-oscillatory features and wavepacket localization during free propagation, within a specific interference timescale.

Contribution

It demonstrates the emergence of local momenta with super- and sub-oscillatory characteristics and wavepacket localization, establishing a novel quantum-optical analogy and analyzing the interference timescale.

Findings

Local momenta with super-oscillatory features can arise during free propagation.

Wavepacket localization occurs within a defined interference timescale.

Quantum and classical analyses complement each other in explaining the phenomena.

Abstract

Can the interplay between quantum mechanics and classical optics offer new perspectives on wavepacket dynamics? Building on this connection, we show that local momenta with both super-oscillatory and suboscillatory characteristics can arise during the free propagation of a quantum particle. This behavior is mathematically analogous to the propagation of a superbandwidth laser pulse in a dispersive medium, where the instantaneous frequencies of the electric field exhibit similar sub- and super-oscillatory features. This analogy is rooted in the equivalence between the wave description of light in classical optics and the formalism of quantum mechanics. In addition, we explore the phenomenon of wavepacket localization during free propagation within a bounded region. This localization is directly linked to the distribution of local momenta within the confined wavepacket region. To complement our quantum mechanical analysis, we also perform a classical analysis to provide further insights into this phenomenon. Our findings reveal that both the emergence of local momenta with sub- and super-oscillatory features and the wavepacket localization occur within a distinct timescale, which we define as the interference time.