Temporal dynamics of tunneling. Hydrodynamic approach

TL;DR

This paper investigates the dynamics of macroscopic tunneling using a hydrodynamic approach, revealing wave breaking, shock formation, and conditions for soliton ejection during the tunneling process.

Contribution

It introduces a hydrodynamic method to analyze tunneling, identifying conditions for soliton formation and providing a criterion for ejection based on the nonlinearity parameter.

Findings

Shock formation occurs during early tunneling stages.

A density blip can transform into a bright soliton under certain conditions.

Soliton ejection can be controlled by initial state parameters.

Abstract

We use the hydrodynamic representation of the Gross -Pitaevskii/Nonlinear Schroedinger equation in order to analyze the dynamics of macroscopic tunneling process. We observe a tendency to a wave breaking and shock formation during the early stages of the tunneling process. A blip in the density distribution appears in the outskirts of the barrier and under proper conditions it may transform into a bright soliton. Our approach, based on the theory of shock formation in solutions of Burgers equation, allows us to find the parameters of the ejected blip (or soliton if formed) including the velocity of its propagation. The blip in the density is formed regardless of the value and sign of the nonlinearity parameter. However a soliton may be formed only if this parameter is negative (attraction) and large enough. A criterion is proposed. An ejection of a soliton is also observed numerically. We demonstrate, theoretically and numerically, controlled formation of soliton through tunneling. The mass of the ejected soliton is controlled by the initial state.