Wavepacket Dynamics in Nonlinear Schr\"odinger Equations

TL;DR

This paper investigates the dynamics of nonlinear coherent states in one-dimensional nonlinear Schrödinger equations, revealing simple classical-like motion in harmonic potentials and complex behaviors like packet splitting under anharmonic perturbations.

Contribution

It demonstrates that only the parabolic potential preserves the shape of nonlinear coherent states during evolution and explores their behavior under various perturbations, including anharmonicities.

Findings

Ground state translations follow classical trajectories in harmonic potentials.

Anharmonicities cause the wave packet to split into coherent and incoherent parts.

Damped oscillations and transfer to high-energy modes observed under perturbations.

Abstract

Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schr\"odinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: the centre follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape, or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the condensate is transferred towards uncoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped towards the bottom of the well.