Gross Pitaevskii Equation with a Morse potential: bound states and evolution of wave packet

TL;DR

This paper investigates the behavior of wave packets in the Gross Pitaevskii equation with a Morse potential, revealing how bound states and escape dynamics depend on the coupling constant and initial conditions.

Contribution

It provides a detailed analysis of bound state existence and wave packet evolution in GPE with Morse potential, highlighting critical coupling and escape conditions.

Findings

Critical coupling constant scales as D^{3/4} for large D.

Wave packets require a critical momentum to escape when g<g_c.

For g>g_c, all wave packets escape, mimicking free particle dynamics.

Abstract

We consider systems governed by the Gross Pitaevskii equation (GPE) with the Morse potential $V(x)=D(e^{-2ax}-2e^{-ax})$ as the trapping potential. For positive values of the coupling constant $g$ of the cubic term in GPE, we find that the critical value $g_c$ beyond which there are no bound states scales as $D^{3/4}$ (for large $D$). Studying the quantum evolution of wave packets, we observe that for $g<g_c$, the initial wave packet needs a critical momentum for the packet to escape from the potential. For $g>g_c$, on the otherhand, all initial wave packets escape from the potential and the dynamics is like that of a quantum free particle. For $g<0$, we find that there can be initial conditions for which the escaping wave packet can propagate with very little change in width i,e., it remains almost shape invariant.