Gravity, Bose-Einstein Condensates and Gross-Pitaevskii Equation

TL;DR

This paper investigates how mutual gravitational interactions influence the dynamics and stability of Bose-Einstein condensates, especially at high occupation numbers, using a variational approach to the Gross-Pitaevskii equation.

Contribution

It introduces a method to analyze gravitational effects on BECs by reducing the Gross-Pitaevskii equation to a particle in a potential, revealing a critical occupation number for stability.

Findings

Identifies a critical occupation number for BEC stability under gravity.

Shows that gravity can cause instability in high-occupation BECs with short-range 1/r^3 interactions.

Demonstrates the use of variational techniques to incorporate gravity into BEC dynamics.

Abstract

We explore the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose-Einstein condensates (BEC). Small amplitude oscillation of BEC is studied by applying variational technique to reduce the Gross-Pitaevskii equation, with gravity included, to the equation of motion of a particle moving in a potential. According to our analysis, if the s-wave scattering length can be tuned to zero using Feshbach resonance for future BEC with occupation numbers as high as $\approx 10^{20}$, there exists a critical ground state occupation number above which the BEC is unstable, provided that its constituents interact with a $1/r^3 $ gravity at short scales.