# Dynamics of binary Bose-Einstein condensate via Ehrenfest like equations: Appearance of almost shape invariant states

## TL;DR

This paper introduces an Ehrenfest-like approach to analyze binary Bose-Einstein condensate dynamics, revealing conditions for almost shape invariant states in free and trapped regimes, including phase separation phenomena.

## Contribution

It proposes a novel Ehrenfest-based method to study CGPE dynamics, uncovering conditions for shape preservation and phase separation in binary BECs beyond traditional techniques.

## Key findings

- Almost shape invariant states can propagate with minimal width change.
- Tuning inter-atomic interactions controls shape retention and phase separation.
- Collapse of Gaussian nature occurs due to collisions in phase-separated states.

## Abstract

We derive Ehrenfest like equations for the coupled Gross Pitaevskii equations (CGPE) which describe the dynamics of the binary Bose-Einstein condensate (BBEC) both in the free particle regime and in the regime where condensate is well trapped. Instead of traditional variational technique, we propose a new Ehrenfest based approach to explore so far unrevealed dynamics for CGPE and illustrate the possibility of almost shape invariant states in both the regimes. In absence of trapping potential, when all the interactions present in the system are attractive, it is possible for an initially mixed Gaussian state to propagate with almost no change in width if the proper initial condition is satisfied. Even for repulsive intra-atomic and attractive inter-atomic interaction ($g_{\alpha\beta}$) one can tune $|g_{\alpha\beta}|$ such that the width of the propagating wave packet remains bounded within almost about $10\%$. We also discuss the dynamics of the initially phase separated condensate and have shown the breakdown of Gaussian nature of the wave packets due to collisions. However, when BEC is trapped in simple harmonic oscillator(SHO) potential, for $g_{\alpha\beta}>0$, it is possible for an initially overlapping state to retain its initial shape if $g_{\alpha\beta}$ is less than a critical value ($g_{\alpha\beta}^c$). If $g_{\alpha\beta}$ exceeds $g_{\alpha\beta}^c$, an overlapping state can become phase separated while keeping its shape unchanged.