Numerical study of localized impurity in a Bose-Einstein condensate

TL;DR

This study numerically explores how a single impurity interacts with a Bose-Einstein condensate, revealing conditions for localization, impurity effects on the condensate, and dynamic behaviors after quenches.

Contribution

It introduces a coupled differential equation model for impurity-condensate interactions and analyzes impurity localization, effective mass, and post-quench dynamics within a mean-field framework.

Findings

Impurity localization depends on intra- and inter-species coupling strengths.

Impurity induces a bump or dip in the condensate density.

Post-quench dynamics include decay of impurity imprint and emergence of shock waves or bi-solitons.

Abstract

Motivated by recent experiments, we investigate a single $^{133}\text{Cs}$ impurity in the center of a trapped $^{87}\text{Rb}$ Bose-Einstein condensate. Within a zero-temperature mean-field description we provide a one-dimensional physical intuitive model which involves two coupled differential equations for the condensate and the impurity wave function, which we solve numerically. With this we determine within the equilibrium phase diagram spanned by the intra- and inter-species coupling strength, whether the impurity is localized at the trap center or expelled to the condensate border. In the former case we find that the impurity induces a bump or dip on the condensate for an attractive or a repulsive Rb-Cs interaction strength, respectively. Conversely, the condensate environment leads to an effective mass of the impurity which increases quadratically for small interspecies interaction strength. Afterwards, we investigate how the impurity imprint upon the condensate wave function evolves for two quench scenarios. At first we consider the case that the harmonic confinement is released. During the resulting time-of-flight expansion it turns out that the impurity-induced bump in the condensate wave function starts decaying marginally, whereas the dip decays with a characteristic time scale which decreases with increasing repulsive impurity-BEC interaction strength. Secondly, once the attractive or repulsive interspecies coupling constant is switched off, we find that white-shock waves or bi-solitons emerge which both oscillate within the harmonic confinement with a characteristic frequency.