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.Comment: arXiv admin note: text overlap with arXiv:1508.0548