Statics and dynamics of quasi one-dimensional Bose-Einstein condensate in harmonic and dimple trap

We investigate a quasi one-dimensional $^{87}\text{Rb}$ Bose-Einstein condensate in a harmonic trap with an additional dimple trap (dT) in the center. Within a zero-temperature Gross-Pitaevskii mean-field description we provide a one-dimensional physical intuitive model, which we solve by both a time-independent variational approach and numerical calculations. With this we obtain at first equilibrium results for the emerging condensate wave function which reveal that a dimple trap potential induces a bump or a dip in case of a red- or a blue-detuned Gaussian laser beam, respectively. Afterwards, we investigate how this dT induced bump/dip-imprint upon the condensate wave function evolves for two quench scenarios. At first we consider the generic case that the harmonic confinement is released. During the resulting time-of-flight expansion it turns out that the dT induced bump in the condensate wave function remains present, whereas the dip starts decaying after a characteristic time scale which decreases with increasing blue-detuned dT depth. Secondly, once the red- or blue-detuned dT is switched off, we find that bright shock-waves or gray/dark bi-soliton trains emerge which oscillate within the harmonic confinement with a characteristic frequency.