Real-time dynamics of an impurity in an ideal Bose gas in a trap

We investigate the behavior of a harmonically trapped system consisting of an impurity in a dilute ideal Bose gas after the boson-impurity interaction is suddenly switched on. As a theoretical framework, we use a field-theory approach in the space-time domain within the T-matrix approximation. We establish the form of the corresponding T matrix and address the dynamical properties of the system. As a numerical application, we consider a simple system of a weakly interacting impurity in one dimension where the interaction leads to oscillations of the impurity density. Moreover, we show that the amplitude of the oscillations can be driven by periodically switching the interaction on and off.