Quench dynamics of interacting impurity boson in boson sea

We use the Yudson Representation to study the quench dynamics of a system consisting of a one dimensional gas of interacting bosons and a mobile impurity boson. We are able to get an exact solution for finite coupling constant and finite time for two particles. To solve for more particles, long time approximation is made to simplify the calculation. We calculate the time evolution of impurity density and noise correlation and compare the results with experimental data.