Scattering approach to parametric pumping

A dc current can be pumped through a quantum dot by periodically varying two independent parameters $X_1$ and $X_2,$ like a gate voltage or magnetic field. We present a formula that relates the pumped current to the parametric derivatives of the scattering matrix ${S(X}_1{,X}_2)$ of the system. As an application we compute the statistical distribution of the pumped current in the case of a chaotic quantum dot.