Effective Hamiltonian approach for simulation of AC Josephson effect

When a voltage difference is maintained across a Josephson junction the superconducting phase difference will evolve in time giving rise to an alternating current across the junction with a frequency determined by the voltage applied. This phenomenon, known as the AC Josephson effect, provides the ideal platform to study the effects of Floquet dynamics. In this work, we study a 2D S-N-S junction in the presence of a periodic time-dependent pairing potential. The system is described by a Hamiltonian that can be separated into two components: a time-independent and time-dependent term. We show an effective Hamiltonian method that allows us to simulate the realistic size system and study its physical properties such as time-averaged density of states (DOS) and current. Our predictions for DOS can be measured experimentally in tunneling conductance. In this effective Hamiltonian, we use the time-independent Hamiltonian in a tight-binding approximation to truncate the system to the lowest N eigenvalues. We study the time-periodic drive using the extended space formalism of the Floquet theory on the truncated basis. Our method shows an efficient alternative to studying Floquet dynamics in a Josephson junction without translational invariance.