Quantum modeling and control of Josephson junction by quantum Hamilton mechanics

In this paper, we apply quantum Hamilton mechanics to describe the dynamical motion of an electronic Cooper pair tunneling through a Josephson junction. Quantum Hamilton equations provide us a set of canonical equations q̇ = f(q, p) and ṗ = g(q, p) to model the tunneling dynamics of a Cooper pair. Instead of using the conventional probabilistic description, we solve complex quantum trajectory q(t) = q_R + q_I·i from the Hamilton equations to demonstrate the tunneling dynamics on a geometrical phase plane. In order to control the dynamics of a cooper pair, we add a gate voltage parameter n_g to the quantum Hamiltonian. By adjusting the magnitude of n_g, we successfully control the tunneling dynamic of the Cooper pair such that the predicted current-voltage relation is in excellent agreement with the experimental measurements.