Theory of the Bloch-wave oscillations in small Josephson junctions

Abstract

A quantum-statistical theory of the low-temperature behavior of Josephson junctions with very small capacitanceC and quasiparticle conductivityG, driven by a small currentI(t), is developed. In such junctions the “secondary” quantum macroscopic effects (tunneling and interference) are significant for all values of the Josephson phase difference ϕ, so that new features in the junction dynamics arise, including quantum “Bloch-wave” oscillations. Here the junction dynamics is analyzed in detail starting from a simple macroscopic Hamiltonian. The simplest way to analyze the Bloch-wave oscillations turns out to be a Langevin-type equation for the operator of the junction “quasicharge”q. In particular, this equation shows that the frequencyfB of these oscillations is related by the fundamental equation\(f_B = (\bar I - G\bar V)/2e\) to the dc current\(\bar I\) and voltage\(\bar V\). The main effects suppressing or masking the Bloch-wave oscillations can be analyzed using the equation for the density matrix of the system traced over the states of the quasiparticles. This analysis has made it possible to establish the main conditions for the experimental observation of the predicted effects and to present a general picture of the low temperature dynamics of Josephson junctions.