Quantum effects in the current-voltage characteristic of a small Josephson junction

We study a simple quantum-mechanical generalization of the resistively shunted junction model to describe the current-voltage characteristic of a single Josephson junction. An exact series representation is given for the nonlinear resistance, which is valid for arbitrary temperature and dissipation and includes both classical and quantum phase slips. For sufficiently high temperature, the quantum effects disappear and \ensuremath{\Elzxh} cancels from the corresponding series. The resulting expression is equivalent to classical Brownian motion in a periodic potential with friction coefficient \ensuremath{\gamma}. We derive a rigorous expression for the leading quantum corrections to the classical result which are of order ${\ensuremath{\Elzxh}}^{2}$. In the overdamped limit they are equivalent to a renormalization of the barrier, whereas for \ensuremath{\gamma}\ensuremath{\rightarrow}0 they are bounded below by the effect of an additional bias of order \ensuremath{\Elzxh}\ensuremath{\gamma}. An approximate continued fraction is derived for the resistance in the general case, which becomes exact in the classical high-damping limit. It is evaluated numerically and leads to reasonable qualitative results for small barriers, unless the temperature approaches zero. We discuss the possibility of seeing the quantum effects in junctions with small capacity, in particular the existence of a regime with negative differential resistance and the implications of our results to the quasireentrant behavior observed in thin superconducting granular films.