Effects of the phase periodicity on the quantum dynamics of a resistively shunted Josephson junction

A phenomenological model is introduced for the dissipative quantum dynamics of the phase $\ensuremath{\varphi}$ across a current-biased Josephson junction. The model is invariant under $\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath{\varphi}+2\ensuremath{\pi}$. This enables us to restrict $\ensuremath{\varphi}$ to the interval 0 to $2\ensuremath{\pi}$, equating $\ensuremath{\varphi}+2\ensuremath{\pi}$ with $\ensuremath{\varphi}$, and study the role played by the resulting nontrivial topology. Using Feynman's influence functional theory it is shown that the dissipation suppresses interference between paths with different winding numbers. For Ohmic dissipation this interference is completely destroyed, and $\ensuremath{\varphi}$ can effectively be treated as an extended coordinate. This justifies the use of the usual washboard potential description of a current-biased junction even in the quantum case, provided on Ohmic dissipation mechanism is present.