The effect of quantum mechanical uncertainty on punchthrough probability in a Josephson junction

Using a quantum mechanical analog of the Stewart-McCumber model of a Josephson junction we have calculated the time evolution of the minimum uncertainty wavepacket for a particle localized near a maximum in a Josephson-like potential. This is used to calculate the punchthrough probability as the probability of finding a particle, initially localized (at t = 0) in the vicinity of the peak of the potential, still within a certain distance of the peak at a time t
d
later (where
t_{d}=\frac{2I\min}{|\dot{I}|} - RC
). Our main prediction is that there is a tail to the punchthrough probability, but that it is substantially quantum mechanical in origin, since the quantum mechanical spread of the wavepacket describing the particle contributes to the punchthrough probability tail and reduces it, in any given region near the peak of the Josephson potential, from what would be expected classically.