  We propose to use a time-dependent imaginary potential to describe quantum
mechanical tunneling through time-varying potential barriers. We use Gamow
solutions for stationary tunneling problems to justify our choice of potential,
and we apply our method to describe tunneling of a mesoscopic quantum variable:
the phase change across a Josephson junction. The Josephson junction phase
variable behaves as the position coordinate of a particle moving in a tilted
washboard potential, and our general solution to the motion in such a potential
with a time-dependent tilt reproduces a number of features associated with
voltage switching in Josephson junctions. Apart from applications as artificial
atoms in quantum information studies, the Josephson junction may serve as an
electric field sensitive detector, and our studies provide a detailed
understanding of how the voltage switching dynamics couples to the electric
field amplitude.
