We present measurements of the supercurrent decay in high quality Nb/AlOx/N
b Josephson junctions which show evidence of energy-level quantization with
in the washboard potential describing the junction. The effect of discrete
energy levels leads to oscillations in the rate of escape from the zero vol
tage state as a function of the bias current, observed at different tempera
tures. The oscillation spacing fits that expected from the energy level qua
ntization. Moreover, a fit with the quantum theory allows us to derive inte
resting conclusions about the effective dissipation of the system at low te
mperatures. [S0163-1829(99)08701-9].