  Quantum phase diffusion in a small underdamped Nb/AlO$_x$/Nb junction ($\sim$
0.4 $\mu$m$^2$) is demonstrated in a wide temperature range of 25-140 mK where
macroscopic quantum tunneling (MQT) is the dominant escape mechanism. We
propose a two-step transition model to describe the switching process in which
the escape rate out of the potential well and the transition rate from phase
diffusion to the running state are considered. The transition rate extracted
from the experimental switching current distribution follows the predicted
Arrhenius law in the thermal regime but is greatly enhanced when MQT becomes
dominant.
