  Low-temperature characters of superconducting devices yield definite probes
for different superconducting phenomena. We study the macroscopic quantum
tunneling (MQT) in a Josephson junction, composed of a single-gap
superconductor and a two-gap superconductor. Since this junction has two kinds
to the superconducting phase differences, calculating the MQT escape rate
requires the analysis of quantum tunneling in a multi-dimensional configuration
space. Our approach is the semi-classical approximation along a 1D curve in a
2D potential- energy landscape, connecting two adjacent potential (local)
minimums through a saddle point. We find that this system has two plausible
tunneling paths; an in-phase path and an out-of-phase path. The former is
characterized by the Josephson-plasma frequency, whereas the latter is by the
frequency of the characteristic collective mode in a two-band superconductor,
Josephson- Leggett mode. Depending on external bias current and inter-band
Josephson-coupling energy, one of them mainly contributes to the MQT. Our
numerical calculations show that the difference between the in-phase path and
the out-of-phase path is manifest, with respect to the bias- current-dependence
of the MQT escape rate. This result suggests that our MQT setting be an
indicator of the Josephson-Leggett mode.
