  Bose gas on a two-leg ladder exhibits an interesting topological phase. We
show the presence of a bosonic symmetry-protected-topological (SPT) phase
protected by $Z_2\times Z_2$ symmetry. This symmetry leads to $Z_4$ fractional
quantization of $Z_4$ Berry phase, that is a topological order parameter to
identify the bulk. Using the $Z_4$ Berry phase, we have shown that the
interacting bosonic system possesses rich topological phases depending on
particle density and strength of interaction. Based on the bulk-edge
correspondence, each edge state of the SPT phases is discussed in relation to
the $Z_4$ Berry phases. Especially we have found an intermediate phase that is
not adiabatically connected to a simple adiabatic limit, that possesses
unconventional edge states, which we have numerically demonstrate by employing
the density-matrix-renormalization group algorithm.
