  Quantum phases provide us with important information for understanding the
fundamental properties of a system. However, the observation of quantum phases,
such as Berry's phase and the sign of the matrix element of the Hamiltonian
between two non-equivalent localized orbitals in a tight-binding formalism, has
been challenged by the presence of other factors, e.g., dynamic phases and
spin/valley degeneracy, and the absence of methodology. Here, we report a new
way to directly access these quantum phases, through polarization-dependent
angle-resolved photoemission spectroscopy (ARPES), using graphene as a
prototypical two-dimensional material. We show that the momentum- and
polarization-dependent spectral intensity provides direct measurements of (i)
the phase of the band wavefunction and (ii) the sign of matrix elements for
non-equivalent orbitals. Upon rotating light polarization by \pi/2, we found
that graphene with a Berry's phase of n\pi (n=1 for single- and n=2 for
double-layer graphene for Bloch wavefunction in the commonly used form)
exhibits the rotation of ARPES intensity by \pi/n, and that ARPES signals
reveal the signs of the matrix elements in both single- and double-layer
graphene. The method provides a new technique to directly extract fundamental
quantum electronic information on a variety of materials.
