  We show that the phase of a Bose-Einstein condensate wave-function of
ultra-cold atoms in an optical lattice potential in two-dimensions can be
detected. The time-of-flight images, obtained in a free expansion of initially
trapped atoms, are related to the initial distribution of atomic momenta but
the information on the phase is lost. However, the initial atomic cloud is
bounded and this information, in addition to the time-of-flight images, is
sufficient in order to employ the phase retrieval algorithms. We analyze the
phase retrieval methods for model wave-functions in a case of a Bose-Einstein
condensate in a triangular optical lattice in the presence of artificial gauge
fields.
