  We design a computational experiment in which a quantum particle tunnels into
a billiard of variable shape and scatters out of it through a double-slit
opening on the billiard's base. The interference patterns produced by the
scattered probability currents for a range of energies are investigated in
relation to the billiard's geometry which is connected to its classical
integrability. Four billiards with hierarchical integrability levels are
considered: integrable, pseudo-integrable, weak-mixing and strongly chaotic. In
agreement with the earlier result by Casati and Prosen [1], we find the
billiard's integrability to have a crucial influence on the properties of the
interference patterns. In the integrable case most experiment outcomes are
found to be consistent with the constructive interference occurring in the
usual double-slit experiment. In contrast to this, non-integrable billiards
typically display asymmetric interference patterns of smaller visibility
characterized by weakly correlated wave function values at the two slits. Our
findings indicate an intrinsic connection between the classical integrability
and the quantum dephasing, responsible for the destruction of interference.
