  Multi-terminal superconducting Josephson junctions based on the proximity
effect offer the bright opportunity to tailor non trivial quantum states in
nanoscale weak-links. These structures can realize exotic topologies in
multidimensions as, for example, artificial topological superconductors able to
support Majorana bound states, and pave the way to emerging quantum
technologies and future quantum information schemes. Here, we report the first
realization of a three-terminal Josephson interferometer based on a proximized
nanosized weak-link. Our tunneling spectroscopy measurements reveal transitions
between gapped (i.e., insulating) and gapless (i.e., conducting) states, those
being controlled by the phase configuration of the three superconducting leads
connected to the junction. We demonstrate the $topological$ nature of these
transitions: a gapless state necessarily occurs between two gapped states of
different topological index, very much like the interface between two
insulators of different topology is necessarily conducting. The topological
numbers characterizing such gapped states are given by superconducting phase
windings over the two loops forming the Josephson interferometer. Since these
gapped states cannot be transformed to one another continuously withouth
passing through a gapless condition, these are topologically $protected$. Our
observation of the gapless state is pivotal for enabling phase engineering of
more sophisticated artificial topological materials realizing Weyl points or
the anomalous Josephson effect.
