We present an exhaustive theoretical analysis of a double-loop Josephson proximity interferometer, such as the one recently realized by Strambini et al. for control of the Andreev spectrum via an external magnetic field. This system, called ω-SQUIPT, consists of a T-shaped diffusive normal metal (N) attached to three superconductors (S) forming a double-loop configuration. By using the quasiclassical Green-function formalism, we calculate the local normalized density of states, the Josephson currents through the device, and the dependence of the former on the length of the junction arms, the applied magnetic field, and the S/N interface transparencies. We show that by tuning the fluxes through the double loop, the system undergoes transitions from a gapped to a gapless state. We also evaluate the Josephson currents flowing in the different arms as a function of magnetic fluxes, and we explore the quasiparticle transport by considering a metallic probe tunnel-coupled to the Josephson junction and calculating its I-V characteristics. Finally, we study the performances of the ω-SQUIPT and its potential applications by investigating its electrical and magnetometric properties.

Multi-terminal superconducting Josephson junctions based on the proximity effect offer the opportunity to tailor non-trivial quantum states in nanoscale weak links. These structures can realize exotic topologies in several dimensions, for example, artificial topological superconductors that are able to support Majorana bound states, and pave the way to emerging quantum technologies and future quantum information schemes. Here we report the realization of a three-terminal Josephson interferometer based on a proximized nanosized weak link. Our tunnelling spectroscopy measurements reveal transitions between gapped (that is, insulating) and gapless (conducting) states that are 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 indices, in much the same way that the interface between two insulators of different topologies is necessarily conducting. The topological numbers that characterize such gapped states are given by superconducting phase windings over the two loops that form the Josephson interferometer. As these gapped states cannot be transformed to one another continuously without passing through a gapless condition, they are topologically protected. The same behaviour is found for all of the points of the weak link, confirming that this topology is a non-local property. Our observation of the gapless state is pivotal for enabling phase engineering of different and more sophisticated artificial topological materials.