  We present an exhaustive theoretical analysis of a double-loop Josephson
proximity interferometer, as the one recently realized by Strambini et al. for
the control of the Andreev spectrum via an external magnetic field. This
system, called $\omega$-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 explore the quasi-particle
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 $\omega$-SQUIPT and its potential applications, by
investigating its electrical and magnetometric properties.
