We study the current-phase relation (CPR) of an InSb-Al nanowire Josephson junction in parallel magnetic fields up to 700 mT. At high magnetic fields and in narrow voltage intervals of a gate under the junction, the CPR exhibits π shifts. The supercurrent declines within these gate intervals and shows asymmetric gate voltage dependence above and below them. We detect these features sometimes also at zero magnetic field. The observed CPR properties are reproduced by a theoretical model of supercurrent transport via interference between direct transmission and a resonant localized state.

We show that the quasicontinuous gapless spectrum of Andreev bound states in multiterminal semi-classical superconducting nanostructures exhibits a large number of topological singularities. We concentrate on Weyl points in a four-terminal nanostructure and compute their density and correlations in three-dimensional parameter space for a universal random matrix theory model as well as for the concrete nanostructures described by the quantum circuit theory. We mention the opportunities for experimental observation of the effect in a quasicontinuous spectrum.

The recent proposals of devices with overlapping Andreev bound states (ABS) open up opportunities to control and fine tune their spectrum that can be used in various applications in quantum sensing and manipulation. In this paper, we study the ABS in a device consisting of a semiconducting nanowire covered with three superconducting leads. The ABS are formed at two junctions where the wire is not covered. They overlap in the wire where the electron propagation is 1D and in one of the leads where the propagation is 3D. We identify a number of regimes where these two overlaps either dominate or compete, depending on the junction separation L as compared to the correlation lengths ζw,ζs in the wire and in the lead, respectively. We utilize a simple model of 1D electron spectrum in the nanowire and take into account the quality of the contact between the nanowire and the superconducting lead. We present the spectra for different L, detailing the transition from a single ABS in the regime of strong 1D hybridization to two almost independent ABS hybridized at the degeneracy points, in the regime of weak 1D hybridization. We present the details of merging the upper ABS with the continuous spectrum upon decreasing L. We study in detail the effect of quantum interference due to the phase accumulated during the electron passage between the junctions. We develop a perturbation theory for analytical treatment of hybridization. We address an interesting separate case of fully transparent junctions. We derive and exemplify a perturbation theory suitable for the competition regime demonstrating the interference of 1D and two 3D transmission amplitudes.

The recent proposals of experiments with single Andreev bound states make relevant a detailed analysis of these states in multiterminal superconducting nanostructures. We evaluate the energy splitting of degenerate Andreev bound states that overlap in a superconducting lead, and find that the splitting is reduced in comparison with their energy by a small factor RGQ, with RGQ being the dimensionless resistance of the overlap region in the normal state. This permits quantum manipulation of the quasiparticles in these states. We provide a simple scheme of such manipulation.