The recent proposals of experiments with single Andreev bound states make relevant a detailed analysis of these states in multi-terminal 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, 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.

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 ξ w , ξ s 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.

We show that the quasi-continuous gapless spectrum of Andreev bound states in multi-terminal semi-classical superconducting nanostructures exhibits a big number of topological singularities. We concentrate on Weyl points in a 4-terminal nanostructure, compute their density and correlations in 3D parameter space for a universal RMT 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 quasi-continuous spectrum.

Inspired by recent experimental findings that will be presented elsewhere, we formulate and investigate a model of a superconducting junction that combines the electron propagation in a quantum channel with an arbitrary transmission, and that through a localized state. Interesting situation occurs if the energy of the localized state is close to Fermi level, that is, the state is in resonant tunnelling regime. Since this energy is affected by the gate voltage, we expect a drastic modification of transport properties of the junction in a narrow interval of the gate voltages where the energy distance to Fermi level is of the order of Γ, ∆, Γ being the energy broadening of the localized state, ∆ being the superconducting energy gap.We consider the model neglecting the interaction in the localized state, as well as accounting for the interaction in a simplistic mean-field approach where it manifests itself as a spin-splitting. This spin splitting is also contributed by external magnetic field. We also take into account the spin-orbit interaction that can be significant in realistic experimental circumstances.In normal state, we find that the model may describe both peak and dip in the transmission at resonant gate voltage. Spin splitting splits the positions of these peculiarities. Fano interference of the transmission amplitudes results in an asymmetric shape of the peaks/dips. In superconducting state, the spin splitting results in a complex dependence of the superconducting current on the superconducting phase. In several cases, this is manifested as a pair of 0 − π transitions in the narrow interval of gate voltages.

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.