A generic semiclassical superconducting nanostructure connected to multiple superconducting terminals hosts a quasicontinuous spectrum of Andreev states. The spectrum is sensitive to the superconducting phases of the terminals. It can be either gapped or gapless depending on the point in the multidimensional parametric space of these phases. Special points in this space correspond to setting some terminals to the phase 0 and the rest to the phase of π. For a generic nanostructure, three distinct spectra come together in the vicinity of a special point: two gapped phases of different topology and a gapless phase separating the two by virtue of topological protection. In this paper, we show that a weak interaction manifesting as quantum fluctuations of superconducting phases drastically changes the spectrum in a narrow vicinity of a special point. We develop an interaction model and derive a universal generic quantum action that describes this situation. The action is complicated incorporating a nonlocal in time matrix order parameter, and its full analysis is beyond the scope of the present paper. Here, we identify and address two limits: the semiclassical one and the quantum one, concentrating on the first-order interaction correction in the last case. In both cases, we find that the interaction squeezes the domain of the gapless phase in the narrow vicinity of the point so the gapped phases tend to contact each other immediately defying the topological protection. We identify the domains of strong coupling where the perturbation theory does not work. In the gapless phase, we find the logarithmic divergence of the first-order corrections. This leads us to an interesting hypothesis: weak interaction might induce an exponentially small gap in the formerly gapless phase.

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We propose to synchronize Bloch oscillations in a double phase-slip junction by modulating the gate voltage rather than the bias voltage. We show this is advantageous, and the relatively small ac modulation of the gate voltage gives rise to the pronounced plateaus of quantized current of the width of the order of Coulomb blockade threshold. We theoretically investigate the setup distinguishing three regimes of strong, weak, and intermediate coupling, defined by the ratio of the gate capacitance C and the effective capacitance of the phase-slip junctions. An important feature of the intermediate-coupling regime is the occurrence of the fractional plateaus of the quantized current. We investigate the finite temperature effects, finding an empirical scaling for the smoothing of integer plateaus.

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It has been shown that a Weyl point in a superconducting nanostructure may give rise to a Weyl disk where two quantum states are almost degenerate in a two-dimensional manifold in the parametric space. This opens up the possibility of a holonomic quantum manipulation: A transformation of the wave function upon an adiabatic change of the parameters within the degenerate manifold. In this paper, we investigate in detail the opportunities for holonomic manipulation in Weyl disks. We compute the connection at the manifold in quasiclassical approximation to show it is Abelian and can be used for a phase gate. To provide a closed example of quantum manipulation that includes a state preparation and readout, we augment the holonomic gate with a change of parameters that brings the system out of the degenerate subspace. For numerical illustrations, we use a finite value of quasiclassical parameter and exact quantum dynamics. We investigate the fidelity of an example gate for different execution times. We evaluate the decoherence rate and show it can be made small to ensure a wide frequency range where an adiabatic manipulation remains coherent.

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The microscopic theory of superconductivity proposed by John Bardeen, Leon Cooper, and John Robert Schrieffer has been a vital milestone of condensed matter physics and the basis of development of new quantum technologies. It explained the superconductivity as an emergent phenomenon arising from weak phonon-mediated attraction of individual electrons and giving raise to what we know as the superconducting condensate characterized by a complex order parameter that has a modulus and a phase. In many superconductors, the modulus defines a spectral gap. Because of the gap, the superconductor cannot host low-energy single-electron excitations. When an electron excitation comes from a normal metal to a superconductor, it is reflected as a hole, and an incoming hole is reflected as an electron. This process is known as Andreev reflection. In the last 60 years, the superconducting heterostructures have been extensively studied, Josephson junction being the most prominent example. The electrons and holes in such structures perform never-ending roundtrips between the super-conductor interfaces. This gives raise to Andreev bound state spectrum, that determines the supercurrent-phase relation of the nanostructure. There is a recent upheaval of interest in nanostructures with multiple superconducting terminals. One of the subjects of interest is Weyl points: for four or more terminals, the Andreev bands can cross zero energy at a point in the space of superconducting phases space. This is a direct analogy to band crossings in the Brillion zone of topological materials. Another subject is the peculiarities of the spectrum under conditions of the superconducting proximity effect, where the gapped-gapless transition in the space of superconducting phase has been observed. It has been long known that the superconducting phase is, in fact, a quantum variable that is canonically conjugated to charge. It is the basis of all applications for the development of superconducting quantum computing. The 2π periodicity of the superconducting phase also manifests itself in a non-trivial way. It enables events known as quantum phase slips, those promise metrological applications, for instance, a metrological standard of cur-rent. Thus it is of great interest to understand the quantum effects of the superconducting phase in novel and more complex setups. This is the main theme of this thesis.@en

A variety of quantum systems exhibit Weyl points in their spectra where two bands cross in a point of three-dimensional parameter space with conical dispersion in the vicinity of the point. We consider theoretically the soft constraint regime where the parameters are dynamical quantum variables. We have shown that in general the soft constraints, in the quasiclassical limit, result in Weyl disks where two states are (almost) degenerate in a finite two-dimensional region of the three-dimensional parameter space. We provide concrete calculations for two setups: Weyl point in a four-terminal superconducting structure and a Weyl exciton, i.e., a bound state of Weyl electron and a massive hole.

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