Quantum effects of superconducting phase

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.