Kvantne geometrijske faze v sistemih s sklopitvijo spin-tir

This Thesis explores the interplay between quantum geometric phases, spin-orbit interaction (SOI), topology, and quantum dynamics in three distinct systems: open quantum spin-boson models, nonlinear optical response in topological semimetals, and synthetic topological phases in Josephson junctions. In the first part, we consider the spectral and initial value problem for the Lindblad-Gorini-Kossakowski-Sudarshan master equation describing an open quantum system of bosons and spins, where the bosonic parts of the Hamiltonian and Lindblad jump operators are quadratic and linear respectively, while the spins couple to bosons via mutually commuting spin operators. As a concrete example, we examine a spin qubit that is confined in a driven harmonic trap, influenced by Rashba SOI, and coupled to a thermal bath. In the band theory, SOI lifts band degeneracy, a crucial factor in the search for topological states of matter, including Weyl semimetals (WSMs). In the second part of this Thesis, we investigate anomalous high-harmonic generation in a model undergoing a transition from a time-reversal symmetry-broken WSM to a semi-Dirac regime. The latter is a gapless semimetal with a dispersion that is parabolic in one direction and conical in the other two. We demonstrate that the intensity of the induced anomalous high harmonics is significantly enhanced in the semi-Dirac regime. In WSMs, we show that anomalous high harmonics primarily originate from excitations at momenta where the dispersion deviates from strict linearity. Furthermore, we establish that within the linearized low-energy theory, the anomalous response remains purely harmonic, highlighting the role of nonlinear band dispersion in driving higher-order harmonics. In the third part, we study a two-terminal Josephson junction with conventional superconductors and a normal region with Rashba SOI, characterized by two Aharonov-Casher (AC) fluxes. When the superconducting phase difference equals $pi$, the Andreev subgap spectrum may host zero-energy Weyl singularities associated with a vanishing normal-state reflection eigenvalue. With one of the AC fluxes playing the role of a quasimomentum, the junction can be viewed as an artificial one-dimensional chiral topological insulator. Its topological phase can be tuned by crossing a Weyl singularity by means of varying the remaining AC flux. By associating an additional component of the quasimomentum with the superconducting phase difference, an artificial Chern insulator is realized. Lastly, we devise a driving protocol to determine the enclosed topological charge by measuring superconducting and spin currents through the system.