This thesis discusses the use of subgap and boundary modes for quantum engineering of novel phases, devices and response characteristics. It is comprised of four separate topics: quantum magnetism in Yu-Shiba-Rusinov chains, single-atom Josephson diodes, readout ofMajorana qubits, and surface photogalvanic response inWeyl semimetals. Chains of magnetic adatoms on superconductors have been discussed as promising systems for realizingMajorana end states. Here,we showthat dilute Yu-Shiba-Rusinov (YSR) chains are also a versatile platform for quantum magnetism and correlated electron dynamics, with widely adjustable spin values and couplings. Focusing on subgap excitations, we derive an extended t − J model for dilute quantum YSR chains and use it to study the phase diagram as well as tunneling spectra. We explore the implications of quantum magnetism for the formation of a topological superconducting phase, contrasting it to existing models assuming classical spin textures. Current-biased Josephson junctions exhibit hysteretic transitions between dissipative and superconducting states as characterized by switching and retrapping currents. Here, we develop a theory for diode-like effects in the switching and retrapping currents ofweakly-damped Josephson junctions. We find that while the diode-like behavior of switching currents is rooted in asymmetric current-phase relations, nonreciprocal retrapping currents originate in asymmetric quasiparticle currents. These different origins also imply distinctly different symmetry requirements. We illustrate our results by a microscopic model for junctions involving YSR subgap states. Our theory provides significant guidance in identifying the microscopic origin of nonreciprocities in Josephson junctions. Schemes for topological quantum computation withMajorana bound states rely heavily on the ability to measure products ofMajorana operators projectively. Here,weemployMarkovian quantum measurement theory, including the readout device, to analyze such measurements. Specifically, we focus on the readout of Majorana qubits via continuous charge sensing of a tunnel-coupled quantum dot by a quantum point contact. We show that projective measurements of Majorana products can be implemented by continuous charge sensing under quite general circumstances. Essential requirements are that a combined local parity ˆπ, involving the quantum dot charge along with the Majorana product of interest, be conserved, and that the two eigenspaces of the combined parity ˆπ generate distinguishable measurement signals. The photogalvanic effect requires the intrinsic symmetry of the medium to be sufficiently low, which strongly limits candidate materials for this effect.We explore how inWeyl semimetals the photogalvanic effect can be enabled and controlled by design of Fermi arc states at the material surface. Specifically, we provide a theory of ballistic photogalvanic current in a Weyl semimetal slab. We show that the confinement-induced response is tightly linked to the configuration of Fermi-arc surface states, thus inheriting the same directionality and sensitivity to boundary conditions. In principle this enables the control of the photogalvanic response through manipulation at the surface only.
