Spin Sensitive Behavior of Quantum Dots Coupled to Topological Insulators or Superconductors

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In this thesis we discuss a quantum dot coupled to a helical Luttinger liquid as well as a double dot Josephson junction. In both setups we define a reduced system. We calculate the state of these systems as well as the transport through these systems. Perturbation theory then allows us to identify the relevant processes. First, we consider a Zeeman-split quantum dot coupled to the edge states of a quantum spin Hall insulator. The quantum dot is assumed to be in the cotunneling regime containing a single spin 1/2 electron such that it can be described by the Kondo model. We describe this system using a master equation. Applying a bias voltage to the helical edge states induces a magnetic field on the quantum dot. If this field is not parallel to the Zeeman field the spin polarization on the quantum dot can be manipulated using the bias voltage applied to the edge states. A resonance in the backscattering conductance shows an asymmetry as a function of the bias voltage. The strength of this asymmetry is directly related to the relative orientation of the induced and the Zeeman field. Using full counting statistics the single events bunch or antibunch depending on the polarization of the bias voltage. By using bosonization we are able to include electron-electron interaction in the helical edge states. Finally, we study the critical Josephson current of a parallel double quantum dot Josephson junction. In contrast to previous studies we include all charging states of the quantum dots and do not restrict the discussion to the limit of infinite superconducting gaps. We use analytical as well as numerical methods to calculate the ground state of the system. In the limit of infinite superconducting gaps we find that local transport is suppressed and resonant features are clear indicators for nonlocal behavior. We show that reducing the superconducting gaps can lead to a nonlocal singlet-triplet transition in the ground state which leads to an asymmetric peak structure in the critical current. The relevant processes are identified using perturbation theory. To organize the processes used in the perturbative treatment we introduce a diagrammatic scheme. Our findings support the interpretation of recent experiments and also suggest new signatures of nonlocal Cooper pair transport.