@phdthesis{Fuchs2016,
  author    = {Fuchs, Moritz Jakob},
  title     = {Spin dynamics in the central spin model: Application to graphene quantum dots},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-136079},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {Due to their potential application for quantum computation, quantum dots have attracted a lot of interest in recent years. In these devices single electrons can be captured, whose spin can be used to define a quantum bit (qubit). However, the information stored in these quantum bits is fragile due to the interaction of the electron spin with its environment. While many of the resulting problems have already been solved, even on the experimental side, the hyperfine interaction between the nuclear spins of the host material and the electron spin in their center remains as one of the major obstacles. As a consequence, the reduction of the number of nuclear spins is a promising way to minimize this effect. However, most quantum dots have a fixed number of nuclear spins due to the presence of group III and V elements of the periodic table in the host material. In contrast, group IV elements such as carbon allow for a variable size of the nuclear spin environment through isotopic purification. Motivated by this possibility, we theoretically investigate the physics of the central spin model in carbon based quantum dots. In particular, we focus on the consequences of a variable number of nuclear spins on the decoherence of the electron spin in graphene quantum dots. Since our models are, in many aspects, based upon actual experimental setups, we provide an overview of the most important achievements of spin qubits in quantum dots in the first part of this Thesis. To this end, we discuss the spin interactions in semiconductors on a rather general ground. Subsequently, we elaborate on their effect in GaAs and graphene, which can be considered as prototype materials. Moreover, we also explain how the central spin model can be described in terms of open and closed quantum systems and which theoretical tools are suited to analyze such models. Based on these prerequisites, we then investigate the physics of the electron spin using analytical and numerical methods. We find an intriguing thermal flip of the electron spin using standard statistical physics. Subsequently, we analyze the dynamics of the electron spin under influence of a variable number of nuclear spins. The limit of a large nuclear spin environment is investigated using the Nakajima-Zwanzig quantum master equation, which reveals a decoherence of the electron spin with a power-law decay on short timescales. Interestingly, we find a dependence of the details of this decay on the orientation of an external magnetic field with respect to the graphene plane. By restricting to a small number of nuclear spins, we are able to analyze the dynamics of the electron spin by exact diagonalization, which provides us with more insight into the microscopic details of the decoherence. In particular, we find a fast initial decay of the electron spin, which asymptotically reaches a regime governed by small fluctuations around a finite long-time average value. Finally, we analytically predict upper bounds on the size of these fluctuations in the framework of quantum thermodynamics.},
  subject      = {Elektronenspin},
  language  = {en}
}