Nonequilibrium Transport and Dynamics in Conventional and Topological Superconducting Junctions
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This dissertation contains work that can be divided into three major topics. Before we discuss the outcome of this research, we need to set the basis by introducing several topics that help to understand what is about to come. In each of these introductory topics, we provide small examples that should make the theoretical tools more accessible and, at the same, already provide the basic ingredients for the actual systems that we investigate. We will first start by reviewing the field of mesoscopic superconductivity in Chapter 2, in which we will discuss the Bogoliubov-de Gennes formalism, the dc and ac Josephson effect, Andreev bound states in superconducting junctions, and (multiple) Andreev reflections, which sets the basis for all superconducting systems that we are investigating in this dissertation. After this, we will discuss the Green function formalism in Chapter 3, which will be the proper tool used to derive the effective model Hamiltonians of the investigated systems, and the Keldysh formalism that provides a powerful toolbox to determine the electronic transport through our systems. After these topics, which involve a lot of physics, we will switch gears a bit in Chapter 4 that has the purpose to serve as an introduction to the field of differential topology. After briefly reviewing the famous quantum Hall effect, which more or less started the investigation of topology in condensed matter systems in the early 1980’s, we will give an illustrating introduction to the rather sophisticated mathematical topic of fiber bundles that sets the necessary basis for the rest of Chapter 4. We will discuss the emergence of (non-)Abelian gauge connections and curvatures on principal bundles, their topological classification based on Chern classes, and their topological invariants, the Chern numbers, which eventually are responsible for the perfect quantization of observables even in nonideal systems. For the readers not familiar with differential forms and its integration, we provide in Appendix A a short overview about these concepts. Part II of this dissertation (Chapter 5) deals with the comparison of one-dimensional short conventional and topological Josephson junctions, which are embedded in a dissipative environment described by a damped Josephson resonator. We will study the nonequilibrium occupation of the Andreev bound states both without and with an additional small microwave drive of the superconducting phase difference. The presence of this drive eventually leads to characteristic transitions that are different in both types of junctions, which results in a different supercurrent. Inspired by the recent prediction of nontrivial topology in multiterminal Josephson junctions mentioned above, we systematically develop and study effective low-energy models in quantum-dot-based multiterminal Josephson junctions in Part III (Chapter 6). We will first consider a single quantum dot coupled to conventional s-wave superconducting leads and work out the conditions under which the low-energy Andreev bound states are topologically nontrivial. We further show how to measure the local quantum geometry of these Andreev bound states using an extended microwave spectroscopy measurement scheme. We extend this scheme to a chain of quantum dots connected to superconducting leads, which should represent an experimentally feasible system to study the emerging noninteracting ground-state quantum geometry and topology of the occupied Andreev bound states. Finally, we also develop a double-dot system that is connected to an engineered multiterminal environment. This system hosts a single pair of twofold degenerate Andreev bound states for which we predict a nontrivial second Chern number. We develop a suitable holonomic state-manipulation protocol and propose how to statistically measure the emergent non-Abelian Berry curvature by microwave spectroscopy. In Part IV, we consider a system that hosts the aforementioned YSR states in magnetic impurities on superconducting substrates. Following the recent experimental developments of scanning tunneling microscopy (STM) of YSR states, we first study the interplay between multiple Andreev reflections (MARs) and a single pair of fully spin-polarized YSR states and determine the current-voltage characteristics and the differential conductance. After identifying all possible MAR processes and their characteristic threshold energies, we consider a novel type of STM system, a YSR-STM, in which a magnetic impurity is located at the tip of the superconducting STM. This YSR-STM is used to probe another magnetic impurity on the superconducting substrate, and this motivates to study and identify the YSR-YSR-mediated MARs. We find a whole new series of MARs at characteristic YSR-YSR energies, which are combinations of the individual YSR energies of both the substrate and tip impurities. Finally, we present a few concluding remarks and a summary of our findings in Part V. We will further give a short summary of remaining interesting research directions based on the systems that we investigate in this dissertation. Further extended calculations that provide supplementary details for the calculations of Parts II, III, and IV are provided in Appendices B, C, and D, respectively.
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KLEES, Raffael L., 2021. Nonequilibrium Transport and Dynamics in Conventional and Topological Superconducting Junctions [Dissertation]. Konstanz: University of KonstanzBibTex
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<dcterms:abstract xml:lang="eng">This dissertation contains work that can be divided into three major topics. Before we discuss the outcome of this research, we need to set the basis by introducing several topics that help to understand what is about to come. In each of these introductory topics, we provide small examples that should make the theoretical tools more accessible and, at the same, already provide the basic ingredients for the actual systems that we investigate. We will first start by reviewing the field of mesoscopic superconductivity in Chapter 2, in which we will discuss the Bogoliubov-de Gennes formalism, the dc and ac Josephson effect, Andreev bound states in superconducting junctions, and (multiple) Andreev reflections, which sets the basis for all superconducting systems that we are investigating in this dissertation. After this, we will discuss the Green function formalism in Chapter 3, which will be the proper tool used to derive the effective model Hamiltonians of the investigated systems, and the Keldysh formalism that provides a powerful toolbox to determine the electronic transport through our systems. After these topics, which involve a lot of physics, we will switch gears a bit in Chapter 4 that has the purpose to serve as an introduction to the field of differential topology. After briefly reviewing the famous quantum Hall effect, which more or less started the investigation of topology in condensed matter systems in the early 1980’s, we will give an illustrating introduction to the rather sophisticated mathematical topic of fiber bundles that sets the necessary basis for the rest of Chapter 4. We will discuss the emergence of (non-)Abelian gauge connections and curvatures on principal bundles, their topological classification based on Chern classes, and their topological invariants, the Chern numbers, which eventually are responsible for the perfect quantization of observables even in nonideal systems. For the readers not familiar with differential forms and its integration, we provide in Appendix A a short overview about these concepts. Part II of this dissertation (Chapter 5) deals with the comparison of one-dimensional short conventional and topological Josephson junctions, which are embedded in a dissipative environment described by a damped Josephson resonator. We will study the nonequilibrium occupation of the Andreev bound states both without and with an additional small microwave drive of the superconducting phase difference. The presence of this drive eventually leads to characteristic transitions that are different in both types of junctions, which results in a different supercurrent. Inspired by the recent prediction of nontrivial topology in multiterminal Josephson junctions mentioned above, we systematically develop and study effective low-energy models in quantum-dot-based multiterminal Josephson junctions in Part III (Chapter 6). We will first consider a single quantum dot coupled to conventional s-wave superconducting leads and work out the conditions under which the low-energy Andreev bound states are topologically nontrivial. We further show how to measure the local quantum geometry of these Andreev bound states using an extended microwave spectroscopy measurement scheme. We extend this scheme to a chain of quantum dots connected to superconducting leads, which should represent an experimentally feasible system to study the emerging noninteracting ground-state quantum geometry and topology of the occupied Andreev bound states. Finally, we also develop a double-dot system that is connected to an engineered multiterminal environment. This system hosts a single pair of twofold degenerate Andreev bound states for which we predict a nontrivial second Chern number. We develop a suitable holonomic state-manipulation protocol and propose how to statistically measure the emergent non-Abelian Berry curvature by microwave spectroscopy. In Part IV, we consider a system that hosts the aforementioned YSR states in magnetic impurities on superconducting substrates. Following the recent experimental developments of scanning tunneling microscopy (STM) of YSR states, we first study the interplay between multiple Andreev reflections (MARs) and a single pair of fully spin-polarized YSR states and determine the current-voltage characteristics and the differential conductance. After identifying all possible MAR processes and their characteristic threshold energies, we consider a novel type of STM system, a YSR-STM, in which a magnetic impurity is located at the tip of the superconducting STM. This YSR-STM is used to probe another magnetic impurity on the superconducting substrate, and this motivates to study and identify the YSR-YSR-mediated MARs. We find a whole new series of MARs at characteristic YSR-YSR energies, which are combinations of the individual YSR energies of both the substrate and tip impurities. Finally, we present a few concluding remarks and a summary of our findings in Part V. We will further give a short summary of remaining interesting research directions based on the systems that we investigate in this dissertation. Further extended calculations that provide supplementary details for the calculations of Parts II, III, and IV are provided in Appendices B, C, and D, respectively.</dcterms:abstract>
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