Interferometric schemes: a probe for topological edge states, a tool in weak measurements

In the first part of this thesis we propose interferometric schemes to probe the properties of edge states of topological insulators and superconductors. First, we consider two helical liquids on opposite edges of a narrow two-dimensional topological insulator, which are connected by one or several local tunnel junctions. In the presence of spatially inhomogeneous Rashba spin-orbit coupling, the spin textures of the helical states on opposite edges are different. We demonstrate that this has a strong impact on the electron transport between the edges. In particular, in the case of many random tunnel contacts, the localization length depends strongly on the spin textures of the edge states. We also propose to realize a Fabry-P\'erot interferometer to measure the spin texture.
Second, we consider domain walls between superconducting and magnetic regions placed on top of a topological insulator, that were predicted support transport channels for Majorana fermions. We propose to study noise correlations in a Hanbury Brown-Twiss type interferometer and find three signatures of the Majorana nature of the channels. First, the average charge current in the outgoing leads vanishes. Furthermore, we predict an anomalously large shot noise in the output ports for a vanishing average current signal. Adding a quantum point contact to the setup, we find a surprising absence of partition noise which can be traced back to the Majorana nature of the carriers. Finally, we calculate the full counting statistics of this structure. At zero bias, we find an interpretation of Majorana-mediated charge transport in terms of two independent half-charge processes.
In the second part of this thesis, we explain how the quantum theory of weak measurements inspired a new method for the measurement of small effects and precision metrology. Many successful implementations of the weak-value amplification scheme have been recently reported. We review this scheme in some details with an emphasis on its benefits and limitations. We then generalize the method, and propose to use weak measurements away from the weak-value amplification regime to carry out precision measurements of time delays of light. Our scheme is robust to several sources of noise that are shown to only limit the relative precision of the measurement. Thus, they do not set a limit on the smallest measurable phase shift contrary to standard interferometry and weak-value based measurement techniques. Our idea is not restricted to phase-shift measurements and could be used to measure other small effects using a similar protocol.