In the presence of interactions, helical edge states have successfully been modeled using the helical Luttinger liquid theory. We aim to develop a better understanding of the interplay between electron-electron interactions and the spin-orbit coupling, which is responsible for the helicity, and find experimentally detectable signatures of this interplay.

However, the Luttinger liquid theory allows only a low-energy description because it assumes that the single-particle spectrum is strictly linear. If non-linearities are taken into account, an exact solution is generally no longer possible. We calculated the dynamic response functions of interacting spinful fermionic systems beyond the approximation of the Luttinger liquid theory.

The departure from the Luttinger liquid theory away from the limit of low energies leads to notable modifications of the dynamic correlation functions and also introduces qualitatively new effects, like relaxation, which are entirely absent in the Luttinger liquid theory. We study those effects which require extending Luttinger liquid theory in various directions. Recently, we have focused in particular on one-dimensional systems with spin-orbit coupling, which can be realized for instance in indium arsenide or indium antimonide quantum wires.

The simplest quantity we can study is the electric current in response to an applied bias voltage. More information can be gained from the study of the full counting statistics, which contains also information about the current noise and higher order correlation functions of the quantum mechanical current operators. Moreover, energy transport still leaves open questions which are important for an understanding of the thermodynamic properties of small quantum systems.

An exciting prospect is the use of MBS as building blocks for topological qubits, on which certain operations can be performed in a decoherence-free way using braiding. However, since these protected operations are insufficient to allow for universal quantum computing, other methods to manipulate MBS need to be developed. We recently investigated the coupling between MBS and microwave photons and showed that MBS can be efficiently manipulated in microwave cavities.

Another topic of interest is the backaction of the electronic or optical read-out device on the resonator. It turns out that for suitable coupling, the measurement can cause a strong backaction on the resonator, thus making it possible to control its motion. This can be exploited, e.g., to cool the resonator to very low temperatures in order to bring it to the quantum regime.

Recently, there has been growing interest in bringing mechanical systems to quantum states, e.g., Fock states or superpositions. One of the long-term goals of this type of research is to develop a better understanding of the transition between classical physics and quantum mechanics.