|Superconductivity in its various manifestations has been stimulating both experimental and theoretical progress in condensed-matter physics for more than a hundred years.
The remarkable property of electrons to pair up and form quasi-particles gives rise to a plethora of phenomena featuring important practical applications not only in science, but, for instance, also in medicine and metrology.
Recently, new directions in investigating this fascinating subject emerged, such as superconductivity out-of equilibrium and topological superconductors.
Providing experimental evidence for enhanced superconducting correlations in optically pumped copper oxides at temperatures far above the equilibrium transition temperature, the first issue caused considerable excitement.
On the other hand, topological superconductors are believed to provide realizations of highly fault-tolerant qubits by means of hosting non-Abelian quasi-particles, which can be the building blocks of scalable quantum computers.
Experimentally verifying the emergence of these Majorana edge modes, exotic quasi-particles in heterostructures consisting of a conventional superconductor and semiconductors or topological insulators, is one of the most urgent questions to be answered right now.
Both subjects cannot be accounted for with analytically solvable approximations only, and also provide very challenging numerical problems. We implemented a matrix-product state (MPS) based toolkit exploiting $U(1)$ symmetries, providing a flexible and efficient platform to study these complex systems.
In order to efficiently simulate out-of equilibrium setups we studied, compared, and developed time-evolution algorithms for MPS enabling us to choose the most suitable method for a given task.
We also developed a new framework to represent operators in an enlarged Hilbert space so that benefits from conserving $U(1)$ symmetries can also be exploited in systems that originally break such symmetries (projected purification).
Using this method we could efficiently model mesoscopic phenomena such as a charging energy controlled by a gate electrode without further approximations.
Equipped with this techniques we studied out-of equilibrium spectral functions to explore how to identify superconducting correlations more reliably on ultra-short timescales.
We found conclusive evidence that in particular two-particle spectral functions yield excellent probes for the formation of a (quasi-)condensate out-of equilibrium.
Furthermore, we also investigated the question whether in a particular model system there is the possibility of true long-range order out-of equilibrium by studying correlation matrices and the scaling of their eigenvalues.
Here, we observe a change in the algebraic decay of the correlations, even though the extrapolated order parameter is still zero within the error bounds.
Furthermore, we also investigated the effects of coupling a superconductor-semiconductor heterostructure, which is subject to an in-plane magnetic field and a charging energy controlled by a gate voltage, to normal leads.
In the context of experimentally verifying the existence of Majorana edge modes, such systems are believed to be the most promising and recent studies seem to underline this expectation.
However, in order to consistently analyze the experimental data, the effects of quantum fluctuations caused by hybridization of the heterostructure with the leads have to be understood.
Here, only perturbative limits are available so far, i.e., the weak and strong tunneling limit, while the experimentally relevant regime is expected to be somewhere inbetween.
We aimed to fill this gap using the projected purification method to calculate the ground state phase diagram over a wide parameter regime.
Our results indicate that the experimental situation is much more involved than what is predicted from perturbative analysis.