We started with the study of strongly interacting Rydberg slow light polaritons. We achieved some very important results on the connection between strongly interacting polaritons and the classical description of light fields within a media with a Kerr non-linearity. This especially demonstrated that previous claims of a no-go theorem for phase gates with photons do not apply for Rydberg slow light polaritons. We analyzed in detail the requirements on conditions for the realization of a two-qubit phase gate based on Rydberg polaritons. In the strongly interacting regime, we paved the way for the proper description of effective three-body interactions between appearing between Rydberg polaritons. A very interesting regime emerged, where the blockade radius exceeds the size of the atomic cloud, and gives rise to a so-called Rydberg superatom. Within this regime, we demonstrated, that the collective light matter coupling strongly enhances the spontaneous emission in the forward direction such that even a free-space setup turns essentially one-dimensional giving rise to many intriguing phenomena. Especially, within a strong experimental collaboration, we managed to detect Rabi oscillations in the out-going field, two-photon correlations, as well as even three-body correlations; phenomena, which are usually expected to be only accessible within cavity quantum electrodynamics. Most intriguingly, the decay dynamics deviates from a simple exponential decay due to the internal structure of the Rydberg superatom, where photon exchange couples bright and dark states. These achievements all belong the the two main tasks of the project.
In a second project, we presented the first observation of a topological phase in artificial matter within a experimental collaboration. These results have only been possible by experimental break throughs in deterministic loading of atoms into an array of optical tweezers, as well as the determination of the microscopic interaction potentials as well as the theoretical understanding of the topological phase. The observed topological phase belongs to the class of symmetry protected topological phases in one-dimension. This research and the collaboration is still extremely active an we are in the process to establish the requirements for the realization of true long-range entangled topological phase in two-dimensions. The first steps have been the demonstration of the intrinsic spin-orbit coupling for the dipolar exchange interaction, which on a single particle level gives rise to topological band structures.
Finally, a strong focus in the project was on understanding the general properties of topological phases in artificial matter such a Rydberg atoms and the study of potential applications. Especially, we demonstrated the efficient coupling between distant qubits by topological networks, as well as presented a novel entanglement transition in a quantum circuit with competing measurements; the latter is closely connected to error-correction schemes based on topological phases such a Kitaev Majorana chain in one-dimension and the toric code in three-dimension.
