Periodic Reporting for period 1 - ToPIKS (Topological P-wave superfluids with Isotopic K mixtureS) Reporting period: 2021-04-01 to 2023-03-31 Summary of the context and overall objectives of the project Fermionic superfluidity and superconductivity are paradigmatic many-body phenomena arising at all energy scales in nature – from ultracold atoms to quarks. At the basis of these phenomena lies pairing between fermionic particles. If pairing happens between particles with non-zero relative angular momentum, the superfluid phase acquires unconventional features, as for the A phase in 3He or electrons in high-Tc superconductors. More specifically, p-wave pairing in 2D is expected to induce superfluid phases with non-trivial topological features, as chirality and anyonic vortex excitations with non-Abelian braiding statistics: the investigation of such topological properties within an ultracold atomic platform is the main goal of the project ToPIKS.The interest in p-wave topological superfluids is motivated by the fact that their anyonic excitations are the zero-magnetic field equivalent of non-Abelian anyonic excitations in 5/2 fractional quantum Hall systems, the unique systems where non-Abelian quasiparticles have been unarguably witnessed so far. The experimental realization of anyonic excitations with non-Abelian statistics under controllable conditions is a paramount goal of nowadays research, due to the promising practical application in quantum computing as fault-tolerant qubits. The high level of control over the system parameters, together with probing capabilities down to the single atom level, makes of ultracold atomic gases an ideal platform towards this achievement.The ToPIKS project activity is structured in three main sequential objectives:1) The realization of ultracold isotopic mixtures of 40K and 39K potassium atoms;2) The experimental investigation of the pairing mechanism with p-wave character predicted for such mixtures in 2D by the seminal work by B. Bazak and D. S. Petrov (Phys. Rev. Lett. 121, 263001 (2018));3) The realization and investigation of p-wave fermionic superfluids and their topological nature. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The report period covers the first 9 months of the ToPIKS activity. During this time, we implemented the quantum simulation of the chiral BF theory in the continuum, a topological field theory which corresponds to the 1D dimensional reduction of the celebrated Chern-Simons theory. The Chern-Simons theory is a powerful theory able to describe anyons within a simple effective picture of weakly interacting particles. Its full implementation in the continuum represents a milestone goal to the scientific community, as it would allow the investigation of anyons in a weakly interacting setting, and allow testing possible detection schemes and understanding their physics without the need for strong correlations.In ToPIKs, we focused on the more accessible goal of performing a quantum simulation of the 1D dimentional reduction of the Abelian Chern-Simons theory, i.e. the chiral BF theory. In this way, we became familiar with the physics of anyons via their field-theoretical description. This work has provided us with very valuable theoretical insight on topological field theories, which lie at the basis of effective descriptions of topological systems with anyonic excitations such as the one that the project ToPIKS aims to realize (objective 3).In parallel to this work, we tackled a new laser setup design for the experimental platform available at ICFO towards the production of ultracold fermionic gases of 40K, on top of the currently available 39K Bose-Einstein Condensates.The applied strategy allowed us keeping the experimental apparatus and toolbox of techniques fully functional without relying in a complete reconstruction of the experiment which would have been complicated to carry out with the limitations of the supply chain caused by the COVID pandemic outbreak.The results on the quantum simulation of topological field theories obtained during the report period are the subject of two publications. Moreover, they were presented by the Fellowship beneficiary E. Neri in three national and international workshops and in one international conference. The beneficiary E. Neri also participated during the project to the formation of two PhD student, and she was involved in a direct knowledge-transfer activity by personally supervising an undergraduate conducting a Summer Student Fellowship project at ICFO. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The project was earlier terminated on 31.12.2021.Here below the planned activities for the natural development of the project beyond the termination date:The next steps of the project tackle the final extension and modification of the apparatus for the production of the first ultracold mixtures of 40K and 39K potassium atoms. This achievement constitutes the first next milestone. The planned activity proceeds with the investigation of the few-body properties of the mixture in 2D, and the p-wave pairing mechanism. These achievements are expected to lead both to technical and scientific publications. The third step of the project tackles the investigation of the 2D system at the many-body level. The goal of the project at this stage it to witness and characterize the emergence of the topological superfluid. This activity, constituting the final milestone of the project, will benefit from the knowledge on topological field theories gained during the first months of the ToPIKS activity covered by this report.The outputs of the ToPIKS project - both predicted and already achieved – are of high impact and interdisciplinary character. On the long term, they are expected to have fundamental out-turns on condensed matter science: indeed, they constitute key advances towards the experimental realization of anyonic excitations with non-Abelian statistics under controllable conditions, and therefore towards their practical application as fault-tolerant qubits in quantum communications and computing technologies.