This dataset presents data on a supersolid phase of matter. The supersolid arises from synthetic spin-orbit coupling coupling of ultracold potassium atoms. The data was obtained from absorption imaging, where absorption of photons is measured with an electron multiplying CCD camera and using a high-numerical-aperture objective. Two different kinds of measurements were performed: 1. In-situ images, where the absorption imaging is performed with the atomic cloud in the optical trap and 2. Time-of-flights images, measuring the momentum distribution of the cloud by mapping the momentum distribution to density via a time-of-flight expansion. The in-situ images are characterized by a modulated Gaussian with a certain width, spatial modulation phase and modulation period, whereas the time-of-flight images are characterized by the positions of four separate momentum clouds.

We acknowledge funding from the European Union (ERC CoG-101003295 SuperComp), the Spanish Ministry of Science and Innovation MCIN/AEI/10.13039/501100011033 (LIGAS projects PID2020-112687GB-C21 at ICFO and PID2020112687GB-C22 at UAB, and Severo Ochoa CEX2019000910-S), the EU QuantERA project DYNAMITE (funded by MICN/AEI/ 10.13039/501100011033 and by the European Union NextGenerationEU/PRTR PCI2022-132919 (Grant No. 101017733)), Deutsche Forschungsgemeinschaft (Research Unit FOR2414, Project No. 277974659), Generalitat de Catalunya (AGAUR SGR 2021-SGR-01448 at ICFO and 2021-SGR-00138 at UAB, CERCA program), Fundació Cellex and Fundació Mir-Puig. C.S.C., S.H., and R.R. acknowledge support from the European Union (Marie Skłodowska-Curie–713729, Marie Skłodowska Curie–101149245 Epiquant, and Marie Skłodowska-Curie–101030630 UltraComp). V.B.M. acknowledges support from the Beatriu de Pinós Program and the Ministry of Research and Universities of the Government of Catalonia (Grant No. 2019-BP-00228). J.C. acknowledges support from the Ministry of Economic Affairs and Digital Transformation of the Spanish Government through the QUANTUM ENIA project call – Quantum Spain project, and A.C. acknowledges support from the UAB Talent Research program during the early stages of this work.