Self-calibrated multiparameter measurement of three-dimensional microwave fields

Rydberg atoms are promising for microwave (MW) sensing and control, but full local MW characterization remains difficult. Existing methods generally do not provide self-calibrated reconstruction of the three-dimensional vector field, which is valuable for both atom-based sensing and in-situ field characterization in complex electromagnetic environments. We propose and implement multi-level, Zeeman-resolved Rydberg electromagnetically induced transparency (EIT) spectroscopy in a laser-cooled atomic ensemble. We extract the three polarization amplitudes from a single spectrum and show that the MW polarization components give rise to closed interferometric loops within the atoms' internal Hilbert space, enabling extraction of their relative phases. Moreover, it is self-calibrated and requires no external reference MW fields, with MW parameters largely separable from one another and from other experimental parameters. These features make it broadly applicable to dedicated sensing platforms as well as quantum optics and quantum information experiments.