Complex-valued 3D atomic spectroscopy with Gaussian-assisted inline holography

When a laser-cooled atomic sample is optically excited, the envelope of coherent forward scattering can often be decomposed into a few complex Gaussian profiles. The convenience of Gaussian propagation helps addressing key challenges in digital holography. In this work, we develop a Gaussian-decomposition-assisted approach to inline holography, for single-shot, simultaneous measurements of absorption and phase-shift profiles of small atomic samples sparsely distributed in 3D. The samples' axial positions are resolved with micrometer resolution, and their spectroscopy are extracted from complex-valued images recorded at various probe frequencies. The phase-angle readout is not only robust against transition saturation, but also insensitive to atom-number and optical-pumping-induced interaction-strength fluctuations. Benefiting from such features, we achieve hundred-kHz-level single-shot resolution to the transition frequency of a $^{87}$Rb D2 line, with merely hundreds of atoms. We further demonstrate single-shot 3D field sensing by measuring local light shifts to the atomic array with micrometer spatial resolution.