Congratulations to all! It is a collective effort but thanks Huang Xing in particular for many works, and for remarkable perseverance along this way -- Let's recall the Gaussian business starts at a time when the school is closed!

Special thanks to Prof. Haidong Yuan for insights on Cramer-Rao bounds, and to Mr Wang, Dong for supportive calculations on phase-angle spectroscopy!

Now, a brief intro to this work:

From precision spectroscopy to quantum simulation, precise optical imaging is crucially important for advancing ultracold atomic physics research at frontiers. The prevailing imaging method in these researches is to directly record fluorescence, absorption, and phase shift.  While holographic imaging techniques are rapidly developing for applications in other fields, their utilities in atomic physics are rare. In this work, we argue that holographic measurement is highly useful for spectroscopy with cold atomic ensembles to overcome intrinsic atom-number and interaction-strength fluctuations, fundamentally efficient, and offers the unique opportunity of single-shot phase-angle readouts with power-broadening resilience. By extending the spectroscopy data to complex numbers and by enabling diffraction-limited 3D resolution, the holographic method may help unlock highly exciting opportunities associated with precise spectroscopic imaging and quantum sensing with cold atomic ensembles on strong optical transitions.

We develop a systematic approach to address key challenges in holographic imaging of cold atoms. Comparing with {\it e.g.} biological applications, our choices of schemes and diversities in holographic measurements are severely constrained by aspects of cold-atom setups. Our philosophy is to fully characterize the imaging setup and to efficiently utilize all the available knowledge from single-shot inline holography. The efficient application of prior knowledge is numerically assisted by Gaussian beam propagation, a simple application of the well-established Gaussian-decomposition technique. With the method, we successfully demonstrate complex-valued spectroscopic imaging of axially displaced microscopic $^{87}$Rb samples in a sparse lattice with micrometer-level 3D resolution. Atom numbers and transition frequencies are simultaneously inferred on each lattice site, within single shot. We achieve hundred-kHz-level single-shot frequency resolution out of the $\Gamma/2\pi=6~$MHz natural linewidth, with merely hundreds of atoms, a result that appears extremely difficult to realize with traditional spectroscopy in presence of the atom number and interaction strength uncertainties.

Of course, there are a few limitations in this work. The spatial resolution could be higher to fascilitate imaging density correlations. The imaging depth-of-view could in principle be extended over a few hundred microns. These aspects appear improvable in a straightforward manner. We are also working on ways to correct for aplanatic aberrations using Gaussian optics.  Let's move forward to more exciting developments!