Observation of mHz-level cooperative Lamb shifts in an optical atomic clock

We report on the direct observation of resonant electric dipole-dipole interactions in a cubic array of atoms in the many-excitation limit. The interactions, mediated by single-atom couplings to the shared electromagnetic vacuum, are shown to produce spatially-dependent cooperative Lamb shifts when spectroscopically interrogating the mHz-wide optical clock transition in strontium-87. We show that the ensemble-averaged shifts can be suppressed below the level of evaluated systematic uncertainties for state-of-the-art optical atomic clocks. Additionally, we demonstrate that excitation of the atomic dipoles near a Bragg angle can enhance these effects by nearly an order of magnitude compared to non-resonant geometries. Given the remarkable precision of frequency measurements and the high accuracy of the modeled response, our work demonstrates that such a clock is a novel platform for studies of the quantum many-body physics of spins with long-range interactions mediated by propagating phot..., , , # Observation of mHz-level cooperative Lamb shifts in an optical atomic clock

[https://doi.org/10.5061/dryad.wpzgmsbtj](https://doi.org/10.5061/dryad.wpzgmsbtj)

The python script \"make\_figures.py\" produces the plots in Figs. 2-4, S1-S3 given the images and control parameters contained in \"cls.h5\"

## Description of the data and file structure

The data in \"cls.h5\" is saved in HDF5 format and contains the following datasets

* \"timestamp\_\": array of timestamps corresponding to the (UNIX) time at which the corresponding experimental shot began.
* \"runpath\_\": unique strings distinguishing runs, between which realignment/calibration procedures were performed.
* \"φ\_\": array specifying pulse phases sin(φ\_out)
* \"Z\_\": array specifying initial pulse area 4 sqrt(2) cos(θ\_in)
* \"q\_\": array specifying probed transition
* \"ψ\_\": array specifying angle of incidence
* \"counts\_g\_\": array of ground state images
* \"counts\_e\_\": array of excited state images

## Code/Software

Running the py...