Resolving the gravitational redshift in a millimetre-scale atomic sample

In this talk I discuss recent progress on the accuracy and precision of state-of-the-art optical atomic clocks. The improved measurement stability of this system enables the resolution of a linear frequency gradient consistent with the gravitational redshift within a single millimetre-scale sample of ultracold strontium. Our result is enabled by improving the fractional frequency measurement uncertainty by more than a factor of 10, now reaching 7.6x10^-21. This heralds a new regime of clock operation necessitating intra-sample corrections for gravitational perturbations. In addition, I discuss the ability to tune the relative strength of the on-site and off-site interactions to achieve a zero density shift at a `magic' lattice depth. This mechanism, together with a large number of atoms, enables the demonstration of the most stable atomic clock while minimizing a key systematic uncertainty related to atomic density. Interactions can also be maximized by driving off-site Wannier-Stark transitions, realizing a ferromagnetic to paramagnetic dynamical phase transition.