Detecting dark matter waves with precision measurement tools

Virialized Ultra-Light Fields (VULFs) are viable cold dark matter candidates and include scalar and pseudo-scalar bosonic fields, such as axions and dilatons. Direct searches for VULFs rely on low-energy precision measurement tools. While the previous proposals have focused on detecting coherent oscillations of the VULF signals at the VULF Compton frequencies at individual devices, here I consider a network of such devices. VULFs are essentially dark matter {\em waves} and as such they carry both temporal and spatial phase information. Thereby, the discovery reach can be improved by using networks of precision measurement tools. To formalize this idea, I derive a spatio-temporal two-point correlation function for the ultralight dark matter fields in the framework of the standard halo model. Due to VULFs being Gaussian random fields, the derived two-point correlation function fully determines $N$-point correlation functions. For a network of $N_{d}$ devices within the coherence length of the field, the sensitivity compared to a single device can be improved by a factor of $\sqrt{N_{d}}$. Further, I derive a VULF dark matter signal profile for an individual device. The resulting line shape is strongly asymmetric due to the parabolic dispersion relation for massive non-relativistic bosons. I discuss the aliasing effect that extends the discovery reach to VULF frequencies higher than the experimental sampling rate. I present sensitivity estimates and develop a stochastic field SNR statistic. Finally, I consider an application of the developed formalism to atomic clocks and their networks.