A Quantum Description of Wave Dark Matter

We outline a fundamentally quantum description of bosonic dark matter (DM) from which the conventional classical-wave picture emerges in the limit m ll 10 textrm{eV}. As appropriate for a quantum system, we start from the density matrix which encodes the full information regarding the possible measurements we could make of DM and their fluctuations. Following fundamental results in quantum optics, we argue that for DM it is most likely that the density matrix takes the explicitly mixed form of a Gaussian over the basis of coherent states. Deviations from this would generate non-Gaussian fluctuations in DM observables, allowing a direct probe of the quantum state of DM. Our quantum optics inspired approach allows us to rigorously define and interpret various quantities that are often only described heuristically, such as the coherence time or length. The formalism further provides a continuous description of DM through the wave-particle transition, which we exploit to study how density fluctuations over various physical scales evolve between the two limits and to reveal the unique behavior of DM near the boundary of the wave and particle descriptions.