Dark Matter Detection With Bound Nuclear Targets: The Poisson Phonon
  Tail
  Dark matter (DM) scattering with nuclei in solid-state systems may produce
elastic nuclear recoil at high energies and single-phonon excitation at low
energies. When the dark matter momentum is comparable to the momentum spread of
nuclei bound in a lattice, $q_0 = \sqrt{2 m_N \omega_0}$ where $m_N$ is the
mass of the nucleus and $\omega_0$ is the optical phonon energy, an
intermediate scattering regime characterized by multi-phonon excitations
emerges. We study a greatly simplified model of a single nucleus in a harmonic
potential and show that, while the mean energy deposited for a given momentum
transfer $q$ is equal to the elastic value $q^2/(2m_N)$, the phonon occupation
number follows a Poisson distribution and thus the energy spread is $\Delta E =
q\sqrt{\omega_0/(2m_N)}$. This observation suggests that low-threshold
calorimetric detectors may have significantly increased sensitivity to sub-GeV
DM compared to the expectation from elastic scattering, even when the energy
threshold is above the single-phonon energy, by exploiting the tail of the
Poisson distribution for phonons above the elastic energy. We use a simple
model of electronic excitations to argue that this multi-phonon signal will
also accompany ionization signals induced from DM-electron scattering or the
Migdal effect. In well-motivated models where DM couples to a heavy,
kinetically-mixed dark photon, we show that these signals can probe
experimental milestones for cosmological DM production via thermal freeze-out,
including the thermal target for Majorana fermion DM.
