Universal Long-distance Tail of Particle-number Density near a Wall in Superfluids

We investigate the long-distance recovery of the particle-number density near a static, impermeable wall in a d-dimensional Galilean-invariant superfluid. Using superfluid effective field theory (EFT) on a half-space, we show that quantum phonon fluctuations produce a z-(d+1) tail in the smooth density profile at zero temperature, where z is the distance from the wall. The tail coefficient is fixed by the bulk equation of state and is independent of microscopic wall details. At low temperature T, the thermal length c s/T, with c s the bulk sound speed, separates the zero-temperature behavior at intermediate distances from a thermal Tz-d tail at larger distances for d>1. For d=1, the long-distance phonon contribution instead decays as (-4πTz/c s). The EFT applies to both Bose and Fermi superfluids, including strongly interacting Fermi gases at unitarity. We estimate the normalized tail amplitude in three dimensions across the BCS--BEC crossover using experimental sound-speed data.