The Surface Region of Superfluid He as a Dilute Bose-Condensed Gas

In the low-density surface region of superfluid 4He, the atoms are far apart and collisions can be ignored. The only effect of the interactions is from the long-range attractive Hartree potential produced by the distant high-density bulk liquid. As a result, at T = 0, all the atoms occupy the same singleparticle state in the low-density tail. Striking numerical evidence for this 100% surface BEC was given by Pandharipande and coworkers in 1988. We derive a generalized Gross-Pitaevskii equation for the inhomogeneous condensate wave function Φ(z) in the low-density region valid at all temperatures. The overall amplitude of Φ(z) is fixed by the bulk liquid, which ensures that it vanishes everywhere at the bulk transition temperature.