Zero-point motion and superfluid helium

Abstract

We propose that He II exhibits macroscopic [σ _P /N ∼O(1)] quantum zero-point motion in momentum space, i.e., that a nonzero root-mean-square superfluid velocity exists even in an equilibrium superfluid system at rest. At absolute zero, using coherent states, we relate the uncertainty σ _P /N in the total momentumP (per particle) to the long-range-order (LRO) part of the phase gradient correlation function, which is proposed as an order parameter. The local equilibrium equation for the superfluid velocity potential derived by Biswas and Rama Rao yields, in the strict equilibrium limit, the equation determining this order parameter in terms of fluctuation correlations that remain to be determined. The order parameter is interaction dependent, nonzero atT=0 if$\tilde \mu$(0)−ρ₀V₀>0, and can vanish at some transition temperatureT _λ when fluctuation terms become comparable to theT=0 value. (HereV ₀ ρ₀, and$\tilde \mu$(0) are the uniform parts of the potential, density, and chemical potential with shifted zero of energy, respectively.) A characteristic length Λ(T), diverging atT=T _λ, appears naturally, with its defining relation reducing to a macroscopic uncertainty relation (σ _P /N)Λ(0)=ħ/2 atT=0. With certain assumptions it is shown that atT=0, LRO in the phase gradient correlation function is incompatible with off-diagonal long-range order (ODLRO) in the 〈ω^†(r′)ω(r)〉 correlation function, and with nonzero condensate function.