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 momentum P (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 at T=0 if μ^~(0)-ρ₀V₀ > 0, and can vanish at some transition temperature T_λ when fluctuation terms become comparable to the T=0 value. (Here V₀ ρ₀, and μ^~(0) are the uniform parts of the potential, density, and chemical potential with shifted zero of energy, respectively.) A characteristic length Λ(T), diverging at T= T_λ, appears naturally, with its defining relation reducing to a macroscopic uncertainty relation (ρ_P/N)Λ(0)=ℏ/2 at T=0. With certain assumptions it is shown that at T=0, LRO in the phase gradient correlation function is incompatible with off-diagonal long-range order (ODLRO) in the < ψ(†r')ψ(r) > correlation function, and with nonzero condensate function.