Universal van der Waals Force between Heavy Polarons in Superfluids

Abstract

Here, we investigate the long-range behavior of the induced Casimir interaction between two spinless heavy impurities, or polarons, in superfluid cold atomic gases. With the help of effective field theory (EFT) of a Galilean invariant superfluid, we show that the induced impurity-impurity potential at long distance universally shows a relativistic van der Waals-like attraction (~1/r 7 ) resulting from the exchange of two superfluid phonons. We also clarify finite temperature effects from the same two-phonon exchange process. The temperature $\mathrm{T}$ introduces the additional length scale c s /$\mathrm{T}$ with the speed of sound c s . Leading corrections at finite temperature scale as $\mathrm{T}$ 6 /r for distances r$\ll$c s /$\mathrm{T}$ smaller than the thermal length. For larger distances the potential shows a nonrelativistic van der Waals behavior (~$\mathrm{T}$/r 6 ) instead of the relativistic one. Our EFT formulation applies not only to weakly coupled Bose or Fermi superfluids but also to those composed of strongly correlated unitary fermions with a weakly coupled impurity. The sound velocity controls the magnitude of the van der Waals potential, which we evaluate for the fermionic superfluid in the BCS-BEC crossover.