Universal van der Waals Force between Heavy Polarons in Superfluids

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 ($\sim 1/r^7$) resulting from the exchange of two superfluid phonons. We also clarify finite temperature effects from the same two-phonon exchange process. The temperature $T$ introduces the additional length scale $c_s/T$ with the speed of sound $c_s$. Leading corrections at finite temperature scale as $T^6/r$ for distances $r\ll c_s/T$ smaller than the thermal length. For larger distances the potential shows a nonrelativistic van der Waals behavior ($\sim T/r^6$) 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.

Figure 1

Schematic picture of the scaling regimes of the induced attractive Casimir interaction $V(r)$.

Figure 2

The exchange of two superfluid phonons (dashed lines) gives rise to the induced potential between two impurities (amputated solid lines).

Figure 3

Strength of the van der Waals interaction $V(r)$ in Eq. (8) compared to the Yukawa potential $V_{\text{Yukawa}}(r)=-g^{2}m\overline{n}{\mathrm{e}}^{-\sqrt{2}r/\xi }/(2\pi r)$ at fixed $r=8\xi$. Our EFT result is evaluated using the experimental data [54] for the speed of sound $c_s/v_F$ as a function of the interaction parameter $-(k_{F}a{)}^{-1}$ with $s$-wave scattering length $a$ and Fermi momentum $k_F$. The importance of the van der Waals potential grows from the weakly coupled molecular BEC (left) to dominate in the unitary and BCS regimes (right).