Two-Body and Three-Body Contacts for Identical Bosons near Unitarity

In a recent experiment with ultracold trapped ${}^{85}\mathrm{Rb}$ atoms, Makotyn et al. studied a quantum-degenerate Bose gas in the unitary limit where its scattering length is infinitely large. We show that the observed momentum distributions are compatible with a universal relation that expresses the high-momentum tail in terms of the two-body contact $C_2$ and the three-body contact $C_3$. We determine the contact densities for the unitary Bose gas with number density $n$ to be ${\mathcal{C}}_2\approx 20n^{4/3}$ and ${\mathcal{C}}_3\approx 2n^{5/3}$. We also show that the observed atom loss rate is compatible with that from 3-atom inelastic collisions, which gives a contribution proportional to $C_3$, but the loss rate is not compatible with that from 2-atom inelastic collisions, which gives a contribution proportional to $C_2$. We point out that the contacts $C_2$ and $C_3$ could be measured independently by using the virial theorem near and at unitarity, respectively.

Figure 1

Momentum distributions for the unitary Bose gas. The dimensionless quantity $k^{4}n(k)/N{k}_F$, where $k_F={(6{\pi }^2⟨n⟩)}^{1/3}$, is plotted as a function of $k/k_F$. The data from the JILA group in Ref. [7] are for two average densities: $⟨n⟩=5.5\times {10}^{12}/{\mathrm{cm}}^3$ (red line with lower tail) and $1.6\times {10}^{12}/{\mathrm{cm}}^3$ (blue line with higher tail) [7]. The solid curve through the higher-$⟨n⟩$ data is a two-parameter fit obtained by adjusting $C_2$ and $C_3$. The dashed curve through the lower-$⟨n⟩$ data is a parameter-free prediction obtained by scaling $C_2$ and $C_3$ from the higher-$⟨n⟩$ fit. The horizontal dotted line is the contribution to both distributions from $C_2$.