Figure 2

Population of momentum tails, including excess compared to Tan–Bogoliubov theory. (a) The product \(N_0n_0\) is a linear predictor of the number of counts within the region \((k_{{\text {min}}}=6~\mu {{\text {m}}}^{-1},k_{{\text {max}}}=10~\mu {{\text {m}}}^{-1})\), consistent with Eq. (10) (solid orange line, dashed lines 95% CI). The gradient \(\Lambda\) in Eq. (15) can be predicted using Eq. (10) (\(\Lambda _{\mathrm{pred}}\) solid purple line) but this disagrees with the experiment by a factor of about 8. Our simulations (dashed line, CE in Fig. 3a) show an increase in counts after release but by less than in the experiment. In (b,c) linear fits to the experimental data yield \(\Lambda _{\mathrm{fit}}\) (points) which vary with the choice of k bounds (fixing \(k_{{\text {max}}}=10\,\upmu {{\text {m}}}^{-1}\) in (b) and \(k_{{\text {min}}}=6\,\upmu {{\text {m}}}^{-1}\) in (c)). For comparison, we show predictions of \(\Lambda\) based directly on Eq. (10) (\(\Lambda _{\mathrm{pred}}\), blue, \(n(k)={\mathscr {C}}/k^4\)), along with the predictions from Eq. (15) using a density function \(n(k)={\mathscr {A}\mathscr {C}/k^4}\) that has an additional prefactor \({\mathscr {A}}=8(3)\) (green) and one that has a modified exponent of \(\alpha =3.86(2)\) via \(n(k)={\mathscr {C}}/k^{\alpha }\) (yellow). A log-normal distribution produces nearly identical predictions (red, offset vertically for visibility). Quoted error estimates correspond to 95% CI of the fit parameters. In (b), the deviation from the predictions at \(k_{{\text {min}}}\lesssim \,6~\mu {{\text {m}}}^{-1}\) is because the collection area starts to overlap with the thermal cloud.