Figure 1

Lack of saturation of a Bose gas. (a) Experimental scheme. The temperature $T$ of a ${}^{39}\mathrm{K}$ gas is fixed by the depth of a crossed optical dipole trap, and the scattering length $a$ is controlled via a Feshbach resonance. The number of thermal ($N^{\prime }$) and condensed ($N_0$) atoms is extracted from bimodal fits to the density distribution after 18 ms of time-of-flight expansion from the trap. (b) $N^{\prime }$ (red points) and $N_0$ (blue points) versus the total atom number $N_{\mathrm{tot}}$ at $T=177\mathrm{nK}$ and $a=135a_0$. The corresponding predictions for a saturated gas are shown by red and blue solid lines. The critical point $N_{\mathrm{tot}}=N_c$ is marked by a vertical dashed line.Reuse & Permissions

Figure 2

Deviation from the saturation picture at a range of interaction strengths and temperatures. We plot $N_0$ versus $N_{\mathrm{tot}}-N_c$ for 18 experimental series, each at fixed $a$ and $T$. The values of the scattering length ($40–356a_0$) and the temperature (115–284 nK) are encoded in the color of the data points. The solid line is the prediction for a saturated gas: $N_0=N_{\mathrm{tot}}-N_c$.Reuse & Permissions

Figure 3

Quantifying the lack of saturation. Here $N^{\prime }$ is plotted as a function of $N_0^{2/5}$ for the same series as in Fig. 1b. The horizontal dotted line is the saturation prediction $N^{\prime }=N_c$. The blue line is the mean-field Hartree-Fock result for a harmonically trapped gas (see the text), with a slope $S_{\mathrm{HF}}=699$. The red line is a linear fit to the data in the range corresponding to $0.1<{\mu }_0/k_{\mathrm{B}}T<0.3$, which gives a nonsaturation slope $S=1283\pm 84$. The solid black line is a guide to the eye based on a second-order polynomial fit. The initial slope of this line is indistinguishable from HF theory.Reuse & Permissions

Figure 4

Saturation in the noninteracting limit. The nonsaturation slopes $S_0$ and $S$ are plotted versus the dimensionless parameter $X=\xi T^2{a}^{2/5}$ (see the text). The $S_0$ data are directly compared with Hartree-Fock theory, $S_{\mathrm{HF}}=1.37X$ (blue line), with no free parameters. For the $S$ data, a linear fit (red line) gives $dS/dX=2.6\pm 0.3$ and an intercept $S(0)=-20\pm 100$, consistent with complete saturation in the ideal gas limit. The $S$ data are based on the 18 ${}^{39}\mathrm{K}$ series shown with the same symbol code in Fig. 2 and two additional series taken with ${}^{87}\mathrm{Rb}$ (black squares). All vertical error bars are statistical. The systematic uncertainty in atom numbers $N^{\prime }$ and $N_0$ is $<10\%$, corresponding to $<6\%$ uncertainty in $S_0$ and $S$ values. The horizontal error bars include the 3 Hz uncertainty in the trapping frequencies $\overline{\omega }/2\pi$ and (for potassium) the 0.1 G uncertainty in the position of the Feshbach resonance.Reuse & Permissions