Effects of Interactions on the Critical Temperature of a Trapped Bose Gas

We perform high-precision measurements of the condensation temperature of a harmonically trapped atomic Bose gas with widely tunable interactions. For weak interactions we observe a negative shift of the critical temperature in excellent agreement with mean-field theory. However for sufficiently strong interactions we clearly observe an additional positive shift, characteristic of beyond-mean-field critical correlations. We also discuss nonequilibrium effects on the apparent critical temperature for both very weak and very strong interactions.

Figure 1

Opposing effects of interactions on the critical point of a Bose gas in a harmonic potential $V(r)$. Compared to an ideal gas (dotted blue line) with the same $T_c$, repulsive interactions reduce the critical density, but also broaden the density distribution (solid red line). Mean-field theory (dashed line) captures only the latter effect, and predicts an increase of the critical atom number $N_c$ at fixed temperature $T$, equivalent to a decrease of $T_c$ at fixed $N$.

Figure 2

Determination of the critical point and differential interaction shift. (a) An absorption image of a cloud with 450 000 atoms and a 0.14% condensed fraction. We show an azimuthally averaged cut through the column density for illustrative purposes, while the full 2D distribution is used for fitting. The gas was prepared at a large scattering length, $a=274a_0$, but the interactions were turned off in TOF. (b) Condensed ($N_0$) versus thermal ($N^{\prime }$) atom number for two concurrently taken series with $a=56a_0$ (blue circles) and $a=274a_0$ (black squares). Note that all points correspond to condensed fractions below 2%. The data are scaled to the same temperature ($T=240\mathrm{nK}$) and show the shift of the critical point in the form $\Delta N_c(T)$. The solid point corresponds to the image shown in (a). Solid lines show the extrapolation to $N_0=0$, necessary to accurately determine $N_c$. (c) $N^{\prime }$ is plotted versus $N_0^{2/5}$ for the same data as in (b), showing more clearly the extrapolation procedure.

Figure 3

Interaction shift of $T_c$. Data points were taken with $N\approx 2\times {10}^5$ (blue circles), $4\times {10}^5$ (black squares), and $8\times {10}^5$ (red triangles) atoms. The dashed line is the mean-field result $\Delta T_c/T_c^0=-3.426a/{\lambda }_0$. The solid line shows a second-order polynomial fit to the data (see text). Vertical error bars show statistical errors. Horizontal error bars reflect the 0.1 G uncertainty in the position of the Feshbach resonance.

Figure 4

Nonequilibrium effects. (a) $\Delta T_c/T_c^0$ for $N\approx 4\times {10}^5$ atoms is determined following the procedure which assumes equilibrium. At both low and high $a$ the apparent $T_c$ deviates from the equilibrium curve. (b) Equilibrium criteria (see text): ${\gamma }_{\mathrm{el}}\tau$ (solid squares) is the number of elastic collisions per particle during 1% atom loss; ${\gamma }_{\mathrm{el}}/\overline{\omega }=1$ (open circles) marks the onset of the hydrodynamic regime.