Experimental study of the thermodynamics of an interacting trapped Bose-Einstein condensed gas

We have investigated experimentally the finite-temperature properties of a Bose-Einstein condensed cloud of ${}^{87}\mathrm{Rb}$ atoms in a harmonic trap. Focusing primarily on condensed fraction and expansion energy, we measure unambiguous deviations from ideal-gas thermodynamics and obtain good agreement with a Hartree-Fock description of the mixed cloud. Our results offer clear evidence of the mutual interaction between the condensed and thermal components. To probe the low-temperature region unaccessible to the usual time-of-flight technique, we use coherent Bragg scattering as a filtering technique for the condensate. This allows us to separate spatially the condensed and normal components in time of flight and to measure reliably temperatures as low as $0.2T_c^0$ and thermal fractions as low as 10%. Finally, we observe evidence for the limitations of the usual image analysis procedure, pointing out to the need for a more elaborate model of the expansion of the mixed cloud.

Figure 1

Bragg diffraction as a condensate filter. (a), (b), and (c) correspond to 92%, 85%, and 30% condensed fraction, respectively ($T∕T_{\text{c}}^0\approx 0.2$, 0.35, and 0.8). Two-dimensional absorption images are shown, with a cut along the direction of the trap weakest axis (“axial cut”). The top images show regular absorption images for a 24.27-ms time of flight (dashed lines in the cut). The bottom images, also shown as the solid line in the axial cut, corresponding to the same temperatures and atom numbers within experimental reproducibility, have been taken after applying a moving optical lattice tuned to realize a $\pi$ Bragg pulse, transferring two photon recoils to almost all condensed atoms (velocity 1.1 cm∕s). The distance traveled by the condensate is verified to be $250\mu \text{m}$, corresponding to a free flight of 22.27 ms after the Bragg pulse. As can be seen, the thermal cloud is barely affected, due to its much larger extent in momentum space.

Figure 2

Condensed fraction as a function of reduced temperature. (a),(b) Experimental data after averaging are shown as solid circles, with statistical error bars. Data in (a) were taken using the standard time-of-flight technique, while data in (b) were measured with the Bragg filtering scheme and extend to lower temperature, down to $T\approx 0.2T_{\text{C}0}$, whereas the standard method is limited to $T\approx 0.4T_{\text{C}0}$. Lines show theoretical expectations according to an ideal gas calculation, including finite-size effects (dashed line), a “semi-ideal” model that neglects interactions within the thermal cloud (dotted line), and a self-consistent HF calculation (solid line). Due to a different average atom number, the parameter $\overline{\eta }$ that controls the importance of interactions is different in each case, $\overline{\eta }=0.49$ and $\overline{\eta }=0.47$ for (a) and (b), respectively. The variation in total number due to evaporative cooling across the data set in (a) is shown in (c), and the corresponding variations of $\eta$ in (d) (note the vertical scale, extending over no more than 5% of the average value). The average $\overline{\eta }=0.49$ is shown by the dashed line. In (e), we show an enlargement of (a) around the critical temperature, to highlight the importance of making the full HF calculation to reproduce the trend seen in the data.

Figure 3

Expansion energy in the radial $x$ direction as a function of temperature. Solid circles in (a) correspond to the whole cloud and open circles in (b) to the thermal component only. The data are taken from the same set as in Fig. 2b. The solid lines on the graphs show the same quantities predicted by the self-consistent HF model, with $\eta =0.47$. The dotted line is the expansion energy (kinetic only) of an ideal cloud.

Figure 4

Radial density profile of the thermal cloud. The dashed line in the inset shows the axis along which the profiles are taken. (a) Radial cut of the column density profile of the thermal cloud after 24.3 ms time of flight (solid line) and the best fit to an ideal Bose-Einstein distribution (dashed line). The Bragg filter has been employed to separate the thermal and condensed components. The condensed fraction is indicated in each case. (b) Residual of the fit for each case in (a).

Figure 5

Deviation from ballistic expansion. The aspect ratios of the condensed (a) and thermal (b) components of mixed clouds, after 22.3 ms of free expansion, are plotted as a function of the reduced temperature. The aspect ratios for noncondensed clouds, analyzed in more detail in [27], are also shown for comparison. The horizontal dashed lines indicate the aspect ratio of a TF condensate and an ideal thermal gas, respectively, and the vertical dotted lines show the critical temperature including mean-field and finite-size effects.

Figure 6

Compression of the condensate by the thermal cloud. The axial length of the condensate as measured in the absorption images is shown, with the TF (dashed line) and the HF predictions for the trapped condensate (dotted line) and after rescaling by the same factor as a TF condensate (solid line).