Measurement of the Canonical Equation of State of a Weakly Interacting 3D Bose Gas

Using a multiple-image reconstruction method applied to a harmonically trapped Bose gas, we determine the equation of state of uniform matter across the critical transition point, within the local density approximation. Our experimental results provide the canonical description of pressure as a function of the specific volume, emphasizing the dramatic deviations from the ideal Bose gas behavior caused by interactions. They also provide clear evidence for the nonmonotonic behavior with temperature of the chemical potential, which is a consequence of superfluidity and Bose-Einstein condensation. The measured thermodynamic quantities are compared to mean-field predictions available for the interacting Bose gas. The limits of applicability of the local density approximation near the critical point are also discussed, focusing on the behavior of the isothermal compressibility.

Figure 1

(a) Column density of the trapped sample, reconstructed from different partial extractions. Each pixel in the image results from the average of 5 to 80 images at different extraction ratios. (b) Pressure along the $x$ axis of the sample, obtained integrating $n_1$ along $y$. The error bars on the pressure are smaller than the marker size. (c) Axial density profile of the sample $n(x)$ obtained via the inverse Abel transform. The insets in (b) and (c) show pressure and density of the gas in log scale, highlighting the high dynamic range needed to capture the different regimes of the Bose gas thermodynamics.

Figure 2

(a) Measurement of the canonical EOS of a uniform Bose gas, showing the pressure as a function of the specific volume at constant $T=280\mathrm{nK}$. Pressure and density are derived within the LDA from the in situ distribution of a harmonically trapped sample. The HF prediction (solid) for a uniform system at the same temperature shows good agreement in the whole range without fitting parameters. Predictions from the IG (dotted) and IBG (dashed) models are shown for comparison. (b) Experimental results for the reduced compressibility versus $v/v_c$ compared to HF theory, which predicts a narrow peak at the critical point. The gray solid line includes the effects of the finite imaging resolution applied to a numerical simulation of the HF density profile in our trap.

Figure 3

Grand canonical EOS for an interacting Bose gas at constant temperature $T=280\mathrm{nK}$. The chemical potential, in units of the local interaction term $gn$, goes from 1 to 2 as the reduced temperature $T/T_c$ goes from 0 to the critical point, and decreases in the thermal region ($T/T_c>1$). Black solid and dot-dashed lines show the HF and $1+(T/T_c{)}^{3/2}$ predictions, respectively, while the gray solid line includes the effects of the finite imaging resolution applied to a numerical simulation of the HF density profile in our trap. The dashed line is the IBG law.