Isothermal compressibility determination across Bose-Einstein condensation

We apply the global thermodynamic variables approach to experimentally determine the isothermal compressibility parameter ${\kappa }_T$ of a trapped Bose gas across the phase transition. We demonstrate the behavior of ${\kappa }_T$ around the critical pressure, revealing the second-order nature of the phase transition. Compressibility is the most important susceptibility to characterize the system. The use of global variables shows advantages with respect to the usual local density approximation method and can be applied to a broad range of situations.

Figure 1

Pressure parameter vs temperature for a constant number of atoms $\left(N=1\times {10}^5\right)$ and different volume parameters: ${\mathcal{V}}_1=1.9\times {10}^{-7}{\mathrm{s}}^3,{\mathcal{V}}_2=6.4\times {10}^{-8}{\mathrm{s}}^3,{\mathcal{V}}_3=3.2\times {10}^{-8}{\mathrm{s}}^3,{\mathcal{V}}_4=2.1\times {10}^{-8}{\mathrm{s}}^3,{\mathcal{V}}_5=1.75\times {10}^{-8}{\mathrm{s}}^3,{\mathcal{V}}_6=1.4\times {10}^{-8}{\mathrm{s}}^3$, and ${\mathcal{V}}_7=1\times {10}^{-8}{\mathrm{s}}^3$. Solid lines above $T_c$ represent the ideal gas law, whereas below $T_c$ are empirical exponential fits. The dotted black line marks the transition between the thermal and the condensed regimes. Error bars represent the statistical error on the average.

Figure 2

Isothermal $\mathcal{V}$ vs $\mathrm{\Pi }$. Symbols represent the measured volume parameter vs the pressure parameter for different temperatures. Solid lines in the thermal region (white) represent the ideal gas law, whereas in the BEC region (gray) they are empirical fitting curves known in the literature as the extended Langmuir adsorption isotherm equation [31] (these curves are not used in the analysis). The error bars on the $\mathcal{V}$ axis come from error propagation of the measurement of the frequencies; on the other hand, the error bars on the $\mathrm{\Pi }$ axis are associated with the exponential fit in Fig. 1.

Figure 3

Isothermal compressibility parameter vs pressure parameter for three temperatures: (a) $T=150\mathrm{nK}$, (b) $T=80\mathrm{nK}$, and (c) $T=40\mathrm{nK}$. The inset in (b) is the ${\kappa }_T$ calculated from a simple toy model for the density distribution. Lines are guides for the eye. The error bar on the abscissa is not included so as to pollute the behavior of ${\kappa }_T$. On the other hand, the error bar on the ordinate comes from the extrapolation of the tangent isothermal curve in Fig. 2.