Exploring the Thermodynamics of a Two-Dimensional Bose Gas

Using in situ measurements on a quasi-two-dimensional, harmonically trapped ${}^{87}\mathrm{Rb}$ gas, we infer various equations of state for the equivalent homogeneous fluid. From the dependence of the total atom number and the central density of our clouds with chemical potential and temperature, we obtain the equations of state for the pressure and the phase-space density. Then, using the approximate scale invariance of this 2D system, we determine the entropy per particle and find very low values (below $0.1k_B$) in the strongly degenerate regime. This shows that this gas can constitute an efficient coolant for other quantum fluids. We also explain how to disentangle the various contributions (kinetic, potential, interaction) to the energy of the trapped gas using a time-of-flight method, from which we infer the reduction of density fluctuations in a nonfully coherent cloud.

Figure 1

Absorption imaging of quasi-2D clouds of ${}^{87}\mathrm{Rb}$ atoms. (a) Image obtained with a short pulse ($\sim 2\mu \mathrm{s}$) of an intense probe beam ($I/I_{\mathrm{sat}}=40$). (b) Image obtained with a longer pulse ($50\mu \mathrm{s}$) of a weak probe beam ($I/I_{\mathrm{sat}}=0.5$). The processing of images (a) and (b) is detailed in the Supplemental Material [15]. (c),(d) Radial density profiles for image (a) (○) and image (b) (●) in linear (c) and logarithmic (d) scales. The solid line is the prediction for the total density ($T=133\mathrm{nK}$, $\mu /k_B=47\mathrm{nK}$). It includes the populations of the excited states of the $z$ motion calculated using HFMF theory, and the population of the ground state calculated from (i) the prediction of 6 in the range where it is available [$\alpha \equiv \mu /k_{B}T\in (-0.2,0.6)$], (ii) the HFMF theory for $\alpha <-0.2$, and (iii) the Thomas-Fermi approximation $\mu ={\hslash }^2\tilde{g}n/m$ for $\alpha >0.6$.

Figure 2

Equations of state for (a) the reduced pressure $\mathcal{P}$, (b) the phase-space density $\mathcal{D}$, and (c) the entropy per particle $\mathcal{S}$. The HFMF prediction is plotted in black full line and extended in dotted line beyond the expected superfluid transition. The dashed (red) line is the Thomas-Fermi prediction. In (a) the gray shaded area indicates the region of parameter space accessible to an ideal gas. In (b) the thick gray line indicates the prediction from 6. For $\mu /k_{B}T>0.2$, data obtained for the same control parameters (trap loading time and evaporative cooling ramp) have been grouped and error bars indicate standard deviation of the measurement. For $\mu /k_{B}T<0.2$, data are displayed for individual images, thus with no error bars. The vertical dash-dotted line (blue) indicates the prediction [23] for the superfluid transition.

Figure 3

(a)–(d) Side view of a cloud initially in the 2D regime and expanding along $z$ once the laser providing the confinement in this direction has been switched off. (a) $t=1\mathrm{ms}$, (b) $t=2\mathrm{ms}$, (c) $t=3\mathrm{ms}$, and (d) $t=4\mathrm{ms}$. (e) Time evolution of the potential energy $E_p$. The different lines represent a fit to the data of a parabola (solid black line), the time evolution assuming flattened density fluctuations (dashed red line) and the one expected for a dilute noncondensed gas (dash-dotted green line).