Determination of Scale-Invariant Equations of State without Fitting Parameters: Application to the Two-Dimensional Bose Gas Across the Berezinskii-Kosterlitz-Thouless Transition

We present a general “fit-free” method for measuring the equation of state (EoS) of a scale-invariant gas. This method, which is inspired from the procedure introduced by Ku et al. [Science 335, 563 (2012)] for the unitary three-dimensional Fermi gas, provides a general formalism which can be readily applied to any quantum gas in a known trapping potential, in the frame of the local density approximation. We implement this method on a weakly interacting two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition and determine its EoS with unprecedented accuracy in the critical region. Our measurements provide an important experimental benchmark for classical-field approaches which are believed to accurately describe quantum systems in the weakly interacting but nonperturbative regime.

Figure 1

Determination of the EoS with variables $X_{-1}$ and $X_1$, along with known limits. The simple cases of the ideal Bose gas (Boltzmann gas) are shown as a blue dashed (dotted) line. The known limits of the EoS of the weakly interacting 2D Bose gas are indicated by a black point for the Thomas-Fermi limit and by a black solid line for the Hartree-Fock mean-field theory. The red line results from the averaging over all the separate intensity profiles, with the error bars corresponding to the standard error introduced by the averaging procedure. The data shown here contain $\sim 100$ different values of $X_{-1}$. Inset: Distribution of measured values of $X_{-1}$ and $X_1$. The gray level indicates the number of individual data points falling in each pixel.

Figure 2

(a) and (b): Density distributions of 2D atomic samples of ${}^{87}\mathrm{Rb}$ corresponding to a partially degenerate (a) and a strongly degenerate cloud (b). (c): Corresponding function $n[V(\mathit{r})]$ resulting from azimuthal averaging. The distributions are obtained with high intensity imaging.

Figure 3

(a) Equation of state of the 2D Bose gas, determined with Eqs. (4) and (5) (red points), with a detailed view of the critical region around the BKT transition (blue dash-dotted line) in (b). Statistical error bars are too small to be shown on these plots. We show, for comparison, the classical-field Monte Carlo prediction ${\mathcal{D}}_{\text{th}}$ [7] in black squares, the Thomas-Fermi limit in the black dash-dotted line, and the Hartree-Fock mean-field theory in the black solid line. We provide a quantitative estimate of the difference between measurement and prediction in (c). There, we plot $\mathcal{D}/{\mathcal{D}}_{\text{th}}-1$, where zero indicates perfect agreement. The error bars result from a bootstrap analysis of the experimental data.