Quantitative test of the mean-field description of a trapped two-dimensional Bose gas

We investigate the accuracy of two mean-field theories of the trapped two-dimensional Bose gas at predicting transition region properties by comparison to nonperturbative classical field calculations. To make these comparisons we examine the density profiles and the predictions for the Berezinskii-Kosterlitz-Thouless superfluid transition temperature over a parameter range in which the degree of thermal activation in the tightly trapped direction varies considerably. These results present an important test of these mean-field theories and provide a characterization of their typical accuracy.

Figure 1

(Color online) Phase-space density. Parameters: (a), $N=13.4\times {10}^3$, $T=34.7\text{nK}$ $\left(\approx T_{\text{BKT}}\right)$. (b) $N=45.8\times {10}^3$, $T=67.6\text{nK}$. (c) $N=67.4\times {10}^3$, $T=65.0\text{nK}$ $\left(\approx T_{\text{BKT}}\right)$. Curves: FSC (solid), HCK (dashed), and c-field (dot-dashed) results. Other parameters: ${\omega }_{x,y}/2\pi =9.4\text{Hz}$ for all results, ${\omega }_z/2\pi =1880\text{Hz}$ in (a) and (b), and ${\omega }_z/2\pi =940\text{Hz}$ in (c).

Figure 2

(Color online) Peak phase-space density as a function of $T$. FSC value of $n_0\left(\mathbf{0}\right){\lambda }_{dB}^2$ (pluses), HCK peak value of the total areal phase space density, ${\sum }_j{n}_j\left(\mathbf{0}\right){\lambda }_{dB}^2$ (triangles) and polynomial fits to these results in the transition region are shown as lines. Also displayed (as nearly horizontal solid lines) are the critical phase space densities for the FSC and HCK theories [see Eq. (6)]. The predictions for the transition temperature made by the FSC $\left(T_{\text{BKT}}^{\text{FSC}}\right)$, the HCK $\left(T_{\text{BKT}}^{\text{HCK}}\right)$ and c-field $\left(T_{\text{BKT}}\right)$ theories are indicated as vertical dashed lines. Calculation parameters: ${\omega }_{x,y}=2\pi \times 9.4\text{Hz}$, ${\omega }_z,=2\pi \times 1.88\text{kHz}$ and $N=13.5\times {10}^3$.