Equilibrium state of a trapped two-dimensional Bose gas

We study experimentally and numerically the equilibrium density profiles of a trapped two-dimensional ${}^{87}\mathrm{Rb}$ Bose gas and investigate the equation of state of the homogeneous system using the local density approximation. We find a clear discrepancy between in situ measurements and quantum Monte Carlo simulations, which we attribute to a nonlinear variation of the optical density of the atomic cloud with its spatial density. However, good agreement between experiment and theory is recovered for the density profiles measured after time of flight, taking advantage of their self-similarity in a two-dimensional expansion.

Figure 1

(a) Potential $V$ along the vertical direction $z$ produced by the magnetic trap and the laser beam. (b,c) Side view of the cloud before (b) and after (c) depumping atoms in the side wells. The horizontal stripes are due to diffraction. (d) Top-view in situ image yielding fit parameters $T=132$ nK and $\alpha =0.29$ (for $\xi =0.25$).

Figure 2

(a) Dots, measured in situ optical density profiles $\Delta (r)$. For $\xi =0.25$ the fit with MFHF theory yields $T=126\hspace{0.5em}(6)$ nK, $\alpha =0.34\hspace{0.5em}(9)$, where uncertainties represent standard deviations obtained by fitting individual images. Continuous line, corresponding QMC simulation with $N=73\hspace{0.5em}900$ atoms (inset, same data in log plot). Upper (dash-dotted) and lower (dashed) lines, QMC results obtained assuming $\xi =0.21$ [fit parameters ($T\hspace{0.5em}(\mathrm{nK}),\alpha ,N)=(130,0.39,96\hspace{0.5em}300$)] and $\xi =0.29$ (122,0.29,57 900), respectively. (b) Set of measured density profiles (dots) and corresponding QMC results (lines) for other rf evaporation parameters. From bottom to top: $(T\hspace{0.5em}(\mathrm{nK}),\alpha ,N)=(87,0.49,54\hspace{0.5em}100)$ (black), (109,0.39,63 800) (red), (142,0.28,78 400) (blue),(153,0.23,79 900) (magenta). Each experimental profile in (a) and (b) is an average of nine images.

Figure 3

Measured optical density ${\Delta }_{\mathrm{meas}.}$ as a function of the calculated optical density ${\Delta }_{\mathrm{calc}.}$, averaged over the data shown in Figs. 2a and 2b. The dashed line with a slope of 1 is a guide for the eye. Error bars indicate the standard deviation of the data.

Figure 4

Continuous lines, QMC results for the phase-space density $D$ as a function of ${\alpha }_{\mathrm{local}}=\alpha -m{\omega }^2{r}^2/k_{\mathrm{B}}T$ for the data shown in Figs. 2a and 2b (same color code). Black dashed line, prediction of [8] for the uniform case; dots, measured $D$, averaged over all experimental data shown in Figs. 2a and 2b. Error bars indicate the standard deviation of the data.

Figure 5

Optical density profiles obtained for TOF durations $t=0$ ms (black squares), 3 ms (red circles), 6 ms (green up-pointing triangles), 10 ms (blue down-pointing triangles), 12 ms (cyan diamonds), and 14 ms (magenta left-pointing triangles). Each profile has been rescaled to its in situ value according to (1).

Figure 6

Squares, optical density profile obtained by averaging the experimental data of Fig. 5 for $10\hspace{0.5em}\mathrm{ms}⩽t⩽14$ ms, yielding fit parameters $T=94$ nK, $\alpha =0.36$ for $\xi =0.63$. Lines, QMC results for the same fit parameters (continuous, $N=42\hspace{0.5em}000$) and for those deduced assuming $\xi =0.47$ [dashed red $(T\hspace{0.5em}(\mathrm{nK}),\alpha ,N)=(104,0.39,57\hspace{0.5em}600)$] and $\xi =0.79$ [dash-dotted blue (87,0.33,32 100)].