Evaporative cooling of cold atoms at surfaces

We theoretically investigate the evaporative cooling of cold rubidium atoms that are brought close to a solid surface. The dynamics of the atom cloud are described by coupling a dissipative Gross-Pitaevskii equation for the condensate with a quantum Boltzmann description of the thermal cloud (the Zaremba-Nikuni-Griffin method). We have also performed experiments to allow for a detailed comparison with this model and find that it can capture the key physics of this system provided the full collisional dynamics of the thermal cloud are included. In addition, we suggest how to optimize surface cooling to obtain the purest and largest condensates.

Figure 1

Schematic diagram (not to scale) of the system showing the total potential $V(x,y=0,z=0)$ (red solid curve) for atoms in a harmonic trap in the vicinity of a surface. For short distances, the Casimir-Polder potential (gray dashed curve) dominates. Far away from the surface, the atoms only see the trapping potential (gold dashed curve). In between, the Casimir-Polder potential leads to an opening of the trap. The solid rectangle indicates the surface and the colored oval indicates the atom cloud. The black arrow shows the $x$ axis, with $x_s$ the distance between the trap center and the surface.

Figure 2

Total atom number $N$ against time $t$ for three different trap-surface separations: $x_s=$ $68\mu \mathrm{m}$ (gold solid curve), $30\mu \mathrm{m}$ (red dashed curve), and $15\mu \mathrm{m}$ (black dotted curve). Points correspond to experimental data and the curves correspond to simulations. The dot-dashed gold curve shows a simulation for $x_s=68\mu \mathrm{m}$ without collisions, i.e., $C_{12}=C_{22}=0$. The gray vertical dashed line marks the point when the atom cloud reaches its final hold position at $t=0$. The gray hashed area shows the shift of the curve when the surface position is varied by $\pm 2.5\mu \mathrm{m}$. The inset shows a breakdown of the cloud atom numbers against time for $x_s=68\mu \mathrm{m}$ from the point when the cloud reaches its holding position. The solid curve shows the total atom number, the dashed curve corresponds to thermal atoms, and the dotted curve corresponds to the condensate atom number.

Figure 3

Density profiles of a cooling ZNG gas at (a) the beginning of the simulation before transport, (b) time $t=0.6\mathrm{s}$ hold time, and (c) $t=2.25\mathrm{s}$ hold time. The upper panels show the full cloud density integrated along the $y$ direction, and the lower panels show cross sections of the thermal cloud density through the $y=0$ plane. Arrows indicate axes and bars indicate scale. In the upper panels, the $z$ direction has been stretched by a factor of 4 to improve clarity.

Figure 4

Total atom number $N$ for a cloud held for $600\mathrm{m}\mathrm{s}$ at a surface for varying trap-surface separations as measured experimentally (crosses) or simulated (solid curves). The main red solid curve corresponds to a ZNG simulation with collisions; the gray solid curve with open circles corresponds to a ZNG simulation without collisions ($C_{12}=C_{22}=0$). The gray area around the red solid curve represents the error bounds assuming the surface position is shifted by $\pm 2.5\mu \mathrm{m}$. The black dashed and dotted curves come from the classical model of Eq. (10) applied respectively in 1D and 3D (see text). The black dot-dashed curve is an error function [Eq. (11)] corresponding to the limit of very rapid transport.

Figure 5

(a) Condensate fraction $N/N_c$ and (b) condensate atom number for different hold points, $x_s$, in the case of an isotropic trap, in units of the harmonic oscillator length $a_{\text{ho}}$. We show the results for four different trap frequencies: ${\omega }_1=2\pi \times 40\text{rads}{\text{s}}^{-1}$ (gold diamond points), ${\omega }_2=2\pi \times 80\text{rads}{\text{s}}^{-1}$ (gray triangles), ${\omega }_3=2\pi \times 120\text{rads}{\text{s}}^{-1}$ (open red circles), and ${\omega }_4=2\pi \times 160\text{rads}{\text{s}}^{-1}$ (black stars). Condensate fractions are highest for these parameters between 4 and $7.5a_{\text{ho}}$, whereas the condensate atom number has a maximum between 10 and 15 $a_{\text{ho}}$. The inset in (a) shows condensate fractions plotted against transport velocity for a trap with frequency ${\omega }_3=2\pi \times 120\text{rads}{\text{s}}^{-1}$.