Cloud parameter estimation for interacting BEC after time-of-flight

TL;DR

The paper addresses how repulsive mean-field interactions between a Bose-Einstein condensate and its thermal component distort time-of-flight expansion profiles, biasing temperature and atom-number estimates if ignored. It develops a numerical ballistic-expansion model that includes the mean-field potentials and computes the evolving thermal density, then contrasts fitting with a Bose-enhanced distribution to fitting with the simulated distribution. The authors quantify systematic errors across trap geometries, temperatures, and TOF, and provide a phenomenological explanation for the observed patterns—chiefly that the BEC shifts the central density and enlarges the usable fit region, trading accuracy in $N_{th}$ for accuracy in $T$. They demonstrate that simulation-based fits yield better agreement with the semi-ideal model Naraschewski1998 and can be used as a fitting model for experimental data, albeit at higher computational cost.

Abstract

Experiments on Bose-Einstein condensates at finite temperature typically extract the system parameters, such as temperature, atom number, and condensed fraction from time-of-flight images taken after a free expansion time. This paper systematically examines the effect of repulsive interactions between the condensed and thermal atoms in partially condensed clouds on the expansion profile of the thermal cloud. An analytical expression for the expansion can be obtained only if the interactions between the Bose-Einstein condensate and thermal atoms are neglected, resulting in a Bose-enhanced distribution for the thermal component. Here, the deformation of the cloud due to interactions and the effects on estimated parameters are investigated by simulating the expansion using a ballistic approximation. By fitting the simulated expansion profiles with a Bose-enhanced distribution, the errors of using such a fit are estimated, and the results are explained phenomenologically. The simulation was also used as a fitting function for experimental data, showing better agreement of the extracted condensed fraction with the semi-ideal model than results from a Bose-enhanced fit.

Figures (10)

• Figure 1: Illustration of the calculation of the thermal cloud profile after time-of-flight. The initial cloud, as well as the initial grid, is illustrated in green (small). The final cloud, as well as the expansion grid, is shown in grey (large). A black point in the expansion grid is highlighted. To calculate the density of atoms at this point, the probability of starting at a specific initial grid point, with the correct momentum to end up at the black point, is calculated for each initial grid point and then added. This is done for all expansion grid points to obtain the thermal atom distribution. The grids and clouds shown here are only an illustration of the method. The clouds are not assumed to be spherical, and the calculations are performed in three dimensions.
• Figure 2: Relative errors in thermal atom number ($\bullet$) and temperature () as a function of the grid size $\nu$ with $\xi=60$ held constant and for the cloud parameters specified in text. Exponential fits are provided as a guide to the eye.
• Figure 3: Relative errors in thermal atom number ($\bullet$) and temperature () as a function of the grid size $\xi$ with $\nu=40$ held constant and for the cloud parameters specified in text. Exponential fits are provided as a guide to the eye.
• Figure 4: Thermal column densities in a spherical trap configuration with $N_s=200k$, $T_s/T_{c,s}=0.2$, $\omega_i=2\pi\cdot100Hz$, and $N_{0,s}/N_s=0.97$, for two different times of flight, normalized to the highest column density. The $x$-axis has been rescaled between the two plots. The BEC distribution and its fit are not plotted. The top has a time of flight of $t= 1ms$ and the bottom has $t= 30ms$. Slices through the center of the simulated column densities are shown as solid blue lines, and slices through the center of a Bose-enhanced fit to the simulated data (fitted outside the BEC region illustrated by vertical dashed lines) are shown as dashed red lines. Inserts in both figures show the simulated column densities.
• Figure 5: Errors in thermal atom number estimation as a function of reduced temperature $T_s/T_{c,s}$ and total atom number for different trap configurations and times of flight. (a) Cigar shape: $\omega_x = 2\pi\cdot10Hz$, $\omega_{y,z} = 2\pi\cdot100Hz$ and $t=3ms$. (b) Cigar shape: $t=30ms$. (c) Spherical shape: $\omega_{x,y,z} = 2\pi\cdot100Hz$ and $t=3ms$. (d) Spherical shape: $t=30ms$. (e) Pancake shape: $\omega_x = 2\pi\cdot1000Hz$, $\omega_{y,z} = 2\pi\cdot100Hz$ and $t=3ms$. (f) Pancake shape: $t=30ms$. A solid line, a dashed line, and a solid circle indicate regions further explored in FIGs. \ref{['fig:EvapConstN']}, \ref{['fig:TOFScan']}, and \ref{['fig:aspRatio']}.