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American Journal of Physics

A calculation of the time-of-flight distribution of trapped atoms

Itay Yavin, M. Weel, A. Andreyuk, Anantharaman Kumarakrishnan

Published Feb 1, 2002• 33 citations
Publisher

Abstract

We consider the ballistic expansion of a cloud of trapped atoms falling under the influence of gravity. Using a simple coordinate transformation, we derive an analytical expression for the time-of-flight signal. The properties of the signal can be used to infer the initial temperature of the cloud. We first assume a point size cloud with an isotropic velocity distribution to explain the physical basis of the calculation. The treatment is then generalized to include a finite-size cloud with an anisotropic velocity distribution, and an exact result for the signal is derived. The properties of the signal are discussed, and an intuitive picture is presented to explain how initial conditions determine the features of the signal.

DOI
10.1119/1.1424266

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We consider the ballistic expansion of a cloud of trapped atoms falling under the influence of gravity. Using a simple coordinate transformation, we derive an analytical expression for the time-of-flight signal. The properties of the signal can be used to infer the initial temperature of the cloud. We first assume a point size cloud with an isotropic velocity distribution to explain the physical basis of the calculation. The treatment is then generalized to include a finite-size cloud with an anisotropic velocity distribution, and an exact result for the signal is derived. The properties of the signal are discussed, and an intuitive picture is presented to explain how initial conditions determine the features of the signal.
The time-of-flight distribution for a cloud of cold atoms falling freely under gravity is considered. We generalize the probability current density approach to calculate the quantum arrival time distribution for the mixed state describing the Maxwell-Boltzmann distribution of velocities for the falling atoms. We find an empirically testable difference between the time-of-flight distribution calculated using the quantum probability current and that obtained from a purely classical treatment which is usually employed in analyzing time-of-flight measurements. The classical time-of-flight distribution matches with the quantum distribution in the large mass and high temperature limits.
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.
A low noise time-of-flight detection system for laser cooled atoms has been constructed and incrementally optimized. Here, a thorough description of the construction is presented along with an analysis of the capabilities of the system. The quality of the detection (the resolution, the reproducibility, the sensitivity, etc.) is crucial for, e.g., the ability to see details in the velocity distribution profile, which is of interest for fundamental studies of statistical physics and of the laser cooling processes, and for detection of small initial velocities of an atomic cloud, important, e.g., when studying small drifts induced by Brownian motors and ratchets. We estimate the signal-to-noise ratio of our signal to be better than 1000:1 for a typical single shot, and we discuss the effect of the initial atomic cloud size, the probe size, and the effects of the wave packet spread during the fall time on the measured quantities. We show that the shape of the velocity distribution is well conserved during the mapping done in the detection, i.e., in the convolution with the probe beam, and that velocities as small as a few percent of the single photon recoil velocity can be resolved.
We analyse the time-of-flight method of measuring the temperature of cold trapped atoms in the specific case of short distances of the probe beam from the trap centre and finite atomic cloud size. We theoretically examine the influence of the probe beam shape and its distance from the initial position of the cloud on the temperature evaluation. These results are then verified with a three-dimensional Monte Carlo simulation and applied to our experimental data to show that the proposed procedure allows accurate and reliable determination of the temperature.
We consider the leading corrections to the Thomas-Fermi approach to the description of the properties of a magnetically trapped, Bose-condensed cloud of atoms. Simple analytical expressions are derived for the kinetic energy in terms of an effective cutoff length, which we calculate numerically by considering the one-dimensional problem for a linear-ramp potential. We also determine the lowest angular velocity at which it is energetically favorable for a vortex to enter the cloud. For large clouds, our results are in excellent agreement with available numerical calculations.