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Physical Review A

Rate of runaway evaporative cooling

Jorik van de Groep, P. van der Straten, J. M. M. J. Vogels

Published Sep 23, 2011• 2 citations
Publisher

Abstract

Evaporative cooling is a process that is essential in creating Bose-Einstein condensates in dilute atomic gasses. This process has often been simulated based on a model using a truncated Boltzmann distribution. This model assumes that the energy distribution up to the threshold energy can still be described by a Boltzmann distribution: it assumes detailed balance up to the threshold energy. However, the evolution of the distribution function in time is not taken into account. Here we solve the kinetic Boltzmann equation for a gas undergoing evaporative cooling in a harmonic and linear trap in order to determine the evolution of the energy distribution. The magnitude of the discrepancy with the truncated Boltzmannmodel is calculated by including a polynomial expansion of the distribution function. We find that up to 35% fewer particles are found in the high-energy tail of the distribution with respect to the truncated Boltzmann distribution and up to 15% more collisions are needed to reach quantum degeneracy. Supported by a detailed investigation of the particle loss rate at different energies, we conclude that the limited occupation of high-energy states during the evaporation process causes the lowering of the evaporation speed and efficiency.

DOI
10.1103/physreva.84.033628

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Evaporative cooling is a process that is essential in creating Bose-Einstein condensates in dilute atomic gasses. This process has often been simulated based on a model using a truncated Boltzmann distribution. This model assumes that the energy distribution up to the threshold energy can still be described by a Boltzmann distribution: it assumes detailed balance up to the threshold energy. However, the evolution of the distribution function in time is not taken into account. Here we solve the kinetic Boltzmann equation for a gas undergoing evaporative cooling in a harmonic and linear trap in order to determine the evolution of the energy distribution. The magnitude of the discrepancy with the truncated Boltzmannmodel is calculated by including a polynomial expansion of the distribution function. We find that up to 35% fewer particles are found in the high-energy tail of the distribution with respect to the truncated Boltzmann distribution and up to 15% more collisions are needed to reach quantum degeneracy. Supported by a detailed investigation of the particle loss rate at different energies, we conclude that the limited occupation of high-energy states during the evaporation process causes the lowering of the evaporation speed and efficiency.
We have simulated the evaporative cooling of a dilute gas of Bose particles including quantum statistics using a Monte Carlo method. This approach can model situations which are far away from quasiequilibrium such as occur during forced evaporative cooling. We have also simulated the dynamical formation of Bose-Einstein condensate for homogeneous and inhomogeneous Bose gases under the random-phase approximation. It was found that the rate of accumulation of Bose particles into low-energy states through collisions is increased by forced evaporation; and a macroscopic population at ground state can be reached at a time scale characterized by classical collision time for an inhomogeneous gas in a harmonic potential. We present the results of simulations for the evaporative cooling and formation of a Bose-Einstein condensate in one-, two-, and three-dimensional position cuts and energy cuts.
Evaporative cooling of trapped atoms is studied based on the quantum kinetic theory. We extend the classical analysis of Luiten et al. [Phys. Rev. A 53, 381 (1996)] to treat Bose statistics of alkali-metal atoms. Dynamics in evaporative cooling process is described by a kinetic equation for the truncated Bose-Einstein distribution function under the assumption that the system is close to the thermal equilibrium. This approach is applicable to the ``slow'' evaporative cooling normally adopted in most experiments for the efficient production of Bose-Einstein condensates. Time-evolution calculations explain the current experiment well with sodium atoms using a slow evaporative cooling and demonstrate a rapid growth of condensate in the cooling process.
We discuss the kinetic theory for evaporative cooling of a dilute collisional gas in a trap. The analysis in O. J. Luiten et al., Phys. Rev. A 53, 381 (1996) is extended to incorporate a time-dependent trap strength and truncation energy. We employ the equations obtained to predict the efficiency of evaporative cooling of sodium atoms in a Ioffe trap and realize that a time-dependent trap strength can be crucial for the cooling. The calculations are classical in the sense that no effects of quantum statistics are taken into account: nevertheless the results of the calculations are useful in order to reveal experimental routes towards a Bose-Einstein condensate of sodium atoms.
We apply kinetic theory to the problem of evaporative cooling of a dilute collisional gas in a trap. Assuming ``sufficient ergodicity'' (phase-space distribution only a function of energy) and s-wave collisions with an energy-independent cross section, an equation for the evolution of the energy distribution of trapped atoms is derived for arbitrary trap shapes. Numerical integration of this kinetic equation demonstrates that during evaporation the gas is accurately characterized by a Boltzmann distribution of atom energies, truncated at the trap depth. Adopting the assumption of a truncated Boltzmann distribution, closed expressions are obtained for the thermodynamic properties of the gas as well as for the particle and energy loss rates due to evaporation. We give analytical expressions both for power-law traps and for a realistic trapping potential (Ioffe quadrupole trap). As an application, we discuss the evaporative cooling of trapped atomic hydrogen gas. \textcopyright{} 1996 The American Physical Society.
Summary form only given.The successful demonstrations of Bose-Einstein condensation (BEG) with alkali atoms in a magnetic trap have opened up a new field of quantum physics. The evaporative cooling that enables six-order increase of phase space density is the essential technique for achieving the BEC transition in these experiments. However, at low temperatures, this cooling method is strongly affected by Bose statistics of alkali atoms, since evaporation occurs through the scattering process between trapped atoms. The quantum kinetics of nonequilibrium gas has been the subject of considerable interest for classic gas on the basis of truncated Boltzman distribution. We extend this method to treat the quantum statistics.
We study the optimization of evaporative cooling in trapped bosonic atoms on the basis of quantum kinetic theory of a Bose gas. The optimized cooling trajectory for 87Rb{}^{87}\mathrm{Rb}87Rb atoms indicates that the acceleration of evaporative cooling around the transition point of Bose-Einstein condensation is very effective against loss of trapped atoms caused by three-body recombination. The number of condensed atoms is largely enhanced by the optimization, more than two orders of magnitude in our present calculation using relevant experimental parameters, as compared with the typical value given by the conventional evaporative cooling where the frequency of radio-frequency magnetic field is swept exponentially. In addition to this optimized cooling, it is also shown that highly efficient evaporative cooling can be achieved by an initial exponential and then a rapid linear sweep of frequency.