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Physical Review A

Scaling laws for evaporative cooling in time-dependent optical traps

Kenneth M. O’Hara, Michael E. Gehm, S. R. Granade, John E. Thomas

Published Oct 12, 2001• 145 citations
Publisher

Abstract

We derive scaling laws for the number of atoms, collision rate, and phase-space density as a function of trap depth for evaporative cooling in an adiabadically lowered optical trap. The results are in excellent agreement with a Boltzmann equation model and show that very large increases in phase-space density can be obtained without excessive slowing of the evaporation rate. Predictions are in reasonable agreement with a recent experiment that achieves Bose-Einstein condensation by evaporation in an optical trap. We also discuss evaporation of fermionic mixtures and explain why Pauli blocking does not strongly inhibit cooling.

DOI
10.1103/physreva.64.051403

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We derive scaling laws for the number of atoms, collision rate, and phase-space density as a function of trap depth for evaporative cooling in an adiabadically lowered optical trap. The results are in excellent agreement with a Boltzmann equation model and show that very large increases in phase-space density can be obtained without excessive slowing of the evaporation rate. Predictions are in reasonable agreement with a recent experiment that achieves Bose-Einstein condensation by evaporation in an optical trap. We also discuss evaporation of fermionic mixtures and explain why Pauli blocking does not strongly inhibit cooling.
Summary form only given. Evaporative cooling of atoms in an optical trap has recently received renewed attention as a means for obtaining degeneracy in an optically trapped bosonic Cs vapor as well as a fermionic Li gas. We have developed a new model describing the evaporative cooling process for atoms confined in a time-dependent optical potential formed by a single focused Gaussian laser beam. We find that a substantial increase in the phase-space density can be obtained by adiabatically reducing the trap laser power and thereby the trap well depth as a function of time. Since we are specifically interested in the evaporative cooling of fermionic /sup 6/Li atoms, we have included the effect of Fermi statistics in the model. Although the collision rate is suppressed as the temperature T is reduced below the Fermi temperature T/sub F/, we find that values of T/T/sub F//spl Lt/1 can be achieved for suitable initial conditions.
We measure the scaling laws for the number of atoms and the cloud size as a function of trap depth for evaporative cooling of a unitary Fermi gas in an optical trap. A unitary Fermi gas comprises a trapped mixture of atoms in two hyperfine states which is tuned to a collisional (Feshbach) resonance using a bias magnetic field. Near resonance, the zero energy s-wave scattering length diverges, and the s-wave scattering cross-section is limited by unitarity to be 4π/ k 2 , where k is the relative wavevector of the colliding particles. In this case, the collision cross-section for evaporation scales inversely with the trap depth, enabling runaway evaporation under certain conditions. We demonstrate high evaporation efficiency, which is achieved by maintaining a high ratio η of trap depth to thermal energy as the trap depth is lowered. We derive and demonstrate a trap lowering curve which maintains η constant for a unitary gas. This evaporation curve yields a quantum degenerate sample from a classical gas in a fraction of a second, with only a factor of three loss in atom number.
We present a combined computational and experimental study to optimize the efficiency of evaporative cooling for atoms in optical dipole traps. By employing a kinetic model of evaporation, we provide a strategy for determining the optimal relation between atom temperature, trap depth, and average trap frequency during evaporation given experimental initial conditions. We then experimentally implement a highly efficient evaporation process in an optical dipole trap, showing excellent agreement between the theory and experiment. This method has allowed the creation of pure Bose-Einstein condensates of 87{}^{87}87Rb with $2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ atoms starting from only $5\ifmmode\times\else\texttimes\fi{}{10}^{5}$ atoms initially loaded in the optical dipole trap, achieving an evaporation efficiency ${\ensuremath{\gamma}}_{\mathrm{eff}}$ of 4.0 during evaporation.
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.
Recent experiments have used forced evaporative cooling to produce Bose-Einstein condensation in dilute gases. The evaporative cooling process can be optimized to provide the maximum phase-space density with a specified number of atoms remaining. We show that this global optimization is approximately achieved by locally optimizing the cooling efficiency at each instant. We discuss how this method can be implemented, and present the results for our 7Li^{7}\mathrm{Li}7Li trap. The predicted behavior of the gas is found to agree well with experiment.
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.