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arXiv:cond-mat/0610116 · Physical Review A

Resonant dimer relaxation in cold atoms with a large scattering length

Eric Braaten, H.‐W. Hammer

Published May 15, 2007• 24 citations
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Abstract

Efimov physics refers to universal phenomena associated with a discrete scaling symmetry in the three-body problem with a large scattering length. The first experimental evidence for Efimov physics was the recent observation of a resonant peak in the three-body recombination rate for $^{133}\mathrm{Cs}$ atoms with large negative scattering length. There can also be resonant peaks in the atom-dimer relaxation rate for large positive scattering length. We calculate the atom-dimer relaxation rate as a function of temperature and show how measurements of the relaxation rate can be used to determine accurately the parameters that govern Efimov physics.

DOI
10.1103/physreva.75.052710

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Efimov physics refers to universal phenomena associated with a discrete scaling symmetry in the three-body problem with a large scattering length. The first experimental evidence for Efimov physics was the recent observation of a resonant peak in the three-body recombination rate for 133Cs^{133}\mathrm{Cs}133Cs atoms with large negative scattering length. There can also be resonant peaks in the atom-dimer relaxation rate for large positive scattering length. We calculate the atom-dimer relaxation rate as a function of temperature and show how measurements of the relaxation rate can be used to determine accurately the parameters that govern Efimov physics.
Three-body systems with large scattering length display universal phenomena associated with a discrete scaling symmetry. These phenomena include resonant enhancement of three-body loss rates when an Efimov three-body resonance is at the scattering threshold. In particular, there can be resonant peaks in the atom-dimer relaxation rate for large positive scattering length. We improve upon earlier studies and calculate the atom-dimer relaxation rate as a function of temperature using a Bose-Einstein distribution for the thermal average. As input, we use calculations of the atom-dimer scattering phase shifts from effective field theory.
Few-body effects play an important role for the understanding of ultracold quantum gases. We make use of an effective field theory approach to investigate various aspects of universal few-body physics close to a Feshbach resonance. That is the regime where the scattering length is large compared to all other length scales of the system and thus determines the observables. It is also the regime where the Efimov effect whose main characteristic is the occurrence of a sequence of three-body bound states becomes important. At unitarity, i.e., for diverging scattering length, the ratio of binding energies of neighboring states approaches a constant. The existence of those trimers can be captured by a single three-body parameter and influences observables such as recombination rates. Starting from adequate Lagrangians, we derive atom-dimer scattering amplitudes that contain all information of interest about the three-particle systems. With this method, we first investigate the scattering of atoms and dimers at finite temperature in the presence of the Efimov effect. We calculate the dimer relaxation rate and obtain good agreement with available experimental data. Furthermore, heteronuclear mixtures exhibiting large interspecies scattering length are studied in detail. If bosons are the majority species, the Efimov effect occurs in an S-wave channel and we are able to compute three-body recombination and dimer relaxation rates. The results are compared to the outcome of two existing experiments. For mainly fermionic systems, the Efimov effect is only present in an overall P-wave and three-body recombination at threshold vanishes. However, in mixtures of atoms and dimers, scattering cross sections and dimer relaxation show the typical Efimov behavior of log-periodicity. We also consider two-dimensional Bose gases, where the Efimov effect does not occur. For those systems, we derive an equation including the first non-universal corrections and deduce three-body observables such as binding energies and atom-dimer scattering parameters.
We demonstrate the implications of Efimov physics in the recently measured recombination rate of 133Cs^{133}\mathrm{Cs}133Cs atoms. By employing previously calculated results for the energy dependence of the recombination rate of 4He^{4}\mathrm{He}4He atoms, we obtain three independent scaling functions that are capable of describing the recombination rates over a large energy range for identical bosons with large scattering length. We benchmark these and previously obtained scaling functions by successfully comparing their predictions with full atom-dimer phase shift calculations with artificial 4He^{4}\mathrm{He}4He potentials yielding large scattering lengths. Exploiting universality, we finally use these functions to determine the three-body recombination rate of 133Cs^{133}\mathrm{Cs}133Cs atoms with large positive scattering length, compare our results to experimental data obtained by Kraemer et al. at Innsbruck and find excellent agreement.
6Li^{6}\mathrm{Li}6Li atoms in the three lowest hyperfine states display universal properties when the SSS-wave scattering length between each pair of states is large. Recent experiments reported four pronounced features arising from Efimov physics in the atom-dimer relaxation rate, namely two resonances and two local minima. We use the universal effective-field theory to calculate the atom-dimer relaxation rate at zero temperature. Our results describe the four features qualitatively and imply there is a hidden local minimum. In the vicinity of the resonance at 685 G, we perform a finite temperature calculation which improves the agreement of theory and experiment. We conclude that finite temperature effects cannot be neglected in the analysis of the experimental data.
For atoms with large scattering length, the dependence of the three-body recombination rate on the collision energy is determined by the scattering length and the Efimov three-body parameters and can be expressed in terms of universal functions of a single scaling variable. We use published results on the three-body recombination rate for 4He^{4}\mathrm{He}4He atoms to constrain the universal functions. We then use those universal functions to calculate the three-body recombination rate for other atoms with large scattering length at nonzero temperature. The constraints from the 4He^{4}\mathrm{He}4He results are strong if the scattering length is near the minimum of the three-body recombination rate at threshold. We apply our results to 133Cs^{133}\mathrm{Cs}133Cs atoms with a large positive scattering length, and compare them with experimental results from the Innsbruck group.
A new narrow three-atom loss resonance associated with an Efimov trimer crossing the three-atom threshold has recently been discovered in a many-body system of ultracold 6Li^{6}\mathrm{Li}6Li atoms in the three lowest hyperfine spin states at a magnetic field near 895 G. O'Hara and coworkers have used measurements of the three-body recombination rate in this region to determine the complex three-body parameter associated with Efimov physics. Using this parameter as the input, we calculate the universal predictions for the spectrum of Efimov states and for the three-body recombination rate in the universal region above 600 G where all three scattering lengths are large. We predict an atom-dimer loss resonance at $672\ifmmode\pm\else\textpm\fi{}2$ G associated with an Efimov trimer disappearing through an atom-dimer threshold. We also predict an interference minimum in the three-body recombination rate at $759\ifmmode\pm\else\textpm\fi{}1$ G where the three-spin mixture may be sufficiently stable to allow experimental study of the many-body system.