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arXiv:cond-mat/0303249 · Physical Review A

Enhanced dimer relaxation in an atomic and molecular Bose-Einstein condensate

Eric Braaten, H.‐W. Hammer

Published Oct 12, 2004• 67 citations
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Abstract

We derive a universal formula for the rate constant $\ensuremath{\beta}$ for relaxation of a shallow dimer into deeply-bound diatomic molecules in the case of atoms with a large scattering length $a$. We show that $\ensuremath{\beta}$ is determined by $a$ and by 2 three-body parameters that also determine the binding energies and widths of Efimov states. The rate constant $\ensuremath{\beta}$ scales like $\ensuremath{\hbar}a∕m$ near the resonance, but the coefficient is a periodic function of $\mathrm{ln}(a)$ that may have resonant enhancement at values of $a$ that differ by multiples of 22.7.

DOI
10.1103/physreva.70.042706

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We derive a universal formula for the rate constant \ensuremathβ\ensuremath{\beta}\ensuremathβ for relaxation of a shallow dimer into deeply-bound diatomic molecules in the case of atoms with a large scattering length aaa. We show that \ensuremathβ\ensuremath{\beta}\ensuremathβ is determined by aaa and by 2 three-body parameters that also determine the binding energies and widths of Efimov states. The rate constant \ensuremathβ\ensuremath{\beta}\ensuremathβ scales like \ensuremathℏa∕m\ensuremath{\hbar}a∕m\ensuremathℏa∕m near the resonance, but the coefficient is a periodic function of ln(a)\mathrm{ln}(a)ln(a) that may have resonant enhancement at values of aaa that differ by multiples of 22.7.
Atom loss resonances in ultracold trapped atoms have been observed at scattering lengths near atom-dimer resonances, at which Efimov trimers cross the atom-dimer threshold, and near two-dimer resonances, at which universal tetramers cross the dimer-dimer threshold. We propose a new mechanism for these loss resonances in a Bose-Einstein condensate of atoms. As the scattering length is ramped to the large final value at which the atom loss rate is measured, the time-dependent scattering length generates a small condensate of shallow dimers coherently from the atom condensate. The coexisting atom and dimer condensates can be described by a low-energy effective field theory with universal coefficients that are determined by matching exact results from few-body physics. The classical field equations for the atom and dimer condensates predict narrow enhancements in the atom loss rate near atom-dimer resonances and near two-dimer resonances due to inelastic dimer collisions.
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.
We show that in the limit of large and positive atom--atom scattering length the properties of an atomic--molecular Bose--Einstein Condensate (amBEC) are determined by an universal energy density functional (EDF). We find that the optimal conditions for the formation of a stable amBEC are in the regime where there are no shallow trimers and the atom--dimer scattering length is negative and comparable in magnitude with the atom--atom scattering length. At temperatures lower than TcT_cTc​ the chemical potentials for the atoms and molecules can be specified independently. Besides three--body recombinations processes into dimers of large size, inelastic processes involving the formation of deeply bound small size molecular states are possible. These inelastic processes do not lead to an efficient heating of the amBEC and can be used for its mostly non--destructive monitoring.
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.
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 predict the resonance-enhanced magnetic field dependence of atom-dimer relaxation and three-body recombination rates in a 87Rb^{87}\mathrm{Rb}87Rb Bose-Einstein condensate close to 1007G1007\phantom{\rule{0.3em}{0ex}}\mathrm{G}1007G. Our exact treatments of three-particle scattering explicitly include the dependence of the interactions on the atomic Zeeman levels. The Feshbach resonance distorts the entire diatomic energy spectrum, causing interferences in both loss phenomena. Our two independent experiments confirm the predicted recombination loss over a range of rate constants that spans four orders of magnitude.