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arXiv:1008.2260 · Comptes Rendus Physique

The Efimov effect in lithium 6

Pascal Naidon, Masahito Ueda

Published Jan 1, 2011• 39 citations
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Abstract

We analyse the recent experiments investigating the low-energy physics of three lithium 6 atoms in three different internal states with resonant two-body scattering lengths. All observed features are qualitatively consistent with the expected Efimov effect, i.e. the effective universal three-body attraction that arises for large values of the scattering lengths. However, we find that a quantitative description at negative energy requires non-universal two- and three-body corrections due to presently unknown behaviour at short distance. An attempt to implement these corrections is made through energy-dependent parameters that are fitted to the experimental data.

DOI
10.1016/j.crhy.2010.12.002

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We analyse the recent experiments investigating the low-energy physics of three lithium 6 atoms in three different internal states with resonant two-body scattering lengths. All observed features are qualitatively consistent with the expected Efimov effect, i.e. the effective universal three-body attraction that arises for large values of the scattering lengths. However, we find that a quantitative description at negative energy requires non-universal two- and three-body corrections due to presently unknown behaviour at short distance. An attempt to implement these corrections is made through energy-dependent parameters that are fitted to the experimental data.
Universality is a fundamental property of diverse few-body systems such as atomic nuclei, simple molecules or ultracold atoms. It is related to a peculiar quantum mechanical phenomenon which was discovered by Russian theoretician Vitaly Efimov in the early 1970-s. A consequence of his prediction was the existence of an infinite number of bound energy levels for three particles even if a short-range two-body potential is just not deep enough to bind two particles together. This counterintuitive discovery attracted a great deal of interest but lacked an experimental verification for many years. Difficulties in finding a system where the inter-particle potential can be changed easily were answered recently with the experimental demonstration of Feshbach resonances in ultracold atoms. In the limit of zero collision energy the two-body interactions are determined by a single parameter, the scattering length a, which is defined by the low energy s-wave scattering phase-shift. In the vicinity of a Feshbach resonance this parameter diverges and thus can be readily tuned to large values and different signs by an experimentally easy-to-use change of the external magnetic field. A universal behavior is expected when a becomes the largest length scale in the system. In this case the three-body observables depend only on the scattering length and on a three-body parameter which serves as a boundary condition for the short range physics. The first experimental evidence for the existence of Efimov quantum states was reported in 2006 in a system of ultracold Cs atoms by the Innsbruck group. They investigated three-body recombination induced loss of atoms from a shallow optical trap and showed the enhancement of this loss at certain values of the scattering length. It was
Particles with resonant short-range interactions have universal properties that do not depend on the details of their structure or their interactions at short distances. In the three-body system, these properties include the existence of a geometric spectrum of three-body Efimov states and a discrete scaling symmetry, which leads to log-periodic dependence of observables on the scattering length. Similar universal properties appear in the four-body system and possibly higher-body systems as well. For example, universal four-body states have recently been predicted and observed in experiments. These phenomena are often referred to as Efimov physics. We review their theoretical description and discuss applications in different areas of physics with a special emphasis on nuclear and particle physics.
We calculate shallow three-body bound states in the universal regime, defined by Efimov, with inclusion of both scattering length and effective range parameters. We find corrections to the universal scaling laws for large binding energies. For narrow resonances we find a distinct nonmonotonic behavior of the threshold at which the lowest Efimov trimer merges with the three-body continuum. The origin of the three-body parameter is related to the two-body atom-atom interactions in a physically clear model. Our results demonstrate that experimental information from narrow Feshbach resonances and/or mixed systems are of vital importance to pin down the relation of two- and three-body physics in atomic systems.
We present a state-of-the-art reanalysis of experimental results on Efimov resonances in the three-fermion system of 6^66Li. We discuss different definitions of the 3-body parameter (3BP) for Efimov states, and adopt a definition that excludes effects due to deviations from universal scaling for low-lying states. We develop a finite-temperature model for the case of three distinguishable fermions and apply it to the excited-state Efimov resonance to obtain the most accurate determination to date of the 3BP in an atomic three-body system. Our analysis of ground-state Efimov resonances in the same system yields values for the three-body parameter that are consistent with the excited-state result. Recent work has suggested that the reduced 3BP for atomic systems is a near-universal quantity, almost independent of the particular atom involved. However, the value of the 3BP obtained for 6^66Li is significantly (∼20\sim 20∼20%) different from that previously obtained from the excited-state resonance in Cs. The difference between these values poses a challenge for theory.
Armed with a new solution to the four-body problem, we reanalyze the recent Innsbruck observation of an Efimov resonance in three-body collisions of cesium atoms, and show that it provides evidence of new, universal four-body physics as well. Our theoretical treatment of four bosonic atoms in the limit of strong two-body attraction suggests a ubiquitous result: namely, that resonantly-bound states of all four atoms should occur whenever the two-body scattering length is a specific fraction (0.43) of the value where a three-body Efimov resonance occurs. A hitherto unanalyzed feature in the 2006 Innsbruck experiment supports this universal prediction, and it provides the first known evidence to date that a direct four-body recombination process plays a role. While we confirm the conclusion reached previously by some studies that no true Efimov effect exists for four particles, we demonstrate the existence of a universal class of four-body states that are intimately connected with three-body Efimov states.
We discuss several issues important for experimentally observing Efimov physics in ultracold quantum gases. By numerically solving the three-boson Schrödinger equation over a broad range of scattering lengths and energies, and by including model potentials with multiple bound states, we address the complications of relating experimental observations to available analytic expressions. These more realistic potentials introduce features that can mask the predicted Efimov physics at small scattering lengths. They also allow us to verify that positive and negative scattering lengths are universally connected only across a pole, not across a zero. Additionally, we show that the spacing between Efimov features for the relatively small scattering lengths accessible experimentally fails to precisely follow the geometric progression expected for Efimov physics. Finally, we emphasize the importance of the short-range three-body physics in determining the position of Efimov features and show that theoretically reproducing two-body physics is not generally sufficient to predict three-body properties quantitatively.