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Physica Scripta

Efimov states: what are they and why are they important?

J. H. Macek

Published Jul 13, 2007• 6 citations
Publisher

Abstract

The interaction of two atoms at energies in the microkelvin range is mainly characterized by one parameter, namely, the s-wave scattering length. Efimov showed that when the scattering length was effectively infinite, i.e. much larger than the natural length scale of the atoms, a large number of three-body bound states could form even when the two-body system supported only one weakly bound state, or possibly none at all. The states are bound in an effective 1/ R 2 potential and therefore form a series with a characteristic energy spacing. These states are called Efimov states and are conceptually important since they indicate a new way to form aggregates of particles. They bear some relation to vibrationally bound states of three atoms. The theory of these states and their relation to vibrational binding is reviewed in light of recent experimental measurements.

DOI
10.1088/0031-8949/76/2/n01

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In systems of ultracold atoms, the quantum Efimov effect can appear where identical bosons form an infinite tower of bound trimer states in the resonant limit, at the bound dimer dissociation threshold. The most characteristic feature of this effect is that their energy spectrum obey a geometric scaling law, which is universal in the sense that it emerges irrespective of the nature of the two body forces. Using a model potential, constructed to resemble the two body interaction between alkali atoms, which was fine tuned to control the scattering length, energy eigenvalues for the two- and threebody problem were calculated numerically. The results where fitted to the analytic theory and the appearance of the first Efimov state was positioned at a scattering length of -9.23rvdW , which is in good agreement with the universal value -9.2rvdW .

•Jan 1, 2016•0 citations
The interaction of two atoms at energies in the microkelvin range is mainly characterized by one parameter, namely, the s-wave scattering length. Efimov showed that when the scattering length was effectively infinite, i.e. much larger than the natural length scale of the atoms, a large number of three-body bound states could form even when the two-body system supported only one weakly bound state, or possibly none at all. The states are bound in an effective 1/ R 2 potential and therefore form a series with a characteristic energy spacing. These states are called Efimov states and are conceptually important since they indicate a new way to form aggregates of particles. They bear some relation to vibrationally bound states of three atoms. The theory of these states and their relation to vibrational binding is reviewed in light of recent experimental measurements.
The Efimov states are three-body bound states, which appear universally when the s-wave scattering length between the particles is resonantly large.Since the Efimov states and their characteristic discrete scale invariance have recently been observed in ultracold atoms, the Efimov states and their associated few-body phenomena have attracted widening interest both theoretically and experimentally.In this thesis, I theoretically study the Efimov physics and related phenomena, focusing on the following three topics: (i) Universal three-body physics for a mass-imbalanced Fermi system (Chapter 3).For a system of two identical spinless (i.e., spin-polarized) fermions and one distinguishable particle which interact via a short-range potential with a large s-wave scattering length, two classes of universal three-body bound states have been known to appear in different regimes of the mass ratio: the Efimov trimers and the Kartavtsev-Malykh trimers, which feature the discrete and continuous scale invariance, respectively.I have found the third class of universal three-body bound states, which I call the "crossover trimers".The crossover trimers show neither the discrete nor continuous scale invariance.I have identified the regions of these three classes of trimers as a function of the mass ratio and the s-wave scattering length, and shown that the Kartavtsev-Malykh trimers and the Efimov trimers can continuously transform into each other via the crossover trimers as the mass ratio and the s-wave scattering length are varied.Owing to the presence of the crossover trimers, the Kartavtsev-Malykh trimers dissociate into a particle and a dimer when the s-wave scattering length is varied, inducing resonances in the particle-dimer scattering.I have calculated the elastic particle-dimer scattering lengths in arbitrary angular-momentum channels, and shown that the particle-dimer resonances indeed occur.From the resonance positions, I have found accurate values of the critical mass ratios at which the Kartavtsev-Malykh trimers in the higher angular-momentum channels appear. (ii) Universal three-body parameter (Chapter 4).In the Efimov physics, the three-body parameter fixes the short-range three-body phase and hence the scale of the energy spectrum.Until quite recently, it has been widely held that the threebody parameter should depend on the short-range part of the inter-atomic potential, and hence vary almost randomly between different atomic species and hyperfine states.However, mounting evidence in recent experiments suggests otherwise.While some theoretical studies have succeeded in reproducing the universal behavior in the three-body parameter, the underlying physical mechanism has remained unclear and controversial.In this thesis, I elucidate the physical origin of the universal three-body parameter.I propose that a non-adiabatic deformation of the three-body wave function induced by a universal two-body correlation results in a universal three-body repulsion, which prevents three particles from coming close and renders the three-body parameter universal.This mechanism is verified by reproducing the universal three-body repulsion with a simple model wave function.The mechanism found here suggests that the three-body parameter would also be universal for non-van der Waals types of potentials.For various classes of deep two-body potentials, it is shown that the effective range is the relevant length scale to characterize the two-body correlation and sets the three-body parameter.Depending on the shape of the two-body correlation, there
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.
Efimov states are a sequence of shallow three-body bound states that arise when the two-body scattering length is large. Efimov showed that the binding energies of these states can be calculated in terms of the scattering length and a three-body parameter by solving a transcendental equation involving a universal function of one variable. We calculate this universal function using effective field theory and use it to describe the three-body system of 4He{}^{4}\mathrm{He}4He atoms. We also extend Efimov's theory to include the effects of deep two-body bound states, which give widths to the Efimov states.
By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length aaa becomes large. [The number of bound states is then roughly $(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)$, where $\ensuremath{\Lambda}$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.
The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle SSS matrix in the complex energy plane, and we study how they transform from virtual states through bound states to resonances when we change the interaction strength. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings---a behavior known from the nonrelativistic case. We additionally find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this ``missing states'' problem.
In systems of ultracold atoms, the quantum Efimov effect can appear where identical bosons form an infinite tower of bound trimer states in the resonant limit, at the bound dimer dissociation threshold. The most characteristic feature of this effect is that their energy spectrum obey a geometric scaling law, which is universal in the sense that it emerges irrespective of the nature of the two body forces. Using a model potential, constructed to resemble the two body interaction between alkali atoms, which was fine tuned to control the scattering length, energy eigenvalues for the two- and threebody problem were calculated numerically. The results where fitted to the analytic theory and the appearance of the first Efimov state was positioned at a scattering length of -9.23rvdW , which is in good agreement with the universal value -9.2rvdW .