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Journal of Physics Conference Series

Universality in few-body Systems: from few-atoms to few-nucleons

Alejandro Kievsky, M. Gattobigio, Eric Garrido

Published Jul 7, 2014• 7 citations
Publisher

Abstract

Some aspects of universal properties are discussed in few-atom as well as in few- nucleon systems. When the two-body scattering length a is large the three-body system presents some peculiar characteristics such as the Efimov effect. In the present contribution, the zerorange theory deduced by V. Efimov is compared to results obtained using potential models. Accordingly range corrections are introduced. These corrections appear as a shift in the variable K * a with K * the three-body parameter. The three-boson spectrum, atom-dimer scattering and the recombination rate at threshold are analyzed using the finite-range theory proposed. In the case of nuclear systems, n – d scattering is discussed using the universal formula derived by Efimov. Furthermore, the spectrum up to N = 6 is discussed using the central Volkov potential.

DOI
10.1088/1742-6596/527/1/012001

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Modern Physics Letters A•Apr 30, 2009•0 citations
Some aspects of universal properties are discussed in few-atom as well as in few- nucleon systems. When the two-body scattering length a is large the three-body system presents some peculiar characteristics such as the Efimov effect. In the present contribution, the zerorange theory deduced by V. Efimov is compared to results obtained using potential models. Accordingly range corrections are introduced. These corrections appear as a shift in the variable K * a with K * the three-body parameter. The three-boson spectrum, atom-dimer scattering and the recombination rate at threshold are analyzed using the finite-range theory proposed. In the case of nuclear systems, n – d scattering is discussed using the universal formula derived by Efimov. Furthermore, the spectrum up to N = 6 is discussed using the central Volkov potential.
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.
The two-body scattering length a for 4He{}^{4}\mathrm{He}4He atoms is much larger than their effective range rs.{r}_{s}.rs​. As a consequence, low-energy few-body observables have universal characteristics that are independent of the interaction potential. Universality implies that, up to corrections suppressed by rs/a,{r}_{s}/a,rs​/a, all low-energy three-body observables are determined by a and a three-body parameter ${\ensuremath{\Lambda}}_{*}.$ We give simple expressions in terms of a and ${\ensuremath{\Lambda}}_{*}$ for the trimer binding-energy equation, the atom-dimer scattering phase shifts, and the rate for three-body recombination at threshold. We determine ${\ensuremath{\Lambda}}_{*}$ for several 4He{}^{4}\mathrm{He}4He potentials from the calculated binding energy of the excited state of the trimer and use it to obtain the universality predictions for the other low-energy observables. We also use the calculated values for one potential to estimate the effective range corrections for the other potentials.
Few-body systems with large scattering length aaa have universal properties that do not depend on the details of their interactions at short distances. The rate constant for three-body recombination of bosonic atoms of mass mmm into a shallow dimer scales as $\ensuremath{\hbar}{a}^{4}∕m$ times a log-periodic function of the scattering length. We calculate the leading and subleading corrections to the rate constant, which are due to the effective range of the atoms, and study the correlation between the rate constant and the atom-dimer scattering length. Our results are applied to 4He^{4}\mathrm{He}4He atoms as a test case.
Using potential models, we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions, we have observed that at first order, it is necessary to supplement the theory with one finite-range parameter ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ for each specific nnn level [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ depends on the way the potentials are changed to tune the scattering length toward the unitary limit. In this work, we analyze a particular path in which the length ${r}_{B}=a\ensuremath{-}{a}_{B}$, measuring the difference between the two-body scattering length aaa and the energy-scattering length aB{a}_{B}aB​, is almost constant. Analyzing systems with very different scales, such as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of ${\mathrm{\ensuremath{\Gamma}}}_{0}^{3}\ensuremath{\approx}0.87$ for the ground-state level. This observation suggests the possibility of constructing a single universal function that incorporates finite-range effects for this class of paths. The result is used to estimate the three-body parameter ${\ensuremath{\kappa}}_{*}$ in the case of real atomic systems brought to the unitary limit through broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation to the two-body contact C2{C}_{2}C2​ at the unitary limit.
We consider a system of three 4{}^{4}4He atoms, which is so far the simplest realistic three-body system exhibiting the Efimov effect, in order to analyze deviations from the universal Efimov three-body spectrum. We first calculate the bound states using a realistic two-body potential, and then analyze how they can be reproduced by simple effective models beyond Efimov's universal theory. We find that the nonuniversal variations of the ground and first excited states can be well reproduced by models parametrized with only three quantities: the scattering length and effective range of the original potential, and a small three-body force. Furthermore, the three-body parameter, which fixes the origin of the infinite set of three-body levels, is found to be consistent with recent experimental observations in other atomic species.
The universal character of non-relativistic large three-body bosonic systems is addressed, with focus on highly interesting situations that occur in low-energy nuclear and atomic physics. Investigations on the trajectory of the first excited Efimov state, in a renormalized zero-range three-body model for a system with two bound and one virtual two-body subsystems, are reported. The approach is applied to n-n-18C, where the n - n virtual energy and the three-body ground state are kept fixed. It is shown that the real part of the elastic s-wave phase-shift ([Formula: see text]) presents a zero, or a pole in [Formula: see text], when the system has an Efimov excited or virtual state. A brief discussion is given on the relevance of the approach for ultracold atom physics with tunable scattering lengths.