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arXiv:1206.5752 · Oxford University Press eBooks

The few-atom problem

Dmitry S. Petrov

Published Nov 15, 2012• 11 citations
arXivHTMLPDF

Abstract

This chapter discusses the few-body problem, which plays a very important role in the physics of ultracold gases. It describes the structure of molecules and their formation in three-body collisions (three-body recombination), atom-molecule and molecule-molecule collisional properties, structure of trimers and larger clusters, numerous phenomena that go under the name of Efimov physics, and many other problems. The chapter is organized as follows. Section 3.2 discusses the effective range expansion for the two-body scattering amplitude and puts a special emphasis on the role of the resonance width. Section 3.3 applies the zero-range approach to the problem of three atoms. It considers the phenomenon of the three-body collapse, or Thomas collapse, and defines the Efimov effect. Then, it introduces a model system of one light and two heavy atoms, where we can use the Born‐Oppenheimer approximation. Section 3.4 describes the method of Skorniakov and Ter‐Martirosian (STM) for few-body problems with resonant short-range interactions. Section 3.5 considers other rapidly developing directions in theoretical few-atom physics.

DOI
10.1093/acprof:oso/9780199661886.003.0003

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This chapter discusses the few-body problem, which plays a very important role in the physics of ultracold gases. It describes the structure of molecules and their formation in three-body collisions (three-body recombination), atom-molecule and molecule-molecule collisional properties, structure of trimers and larger clusters, numerous phenomena that go under the name of Efimov physics, and many other problems. The chapter is organized as follows. Section 3.2 discusses the effective range expansion for the two-body scattering amplitude and puts a special emphasis on the role of the resonance width. Section 3.3 applies the zero-range approach to the problem of three atoms. It considers the phenomenon of the three-body collapse, or Thomas collapse, and defines the Efimov effect. Then, it introduces a model system of one light and two heavy atoms, where we can use the Born‐Oppenheimer approximation. Section 3.4 describes the method of Skorniakov and Ter‐Martirosian (STM) for few-body problems with resonant short-range interactions. Section 3.5 considers other rapidly developing directions in theoretical few-atom 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.
Abstract The Efimov scenario is a universal three-body effect addressing many areas of modern quantum physics. It plays an important role in the transition between few- and many-body physics and has enabled important breakthroughs in the understanding of the universal few-body theory. We review the basic concepts of the Efimov scenario with specific emphasis on the similarities and differences between homonuclear and heteronuclear systems. In the latter scenario, the existence of a second, independently tunable interaction parameter enables novel few-body phenomena that are universal and have no counterexamples in the homonuclear case. We discuss recent experimental approaches using ultracold atomic gases with magnetically tunable interactions and elucidate the role of short-range interactions in the emergence of universal and non-universal behavior.
This work will introduce the Efimov effect and the resonant and scaling limits and derive the formula for the binding energies of the Efimov states. We use the hyperspherical coordinates for the stationary wave function of three particles and solve the low-energy Faddeev equation with the hyperspherical expansion and use the expansion for solving the channel eigenvalues λn(R). The channel eigenvalues are defined by a constant s0, which is the solution of the resulting transcendental equation. We also solve the scaling-violation parameter Λ0 and finally compile all the results to derive the Efimov states. In the unitary limit we find infinitely many Efimov states, with an accumulation point at zero energy and an asymptotic discrete scaling symmetry with the discrete scaling factor e π s0 ≈ 22.7. In this work, we will also delve into effective field theories, which can be used to numerically analyze and solve Efimov states in different cases. We will first go through the two-body problem which is used as a simpler example on how to solve the three-body problem and to solve the two-body coupling constant g2, which will also appear in the three-body problem. By using the diatom field trick introduced by Bedaque, Hammer and van Kolck, we derive the Skorniakov-Ter-Martirosian equation for the three-body problem. Finally this work will take a quick look at the first experimental evidences for Efimov states that were found since 2006. In the experiment, proper Efimov resonances in measurements of three-body recombination have been observed.
We are developing theoretical and computational methods to study two related three-body processes in ultracold quantum gases: three-body resonances and three-body recombination. Three-body recombination causes the ultracold gas to heat up and atoms to leave the trap where they are confined. Therefore, it is an undesirable effect in the process of forming ultracold quantum gases. Metastable three-body states (resonances) are formed in the ultracold gas. When decaying they also give additional kinetic energy to the gas, that leads to the heating too. In addition, a reliable method to obtain three-body resonances would be useful in a number of problems in other fields of physics, for example, in models of metastable nuclei or to study dissociative recombination of H3 +. Our project consists of employing computer modeling to develop a method to obtain three-body resonances. The method uses a novel two-step diagonalization approach to solve the three-body Schrödinger equation. The approach employs the SVD method of Tolstikhin et al. coupled with a complex absorbing potential. We tested this method on a model system of three identical bosons with nucleon mass and compared it to the results of a previous study. This model can be employed to understand the 3He nucleus . We found one three-body bound state and four resonances. We are also studying Efimov resonances using a 4He-based model. In a system of identical spinless bosons, Efimov states are a series of loosely bound three-body states which begin to appear as the energy of the two-body bound state approaches zero . Although they were predicted 35 years ago, recent evidence of Efimov states found by Kraemer et al. in a gas of ultracold Cs atoms has sparked great interest by theorists and experimentalists. Efimov resonances are a kind of pre-dissociated Efimov trimer. To search for Efimov resonances we tune the diatom interaction potential, V(r): V(r) → λV(r) as Esry et al. did . We calculated the first two values of λ for which there is a "condensation" (infinite number) of Efimov states. They are λEfimov1 = 0.9765 and λEfimov2 = 6.834. We performed calculations for λ = 2.4, but found no evidence of Efimov resonances. For future work we plan to work with λ ≈ 4 and λ ≈ λEfimov2 where we might see d-wave and higher l-wave Efimov resonances. There is also a many-body project that forms part of this thesis and consists of a direct diagonalization of the Bogolyubov Hamiltonian, which describes elementary excitations of a gas of bosons interacting through a pairwise interaction. We would like to reproduce the corresponding energy spectrum. So far we have performed several convergence tests, but have not observed the desired energy spectrum. We show preliminary results.
We examine the physics of two, three, and four heavy fermions interacting with a single light fermion via short-range interactions. Four-particle bosonic Efimov states have proven important experimentally and also been the subject of significant theoretical effort. Similar fermionic systems are just now being investigated. We find that with some simple interactions the four- and five-particle states collapse to the interaction range at smaller mass ratios than the three-body state, and also before larger clusters can collapse. These states and their excitations can be studied in cold atom experiments, providing unique insights into the role of few-body systems in many-body physics.
Generally, the three-body recombination (TBR) of the ultracold helium atoms in the zero-collision-energy limit could be described by the original zero-range (Efimov) universal function [E. Braaten, D. Kang, and L. Platter, Phys. Rev. A 75, 052714 (2007)] using the scattering length aaa of the He-He interaction and a three-body parameter a∗{a}_{*}a∗​ as input, where a∗{a}_{*}a∗​ is associated with the positive scattering length at which an interference minimum in the TBR rate occurs at the threshold. This universal property has been reinvestigated by considering various post-Born-Oppenheimer (post-BO) effects on the relevant scattering process. It is found that a linear correction to 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a should be introduced to give a better description of these considered TBR cases because of the finite-range effects. Particularly, such a correction may be ascribed not only to the general scattering properties associated with the Efimov channel at the short hyperradial distance, but also to the tunneling probability in the lowest incident channel. In addition, a successful fit of the modified universal function that includes correctly the finite-range effects [E. Garrido, M. Gattobigio, and A. Kievsky, Phys. Rev. A 88, 032701 (2013)] or the zero-range universal function with a linearly variational 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a to numerically accurate data also indicates a clear manifestation of the van der Waals universality. Finally, the post-BO effects on the TBR process for the partial wave with the total angular momentum J>0J>0J>0 are also discussed.