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arXiv:0908.0852

Universality in Ultra-Cold Few- and Many-Boson Systems

Thøgersen, Martin

Published Aug 6, 2009• 4 citations
arXivHTMLPDF

Abstract

This dissertation presents theoretical investigations of universality and finite-range corrections in few- and many-boson systems. The main focus is on ultra-cold atomic gases seen both from the three-body, many-body, and mean-field perspective. Topics include corrections to Efimov physics, the N-body Efimov effect, two-body correlations in condensates and higher-order interactions in mean-field BECs.

DOI
10.48550/arxiv.0908.0852

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This dissertation presents theoretical investigations of universality and finite-range corrections in few- and many-boson systems. The main focus is on ultra-cold atomic gases seen both from the three-body, many-body, and mean-field perspective. Topics include corrections to Efimov physics, the N-body Efimov effect, two-body correlations in condensates and higher-order interactions in mean-field BECs.
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.
Systems, such as cold atoms and light halo nuclei, share universal features at low energies, for which details of inter-particle interactions are not essential.We study threebody physics in such systems in an effective-field-theory (EFT) framework.We will discuss universal behavior in three-body systems and higher-order effectiverange corrections beyond universality.We study effective-range effects perturbatively up to the next-to-next-to-leading order (N 2 LO) for the case of three identical bosons.We apply this analysis to recombination features in cold atomic gases and properties of Helium trimers.Effective range corrections also play an important role in the EFT study of fewnucleon systems and halo nuclei.The ground-state of Helium-6 can be treated as a two-neutron halo with an alpha core.This bound state is generated by the resonant neutron-neutron and neutron-alpha interactions.The latter is dominated by a shallow p-wave resonance.We present a calculation of Helium-6 in an EFT approach appropriate for halo systems.
A universal characterization of interactions in few- and many-body quantum systems is often possible without detailed description of the interaction potential, and has become a defacto assumption for cold atom research. Universality in this context is defined as the validity to fully characterize the system in terms of two-body scattering length. We discuss universality in the following three contexts: closed-channel dominated Feshbach resonance, Efimov physics near Feshbach resonances, and corrections to the mean field energy of Bose-Einstein condensates with large scattering lengths. Novel experimental tools and strategies are discussed to study universality in ultracold atomic gases: dynamic control of interactions, run-away evaporative cooling in optical traps, and preparation of few-body systems in optical lattices.
Ultracold atomic systems have been of great theoretical and experimental interest in the past few decades.They provide ideal paradigms to study quantum phenomena, simulate condensed matter systems, and could potentially become a platform for quantum computations.This thesis focuses on the theoretical investigation of universal and non-universal properties of ultracold few-atom systems.The goal of this thesis is to answer or provide insights into several interesting open questions.First, what is the role of the non-universal three-body parameter in weakly-interacting harmonically trapped Bose gases and at which order does it kick in?Second, how large is the ground state energy of small two-component Fermi gases with infinitely strong interactions compared to their non-interacting counterparts?Lastly, what is the interplay between the spin-orbit coupling and the s-wave two-body interaction and how does it affect the energy spectrum of the two-atom system?In order to answer these questions, we develop and employ different analytical and numerical techniques.A perturbative effective field theory treatment to study the weaklyinteracting trapped Bose gas is employed.The treatment is extended to up to fourth-order perturbation theory and techniques to regularize the diverging sums are developed.A logarithmic divergence in the three-body sector is identified and the need for a non-universal three-body parameter is shown.v The explicitly correlated Gaussian method is used to calculate the ground state energy of small trapped two-component Fermi gases with up to ten particles.A code is developed from scratch and parallelized.An improved optimization scheme is developed.A novel technique to remove the finite-range dependence is proposed and tested.Our calculations serve as benchmarks for other numerical techniques.The explicitly correlated Gaussian method is extended to treat systems with periodic boundary conditions for the first time.Standard perturbation theory is used to treat the spin-orbit coupling term in trapped two-atom systems.Our treatment is based on the exact two-body zero-range solution in the trap.The interplay between the spin-orbit coupling and the two-body s-wave interaction is investigated and shown to have a great impact on the energy spectrum for certain parameter combinations.
Advances in trapping and cooling of ultracold gases over the last several decades have made it possible to test many formerly outstanding predictions from disparate branches of physics. This thesis touches on three historical problems that have found new life recently in the context of ultracold Bose gases of alkali atoms. The first problem revolves around an outstanding prediction from Boltzmann over a century and half old that the breathing mode of a isotropically trapped classical gas should oscillate indefinitely. I analyze recent experimental results [Nat. Phys. 11, 1009 (2015)], and attribute observed damping sources to trap imperfections. The second question is about the analogue of first and second sound modes from liquid helium in trapped dilute gases. I present the results of a joint theoretical/experimental investigation of the breathing mode of a finite temperature Bose-Einstein condensate (BEC), attributing a striking collapse revival behavior of the resultant oscillation to in-phase and out-of-phase normal modes of the thermal cloud and condensate. The third problem is that of the formation of Borromean ring-like three-body bound states, referred to as Efimov trimers, in strongly-interacting few-body systems. I extend the predicted spectrum of Efimov states into the realm of many degenerate internal levels, and investigate the difficult three-body elastic scattering problem.\nThese questions are part of the broader theme of this thesis: How can our understanding of few-body physics in the ultracold limit be translated into statements about the bulk behavior of an ultracold gas? For weakly-interacting Bose gases, this translation is well-known: the many-body properties of the gas are well-described by the tracking just the one and two particle correlations. I analyze a generalization of this procedure to higher order correlations, the general connection between few-body physics and correlations in a dilute gas, and results for the emergence of Efimov physics in the magnetic phase of the strongly-interacting Bose gas.
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.