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arXiv:1809.03023 · Physical Review A

Three-body problem of bosons near a d -wave resonance

Juan Yao, Pengfei Zhang, Ran Qi, Hui Zhai

Published Jan 2, 2019• 4 citations
arXivHTMLPDF

Abstract

Motivated by recent experimental progresses, we investigate few-body properties of interacting spinless bosons near a $d$-wave resonance. Using the Skorniakov-Ter-Martirosion (STM) equations, we calculate the scattering length between an atom and a $d$-wave dimer, and we find that the atom-dimer scattering length is positive and is much smaller the result from the mean-field approximation. We also reveal unique properties of the three-body recombination rate for a degenerate Bose condensate near the $d$-wave resonance. We find that the total recombination rate is nearly a constant at the quasibound side, in contrast to the behavior of a thermal gas near a high-partial-wave resonance. We also find that the recombination rate monotonically increases across the unitary point toward the bound side, which is due to the largely enhanced coupling between the atom and the $d$-wave dimer with deeper binding energy. This monotonic behavior is also qualitatively different from that of a degenerate gas near an $s$-wave resonance counterpart.

DOI
10.1103/physreva.99.012701

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Motivated by recent experimental progresses, we investigate few-body properties of interacting spinless bosons near a ddd-wave resonance. Using the Skorniakov-Ter-Martirosion (STM) equations, we calculate the scattering length between an atom and a ddd-wave dimer, and we find that the atom-dimer scattering length is positive and is much smaller the result from the mean-field approximation. We also reveal unique properties of the three-body recombination rate for a degenerate Bose condensate near the ddd-wave resonance. We find that the total recombination rate is nearly a constant at the quasibound side, in contrast to the behavior of a thermal gas near a high-partial-wave resonance. We also find that the recombination rate monotonically increases across the unitary point toward the bound side, which is due to the largely enhanced coupling between the atom and the ddd-wave dimer with deeper binding energy. This monotonic behavior is also qualitatively different from that of a degenerate gas near an sss-wave resonance counterpart.
We consider a three-boson system with resonant binary interactions and show that for sufficiently narrow resonances three-body observables depend only on the resonance width and the scattering length. The effect of narrow resonances is qualitatively different from that of wide resonances revealing novel physics of three-body collisions. We calculate the rate of three-body recombination to a weakly bound level and the atom-dimer scattering length and discuss implications for experiments on Bose-Einstein condensates and atom-molecule mixtures near Feshbach resonances.
For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body sss-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the sss-wave two-body scattering length is $a\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}$, rvdW{r}_{\mathrm{vdW}}rvdW​ being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body ddd-wave dimer, resulting in strong ddd-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the ddd-wave dimer is always formed when $a\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}$. Here we find that a single universal three-body state associated with the ddd-wave dimer is also formed near the three-body breakup threshold at $a\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}$, or alternatively a2=0.902rvdW{a}_{2}=0.902{r}_{\mathrm{vdW}}a2​=0.902rvdW​, where a2{a}_{2}a2​ is the two-body ddd-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.
For identical bosons with a large scattering length, the dependence of the three-body recombination rate on the collision energy is determined in the zero-range limit by universal functions of a single scaling variable. There are six scaling functions for angular momentum zero and one scaling function for each higher partial wave. We calculate these universal functions by solving the Skorniakov-Ter-Martirosian equation. The results for the three-body recombination as a function of the collision energy are in good agreement with previous results from solving the three-body Schr\"odinger equation for 4He^{4}\mathrm{He}4He atoms. The universal scaling functions can be used to calculate the three-body recombination rate at nonzero temperature. We obtain an excellent fit to the data from Grimm and co-workers for 133Cs^{133}\mathrm{Cs}133Cs atoms with a large positive scattering length.
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.
We study the three-body recombination of identical fermionic atoms. Using a zero-range model for the ppp-wave interaction, we show that the rate constant of three-body recombination into weakly bound ppp-wave dimers can be written as ${\ensuremath{\alpha}}_{\mathrm{rec}}\ensuremath{\propto}{v}^{5/2}{R}^{1/2}{k}_{T}^{4}(1+C{k}_{T}^{2}{l}_{d}^{2})$ for large and positive scattering volume vvv. Here RRR is the ppp-wave effective range, kT2{k}_{T}^{2}kT2​ gives the average thermal kinetic energy of the colliding atoms, and ld{l}_{d}ld​ is the size of the ppp-wave dimer. The leading term is different from the usually stated v8/3{v}^{8/3}v8/3-scaling law, but is consistent with an earlier two-channel calculation. For the subleading term, we compute the constant CCC by solving the relevant three-body problem perturbatively when the parameter $\ensuremath{\gamma}\ensuremath{\equiv}R/{v}^{1/3}$ is small. The additional CkT2ld2C{k}_{T}^{2}{l}_{d}^{2}CkT2​ld2​ term provides important corrections for the temperature and interaction dependence of ${\ensuremath{\alpha}}_{\mathrm{rec}}$, especially close to resonance when kTld{k}_{T}{l}_{d}kT​ld​ is relatively large.
We study three-body recombination of fully spin-polarized 6Li^{6}\mathrm{Li}6Li atoms that are interacting resonantly in relative ppp waves. Motivated by a recent experiment, we focus on negative scattering volumes where three atoms recombine into a deep dimer and another atom. We calculate the three-body recombination rate using a Faddeev equation derived from effective field theory. In particular, we study the magnetic field and temperature dependences of the loss rate and use the recombination data to determine the effective range of the ppp-wave atom-atom interaction. We also predict the existence of a shallow three-body resonance state that manifests itself as a prominent feature in the energy-dependent three-body recombination rate.