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arXiv:1807.09992 · Physical Review A

Few-body states of bosons interacting with a heavy quantum impurity

Shuhei M. Yoshida, Zhe-Yu Shi, Jesper Levinsen, Meera M. Parish

Published Dec 11, 2018• 14 citations
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Abstract

We consider the problem of a fixed impurity coupled to a small number $N$ of noninteracting bosons. We focus on impurity-boson interactions that are mediated by a closed-channel molecule, as is the case for tuneable interatomic interactions in cold-atom experiments. We show that this two-channel model can be mapped to a boson model with effective boson-boson repulsion, which enables us to solve the three-body $(N=2)$ problem analytically and determine the trimer energy for impurity-boson scattering lengths $a>0$. By analyzing the atom-dimer scattering amplitude, we find a critical scattering length ${a}^{*}$ at which the atom-dimer scattering length diverges and the trimer merges into the dimer continuum. We furthermore calculate the tetramer energy exactly for $a>0$ and show that the tetramer also merges with the continuum at ${a}^{*}$. Indeed, since the critical point ${a}^{*}$ formally resembles the unitary point $1/a=0$, we find that all higher-body bound states (involving the impurity and $N>1$ bosons) emerge and disappear at both of these points. We show that the behavior at these ``multibody resonances'' is universal, since it occurs for any model with an effective three-body repulsion involving the impurity. Thus we see that the fixed-impurity problem is strongly affected by a three-body parameter even in the absence of the Efimov effect.

DOI
10.1103/physreva.98.062705

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We consider the problem of a fixed impurity coupled to a small number NNN of noninteracting bosons. We focus on impurity-boson interactions that are mediated by a closed-channel molecule, as is the case for tuneable interatomic interactions in cold-atom experiments. We show that this two-channel model can be mapped to a boson model with effective boson-boson repulsion, which enables us to solve the three-body (N=2)(N=2)(N=2) problem analytically and determine the trimer energy for impurity-boson scattering lengths a>0a>0a>0. By analyzing the atom-dimer scattering amplitude, we find a critical scattering length a∗{a}^{*}a∗ at which the atom-dimer scattering length diverges and the trimer merges into the dimer continuum. We furthermore calculate the tetramer energy exactly for a>0a>0a>0 and show that the tetramer also merges with the continuum at a∗{a}^{*}a∗. Indeed, since the critical point a∗{a}^{*}a∗ formally resembles the unitary point 1/a=01/a=01/a=0, we find that all higher-body bound states (involving the impurity and N>1N>1N>1 bosons) emerge and disappear at both of these points. We show that the behavior at these ``multibody resonances'' is universal, since it occurs for any model with an effective three-body repulsion involving the impurity. Thus we see that the fixed-impurity problem is strongly affected by a three-body parameter even in the absence of the Efimov effect.
We investigate the problem of N identical bosons that are coupled to an impurity particle with infinite mass. For noninteracting bosons, we show that a dynamical impurity-boson interaction, mediated by a closed-channel dimer, can induce an effective boson-boson repulsion which strongly modifies the bound states consisting of the impurity and N bosons. In particular, we demonstrate the existence of two universal "multibody" resonances, where all multibody bound states involving any N emerge and disappear. The first multibody resonance corresponds to infinite impurity-boson scattering length, a→+∞, while the second corresponds to the critical scattering length a^{*}>0 beyond which the trimer (N=2 bound state) ceases to exist. Crucially, we show that the existence of a^{*} ensures that the ground-state energy in the multibody bound-state region, ∞>a>a^{*}, is bounded from below, with a bound that is independent of N. Thus, even though the impurity can support multibody bound states, they become increasingly fragile beyond the dimer state. This has implications for the nature of the Bose polaron currently being studied in cold-atom experiments.
Small weakly bound droplets determine a number of properties of ultracold Bose and Fermi gases. For example, Efimov trimers near the atom-atom-atom and atom-dimer thresholds lead to enhanced losses from bosonic clouds. Generalizations to four- and higher-body systems have also been considered. Moreover, Efimov trimers have been predicted to play a role in the Bose polaron with large boson-impurity scattering length. Motivated by these considerations, the present work provides a detailed theoretical analysis of weakly bound NNN-body clusters consisting of $N\ensuremath{-}1$ identical bosons (denoted by ``B'') of mass mmm that interact with a single distinguishable impurity particle (denoted by ``X'') of mass MMM. The system properties are analyzed as a function of the mass ratio $\ensuremath{\kappa}$ (values from $\ensuremath{\kappa}=1$ to 50 are considered), where $\ensuremath{\kappa}$ is equal to m/Mm/Mm/M, and the two-body sss-wave scattering length aBX{a}_{\text{BX}}aBX​ between the bosons and the impurity. To reach the universal Efimov regime in which the size of the BBX trimer as well as those of larger clusters is much larger than the length scales of the underlying interaction model, three different approaches are considered: resonance states are determined in the absence of BB and BBX interactions, bound states are determined in the presence of repulsive three-body boson-boson-impurity interactions, and bound states are determined in the presence of repulsive two-body boson-boson interactions. The universal regime, in which the details of the underlying interaction model become irrelevant, is identified.
We study clusters of the type ANBM{A}_{N}{B}_{M}AN​BM​ with $N\ensuremath{\le}M\ensuremath{\le}3$ in a two-dimensional mixture of AAA and BBB bosons, with attractive ABABAB and equally repulsive AAAAAA and BBBBBB interactions. In order to check universal aspects of the problem, we choose two very different models: dipolar bosons in a bilayer geometry and particles interacting via separable Gaussian potentials. We find that all the considered clusters are bound and that their energies are universal functions of the scattering lengths aAB{a}_{AB}aAB​ and aAA=aBB{a}_{AA}={a}_{BB}aAA​=aBB​, for sufficiently large attraction-to-repulsion ratios aAB/aBB{a}_{AB}/{a}_{BB}aAB​/aBB​. When aAB/aBB{a}_{AB}/{a}_{BB}aAB​/aBB​ decreases below $\ensuremath{\approx}10$, the dimer-dimer interaction changes from attractive to repulsive and the population-balanced AABBAABBAABB and AAABBBAAABBBAAABBB clusters break into ABABAB dimers. Calculating the AAABBBAAABBBAAABBB hexamer energy just below this threshold, we find an effective three-dimer repulsion which may have important implications for the many-body problem, particularly for observing liquid and supersolid states of dipolar dimers in the bilayer geometry. The population-imbalanced ABBABBABB trimer, ABBBABBBABBB tetramer, and AABBBAABBBAABBB pentamer remain bound beyond the dimer-dimer threshold. In the dipolar model, they break up at ${a}_{AB}\ensuremath{\approx}2{a}_{BB}$, where the atom-dimer interaction switches to repulsion.
We consider a few-boson system confined to one dimension with a single distinguishable particle of lesser mass. All particle interactions are modeled with $\ensuremath{\delta}$ functions, but due to the mass imbalance the problem is nonintegrable. Universal few-body binding energies, atom-dimer and atom-trimer scattering lengths, are all calculated in terms of two parameters, namely the mass ratio mL/mH{m}_{\text{L}}/{m}_{\text{H}}mL​/mH​, and ratio gHH/gHL{g}_{\text{HH}}/{g}_{\text{HL}}gHH​/gHL​ of the $\ensuremath{\delta}$-function couplings. We specifically identify the values of these ratios for which the atom-dimer or atom-trimer scattering lengths vanish or diverge. We identify regions in this parameter space in which various few-body inelastic processes become energetically allowed. In the Tonks-Girardeau limit (${g}_{\text{HH}}\ensuremath{\rightarrow}\ensuremath{\infty}$), our results are relevant to experiments involving trapped fermions with an impurity atom.
Motivated by recent experimental investigations of Cs-Cs-Li Efimov resonances, this work theoretically investigates the few-body properties of $N\ensuremath{-}1$ noninteracting identical heavy bosons, which interact with a light impurity through a large sss-wave scattering length. For Cs-Cs-Cs-Li, we predict the existence of universal four-body states with energies E4(n,1){E}_{4}^{(n,1)}E4(n,1)​ and E4(n,2){E}_{4}^{(n,2)}E4(n,2)​, which are universally linked to the energy E3(n){E}_{3}^{(n)}E3(n)​ of the nnnth Efimov trimer. For infinitely large $^{133}\mathrm{Cs}\text{\ensuremath{-}}^{6}\mathrm{Li}$ and vanishing $^{133}\mathrm{Cs}\text{\ensuremath{-}}^{133}\mathrm{Cs}$ scattering lengths, we find $({E}_{4}^{(1,1)}/{E}_{3}^{(1)}{)}^{1/2}\ensuremath{\approx}1.51$ and $({E}_{4}^{(1,2)}/{E}_{3}^{(1)}{)}^{1/2}\ensuremath{\approx}1.01$. The $^{133}\mathrm{Cs}\text{\ensuremath{-}}^{6}\mathrm{Li}$ scattering lengths at which these states merge with the four-atom threshold, the dependence of these energy ratios on the mass ratio between the heavy and light atoms, and selected aspects of the generalized Efimov scenario for N>4N>4N>4 are also discussed. Possible implications of our results for ongoing cold atom experiments are presented.
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.