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arXiv:1508.00156 · Physical Review A

Few-boson processes in the presence of an attractive impurity under one-dimensional confinement

Nirav P. Mehta, Connor Morehead

Published Oct 20, 2015• 9 citations
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Abstract

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 ${m}_{\text{L}}/{m}_{\text{H}}$, and ratio ${g}_{\text{HH}}/{g}_{\text{HL}}$ 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.

DOI
10.1103/physreva.92.043616

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We consider a few-boson system confined to one dimension with a single distinguishable particle of lesser mass. All particle interactions are modeled with \ensuremathδ\ensuremath{\delta}\ensuremathδ 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 \ensuremathδ\ensuremath{\delta}\ensuremathδ-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 (gHH\ensuremath→\ensuremath∞{g}_{\text{HH}}\ensuremath{\rightarrow}\ensuremath{\infty}gHH​\ensuremath→\ensuremath∞), our results are relevant to experiments involving trapped fermions with an impurity atom.
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 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.
The energy spectrum, atom-dimer scattering length, and atom-trimer scattering length for systems of three and four ultracold atoms with $\ensuremath{\delta}$-function interactions in one dimension are presented as a function of the relative mass ratio of the interacting atoms. The Born-Oppenheimer approach is used to treat three-body (``HHL'') systems of one light and two heavy atoms, as well as four-body (``HHHL'') systems of one light and three heavy atoms. Zero-range interactions of arbitrary strength are assumed between different atoms, but the heavy atoms are assumed to be noninteracting among themselves. Fermionic and bosonic heavy atoms with both positive and negative parity are considered.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions and predict the existence of two three-body bound states when both interactions are attractive. Their binding energies are universal functions of the two-body and three-body scattering lengths and have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
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 study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale ${a}_{\ensuremath{\perp}}$. For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length aaa and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths aad{a}_{\mathit{ad}}aad​ and bad{b}_{\mathit{ad}}bad​. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find bad=0{b}_{\mathit{ad}}=0bad​=0 and aad{a}_{\mathit{ad}}aad​ is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal: aad{a}_{\mathit{ad}}aad​ and bad{b}_{\mathit{ad}}bad​ depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter R∗{R}^{*}R∗ related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.