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arXiv:1208.2907 · Physical Review Letters

Universal Four-Body States in Heavy-Light Mixtures with a Positive Scattering Length

D. Blume

Published Dec 5, 2012• 44 citations
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Abstract

The number of four-body states known to behave universally is small. This work adds a new class of four-body states to this relatively short list. We predict the existence of a universal four-body bound state for heavy-light mixtures consisting of three identical heavy fermions and a fourth distinguishable lighter particle with a mass ratio κ>/~9.5 and short-range interspecies interaction characterized by a positive s-wave scattering length. The structural properties of these universal states are discussed, and finite-range effects are analyzed. The bound states can be experimentally realized and probed by utilizing ultracold atom mixtures.

DOI
10.1103/physrevlett.109.230404

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The number of four-body states known to behave universally is small. This work adds a new class of four-body states to this relatively short list. We predict the existence of a universal four-body bound state for heavy-light mixtures consisting of three identical heavy fermions and a fourth distinguishable lighter particle with a mass ratio κ>/~9.5 and short-range interspecies interaction characterized by a positive s-wave scattering length. The structural properties of these universal states are discussed, and finite-range effects are analyzed. The bound states can be experimentally realized and probed by utilizing ultracold atom mixtures.
We study the behavior of the bound-state energy of a system consisting of two identical heavy fermions of mass MMM and a light particle of mass mmm. The heavy fermions interact with the light particle through a short-range two-body potential with positive sss-wave scattering length as{a}_{s}as​. We impose a short-range boundary condition on the logarithmic derivative of the hyperradial wave function and show that, in the regime where Efimov states are absent, a nonuniversal three-body state cuts through the universal three-body states previously described by Kartavtsev and Malykh [O. I. Kartavtsev and A. V. Malykh, J. Phys. B 40, 1429 (2007)]. The presence of the nonuniversal state alters the behavior of the universal states in certain regions of the parameter space. We show that the existence of the nonuniversal state is predicted accurately by a simple quantum defect theory model that utilizes hyperspherical coordinates. An empirical two-state model is employed to quantify the coupling of the nonuniversal state to the universal states.
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.
We treat small trapped unequal-mass two-component Fermi gases at unitarity within a nonperturbative microscopic framework and investigate the system properties as functions of the mass ratio κ, and the numbers N1 and N2 of heavy and light fermions. While equal-mass Fermi gases with infinitely large interspecies s-wave scattering length a(s) are universal, we find that unequal-mass Fermi gases are, for sufficiently large κ and in the regime where Efimov physics is absent, not universal. In particular, the (N₁,N₂) = (2, 1) and (3, 1) systems exhibit three-body and four-body resonances at κ=12.314(2) and 10.4(2), respectively, as well as surprisingly large finite-range effects. These findings have profound implications for ongoing experimental efforts and quantum simulation proposals that utilize unequal-mass atomic Fermi gases.
We show that four heavy fermions interacting resonantly with a lighter atom (4+1 system) become Efimovian at mass ratio 13.279(2), which is smaller than the corresponding 2+1 and 3+1 thresholds. We thus predict the five-body Efimov effect for this system in the regime where any of its subsystem is non-Efimovian. For smaller mass ratios we show the existence and calculate the energy of a universal 4+1 pentamer state, which continues the series of the 2+1 trimer predicted by Kartavtsev and Malykh and 3+1 tetramer discovered by Blume. We also show that the effective-range correction for the light-heavy interaction has a strong effect on all these states and larger effective ranges increase their tendency to bind.
The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as a function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22, we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio, and we map out an energy–mass phase diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.