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arXiv:1103.5157 · Physical Review A

Single-particle momentum distribution of an Efimov trimer

Yvan Castin, Félix Werner

Published Jun 13, 2011• 47 citations
arXivHTMLPDF

Abstract

Experimental progress in the study of strongly interacting ultracold atoms has recently allowed the observation of Efimov trimers. We study theoretically a nonconventional observable for these trimer states, which may be accessed experimentally, the momentum distribution $n(\mathbf{k})$ of the constitutive bosonic particles. The large momentum part of the distribution is particularly intriguing: In addition to the expected $1/{k}^{4}$ tail associated with contact interactions, it exhibits a subleading tail $1/{k}^{5}$ which is a hallmark of Efimov physics and leads to a breakdown of a previously proposed expression of the energy as a functional of the momentum distribution.

DOI
10.1103/physreva.83.063614

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Experimental progress in the study of strongly interacting ultracold atoms has recently allowed the observation of Efimov trimers. We study theoretically a nonconventional observable for these trimer states, which may be accessed experimentally, the momentum distribution n(k)n(\mathbf{k})n(k) of the constitutive bosonic particles. The large momentum part of the distribution is particularly intriguing: In addition to the expected 1/k41/{k}^{4}1/k4 tail associated with contact interactions, it exhibits a subleading tail 1/k51/{k}^{5}1/k5 which is a hallmark of Efimov physics and leads to a breakdown of a previously proposed expression of the energy as a functional of the momentum distribution.
We compute the virial coefficients, the contact parameters, and the momentum distribution of a strongly interacting three-dimensional Bose gas by means of a virial expansion up to third order in the fugacity, which takes into account three-body correlations exactly. Our results characterize the nondegenerate regime of the interacting Bose gas, where the thermal wavelength is smaller than the interparticle spacing but the scattering length may be arbitrarily large. We observe a rapid variation of the third virial coefficient as the scattering length is tuned across the three-atom and the atom-dimer thresholds. The momentum distribution at unitarity displays a universal high-momentum tail with a log-periodic momentum dependence, which is a direct signature of Efimov physics. We provide a quantitative description of the momentum distribution at high momentum as measured by P. Makotyn et al. [Nat. Phys. 10, 116 (2014)], and our calculations indicate that the lowest trimer state might not be occupied in the experiment. Our results allow for a spectroscopy of Efimov states in the unitary limit.
We studied the single-particle momentum distribution of mass-imbalanced Efimov states embedded in noninteger dimensions. The contact parameters, which can be related to the thermodynamic properties of the gas, were extracted from the tail of the single-particle momentum densities. We studied the dependence of the contact parameters on the progressive change in the noninteger dimension, ranging from three (D=3D=3D=3) to two (D=2D=2D=2) dimensions. Within this interval, we move from the D=3D=3D=3 regime where the Efimov discrete-scale symmetry drives the physics to close to the critical dimension, which depends on the mass imbalance, where the continuum-scale symmetry takes place. We found that the two- and three-body contacts grow significantly in magnitude with the decrease of the noninteger dimension towards the critical dimension, impacting observables of resonantly interacting trapped Bose gases.
Since the early days of quantum physics, the complex behavior of three interacting particles has been the subject of numerous experimental and theoretical studies. In a recent Letter to Nature, Kraemer et al. [Nature (London) 440, 315 (2006)] report on experimental ``evidence for Efimov quantum states'' in an ultracold gas of cesium atoms. Such quantum states refer to an infinite series of energy levels of three identical Bose particles, accumulating at the threshold for dissociation as the scattering length of each pair is tuned to infinity. Whereas the existence of a single Efimov state has been predicted for three helium atoms, earlier experimental studies concluded that this elusive state had not been found. In this paper we show by an intuitive argument and full numerical calculations that the helium and cesium experiments actually provide evidence of the same, ground state of this trimer spectrum, which the helium experimentalists and pioneering theoretical studies had not associated with Efimov's effect. Unlike the helium trimer, the observed 133Cs3^{133}\mathrm{Cs}_{3}133Cs3​ resonance refers to a Borromean molecular state. We discuss how as yet unobserved, excited Efimov quantum states might be detected in ultracold gases of 85Rb^{85}\mathrm{Rb}85Rb and of 133Cs^{133}\mathrm{Cs}133Cs at magnetic field strengths in the vicinity of 0.08T0.08\phantom{\rule{0.3em}{0ex}}\mathrm{T}0.08T (800G)(800\phantom{\rule{0.3em}{0ex}}\mathrm{G})(800G).
We discuss the collisional formation of Efimov trimers via ultracold four-body recombination. In particular, we consider the reaction A+A+A+B-->A_{3}+B with A and B ultracold atoms. We obtain expressions for the four-body recombination rate up to an overall constant and show that it reflects the three-body Efimov physics either as a function of collision energy or as a function of the two-body s-wave scattering length between A atoms. In addition, we briefly discuss issues important for experimentally observing this interesting and unexplored process.
The Efimov states are three-body bound states, which appear universally when the s-wave scattering length between the particles is resonantly large.Since the Efimov states and their characteristic discrete scale invariance have recently been observed in ultracold atoms, the Efimov states and their associated few-body phenomena have attracted widening interest both theoretically and experimentally.In this thesis, I theoretically study the Efimov physics and related phenomena, focusing on the following three topics: (i) Universal three-body physics for a mass-imbalanced Fermi system (Chapter 3).For a system of two identical spinless (i.e., spin-polarized) fermions and one distinguishable particle which interact via a short-range potential with a large s-wave scattering length, two classes of universal three-body bound states have been known to appear in different regimes of the mass ratio: the Efimov trimers and the Kartavtsev-Malykh trimers, which feature the discrete and continuous scale invariance, respectively.I have found the third class of universal three-body bound states, which I call the "crossover trimers".The crossover trimers show neither the discrete nor continuous scale invariance.I have identified the regions of these three classes of trimers as a function of the mass ratio and the s-wave scattering length, and shown that the Kartavtsev-Malykh trimers and the Efimov trimers can continuously transform into each other via the crossover trimers as the mass ratio and the s-wave scattering length are varied.Owing to the presence of the crossover trimers, the Kartavtsev-Malykh trimers dissociate into a particle and a dimer when the s-wave scattering length is varied, inducing resonances in the particle-dimer scattering.I have calculated the elastic particle-dimer scattering lengths in arbitrary angular-momentum channels, and shown that the particle-dimer resonances indeed occur.From the resonance positions, I have found accurate values of the critical mass ratios at which the Kartavtsev-Malykh trimers in the higher angular-momentum channels appear. (ii) Universal three-body parameter (Chapter 4).In the Efimov physics, the three-body parameter fixes the short-range three-body phase and hence the scale of the energy spectrum.Until quite recently, it has been widely held that the threebody parameter should depend on the short-range part of the inter-atomic potential, and hence vary almost randomly between different atomic species and hyperfine states.However, mounting evidence in recent experiments suggests otherwise.While some theoretical studies have succeeded in reproducing the universal behavior in the three-body parameter, the underlying physical mechanism has remained unclear and controversial.In this thesis, I elucidate the physical origin of the universal three-body parameter.I propose that a non-adiabatic deformation of the three-body wave function induced by a universal two-body correlation results in a universal three-body repulsion, which prevents three particles from coming close and renders the three-body parameter universal.This mechanism is verified by reproducing the universal three-body repulsion with a simple model wave function.The mechanism found here suggests that the three-body parameter would also be universal for non-van der Waals types of potentials.For various classes of deep two-body potentials, it is shown that the effective range is the relevant length scale to characterize the two-body correlation and sets the three-body parameter.Depending on the shape of the two-body correlation, there