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arXiv:0802.0549 · Physical Review Letters

Analytical Solution of the Bosonic Three-Body Problem

Alexander O. Gogolin, Christophe Mora, Reinhold Egger

Published Apr 9, 2008• 118 citations
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Abstract

We revisit the problem of three identical bosons in free space, which exhibits a universal hierarchy of bound states (Efimov trimers). Modeling a narrow Feshbach resonance within a two-channel description, we map the integral equation for the three-body scattering amplitude to a one-dimensional Schrödinger-type single-particle equation, where an analytical solution of exponential accuracy is obtained. We give exact results for the trimer binding energies, the three-body parameter, the threshold to the three-atom continuum, and the recombination rate.

DOI
10.1103/physrevlett.100.140404

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We revisit the problem of three identical bosons in free space, which exhibits a universal hierarchy of bound states (Efimov trimers). Modeling a narrow Feshbach resonance within a two-channel description, we map the integral equation for the three-body scattering amplitude to a one-dimensional Schrödinger-type single-particle equation, where an analytical solution of exponential accuracy is obtained. We give exact results for the trimer binding energies, the three-body parameter, the threshold to the three-atom continuum, and the recombination rate.
We calculate shallow three-body bound states in the universal regime, defined by Efimov, with inclusion of both scattering length and effective range parameters. We find corrections to the universal scaling laws for large binding energies. For narrow resonances we find a distinct nonmonotonic behavior of the threshold at which the lowest Efimov trimer merges with the three-body continuum. The origin of the three-body parameter is related to the two-body atom-atom interactions in a physically clear model. Our results demonstrate that experimental information from narrow Feshbach resonances and/or mixed systems are of vital importance to pin down the relation of two- and three-body physics in atomic systems.
We investigate finite-range effects in systems with three identical bosons. We calculate recombination rates and bound state spectra using two different finite-range models that have been used recently to describe the physics of cold atomic gases near Feshbach resonances where the scattering length is large. The models are built on contact potentials which take into account finite-range effects; one is a two-channel model and the other is an effective range expansion model implemented through the boundary condition on the three-body wavefunction when two of the particles are at the same point in space. We compare the results with the results of the ubiquitous single-parameter zero-range model where only the scattering length is taken into account. Both finite-range models predict variations of the well-known geometric scaling factor 22.7 that arises in Efimov physics. The threshold value at negative scattering length for creation of a bound trimer moves to higher or lower values depending on the sign of the effective range compared to the location of the threshold for the single-parameter zero-range model. Large effective ranges, corresponding to narrow resonances, are needed for the reduction to have any noticeable effect.
We construct a simple model to calculate the trimer bound state energy ET(n){E}_{T}^{(n)}ET(n)​ and three-body recombination rate K30{K}_{3}^{0}K30​ of three identical bosons with van der Waals interaction without using any two- or three-body fitting parameter. Using this simple model, we investigate the influence of the van der Waals finite-range effect on ET(n){E}_{T}^{(n)}ET(n)​ and K30{K}_{3}^{0}K30​. Our calculation shows that the finite-range effect leads to the ground trimer state energy ET(0){E}_{T}^{(0)}ET(0)​ not crossing the atom-dimer threshold, and the scaled three-body recombination rate K30/a4{K}_{3}^{0}/{a}^{4}K30​/a4 deviating from the universal three-body theory. The results of our simple model agree within a few percent with other theoretical works with van der Waals interaction and also the experimental data.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.
Three interacting particles form a system which is well known for its complex physical behavior. A landmark theoretical result in few-body quantum physics is Efimov's prediction of a universal set of weakly bound trimer states appearing for three identical bosons with a resonant two-body interaction. Surprisingly, these states even exist in the absence of a corresponding two-body bound state and their precise nature is largely independent of the particular type of the two-body interaction potential. Efimov's scenario has attracted great interest in many areas of physics; an experimental test however has not been achieved. We report the observation of an Efimov resonance in an ultracold thermal gas of cesium atoms. The resonance occurs in the range of large negative two-body scattering lengths and arises from the coupling of three free atoms to an Efimov trimer. We observe its signature as a giant three-body recombination loss when the strength of the two-body interaction is varied near a Feshbach resonance. This resonance develops into a continuum resonance at non-zero collision energies, and we observe a shift of the resonance position as a function of temperature. We also report on a minimum in the recombination loss for positive scattering lengths, indicating destructive interference of decay pathways. Our results confirm central theoretical predictions of Efimov physics and represent a starting point from which to explore the universal properties of resonantly interacting few-body systems.
A nonrelativistic system of three identical particles will display a rich set of universal features known as Efimov physics if the scattering length aaa is much larger than the range lll of the underlying two-body interaction. An appropriate effective theory facilitates the derivation of both results in the $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit and finite-l∕al∕al∕a corrections to observables of interest. Here we use such an effective-theory treatment to consider the impact of corrections linear in the two-body effective range, rs{r}_{s}rs​, on the three-boson bound-state spectrum and recombination rate for $\ensuremath{\mid}a\ensuremath{\mid}⪢\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$. We do this by first deriving results appropriate to the strict limit $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ in coordinate space. We then extend these results to finite aaa using once-subtracted momentum-space integral equations. We take the cutoff on these equations to be large compared to $1∕\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$, and find that the first-order effects of $\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$ are independent of the cutoff in this regime. We also discuss the implications of our results for experiments that probe three-body recombination in Bose-Einstein condensates near a Feshbach resonance.