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Physical Review Letters

Low-Energy Recombination of Identical Bosons by Three-Body Collisions

Esben Nielsen, J. H. Macek

Published Aug 23, 1999• 234 citations
Publisher

Abstract

The threshold recombination coefficient for three bosons whose two-body scattering length $a$ is positive and large compared with the nominal range of the potentials is shown to proceed via a crossing of hyperspherical adiabatic potential curves at $R\ensuremath{\approx}2.6a$. The hidden crossing theory gives a simple expression for the rate in terms of the scattering length and a phase $\ensuremath{\Delta}$. The phase depends upon the details of the interaction, and varies from species to species, however an upper limit to the rate which depends only upon the scattering length emerges from the hidden crossing theory.

DOI
10.1103/physrevlett.83.1566

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The threshold recombination coefficient for three bosons whose two-body scattering length aaa is positive and large compared with the nominal range of the potentials is shown to proceed via a crossing of hyperspherical adiabatic potential curves at R\ensuremath≈2.6aR\ensuremath{\approx}2.6aR\ensuremath≈2.6a. The hidden crossing theory gives a simple expression for the rate in terms of the scattering length and a phase \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ. The phase depends upon the details of the interaction, and varies from species to species, however an upper limit to the rate which depends only upon the scattering length emerges from the hidden crossing theory.
We investigate universal behavior in the recombination rate of three bosons close to threshold. Using the He-He system as a reference, we solve the three-body Schr\"odinger equation above the dimer threshold for different potentials having large values of the two-body scattering length aaa. To this aim, we use the hyperspherical adiabatic expansion and we extract the SSS matrix through the integral relations recently derived. The results are compared to the universal form, $\ensuremath{\alpha}\ensuremath{\approx}67.1{\mathrm{sin}}^{2}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\gamma}]$, for different values of aaa and selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. A good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections, which have been recently proposed [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. Furthermore, we analyze the validity of the above formula in the description of a very different system: neutron-neutron-proton recombination. Our analysis confirms the universal character of the process in systems of very different scales having a large two-body scattering length.
This work treats resonant collisions between five identical ultracold bosons in the framework of the adiabatic hyperspherical representation. The five-body recombination rate coefficient is quantified using a semiclassical description in conjunction with an analysis of the lowest five-body hyperspherical adiabatic potential curves in a scattering length regime with no universal weakly bound tetramers, trimers, or dimers. A comparison is made between these results and the only existing experimental measurement of five-body loss in an ultracold gas of bosonic cesium atoms and with the lone theoretical estimation of the loss rate. The recombination rate for the process B + B + B + B + B → B 4 + B is also computed in a different regime of scattering lengths where there is one universal bound tetramer by implementing a few-channel quantum scattering calculation based on five-body hyperspherical potential curves and nonadiabatic couplings. Our calculations predict regions where five-body recombination can cause decay of the atom cloud in an ultracold gas that is even faster than 3-body and 4-body recombination, which can ideally be tested by using the current generation of box traps having nearly uniform density.
For identical bosons with a large scattering length, the dependence of the three-body recombination rate on the collision energy is determined in the zero-range limit by universal functions of a single scaling variable. There are six scaling functions for angular momentum zero and one scaling function for each higher partial wave. We calculate these universal functions by solving the Skorniakov-Ter-Martirosian equation. The results for the three-body recombination as a function of the collision energy are in good agreement with previous results from solving the three-body Schr\"odinger equation for 4He^{4}\mathrm{He}4He atoms. The universal scaling functions can be used to calculate the three-body recombination rate at nonzero temperature. We obtain an excellent fit to the data from Grimm and co-workers for 133Cs^{133}\mathrm{Cs}133Cs atoms with a large positive scattering length.
We study three-body recombination in two dimensions for systems interacting via short-range two-body interactions in the regime of large scattering lengths. Using the adiabatic hyperspherical representation, we derive semianalytical formulas for three-body recombination in both weakly and deeply bound diatom states. Our results demonstrate the importance of long-range corrections to the three-body potentials by showing how they alter the low-energy and scattering-length dependence of the recombination rate for both bosonic and fermionic systems, which exhibit suppressed recombination if compared to the three-dimensional case. We verify these results through numerical calculations of recombination for systems with finite-range interactions and supporting a few two-body bound states. We also study finite-range effects for the energies of the universal three-identical-bosons states and find a slow approach to universal predictions as a function of the scattering length.
Recombination of identical, spin-polarized fermions in cold three-body collisions is investigated. We parametrize the mechanisms for recombination in terms of the `scattering volume' V p and another length scale r 0 . Model two-body interactions were used within the framework of the adiabatic hyperspherical representation. We examine the recombination rate K 3 as a function of the collision energy E for various values of V p . Not only do we consider the dominant J Π = 1 + case, but also the next-leading order contributions from J Π = 1 - and 3 - . We discuss the behaviour near a two-body resonance and the expected universality of fermionic recombination. Comparisons with boson recombination are considered in detail.
The scattering length a associated with two-body interactions is the relevant parameter for near threshold processes in cold atom-atom collisions. For this reason zero-range potentials are traditionally used to model collective behaviour of dilute collections of bosons. The model is also used to compute three-body recombination rates, where it gives an a 4 law. In this paper we examine the applicability of the zero-range model to real physical systems. Hyperspherical adiabatic potentials obtained from the zero-range model are compared with published potentials based on realistic two-body interactions. From these comparisons it is possible to determine the regions where the model applies.