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arXiv:0903.4145 · Physical Review Letters

General Theoretical Description of N -Body Recombination

Nirav P. Mehta, Seth T. Rittenhouse, José Paulo D'Incao, J. von Stecher, Chris H. Greene

Published Oct 9, 2009• 61 citations
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Abstract

Formulas for the cross section and event rate constant describing recombination of N particles are derived in terms of general S-matrix elements. Our result immediately yields the generalized Wigner threshold scaling for the recombination of N bosons. A semianalytical formula encapsulates the overall scaling with energy and scattering length, as well as resonant modifications by the presence of N-body states near the threshold collision energy in the entrance channel. We then apply our model to the case of four-boson recombination into an Efimov trimer and a free atom.

DOI
10.1103/physrevlett.103.153201

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Formulas for the cross section and event rate constant describing recombination of N particles are derived in terms of general S-matrix elements. Our result immediately yields the generalized Wigner threshold scaling for the recombination of N bosons. A semianalytical formula encapsulates the overall scaling with energy and scattering length, as well as resonant modifications by the presence of N-body states near the threshold collision energy in the entrance channel. We then apply our model to the case of four-boson recombination into an Efimov trimer and a free atom.
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.
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.
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.
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.
For atoms with large scattering length, the dependence of the three-body recombination rate on the collision energy is determined by the scattering length and the Efimov three-body parameters and can be expressed in terms of universal functions of a single scaling variable. We use published results on the three-body recombination rate for 4He^{4}\mathrm{He}4He atoms to constrain the universal functions. We then use those universal functions to calculate the three-body recombination rate for other atoms with large scattering length at nonzero temperature. The constraints from the 4He^{4}\mathrm{He}4He results are strong if the scattering length is near the minimum of the three-body recombination rate at threshold. We apply our results to 133Cs^{133}\mathrm{Cs}133Cs atoms with a large positive scattering length, and compare them with experimental results from the Innsbruck group.
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 $R\ensuremath{\approx}2.6a$. 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.