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arXiv:1306.1711 · Physical Review A

Recombination rates from potential models close to the unitary limit

Eric Garrido, M. Gattobigio, Alejandro Kievsky

Published Sep 9, 2013• 16 citations
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Abstract

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 $a$. To this aim, we use the hyperspherical adiabatic expansion and we extract the $S$ 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 $a$ 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.

DOI
10.1103/physreva.88.032701

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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, \ensuremathα\ensuremath≈67.1sin2[s0ln(\ensuremathκ∗a)+\ensuremathγ]\ensuremath{\alpha}\ensuremath{\approx}67.1{\mathrm{sin}}^{2}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\gamma}]\ensuremathα\ensuremath≈67.1sin2[s0​ln(\ensuremathκ∗​a)+\ensuremathγ], for different values of aaa and selected values of the three-body parameter \ensuremathκ∗{\ensuremath{\kappa}}_{*}\ensuremathκ∗​. 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.
We investigate universal behavior in elastic atom-dimer scattering below the dimer breakup threshold calculating the atom-dimer effective-range function $ak\mathrm{cot}\ensuremath{\delta}$. Using the He-He system as a reference, we solve the Schr\"odinger equation for a family of potentials having different values of the two-body scattering length aaa and we compare our results to the universal zero-range form deduced by Efimov, $ak\mathrm{cot}\ensuremath{\delta}={c}_{1}(ka)+{c}_{2}(ka)\mathrm{cot}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\phi}(ka)]$, for selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. Using the parametrization of the universal functions ${c}_{1},{c}_{2},\ensuremath{\phi}$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter ${\ensuremath{\kappa}}_{*}$.
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.
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.
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.
We study the three-body problem in one dimension for both zero- and finite-range interactions using the adiabatic hyperspherical approach. Particular emphasis is placed on the threshold laws for recombination, which are derived for all combinations of the parity and exchange symmetries. For bosons, we provide a numerical demonstration of several universal features that appear in the three-body system, and discuss how certain universal features in three dimensions are different in one dimension. We show that the probability for inelastic processes vanishes as the range of the pairwise interaction is taken to zero and demonstrate numerically that the recombination threshold law manifests itself for large scattering length.
Some aspects of universal properties are discussed in few-atom as well as in few- nucleon systems. When the two-body scattering length a is large the three-body system presents some peculiar characteristics such as the Efimov effect. In the present contribution, the zerorange theory deduced by V. Efimov is compared to results obtained using potential models. Accordingly range corrections are introduced. These corrections appear as a shift in the variable K * a with K * the three-body parameter. The three-boson spectrum, atom-dimer scattering and the recombination rate at threshold are analyzed using the finite-range theory proposed. In the case of nuclear systems, n – d scattering is discussed using the universal formula derived by Efimov. Furthermore, the spectrum up to N = 6 is discussed using the central Volkov potential.