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Physical Review A

Exact diagrammatic approach for dimer-dimer scattering and bound states of three and four resonantly interacting particles

Il'ya V. Brodsky, M. Yu. Kagan, Aleksei V. Klaptsov, R. Combescot, X. Leyronas

Published Mar 24, 2006• 123 citations
Publisher

Abstract

We present a diagrammatic approach for the dimer-dimer scattering problem in two or three spatial dimensions, within the resonance approximation where these dimers are in a weakly bound resonant state. This approach is first applied to the calculation of the dimer-dimer scattering length ${a}_{B}$ in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length ${a}_{F}$, and the already known result ${a}_{B}=0.60$ ${a}_{F}$ is recovered exactly. Then we make use of our approach to obtain results in two spatial dimensions for fermions as well as for bosons. Specifically, we calculate bound-state energies for three $bbb$ and four $bbbb$ resonantly interacting bosons in two dimensions. We consider also the case of a resonant interaction between fermions and bosons, and we obtain the exact bound-state energies of two bosons plus one fermion $bbf$, two bosons plus two fermions $b{f}_{\ensuremath{\uparrow}}b{f}_{\ensuremath{\downarrow}}$, and three bosons plus one fermion $bbbf$.

DOI
10.1103/physreva.73.032724

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We present an exact diagrammatic approach for the problem of dimer-dimer scattering in 3D for dimers being a resonance bound state of two fermions in a spin-singlet state, with corresponding scattering length a F . Applying this approach to the calculation of the dimmer-dimer scattering length a B , we recover exactly the already known result a B = 0.6 a F . We use the developed approach to obtain new results in 2D for fermions and bosons. Namely, we calculate bound state energies for three bbb and four bbbb resonantly interacting bosons in 2D. For the case of resonance interaction between fermions and bosons, we exactly calculate bound state energies of the following complexes: two bosons plus one fermion bbf, two bosons plus two fermions bf↑bf↓, and three bosons plus one fermion bbbf.
We present an exact diagrammatic approach for the problem of dimer-dimer scattering in 3D for dimers being a resonant bound state of two fermions in a spin singlet state, with corresponding scattering length a. We recover exactly the previously known result a B = 0.60a, where a B is the dimer-dimer scattering length. A detailed discussion of how one can “sum all the diagrams” in this case is presented. Applications to the study of 4-particle bound states of various complexes in 2D are briefly presented.
We discuss the three-body properties of identical bosons exhibiting large scattering length in two spatial dimensions. Within an effective field theory for resonant interactions, we calculate the leading nonuniversal corrections from the two-body effective range to bound-state and scattering observables. In particular, we compute the three-body binding energies, the boson-dimer scattering properties, and the three-body recombination rate for finite energies. We find significant effective range effects for three-body observables in the vicinity of the unitary limit. The implications of this result for future experiments are briefly discussed.
We use the diagrammatic TTT-matrix approach to analyze the three-body scattering problem between two identical fermions and a third particle (which could be a different species of fermion or a boson). We calculate the sss-wave dimer-atom scattering length for all mass ratios, and our results exactly match the results of Petrov. In particular, we list the exact dimer-atom scattering lengths for all available two-species Fermi-Fermi and Bose-Fermi mixtures. In addition, unlike that of the equal-mass-particles case where the three-body scattering TTT matrix decays monotonically as a function of the outgoing momentum, we show that, after an initial rapid drop, this function changes sign and becomes negative at large momenta and then decays slowly to zero when the mass ratio of the fermions to the third particle is higher than a critical value (around 6.56.56.5). As the mass ratio gets higher, modulations of the TTT matrix become more apparent with multiple sign changes, related to the ``fall of a particle to the center'' phenomenon and to the emergence of three-body Efimov bound states.
We consider weakly bound diatomic molecules (dimers) formed in a two-component atomic Fermi gas with a large positive scattering length for the interspecies interaction. We develop a theoretical approach for calculating atom-dimer and dimer-dimer elastic scattering and for analyzing the inelastic collisional relaxation of the molecules into deep bound states. This approach is based on the single-channel zero-range approximation, and we find that it is applicable in the vicinity of a wide two-body Feshbach resonance. Our results draw prospects for various interesting manipulations of weakly bound dimers of fermionic atoms.
We solve the three-boson problem with contact two-and three-body interactions in one dimension and analytically calculate the ground and excited trimer-state energies.Then, by using the diffusion Monte Carlo technique we calculate the binding energy of three dimers formed in a one-dimensional Bose-Bose or Fermi-Bose mixture with attractive interspecies and repulsive intraspecies interactions.Combining these results with our three-body analytics, we extract the three-dimer scattering length close to the dimer-dimer zero crossing.In both considered cases the three-dimer interaction turns out to be repulsive.Our results constitute a concrete proposal for obtaining a one-dimensional gas with a pure three-body repulsion.