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Laser Physics

Four-particle problem using Feynman diagrams

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

Published Apr 1, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s1054660x07040342

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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 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 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 aB{a}_{B}aB​ in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length aF{a}_{F}aF​, and the already known result aB=0.60{a}_{B}=0.60aB​=0.60 aF{a}_{F}aF​ 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 bbbbbbbbb and four bbbbbbbbbbbb 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 bbfbbfbbf, two bosons plus two fermions $b{f}_{\ensuremath{\uparrow}}b{f}_{\ensuremath{\downarrow}}$, and three bosons plus one fermion bbbfbbbfbbbf.
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.
Mixtures of polarized fermions of two different masses can form weakly bound clusters, such as dimers and trimers, that are universally described by the scattering length between the heavy and light fermions.We use the resonating group method to investigate the low-energy scattering processes involving dimers or trimers.The method reproduces approximately the known particle-dimer and dimer-dimer scattering lengths.We use it to estimate the trimer-trimer scattering length, which is presently unknown, and find it to be positive.
We discuss a major theoretical generalization of existing techniques for handling the three-body problem that accurately describes the interactions among four fermionic atoms. Application to a two-component Fermi gas accurately determines dimer-dimer scattering parameters at finite energies and can give deeper insight into the corresponding many-body phenomena. To account for finite temperature effects, we calculate the energy-dependent complex dimer-dimer scattering length, which includes contributions from elastic and inelastic collisions. Our results indicate that strong finite-energy effects and dimer dissociation are crucial for understanding the physics in the strongly interacting regime for typical experimental conditions. While our results for dimer-dimer relaxation are consistent with experiment, they confirm only partially a previously published theoretical result.
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.