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arXiv:1003.0106 · Physical Review A

Dimer-atom scattering between two identical fermions and a third particle

M. Iskin

Published Apr 27, 2010• 18 citations
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Abstract

We use the diagrammatic $T$-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 $s$-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 $T$ 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 $6.5$). As the mass ratio gets higher, modulations of the $T$ 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.

DOI
10.1103/physreva.81.043634

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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.
Using the framework of effective field theory, we present a detailed study of the Efimov effect in higher partial waves for systems of two identical particles and a third distinguishable particle. Depending on the total angular momentum LLL, the two identical particles must be bosons or fermions. We derive analytical expressions for the elastic and inelastic atom-dimer scattering cross sections as well as the atom-dimer relaxation rate at the dimer breakup threshold. For the experimentally most relevant case of P-waves, we numerically calculate the atom-dimer scattering cross sections and relaxation rates as a function of the scattering length, three-body parameter, and mass ratio for energies away from the breakup threshold.
We study near a Feshbach resonance, as a function of the mass ratio, the fermion-dimer scattering amplitude in fermionic mixtures of two fermion species. When masses are equal the physical situation is known to be quite simple. We show that, when the mass ratio is increased, the situation becomes much more complex. For the sss-wave contribution we obtain an analytical solution in the asymptotic limit of very large mass ratio. In this regime the sss-wave scattering amplitude displays a large number of zeros, essentially linked to the known large value of the fermion-dimer scattering length in this regime. We find by an exact numerical calculation that a zero is still present for a mass ratio of 15. For the ppp-wave contribution we make our study below the mass ratio of 8.17, where a fermion-dimer bound state appears. We find that a strong ppp-wave resonance is present at low energy, due to a virtual bound state, in the fermion-dimer system, which is a forerunner of the real bound state. This resonance becomes prominent in the mass ratio range around the one corresponding to the 40{}^{40}40K -6{}^{6}6Li mixtures, much studied experimentally. This resonance should affect a number of physical properties. These include the equation of state of unbalanced mixtures at very low temperature but also the equation of state of balanced mixtures at moderate or high temperature. The frequency and the damping of collective modes should also provide a convenient way to evidence this resonance. Finally it should be possible to modify the effective mass of one of the fermionic species by making use of an optical lattice. This would allow one to study the strong dependence of the resonance as a function of the mass ratio of the two fermionic elements. In particular one could check if the virtual bound state is relevant for the instabilities of these mixtures.
We investigate the three-body properties of two identical ↑ fermions and one distinguishable ↓ atom interacting in a strongly confined two-dimensional geometry.We compute exactly the atom-dimer scattering properties and the three-body recombination rate as a function of collision energy and mass ratio m ↑ /m ↓ .We find that the recombination rate for fermions is strongly energy dependent, with significant contributions from higher partial waves at low energies.For m ↑ m ↓ , the s-wave atom-dimer scattering below threshold is completely described by the scattering length.Furthermore, we examine the ↑↑↓ bound states (trimers) appearing at large m ↑ /m ↓ and find that the energy spectrum for the deepest bound trimers resembles that of a hydrogen atom confined to two dimensions.
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We investigate the three-body properties of two identical "up" fermions and one distinguishable "down" atom interacting in a strongly confined two-dimensional geometry. We compute exactly the atom-dimer scattering properties and the three-body recombination rate as a function of collision energy and mass ratio m_up/m_down. We find that the recombination rate for fermions is strongly energy dependent, with significant contributions from higher partial waves at low energies. For m_up < m_down, the s-wave atom-dimer scattering below threshold is completely described by the scattering length. Furthermore, we examine the "up-up-down" bound states (trimers) appearing at large m_up/m_down and find that the energy spectrum for the deepest bound trimers resembles that of a hydrogen atom confined to two dimensions.
We use the zero-range approximation to study a system of two identical bosons interacting resonantly with a third particle. The method is derived from effective field theory. It reduces the three-body problem to an integral equation which we then solve numerically. We also develop an alternative approach which gives analytic solutions of the integral equation in coordinate representation in the limit of vanishing total energy. The atom-dimer scattering length, the rates of atom-dimer relaxation, and the three-body recombination to shallow and to deep molecular states are calculated either analytically or numerically with a well-controlled accuracy for various energies as functions of the mass ratio, scattering length, and three-body parameter. We discuss in detail the relative positions of the recombination loss peaks, which in the universal limit depend only on the mass ratio. Our results have implications for ongoing and future experiments on Bose-Bose and Bose-Fermi atomic mixtures.