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arXiv:1209.4018 · Physical Review A

Atom-dimer scattering amplitude for fermionic mixtures with different masses: s -wave and p -wave contributions

Florent Alzetto, R. Combescot, X. Leyronas

Published Dec 20, 2012• 10 citations
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Abstract

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 $s$-wave contribution we obtain an analytical solution in the asymptotic limit of very large mass ratio. In this regime the $s$-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 $p$-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 $p$-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 ${}^{40}$K -${}^{6}$Li 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.

DOI
10.1103/physreva.86.062708

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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 show that, near a Feshbach resonance, a strong p-wave resonance is present at low energy in atom-dimer scattering for 6^66Li-40^{40}40K fermionic mixtures. This resonance is due to a virtual bound state, in the atom-dimer system, which is present at this low energy. When the mass ratio between the two fermionic elements is increased, this virtual bound state goes to a known real bound state which appears when the mass ratio reaches 8.17. 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 the fermionic species by making use of an optical lattice. This would allow to study the strong dependence of the resonance as a function of the mass ratio of the two fermionic elements.
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 derive the expressions for the virial coefficients up to the third order for two-species fermion mixtures, and investigate the fugacity of the species and the interaction energy density around the sss-wave Feshbach resonance in 6Li^{6}\mathrm{Li}6Li-40K^{40}\mathrm{K}40K mixtures. The theoretical results show that the magnitude of the virial coefficients depends on the mass ratio in a mass-imbalanced system. We find that the contribution of two-body bound states to high-order virial coefficients is very small at temperatures of microkelvin order. With the decrease of T/TFT/{T}_{F}T/TF​, the positions of the maxima of fugacity and interaction energy density shift away from the resonance position in the third-order virial expansion calculation. The mass ratio determines the relevance between the fugacity and number density of heavy species. Our conclusion is also effective for other mass-imbalanced systems.
We study the dimer-dimer scattering length a4{a}_{4}a4​ for a two-component Fermi mixture in which the different fermions have different masses ${m}_{\ensuremath{\uparrow}}$ and ${m}_{\ensuremath{\downarrow}}$. This is made in the framework of the exact field-theoretic method. In the large mass ratio domain the equations are simplified enough to lead to an analytical solution. In particular we link a4{a}_{4}a4​ to the fermion-dimer scattering length a3{a}_{3}a3​ for the same fermions and obtain the very simple relation a4=a3/2{a}_{4}={a}_{3}/2a4​=a3​/2. The result ${a}_{4}\ensuremath{\simeq}{a}_{3}/2$ is actually valid whatever the mass ratio with quite good precision. As a result we find an analytical expression providing a4{a}_{4}a4​ with fairly good precision for any mass. To dominant orders for large mass ratio it agrees with the literature. We show that in this large mass ratio domain, the dominant processes are the repeated dimer-dimer Born scatterings, considered earlier by Pieri and Strinati [Phys Rev. B 61, 15370 (2000)]. We conclude that their approximation of retaining only these processes is a fairly good one whatever the mass ratio.
We present a joint experimental and theoretical investigation of the lifetime of weakly bound dimers formed near narrow interspecies Feshbach resonances in mass-imbalanced Fermi-Fermi systems, considering the specific example of a mixture of 6Li^{6}\text{Li}6Li and 40K^{40}\text{K}40K atoms. Our work addresses the central question of the increase in the stability of the dimers resulting from Pauli suppression of collisional losses, which is a well-known effect in mass-balanced fermionic systems near broad resonances. We present measurements of the spontaneous dissociation of dimers in dilute samples, and of the collisional losses in dense samples arising from both dimer-dimer processes and from atom-dimer processes. We find that all loss processes are suppressed close to the Feshbach resonance. Our general theoretical approach for fermionic mixtures near narrow Feshbach resonances provides predictions for the suppression of collisional decay as a function of the detuning from resonance, and we find excellent agreement with the experimental benchmarks provided by our $^{6}\text{Li}\ensuremath{-}^{40}\text{K}$ system. We finally present model calculations for other Feshbach-resonant Fermi-Fermi systems, which are of interest for experiments in the near future.
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.