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Atom-dimer scattering and Stability of a mixture of Bose and Fermi superfluids

Xiaoling Cui

Published Jun 5, 2014• 0 citations
OpenAlex

Abstract

Motivated by a recent experiment by ENS group on the mixture of Bose and Fermi superfluids (arxiv:1404.2548), we investigate the effective scattering between a bosonic atom and a molecule(dimer) of fermion atoms. It is found that the mean-field prediction of the atom-dimer scattering length ($a_{ad}$), as simply given by the boson-fermion scattering length ($a_{bf}$), generically fails. Instead, $a_{ad}$ crucially depends on the ratio between $a_{bf}$ and $a_{ff}$ (the fermion-fermion scattering length), and in addition log-periodically depends on the three-body parameter. We identify the universal parameters in characterizing $a_{ad}$ for a wide range of $a_{ff}$ in the molecular side of the fermion-fermion Feshbach resonance, and further demonstrate that the atom-dimer many-body system can become unstable against either phase separation or collapse as tuning $a_{ff}$. Our results have practical implications to the atom loss and collective mode measurement in ENS experiment.

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Motivated by a recent experiment by ENS group on the mixture of Bose and Fermi superfluids (arxiv:1404.2548), we investigate the effective scattering between a bosonic atom and a molecule(dimer) of fermion atoms. It is found that the mean-field prediction of the atom-dimer scattering length (aada_{ad}aad​), as simply given by the boson-fermion scattering length (abfa_{bf}abf​), generically fails. Instead, aada_{ad}aad​ crucially depends on the ratio between abfa_{bf}abf​ and affa_{ff}aff​ (the fermion-fermion scattering length), and in addition log-periodically depends on the three-body parameter. We identify the universal parameters in characterizing aada_{ad}aad​ for a wide range of affa_{ff}aff​ in the molecular side of the fermion-fermion Feshbach resonance, and further demonstrate that the atom-dimer many-body system can become unstable against either phase separation or collapse as tuning affa_{ff}aff​. Our results have practical implications to the atom loss and collective mode measurement in ENS experiment.
Motivated by a recent experiment by the Ecole Normale Sup\'erieure de Lyon (ENS) group on the mixture of Bose and Fermi superfluids [I. Ferrier-Barbut et al., Science 345, 1035 (2014)], we investigate the effective scattering between a bosonic atom and a molecule (dimer) of fermion atoms. It is found that the mean-field prediction of the atom-dimer scattering length (aad)({a}_{\mathrm{ad}})(aad​), as simply given by the boson-fermion scattering length (abf)({a}_{\mathrm{bf}})(abf​), generically fails. Instead, aad{a}_{\mathrm{ad}}aad​ crucially depends on the ratio between abf{a}_{\mathrm{bf}}abf​ and aff{a}_{\mathrm{ff}}aff​ (the fermion-fermion scattering length), and in addition it log-periodically depends on the three-body parameter. We identify the universal parameters in characterizing aad{a}_{\mathrm{ad}}aad​ for a wide range of aff{a}_{\mathrm{ff}}aff​ in the molecular side of the fermion-fermion Feshbach resonance, and further demonstrate that the atom-dimer many-body system can become unstable against either phase separation or collapse as tuning aff{a}_{\mathrm{ff}}aff​. Our results have some implications for the ENS experiment.
In this paper we study the interaction energy in a mixture of Bose and Fermi superfluids realized in recent cold atom experiments. On the Bose-Einstein-condensate (BEC) side of a Feshbach resonance between fermionic atoms, this interaction energy can be directly related to the scattering length between a bosonic atom and a dimer composed of fermions. We calculate the atom-dimer scattering length from a three-body analysis with both a zero-range model and a separable model, including the van der Waals length scale, and we find significant deviation from the result given by a mean-field approach. We also find that the multiple scattering between atom and dimer can account for such a deviation. Our results provide a calibration to the mean-field interaction energy, which can be verified by measuring the shift of collective oscillation frequency.
We study a dilute mixture of degenerate bosons and fermions across a Feshbach resonance of the Fermi-Fermi scattering length aF{a}_{F}aF​. This scattering length is renormalized by the boson-induced interaction between fermions and its value is crucial to determine the phase diagram of the system. For the mixture in a box and a positive Bose-Fermi scattering length, we show that there are three possibilities: A single uniform mixed phase, a purely fermionic phase coexisting with a mixed phase, and a purely fermionic phase coexisting with a purely bosonic one. As 1∕aF1∕{a}_{F}1∕aF​ is increased from a negative value to the Feshbach resonance (1∕aF=0)(1∕{a}_{F}=0)(1∕aF​=0), the region of pure separation increases and the other two regions are strongly reduced. Above the Feshbach resonance (1∕aF>0)(1∕{a}_{F}>0)(1∕aF​>0), pairs of Fermi atoms become Bose-condensed molecules. We find that these molecules are fully spatially separated from the bosonic atoms when 1∕aF1∕{a}_{F}1∕aF​ exceeds a critical value. For a negative Bose-Fermi scattering length we deduce the condition for collapse, which coincides with the onset of dynamical instability of the fully mixed phase. We consider also the mixture in a harmonic trap and determine the conditions for partial demixing, full demixing, and collapse. The experimental implications of our results are investigated by analyzing mixtures of $^{6}\mathrm{Li}\text{\ensuremath{-}}^{23}\mathrm{Na}$ and $^{40}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ atoms.
In the strong-coupling Bose-Einstein condensation (BEC) region where a Feshbach resonance gives rise to tightly bound dimer molecules, we show that a spin-polarized Fermi superfluid reduces to a simple Bose-Fermi mixture of Bose-condensed dimers and the leftover unpaired fermions. Using a many-body functional integral formalism, the Gaussian fluctuations give rise to an induced dimer-dimer interaction mediated by the unpaired fermions, with the dimer-fermion vertex being given by the (mean-field) Born approximation. Treating the pairing fluctuations to quartic order, we show how the action for a spin-polarized Fermi superfluid reduces to one for a Bose-Fermi mixture. This Bose-Fermi action includes an expression for the effective dimer-unpaired fermion interaction in a spin-polarized Fermi superfluid beyond the Born approximation, in the BEC limit. In the low-density limit, we show how this dimer-fermion interaction gives the sss-wave scattering length aBF=1.18aF{a}_{\mathit{BF}}=1.18{a}_{F}aBF​=1.18aF​ (aF{a}_{F}aF​ is the sss-wave fermion scattering length), a result first derived by Skorniakov and Ter-Martirosian in 1957 for three interacting fermions.
We investigate the homogeneous superfluid mixtures of a Bardeen-Cooper-Schrieffer (BCS) superfluid originating from pairing two-species fermionic atoms and superfluidity stemming from condensation of bosonic atoms. By integrating out the freedoms associated with the BCS superfluid, we derive the fermion-mediated interactions between bosons, which is attractive and can be tuned from long range in the BCS region to short range in the region of Bose-Einstein condensation of molecular dimers. By analyzing the Bogoliubov spectrum and the damping rate of bosonic superfluid, we map out the phase diagram spanned by the boson-fermion mass ratio and the boson-fermion coupling strength, which consists of a phase separation region and two phase-mixing regions with and without Landau damping. The three different phases can coexist at a tricritical point, which moves toward a low boson-fermion mass ratio and a high boson-fermion scattering length as the fermion-fermion interaction strength is tuned up on the BCS side.
The phase diagrams of low density Fermi-Fermi mixtures with equal or unequal masses and equal or unequal populations are described at zero and finite temperatures in the strong attraction limit. In this limit, the Fermi-Fermi mixture can be described by a weakly interacting Bose-Fermi mixture, where the bosons correspond to Feshbach molecules and the fermions correspond to excess atoms. First, we discuss the three and four fermion scattering processes, and use the exact boson-fermion and boson-boson scattering lengths to generate the phase diagrams in terms of the underlying fermion-fermion scattering length. In three dimensions, in addition to the normal and uniform superfluid phases, we find two stable nonuniform states corresponding to (i) phase separation between pure unpaired (excess) and pure paired fermions (molecular bosons); and (ii) phase separation between pure excess fermions and a mixture of excess fermions and molecular bosons. Lastly, we also discuss the effects of the trapping potential in the density profiles of condensed and noncondensed molecular bosons, and excess fermions at zero and finite temperatures, and discuss possible implications of our findings to experiments involving mixtures of ultracold fermions.