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arXiv:1409.0282 · Physical Review A

Calibration of the interaction energy between Bose and Fermi superfluids

Ren Zhang, Wei Zhang, Hui Zhai, Peng Zhang

Published Dec 8, 2014• 36 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.90.063614

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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.
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 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 present a theoretical study of the BEC-BCS crossover in the Bose-Einstein-condensate (BEC) regime in the case of an unequal number of fermions of two species. We take full account of the composite nature of the dimers made of fermions. In the limit of low densities, we calculate the ground-state energy of the system, or equivalently the chemical potentials of each species, as well as the one-particle gap and the energy of an ``impurity'' immersed in a Fermi sea. For the chemical potentials we go up to order (density)4/3(\mathrm{density}){}^{4/3}(density)4/3. The results found involve the exact atom-dimer aAD{a}_{\mathrm{AD}}aAD​ and dimer-dimer aDD{a}_{\mathrm{DD}}aDD​ scattering lengths and therefore include the three- and four-body problems in the many-body problem. We briefly comment on the importance of the different mean-field corrections for recent experiments.
With help of a Feshbach scattering resonance it is possible to change the effective two-body interaction of ultra-cold atoms from repulsive to attractive, from weak to strong. Regular field theory for the description of Bose-Einstein condensates is based on a single interaction parameter only, the scattering length. Such a description, however, is not valid anymore for the resonance regime. We formulated a field-theory that incorporated the microscopic energy-dependent two-body interactions, that also takes pair correlations into account. This theory allowed us to explain an experiment at JILA where Ramsey fringes were observed in a ^85Rb Bose-Einstein condensate by making use of a Feshbach resonance. The theory and experiment show that coherent molecules are created, a first step in the direction of molecular BEC. For non-resonant interactions, the self-energy corresponding to the atom-atom interactions is proportional to the scattering length. Close to resonance, however, this relationship no longer applies, and the energy will level off to a finite value on resonance. We investigate this resonance self-energy, and compare it with experimental results from strongly interacting Fermi systems.
We calculate the ground-state energy and the collective excitation frequency of trapped bosons at large scattering length interacting via the realistic two-body van der Waals potential. Our many-body method keeps two-body correlations produced by all interacting pairs. When the scattering length is small compared to the trap size and the number of bosons in the trap is of the order of a few thousands, the mean-field results are in good agreement with the many-body results. However for large particle numbers, even when the condensate is sufficiently dilute, the interatomic correlation comes into the picture. When the scattering length is quite large near the Feshbach resonance, the Bose gas becomes highly correlated. The many-body results are close to the Gross-Pitaevskii results for a small number of bosons, however, large deviations are noted in the large particle limit. We also calculate the lowest collective excitation and the interaction energy for large scattering lengths. The monopole excitation frequency exhibits a pronounced dependence on the scattering length. We also observe a universal behavior for the interaction energy at the limit of large scattering length.