Three-body contact for fermions.  I. General relations

Félix Werner; Xavier Leyronas

doi:10.5802/crphys.181

Research article

Three-body contact for fermions. I. General relations

Félix Werner ¹ ; Xavier Leyronas ²

¹ Laboratoire Kastler Brossel, Ecole Normale Supérieure - Université PSL, CNRS, Collège de France, Sorbonne Université, 75005 Paris, France
² Laboratoire de Physique de l’Ecole Normale Supérieure, ENS - Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 75005 Paris, France

Comptes Rendus. Physique, Volume 25 (2024), pp. 179-218.

Abstract
Résumé

We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for $↑$ and $↓$ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.

Nous considérons le gaz de Fermi résonnant, à savoir des fermions avec deux états internes à trois dimensions avec des interactions à courte portée de grande longueur de diffusion. Nous introduisons une quantité, le contact à trois corps, qui détermine plusieurs observables. Pour le modèle de portée nulle, le nombre de triplets de fermions proches, la queue de la distribution selon l’impulsion du centre de masse des paires de fermions proches, ainsi que la queue de la distribution en impulsion à deux particules, sont exprimées en termes du contact à trois corps. Pour une portée non nulle, le taux de formation de dimères fortement liés par recombinaison à trois corps, ainsi que la contribution à trois corps à la correction de portée finie à l’énergie, sont exprimées en termes du contact à trois corps et d’un paramètre à trois corps. Ce paramètre à trois corps, qui tend vers zéro dans la limite de portée nulle, est défini via le comportement asymptotique de l’état de diffusion d’énergie nulle à des distances intermédiaires entre la portée et la longueur de diffusion à deux corps. En général, le contact à trois corps a différentes contributions repérées par des indices de spin et de moment cinétique, et le paramètre à trois corps peut dépendre de ces indices. Nous incluons aussi la généralisation à des masses différentes pour les particules $↑$ et $↓$ . Par rapport à la relation donnée dans [Petrov, Salomon et Shlyapnikov, PRL 93, 090404 (2004)] entre taux de pertes à trois corps et nombre de triplets de fermions proches, le présent travail ajoute une dérivation, exprime le facteur de proportionnalité en termes du paramètre à trois corps, et inclut le cas général où il y a plusieurs contributions au contact à trois corps et plusieurs paramètres à trois corps.

Received: 2022-11-17
Revised: 2023-12-07
Accepted: 2024-02-26
Published online: 2024-04-22

DOI: 10.5802/crphys.181

Keywords: Unitary gas, fermions, cold atoms
Mot clés : Gaz unitaire, fermions, atomes froids

Author's affiliations:

Félix Werner ¹; Xavier Leyronas ²

¹ Laboratoire Kastler Brossel, Ecole Normale Supérieure - Université PSL, CNRS, Collège de France, Sorbonne Université, 75005 Paris, France
² Laboratoire de Physique de l’Ecole Normale Supérieure, ENS - Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 75005 Paris, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {F\'elix Werner and Xavier Leyronas},
     title = {Three-body contact for fermions.  {I.} {General} relations},
     journal = {Comptes Rendus. Physique},
     pages = {179--218},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {25},
     year = {2024},
     doi = {10.5802/crphys.181},
     language = {en},
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Félix Werner; Xavier Leyronas. Three-body contact for fermions.  I. General relations. Comptes Rendus. Physique, Volume 25 (2024), pp. 179-218. doi : 10.5802/crphys.181. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.181/

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