Exact solution of the three-boson problem at vanishing energy

Christophe Mora; Alexander O. Gogolin; Reinhold Egger

doi:10.1016/j.crhy.2010.11.002

Exact solution of the three-boson problem at vanishing energy
[Solution exacte du problème à trois corps à énergie nulle]

Christophe Mora ¹ ; Alexander O. Gogolin ² ; Reinhold Egger ³

¹ Laboratoire Pierre-Aigrain, ENS, Université Denis-Diderot 7 – CNRS, 24, rue Lhomond, 75005 Paris, France
² Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
³ Institut für Theoretische Physik, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany

Comptes Rendus. Physique, Volume 12 (2011) no. 1, pp. 27-38.

Résumé
Abstract

Les interactions résonantes à deux corps entre trois bosons identiques sont modélisées par une approche de portée nulle. Une phase, paramètre sans dimension, caractérise la condition aux limites à trois corps tandis que la longueur de diffusion entre dans la condition aux limites de Bethe–Peierls. Le modèle est résolu exactement à énergie nulle pour toute valeur de la longueur de diffusion, positive ou négative. Une expression analytique pour le taux de recombinaison à trois corps vers le dimère universel peu profond est extraite de cette solution.

A zero-range approach is used to model resonant two-body interactions between three identical bosons. A dimensionless phase parametrizes the three-body boundary condition while the scattering length enters the Bethe–Peierls boundary condition. The model is solved exactly at zero energy for any value of the scattering length, positive or negative. From this solution, an analytical expression for the rate of three-body recombination to the universal shallow dimer is extracted.

Publié le : 2011-01-01

DOI : 10.1016/j.crhy.2010.11.002

Keywords: Few-body problem, Three-body recombination, Efimov effect
Mot clés : Problème à quelques corps, Recombinaison à trois corps, Phénomène d'Efimov

Affiliations des auteurs :

Christophe Mora ¹ ; Alexander O. Gogolin ² ; Reinhold Egger ³

¹ Laboratoire Pierre-Aigrain, ENS, Université Denis-Diderot 7 – CNRS, 24, rue Lhomond, 75005 Paris, France
² Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
³ Institut für Theoretische Physik, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany

@article{CRPHYS_2011__12_1_27_0,
     author = {Christophe Mora and Alexander O. Gogolin and Reinhold Egger},
     title = {Exact solution of the three-boson problem at vanishing energy},
     journal = {Comptes Rendus. Physique},
     pages = {27--38},
     publisher = {Elsevier},
     volume = {12},
     number = {1},
     year = {2011},
     doi = {10.1016/j.crhy.2010.11.002},
     language = {en},
}

TY  - JOUR
AU  - Christophe Mora
AU  - Alexander O. Gogolin
AU  - Reinhold Egger
TI  - Exact solution of the three-boson problem at vanishing energy
JO  - Comptes Rendus. Physique
PY  - 2011
SP  - 27
EP  - 38
VL  - 12
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crhy.2010.11.002
LA  - en
ID  - CRPHYS_2011__12_1_27_0
ER  -

%0 Journal Article
%A Christophe Mora
%A Alexander O. Gogolin
%A Reinhold Egger
%T Exact solution of the three-boson problem at vanishing energy
%J Comptes Rendus. Physique
%D 2011
%P 27-38
%V 12
%N 1
%I Elsevier
%R 10.1016/j.crhy.2010.11.002
%G en
%F CRPHYS_2011__12_1_27_0

Christophe Mora; Alexander O. Gogolin; Reinhold Egger. Exact solution of the three-boson problem at vanishing energy. Comptes Rendus. Physique, Volume 12 (2011) no. 1, pp. 27-38. doi : 10.1016/j.crhy.2010.11.002. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2010.11.002/

Bibliographie
Cité par

[1] E. Braaten; H.-W. Hammer Universality in few-body systems with large scattering length, Phys. Rep., Volume 428 (2006) no. 5–6, pp. 259-390

[2] I. Bloch; J. Dalibard; W. Zwerger Many-body physics with ultracold gases, Rev. Mod. Phys., Volume 80 (2008) no. 3, pp. 885-964

[3] E. Braaten; A. Nieto Quantum corrections to the energy density of a homogeneous Bose gas, Eur. Phys. J. B, Volume 11 (1999) no. 1, pp. 143-159

[4] P.F. Bedaque; H.-W. Hammer; U. van Kolck Renormalization of the three-body system with short-range interactions, Phys. Rev. Lett., Volume 82 (1999) no. 3, pp. 463-467

[5] E. Braaten; H.-W. Hammer; T. Mehen Dilute Bose–Einstein condensate with large scattering length, Phys. Rev. Lett., Volume 88 (2002) no. 4, p. 040401

[6] D.S. Petrov; C. Salomon; G.V. Shlyapnikov Weakly bound dimers of fermionic atoms, Phys. Rev. Lett., Volume 93 (2004) no. 9, p. 090404

[7] C. Mora; A. Komnik; R. Egger; A.O. Gogolin Four-body problem and BEC-BCS crossover in a quasi-one-dimensional cold fermion gas, Phys. Rev. Lett., Volume 95 (2005) no. 8, p. 080403

[8] E. Nielsen; J.H. Macek Low-energy recombination of identical bosons by three-body collisions, Phys. Rev. Lett., Volume 83 (1999) no. 8, pp. 1566-1569

[9] B.D. Esry; C.H. Greene; J.P. Burke Recombination of three atoms in the ultracold limit, Phys. Rev. Lett., Volume 83 (1999) no. 9, pp. 1751-1754

[10] P.F. Bedaque; E. Braaten; H.W. Hammer Three-body recombination in Bose gases with large scattering length, Phys. Rev. Lett., Volume 85 (2000) no. 5, pp. 908-911

[11] D.S. Petrov Three-boson problem near a narrow Feshbach resonance, Phys. Rev. Lett., Volume 93 (2004) no. 14, p. 143201

[12] M. Jona-Lasinio; L. Pricoupenko; Y. Castin Three fully polarized fermions close to a p-wave Feshbach resonance, Phys. Rev. A, Volume 77 (2008) no. 4, p. 043611

[13] T. Köhler; K. Göral; P.S. Julienne Production of cold molecules via magnetically tunable Feshbach resonances, Rev. Mod. Phys., Volume 78 (2006) no. 4, pp. 1311-1361

[14] L. Platter Low-energy universality in atomic and nuclear physics, Few-Body Syst., Volume 46 (2009) no. 3, pp. 139-171

[15] J.P. D'Incao; B.D. Esry Scattering length scaling laws for ultracold three-body collisions, Phys. Rev. Lett., Volume 94 (2005) no. 21, p. 213201

[16] V. Efimov Weakly-bound states of three resonantly-interacting particles, Sov. J. Nucl. Phys., Volume 12 (1970), p. 1080

[17] T. Kraemer; M. Mark; P. Waldburger; J.G. Danzl; C. Chin; B. Engeser; A.D. Lange; K. Pilch; A. Jaakkola; H.C. Nägerl; R. Grimm Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature, Volume 440 (2006) no. 7082, pp. 315-318

[18] S. Knoop; F. Ferlaino; M. Mark; M. Berninger; H. Schöbel; H.C. Nägerl; R. Grimm Observation of an Efimov-like trimer resonance in ultracold atom–dimer scattering, Nat. Phys., Volume 5 (2009) no. 3, pp. 227-230

[19] M. Zaccanti; B. Deissler; C. D'Errico; M. Fattori; M. Jona-Lasinio; S. Müller; G. Roati; M. Inguscio; G. Modugno Observation of an Efimov spectrum in an atomic system, Nat. Phys., Volume 5 (2009) no. 8, pp. 586-591

[20] G. Barontini; C. Weber; F. Rabatti; J. Catani; G. Thalhammer; M. Inguscio; F. Minardi Observation of heteronuclear atomic Efimov resonances, Phys. Rev. Lett., Volume 103 (2009) no. 4, p. 043201

[21] N. Gross; Z. Shotan; S. Kokkelmans; L. Khaykovich Observation of universality in ultracold ⁷Li three-body recombination, Phys. Rev. Lett., Volume 103 (2009) no. 16, p. 163202

[22] T.B. Ottenstein; T. Lompe; M. Kohnen; A.N. Wenz; S. Jochim Collisional stability of a three-component degenerate Fermi gas, Phys. Rev. Lett., Volume 101 (2008) no. 20, p. 203202

[23] J.H. Huckans; J.R. Williams; E.L. Hazlett; R.W. Stites; K.M. O'Hara Three-body recombination in a three-state Fermi gas with widely tunable interactions, Phys. Rev. Lett., Volume 102 (2009) no. 16, p. 165302

[24] S. Pollack; D. Dries; R. Hulet Universality in three- and four-body bound states of ultracold atoms, Science, Volume 326 (2009) no. 5960, p. 1683

[25] E. Braaten; H.-W. Hammer Efimov physics in cold atoms, Ann. Phys. (N.Y.), Volume 322 (2007) no. 1, pp. 120-163

[26] E. Nielsen; D.V. Fedorov; A.S. Jensen; E. Garrido The three-body problem with short-range interactions, Phys. Rep., Volume 347 (2001) no. 5, pp. 373-459

[27] G. Gasaneo; S. Ovchinnikov; J. Macek Kontorovich–Lebedev representation for zero-range potential eigensolutions, J. Phys. A, Volume 34 (2001), pp. 8941-8954

[28] J.H. Macek; S. Ovchinnikov; G. Gasaneo Solution for boson–diboson elastic scattering at zero energy in the shape-independent model, Phys. Rev. A, Volume 72 (2005) no. 3, p. 032709

[29] J.H. Macek; S.Yu. Ovchinnikov; G. Gasaneo Exact solution for three particles interacting via zero-range potentials, Phys. Rev. A, Volume 73 (2006) no. 3, p. 032704

[30] G.V. Skorniakov; K.A. Ter-Martirosian The three-body problem with short-range forces. Scattering of low-energy neutrons by deuterons, Sov. Phys. JETP, Volume 31 (1956), p. 775

[31] A.O. Gogolin; C. Mora; R. Egger Analytical solution of the bosonic three-body problem, Phys. Rev. Lett., Volume 100 (2008) no. 14, p. 140404

[32] D.S. Petrov Three-body problem in Fermi gases with short-range interparticle interaction, Phys. Rev. A, Volume 67 (2003) no. 1, p. 010703

[33] C. Mora; R. Egger; A.O. Gogolin; A. Komnik Atom–dimer scattering for confined ultracold fermion gases, Phys. Rev. Lett., Volume 93 (2004) no. 17, p. 170403

[34] C. Mora; R. Egger; A.O. Gogolin Three-body problem for ultracold atoms in quasi-one-dimensional traps, Phys. Rev. A, Volume 71 (2005) no. 5, p. 052705

[35] G.S. Danilov On the three-body problem with short-range forces, Sov. Phys. JETP, Volume 40 (1961), p. 498

[36] L.H. Thomas The interaction between a neutron and a proton and the structure of ³H, Phys. Rev., Volume 47 (1935) no. 12, pp. 903-909

[37] R.A. Minlos; L.D. Faddeev Comment on the problem of three particles with point interactions, Sov. Phys. JETP, Volume 41 (1961), p. 1850

[38] R.A. Minlos; L.D. Faddeev On the point interaction for a three-particle system in quantum mechanics, Sov. Phys. Dokl., Volume 141 (1961), p. 1335

[39] E. Nielsen; H. Suno; B.D. Esry Efimov resonances in atom–diatom scattering, Phys. Rev. A, Volume 66 (2002) no. 1, p. 012705

[40] E. Braaten; H.-W. Hammer; M. Kusunoki Universal equation for Efimov states, Phys. Rev. A, Volume 67 (2003) no. 2, p. 022505

[41] D.V. Fedorov; A.S. Jensen Regularization of a three-body problem with zero-range potentials, J. Phys. A, Volume 34 (2001), p. 6003

[42] K. Helfrich; H.-W. Hammer; D.S. Petrov Three-body problem in heteronuclear mixtures with resonant interspecies interaction, Phys. Rev. A, Volume 81 (2010) no. 4, p. 042715

[43] J.P. D'Incao; B.D. Esry Enhancing the observability of the Efimov effect in ultracold atomic gas mixtures, Phys. Rev. A, Volume 73 (2006) no. 3, p. 30703

[44] D.S. Petrov, unpublished results, 2005.

Cité par Sources :

Commentaires - Politique