Abstract
We study the Efimov effect in a harmonic oscillator in the hyperspherical formulation, and show how a reduced model allows for a description that is a generalization of the Efimov effect in free space and leads to results that are easily interpreted. Three-particle states that resemble Efimov trimers for large scattering length, become more similar to independent particles in a harmonic trap when adiabatically decreasing the scattering length. In the transition region, Efimov physics may be observed, while the increased size of the bound states reduces decay into more deeply bound states. The model also allows for the study of non-universal Efimov trimers by including the effective range scattering parameter. While we find that in a certain regime the effective range parameter can take over the role of the three-body parameter, interestingly, we obtain a numerical relationship between these two parameters different from what was found in other models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Efimov V.: Energy levels arising from resonant two-body forces in a three-body system. Phys. Lett. B 33, 563 (1970)
Efimov V.: Weakly bound states of three resonantly-interacting particles. Sov. J. Nucl. Phys. 12, 589 (1971)
Efimov V.: Low-energy properties of three resonantly interacting particles. Sov. J. Nucl. Phys. 29, 546 (1979)
Kraemer T., Mark M., Waldburger P., Danzl J.G., Chin C., Engeser B., Lange A.D., Pilch K., Jaakkola A., Nägerl H.-C., Grimm R.: Evidence for Efimov quantum states in an ultracold gas of caesium atoms. Nature 440, 315 (2006)
Ferlaino F., Grimm R.: Forty years of Efimov physics: how a bizarre prediction turned into a hot topic. Physics 3, 9 (2010)
Greene C.H.: Universal insights from few-body land. Phys. Today 63, 40 (2010)
Knoop S., Ferlaino F., Mark M., Berninger M., Schöbel H., Nägerl H.-C., Grimm R.: Observation of an Efimov-like trimer resonance in ultracold atom–dimer scattering. Nat. Phys. 5, 227 (2009)
Zaccanti M., Deissler B., D’Errico C., Fattori M., Jona-Lasinio M., Müller S., Roati G., Inguscio M., Modugno G.: Observation of an Efimov spectrum in an atomic system. Nat. Phys. 5, 586 (2009)
Gross N., Shotan Z., Kokkelmans S.J.J.M.F., Khaykovich L.: Observation of universality in ultracold 7Li three-body recombination. Phys. Rev. Lett. 103, 163202 (2009)
Pollack S.E., Dries D., Hulet R.G.: Universality in three- and four-body bound states of ultracold atoms. Science 326, 1683 (2009)
Wenz A.N., Lompe T., Ottenstein T.B., Serwane F., Zürn G., Jochim S.: Universal trimer in a three-component Fermi gas. Phys. Rev. A 80, 040702(R) (2009)
Lompe T., Ottenstein T.B., Serwane F., Viering K., Wenz A.N., Zürn G., Jochim S.: Atom–dimer scattering in a three-component fermi gas. Phys. Rev. Lett. 105, 103201 (2010)
Huckans J.H., Williams J.R., Hazlett E.L., Stites R.W., O’Hara K.M.: Three-body recombination in a three-state fermi gas with widely tunable interactions. Phys. Rev. Lett. 102, 165302 (2009)
Williams J.R., Hazlett E.L., Huckans J.H., Stites R.W., Zhang Y., O’Hara K.M.: Evidence for an excited-state Efimov trimer in a three-component fermi gas. Phys. Rev. Lett. 103, 130404 (2009)
Nakajima S., Horikoshi M., Mukaiyama T., Naidon P., Ueda M.: Nonuniversal Efimov atom–dimer resonances in a three-component mixture of 6Li. Phys. Rev. Lett. 105, 023201 (2010)
Barontini G., Weber C., Rabatti F., Catani J., Thalhammer G., Inguscio M., Minardi F.: Observation of heteronuclear atomic Efimov resonances. Phys. Rev. Lett. 103, 043201 (2009)
Feshbach H.: Unified theory of nuclear reactions. Ann. Phys. (NY) 5, 357 (1958)
Feshbach, H.: A unified theory of nuclear reactions. II 19, 287 (1962)
Tiesinga E., Moerdijk A.J., Verhaar B.J., Stoof H.T.C.: Conditions for Bose–Einstein condensation in magnetically trapped atomic cesium. Phys. Rev. A 46, R1167 (1992)
Tiesinga E., Verhaar B.J., Stoof H.T.C.: Threshold and resonance phenomena in ultracold ground-state collisions. Phys. Rev. A 47, 4114 (1993)
Braaten E., Hammer H.-W.: Universality in few-body systems with large scattering length. Phys. Rep. 428, 259 (2006)
Platter L., Hammer H.-W., Meissner U.-G.: Universal properties of the four-Boson system in two dimensions. Few Body Syst. 35, 169 (2004)
Platter L., Hammer H.-W., Meissner U.-G.: Four-Boson system with short-range interactions. Phys. Rev. A 70, 052101 (2004)
von Stecher J., D’Incao J.P., Greene C.H.: Signatures of universal four-body phenomena and their relation to the Efimov effect. Nat. Phys. 5, 417 (2009)
Ferlaino F., Knoop S., Berninger M., Harm W., D’Incao J.P., Nägerl H.-C., Grimm R.: Evidence for universal four-body states tied to an Efimov trimer. Phys. Rev. Lett. 102, 140401 (2009)
Lompe T., Ottenstein T.B., Serwane F., Wenz A.N., Zürn G., Jochim S.: Radio–frequency association of Efimov trimers. Science 330, 940 (2010)
Jonsell S., Heiselberg H., Pethick C.J.: Universal behavior of the energy of trapped few-Boson systems with large scattering length. Phys. Rev. Lett. 89, 250401 (2002)
Fedorov D.V., Jensen A.S.: Efimov effect in coordinate space Faddeev equations. Phys. Rev. Lett. 71, 4103 (1993)
Werner F., Castin Y.: Unitary quantum three-body problem in a harmonic trap. Phys. Rev. Lett. 97, 150401 (2006)
Stoll M., Köhler T.: Production of three-body Efimov molecules in an optical lattice. Phys. Rev. A 72, 022714 (2005)
Thøgersen M., Fedorov D.V., Jensen A.S.: Universal properties of Efimov physics beyond the scattering length approximation. Phys. Rev. A 78, 020501 (2008)
Liu X.J., Hu H., Drummond P.D.: Three attractively interacting fermions in a harmonic trap: exact solution, ferromagnetism and high-temperature thermodynamics. Phys. Rev. A 82, 023619 (2010)
Nielsen E., Fedorov D.V., Jensen A.S., Garrido E.: The three-body problem with short-range interactions. Phys. Rep. 347, 373 (2001)
Fedorov D.V., Jensen A.S.: Regularization of a three-body problem with zero-range potentials. J. Phys. A 34, 6003 (2001)
Platter L., Ji C., Phillips D.R.: Range corrections to three-body observables near a Feshbach resonance. Phys. Rev. A 79, 022702 (2009)
Werner, F.: Trapped cold atoms with resonant interactions: unitary gas and three-body problem, PhD thesis, ENS (2008)
Busch T., Englert B.-G., Rzazewski K., Wilkens M.: Two cold atoms in a harmonic trap. Found. Phys. 28, 549 (1998)
Stöferle T., Moritz H., Günter K., Köhl M., Esslinger T.: Molecules of fermionic atoms in an optical lattice. Phys. Rev. Lett. 96, 030401 (2006)
Petrov D.S.: Three-Boson problem near a narrow Feshbach resonance. Phys. Rev. Lett. 93, 143201 (2004)
Gogolin A.O., Mora C., Egger R.: Analytical solution of the Bosonic three-body problem. Phys. Rev. Lett. 100, 140404 (2008)
Gross N., Shotan Z., Kokkelmans S., Khaykovich L.: Nuclear-spin-independent short-range three-body physics in ultracold atoms. Phys. Rev. Lett. 105, 103203 (2010)
Gross, N., Shotan, Z., Machtey, O., Kokkelmans, S., Khaykovich, L.: Study of Efimov physics in two nuclear-spin sublevels of 7Li. (preprint) (accepted for Comptes rendus Physique) (arXiv:1009.0926)
Ji C., Platter L., Phillips D.R.: Beyond universality in three-body recombination: an effective field theory treatment. EPL 92, 13003 (2010)
Marcelis B., Verhaar B., Kokkelmans S.: Total control over ultracold interactions via electric and magnetic fields. Phys. Rev. Lett. 100, 153201 (2008)
Acknowledgments
The authors gratefully acknowledge useful discussions with J. de Graaf and J. Slot.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Special issue devoted to Efimov physics.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Portegies, J., Kokkelmans, S. Efimov Trimers in a Harmonic Potential. Few-Body Syst 51, 219–234 (2011). https://doi.org/10.1007/s00601-011-0237-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00601-011-0237-6