Efimov state

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The Efimov effect is an effect in the quantum mechanics of few-body systems predicted by the Russian theoretical physicist V. N. Efimov[1][2] in 1970. Efimov’s effect is where three identical bosons interact, with the prediction of an infinite series of excited three-body energy levels when a two-body state is exactly at the dissociation threshold. One corollary is that there exist bound states (called Efimov states) of three bosons even if the two-particle attraction is too weak to allow two bosons to form a pair. A (three-particle) Efimov state, where the (two-body) sub-systems are unbound, are often depicted symbolically by the Borromean rings. This means that if one of the particles is removed, the remaining two fall apart. In this case, the Efimov state is also called a Borromean state.


Halo Effect (hopefully close to scale). This state of balance and "Russian nesting dolls" was predicted by Efimov.

The unusual Efimov state has an infinite number of similar states. These states are completely identical except that their sizes and energy levels scale by a universal factor of approximately 22.7 (in the case of three identical bosonic particles), see OEISA242978.

In 2005, for the first time the research group of Rudolf Grimm and Hanns-Christoph Nägerl from the Institute for Experimental Physics (University of Innsbruck, Austria) experimentally confirmed such a state in an ultracold gas of caesium atoms. In 2006, they published their findings in the scientific journal Nature.[3] Further experimental proof for the existence of the Efimov state has been given recently by independent groups.[4] Almost 40 years after Efimov's purely theoretical prediction, the characteristic periodic behavior of the states has been confirmed.[5][6]


The most accurate experimental value of the scaling factor of the states has been determined by the experimental group of Bo Huang at Innsbruck University as 21.0(1.3),[7] being very close to Efimov's original prediction.

The interest in the "universal phenomena" of cold atomic gases is still growing, especially because of the long-awaited experimental results.[8][9] The discipline of universality in cold atomic gases near the Efimov states are sometimes commonly referred to as "Efimov physics".

In 2014 the experimental group of Cheng Chin of the University of Chicago and the group of Matthias Weidemüller of the University of Heidelberg have observed Efimov states in an ultracold mixture of lithium and caesium atoms,[10][11] which extends Efimov's original picture of three identical bosons.

An Efimov state existing as an excited state of a helium trimer was observed in an experiment in 2015.[12]


The Efimov states are independent of the underlying physical interaction and can in principle be observed in all quantum mechanical systems (i.e. molecular, atomic, and nuclear). The states are very special because of their "non-classical" nature: The size of each three-particle Efimov state is much larger than the force-range between the individual particle pairs. This means that the state is purely quantum mechanical. Similar phenomena are observed in two-neutron halo-nuclei, such as lithium-11. (Halo nuclei could be seen as special Efimov states, depending on the subtle definitions.)


  1. ^ В.И. Ефимов: Слабосвязанные состояния трех резонансно взаимодействующих частиц, Ядерная Физика, т. 12, вып. 5, 1080-1090, 1970 г.
  2. ^ Efimov, V. (1970). "Energy levels arising from resonant two-body forces in a three-body system". Physics Letters B. 33 (8): 563–564. Bibcode:1970PhLB...33..563E. doi:10.1016/0370-2693(70)90349-7. 
  3. ^ 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 (2006). "Evidence for Efimov quantum states in an ultracold gas of caesium atoms". Nature. 440 (7082): 315–318. Bibcode:2006Natur.440..315K. PMID 16541068. arXiv:cond-mat/0512394Freely accessible. doi:10.1038/nature04626. 
  4. ^ Knoop, S.; Ferlaino, F.; Mark, M.; Berninger, M.; Schöbel, H.; Nägerl, H. -C.; Grimm, R. (2009). "Observation of an Efimov-like trimer resonance in ultracold atom–dimer scattering". Nature Physics. 5 (3): 227. Bibcode:2009NatPh...5..227K. arXiv:0807.3306Freely accessible. doi:10.1038/nphys1203. 
  5. ^ Zaccanti, M.; Deissler, B.; D’Errico, C.; Fattori, M.; Jona-Lasinio, M.; Müller, S.; Roati, G.; Inguscio, M.; Modugno, G. (2009). "Observation of an Efimov spectrum in an atomic system". Nature Physics. 5 (8): 586. Bibcode:2009NatPh...5..586Z. arXiv:0904.4453Freely accessible. doi:10.1038/nphys1334. 
  6. ^ "Universality in Three- and Four-Body Bound States of Ultracold Atoms". Science. 326: 1683–1685. Bibcode:2009Sci...326.1683P. arXiv:0911.0893Freely accessible. doi:10.1126/science.1182840. 
  7. ^ http://physics.aps.org/articles/v7/51
  8. ^ Braaten, E.; Hammer, H. (2006). "Universality in few-body systems with large scattering length". Physics Reports. 428 (5–6): 259. Bibcode:2006PhR...428..259B. arXiv:cond-mat/0410417Freely accessible. doi:10.1016/j.physrep.2006.03.001. 
  9. ^ Thøgersen, Martin (2009). "Universality in Ultra-Cold Few- and Many-Boson Systems". arXiv:0908.0852Freely accessible.  Ph.D. thesis.
  10. ^ Shih-Kuang Tung; Karina Jiménez-García; Jacob Johansen; Colin V. Parker; Cheng Chin (2014). "Geometric Scaling of Efimov States in a Li6−Cs133 Mixture". Physical Review Letters. 113 (24): 240402. Bibcode:2014PhRvL.113x0402T. arXiv:1402.5943Freely accessible. doi:10.1103/PhysRevLett.113.240402. 
  11. ^ R. Pires; J. Ulmanis; S. Häfner; M. Repp; A. Arias; E. D. Kuhnle; M. Weidemüller (2014). "Observation of Efimov Resonances in a Mixture with Extreme Mass Imbalance". Physical Review Letters. 112 (25): 250404. Bibcode:2014PhRvL.112y0404P. PMID 25014797. arXiv:1403.7246Freely accessible. doi:10.1103/PhysRevLett.112.250404. 
  12. ^ Kunitski, Maksim; Zeller, Stefan; Voigtsberger, Jörg; Kalinin, Anton; Schmidt, Lothar Ph. H.; Schöffler, Markus; Czasch, Achim; Schöllkopf, Wieland; Grisenti, Robert E.; Jahnke, Till; Blume, Dörte; Dörner, Reinhard (May 2015). "Observation of the Efimov state of the helium trimer". Science. 348 (6234): 551–555. Bibcode:2015Sci...348..551K. PMID 25931554. arXiv:1512.02036Freely accessible. doi:10.1126/science.aaa5601. 

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