Efimov states

M. Beminger, A. Zenesini, B. Huang, W. Harm, H.-C. Nagerl, F. Ferlaino, et al., Universality of the three-body parameter for Efimov states in ultracold cesium, Phys. Rev. Lett. 107(2011) 120401. 

Efimov quantum states in a system of three identical bosons [75,76] are a paradigm for universal few-body physics. These states have attracted considerable interest, fueled by their bizarre and counter-intuitive properties, and by the fact that they had been elusive to experimentahsts for more than 35 years. In 2006, experimental evidence for Efimov states in an ultracold gas of Cs atoms was reported [77]. In the context of ultracold quantum gases, Efimov physics manifests itself in three-body decay properties, such as resonances in the three-body recombination and atom-dimer relaxation loss rates. These resonances appear on top of the nonresonant background scattering behavior, given by the universal scaling laws discussed in Section 9.4.2. 

The Efimov trimers influence the three-body scattering properties. When an Efimov state intersects the continuum threshold for a < 0 three-body recombination loss is enhanced [79,80], as the resonant coupling of three atoms to an Efimov state opens up fast decay channels into deeply bound dimer states plus a free atom. Such an Efimov resonance has been observed in an ultracold, thermal gas of Cs atoms [77]. For fl > 0 a similar phenomenon is predicted, namely an atom-dimer scattering resonance at the location at which an Efimov state intersects the atom-dimer threshold [81,82]. Resonance enhancement of P has been observed in a mixture of Cs atoms and Cs2 halo dimers [78] see Figure 9.15. The asymmetric shape of the resonance can be explained by the background scattering behavior, which here is a linear increase as a function of a. 

For an overcritical mass ratio M/m > 13.6 we have a well-known phenomenon of the fall of a particle to the center in an attractive l/R potential [26]. In this case the shape of the wavefunction at distances of the order of can significantly influence the large-scale behavior, and a short-range three-body parameter is required to describe the system. The wavefunction of heavy atoms x(R) acquires many nodes at short distances R, which indicates the appearance of three-body bound Efimov states. 

Baccarelh I, Delgado-Barrio G, Gianturco F, Gonzalez-Lezana T, Miret-Artes S, Villarreal P. (2000) Searching for Efimov states in triatomic systems The case of LiHe2. Europ. Phys. Lett. 50 567. 

Braaten E, Hammer H, Kusunoki M. (2003) Efimov states in a bose-einstein condensate near a Feshbach resonance. Phys. Rev. Lett. 90 170402. 

Bruhl R, Kahnin A, Kornilov O, Toennies J, Hegerfeldt G, Stoll M. (2005) Matter wave diffraction from an inclined transmission grating Searching for the elusive He-4 trimer Efimov state. Phys. Rev. Lett. 95 063002. 

Skundin, A. M., O. N. Efimov, and O. V. Yarmolenko, The state-of-the-art and prospects for the development of rechargeable lithium batteries, Russ. Chem. Rev., 71, 329 (2002). Vincent, C. A., and B. Scrosati, Modem Batteries An Introduction to Electrochemical Power Sources, Edward Arnold, London, 1997. 

The dipole potential is known to occur even in the asymptotic absence of a charged particle. Thus, a series of dipole-supported bound states and QBSs are possible for systems of three neutral particles. This phenomenon, called the Efimov effect, will be discussed briefly in the last paragraph of Section 3.2.2. 

V. Efimov, Weakly-bound states of three resonantly-interacting particles, Sov. J. Nucl. Phys. 12 (1971) 589 [Yad. Fiz. 12 (1970) 1080],... 

T. Kraemer, M. Mark, P. Waldburger, J.G. Danzl, C. Chin, B. Engeser, et al., Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature 440 (2006) 315. 

Efimov and Erusalimchik (10) have criticized the results of Bohnenkamp and Engell (9), They state that the capacity values of the minimum of the capacity-potential curve reported by Bohnenkamp and Engell are too. small, compared with (heir own measurements, and assume that this is due to a poor contact at the reverse side of the germanium electrodes and an inadequate preparation of the surface. We tested the contact on the reverse side with the help of a sample contacted at both sides and found no capacitive component large enough to in-... 

Klande, T., Efimov, K., Cusenza, S. etal. (2011) Effect of doping, microstructure, and CO2 on La2Ni0444 i -based oxygen-transporting materials. Journal of Solid State Chemistry, 184, 3310-3318. 

The trimer states, which in most cases can be called Efimov trimers, are interesting objects. Their existence can be seen from the Born-Oppenheimer picture for two heavy atoms and one light atom in the gerade state. Within the Born-Oppenheimer approach the three-body problem reduces to the calculation of the relative motion of the heavy atoms in the effective potential created by the light atom. For the light atom in the gerade state, this potential is + (/ ), found in the previous subsection. The Schrodinger equation for the wavefunction of the relative motion of the heavy atoms, Xv(R), reads... 

The three-body parameter ro determines the phase of the wavefunction at small distances and, in principle, depends on 1. The wavefunction 10.56 has infinitely many nodes, which means that in the zero-range approximation there are infinitely many trimer states. This is one of the properties of three-body systems with resonant interactions discovered by Efimov [54], We see that the fall to the center is possible in many angular momentum channels, provided the mass ratio is sufficiently large. However, for practical purposes and for simplicity, it is sufficient to consider the case where the Efimov effect occurs only for the angular momentum channel with the lowest possible / for a given symmetry. This implies that when the heavy atoms are fermions and one has odd I, in order to confine ourselves to / = 1 we should have the mass ratio in the range 14 < M/m < 76. For bosonic heavy atoms where / is even, we set I = 0 and consider M/m < 39 to avoid the Efimov effect for / > 2. In both cases we need a single three-body parameter tq. 

Besides the Efimov trimers, one light and two heavy atoms may form universal trimer states well described in the zero-range approximation without introducing the three-body parameter [69]. In particular, they exist for the orbital angular momentum / = 1 and mass ratios below the critical value, where the Efimov effect is absent and short-range physics drops out of consideration. One of such states emerges at Mfmf S and crosses the trimer formation threshold (cn = —2 eo ) at M/m 12.7. The existence of this state is already seen in the Bom-Oppenheimer picture. It appears as a bound state of fermionic heavy atoms in the potential +(/ ) for / = 1. The other state exists at M/m even closer to the critical mass ratio and never becomes sufficiently deeply bound to be formed in cold dimer-dimer collisions. The universal trimer states also exist for / > 1 and M/m > 13.6 [69]. However, trimer formation in dimer-dimer collisions at such mass ratios is dominated by the contribution of Efimov trimers with smaller /. Therefore, below, we focus on the formation of Efimov trimers. 

The same method was employed to estimate the formation of the universal trimer state with the orbital angular momentum / = 1 at mass ratios M/m > 12.7 but below the critical value for the onset of the Efimov effect [67]. The rate constant increases with M/m and reaches a = 0.2(ha/M) close to the critical mass ratio. This corresponds to the imaginary part of the scattering length Imfldd -4 x 10 a, which is smaller by a factor of300 than the real part of add obtained from four-body calculations... 

Of particular interest are the trimer states of two heavy and one light fermion in an optical lattice. For two-dimensional densities 10 cm therate of trimer formation can be of the order of seconds, and these states can be detected optically. As we mentioned in Section 10.4, the lattice trimers are long-lived, with a lifetime that can be of the order of tens of seconds. Thus, it is interesting to study to what extent these nonconventional states, in which the heavy atoms are localized in different sites and the light atom is delocalized between them, can exhibit the Efimov effect. 

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