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Physical Review C

Adiabatic models of the Efimov effect

Yasuhisa Abe, BERTRAND G. GIRAUD, Yukap Hahn, Hitoshi Inamori, Jean LeTourneux

Published Nov 1, 1997• 2 citations
Publisher

Abstract

Elementary demonstrations of the three-body Efimov effect are provided for several models involving a light particle and two heavy ones, and differing by the nature of the two-body (local or nonlocal) potentials which bind at zero energy the light particle to each of the heavy ones. The proofs rely on Born-Oppenheimer approximations, which are actually projections of the Hamiltonian into suitable subspaces, and they use the correct reduced mass. A numerical estimate of the long-range potential responsible for the accumulation of Efimov bound states is obtained through Pad\'e approximants.

DOI
10.1103/physrevc.56.2557

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Elementary demonstrations of the three-body Efimov effect are provided for several models involving a light particle and two heavy ones, and differing by the nature of the two-body (local or nonlocal) potentials which bind at zero energy the light particle to each of the heavy ones. The proofs rely on Born-Oppenheimer approximations, which are actually projections of the Hamiltonian into suitable subspaces, and they use the correct reduced mass. A numerical estimate of the long-range potential responsible for the accumulation of Efimov bound states is obtained through Pad'e approximants.
The possibility of a zero-energy Efimov effect is investigated in a model consisting of three identical heavy particles and a lighter one, when the light-heavy interaction leads to a zero-energy bound state for the two-heavy---one-light subsystem. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable sss-wave potential of Yamaguchi form. The heavy-heavy interaction is short range and, if attractive, is to be taken weak enough to support no two- or three-heavy-particle bound states. The relevant parameter is the potential strength $\ensuremath{\lambda}$ of the light-heavy interaction which for $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ supports a single zero-energy light-heavy bound state. If the first zero-energy bound state of the two-heavy---one-light subsystem occurs for $\ensuremath{\lambda}={\ensuremath{\lambda}}^{\ensuremath{'}}$ such that $|{\ensuremath{\lambda}}^{\ensuremath{'}}|<|{\ensuremath{\lambda}}_{c}|$, there are no Efimov four-body bound states. On the contrary, if the chosen heavy-heavy potential is repulsive enough to prevent the existence of three-body bound states for $|\ensuremath{\lambda}|<|{\ensuremath{\lambda}}_{c}|$, then as $\ensuremath{\lambda}\ensuremath{\rightarrow}{\ensuremath{\lambda}}_{c}$ a few four-body Efimov states may emerge but their number remains finite. These states disappear for $|\ensuremath{\lambda}|\ensuremath{\gtrsim}|{\ensuremath{\lambda}}_{c}|$ as the three-body cut overrides them.
Abstract We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born–Oppenheimer approximation for an arbitrary dimension D . We restrict D to the interval 2 < D < 4, and derive the heavy–heavy D -dimensional effective potential proportional to 1/ R 2 ( R is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy–heavy effective potential 1/ R 2 approaches the limit − ( D − 2 ) 2 / 4 . We obtained the scaling function for the 133 Cs– 133 Cs– 6 Li system as the limit cycle of the correlation between the energies of two consecutive Efimov states as a function of D and the heavy-light binding energy E D 2 . In addition, we found that the energy of the ( N + 1)th excited state reaches the two-body continuum independently of the dimension D when E 2 D / E 3 ( N ) = 0.89 , where E 3 ( N ) is the N th excited three-body binding energy.
In this note we discuss the Efimov effect emerging in a three-particle quantum system with zero-range interactions. In particular, we consider two non-interacting identical bosons plus a different lighter particle such that the interaction between a boson and the light particle is resonant. We also assume the validity of the Born-Oppenheimer approximation. Under these conditions, we show that the three-particle system exhibits infinitely many negative eigenvalues which accumulate at zero and satisfy the universal geometrical law characterising the Efimov effect. The result we find is a generalisation of previous results recently obtained in [13, 24].
The Efimov effect for a three-body system was studied previously in an adiabatic representation, where the distance between two ``heavy'' particles was represented by x, and the third ``light'' particle was described by coordinate y relative to the center of mass of the heavy ones. When x is held fixed, an exact solution for the light particle can be obtained if the interaction is assumed to be of separable form. The resulting adiabatic potential between the heavy particles shows the critical $\ensuremath{-}{1/x}^{2}$ behavior that leads to an infinite number of bound states. However, subsequently the leading nonadiabatic effect was shown to generate an undesirable correction of 1/x1/x1/x type, the pseudo-Coulomb disease (PCD). To remedy the PCD, the Efimov effect is reexamined in the adiabatic state representation, but with the new Jacobi coordinates (r,R), where r describes the distance between one of the heavy particles and the light one and R is the position of the other heavy particle with respect to the center of mass of the subsystem of light-heavy particles. It is then shown that the adiabatic potential in this distorted cluster state representation behaves as $\ensuremath{-}{1/R}^{2}$ at large R if the pair described by r has a zero-energy bound state. Moreover, in this case the leading nonadiabatic correction term does not manifest the PCD, thus explicitly showing that the PCD is due to the ``wrong'' choice of coordinates (x,y). Alternatively, a particle boost factor is introduced to eliminate the PCD in the treatment with the original (x,y) coordinates. This is shown to be equivalent to the change in coordinates, described above. Such a factor is usually associated with high energy collisions, but for the first time shown here to play also an important role in near zero-energy situations.
Two particles that are just shy of binding may develop an infinite number of shallow bound states when a third particle is added. This counterintuitive effect was first predicted by Efimov for identical bosons interacting with a short-range pairwise potential. The Efimov effect persists for nonidentical particles if at least two of the three bonds are almost bound. The Efimov effect has recently been verified experimentally using ultracold atoms. We explain the origin of this effect using elementary quantum mechanics and summarize the experimental evidence for it.