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Physical Review A

Effective interactions in imperfect gases and enhancement of the Efimov effect

Carmelo Pisani, Bruce H. J. McKellar

Published Mar 1, 1989• 1 citation
Publisher

Abstract

In this paper we present a realization of the concept, first put forth by V. Efimov [Few Body Dynamics, edited by A. N. Mitra et al. (North-Holland, Amsterdam, 1976)], of enhancing the Efimov effect by placing the three particles in a medium whose parameters vary. We do this in the context of equilibrium statistical mechanics by incorporating three-body potentials and subsequently writing their contribution to the cluster series in terms of effective, density-dependent, two-body potentials. We then perform approximate partial summations of the cluster series expressing the sums in terms of cluster coefficients involving the effective potentials. From the high-temperature expansion of the corresponding equation of state for a model system, we find that, at the Efimov point, behavior resembling that of a first-order phase transition appears.

DOI
10.1103/physreva.39.2297

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In this paper we present a realization of the concept, first put forth by V. Efimov [Few Body Dynamics, edited by A. N. Mitra et al. (North-Holland, Amsterdam, 1976)], of enhancing the Efimov effect by placing the three particles in a medium whose parameters vary. We do this in the context of equilibrium statistical mechanics by incorporating three-body potentials and subsequently writing their contribution to the cluster series in terms of effective, density-dependent, two-body potentials. We then perform approximate partial summations of the cluster series expressing the sums in terms of cluster coefficients involving the effective potentials. From the high-temperature expansion of the corresponding equation of state for a model system, we find that, at the Efimov point, behavior resembling that of a first-order phase transition appears.
This chapter discusses the few-body problem, which plays a very important role in the physics of ultracold gases. It describes the structure of molecules and their formation in three-body collisions (three-body recombination), atom-molecule and molecule-molecule collisional properties, structure of trimers and larger clusters, numerous phenomena that go under the name of Efimov physics, and many other problems. The chapter is organized as follows. Section 3.2 discusses the effective range expansion for the two-body scattering amplitude and puts a special emphasis on the role of the resonance width. Section 3.3 applies the zero-range approach to the problem of three atoms. It considers the phenomenon of the three-body collapse, or Thomas collapse, and defines the Efimov effect. Then, it introduces a model system of one light and two heavy atoms, where we can use the Born‐Oppenheimer approximation. Section 3.4 describes the method of Skorniakov and Ter‐Martirosian (STM) for few-body problems with resonant short-range interactions. Section 3.5 considers other rapidly developing directions in theoretical few-atom physics.
This work will introduce the Efimov effect and the resonant and scaling limits and derive the formula for the binding energies of the Efimov states. We use the hyperspherical coordinates for the stationary wave function of three particles and solve the low-energy Faddeev equation with the hyperspherical expansion and use the expansion for solving the channel eigenvalues λn(R). The channel eigenvalues are defined by a constant s0, which is the solution of the resulting transcendental equation. We also solve the scaling-violation parameter Λ0 and finally compile all the results to derive the Efimov states. In the unitary limit we find infinitely many Efimov states, with an accumulation point at zero energy and an asymptotic discrete scaling symmetry with the discrete scaling factor e π s0 ≈ 22.7. In this work, we will also delve into effective field theories, which can be used to numerically analyze and solve Efimov states in different cases. We will first go through the two-body problem which is used as a simpler example on how to solve the three-body problem and to solve the two-body coupling constant g2, which will also appear in the three-body problem. By using the diatom field trick introduced by Bedaque, Hammer and van Kolck, we derive the Skorniakov-Ter-Martirosian equation for the three-body problem. Finally this work will take a quick look at the first experimental evidences for Efimov states that were found since 2006. In the experiment, proper Efimov resonances in measurements of three-body recombination have been observed.
Few-body effects play an important role for the understanding of ultracold quantum gases. We make use of an effective field theory approach to investigate various aspects of universal few-body physics close to a Feshbach resonance. That is the regime where the scattering length is large compared to all other length scales of the system and thus determines the observables. It is also the regime where the Efimov effect whose main characteristic is the occurrence of a sequence of three-body bound states becomes important. At unitarity, i.e., for diverging scattering length, the ratio of binding energies of neighboring states approaches a constant. The existence of those trimers can be captured by a single three-body parameter and influences observables such as recombination rates. Starting from adequate Lagrangians, we derive atom-dimer scattering amplitudes that contain all information of interest about the three-particle systems. With this method, we first investigate the scattering of atoms and dimers at finite temperature in the presence of the Efimov effect. We calculate the dimer relaxation rate and obtain good agreement with available experimental data. Furthermore, heteronuclear mixtures exhibiting large interspecies scattering length are studied in detail. If bosons are the majority species, the Efimov effect occurs in an S-wave channel and we are able to compute three-body recombination and dimer relaxation rates. The results are compared to the outcome of two existing experiments. For mainly fermionic systems, the Efimov effect is only present in an overall P-wave and three-body recombination at threshold vanishes. However, in mixtures of atoms and dimers, scattering cross sections and dimer relaxation show the typical Efimov behavior of log-periodicity. We also consider two-dimensional Bose gases, where the Efimov effect does not occur. For those systems, we derive an equation including the first non-universal corrections and deduce three-body observables such as binding energies and atom-dimer scattering parameters.
We are concerned with few-particle correlations in a fermionic system at finite temperature and density. Within the many-body Green functions formalism the description of correlations is provided by the Dyson equation approach that leads to effective few-body equations. They contain the dominant medium effects, which are self energy corrections and the Pauli blocking. Hence the effective two-body interactions between quasiparticles are momentum/energy-dependent and therefore they can be usesed in the medium modified, momentum space, integral AGS equations for three- and four-body systems. To investigate correlations and clusters beyond four-body, we employ, instead, the configuration space two-variable integro-differential equations (IDEA) for AAA-body bound systems which are based on Hyperspherical Harmonics and the Faddeev decomposition of the wave function in two-body amplitudes. This requires the transformation of the energy dependent two-body interactions to equivalent local, energy independent, ones. To achieve this we use inverse scattering techniques the resulting interactions being, on-- and (to all practical purposes) off--shell equivalent to the energy dependent potentials. In this way we obtain binding energy results for the 2--, 3--, 4--, and 16--particle in a medium at a finite temperature and various densities. Several aspects of the problem are discussed and the behavior of the potential surfaces obtained in the extreme adiabatic approximation, below and above the Mott transition, is investigated.
A formulation of the energy eigenvalue problem for a many-particle system is presented in several alternative forms and, following Brueckner and his collaborators, the energy of the system is expressed in terms of solutions to the two-body problem. This work bears considerable resemblance to Brueckner's "linked cluster" expansion. The present expansion is derived with the aid of an ordering operator and involves a sequence of "nearest neighbor" interactions. The particles tend to form "chains," the sequence of approximations involving "two-particle chains," "three-particle chains," etc. The rapidity of convergence of the method depends on the particle density and "temperature" and on the nature of the force between particles. The method lends itself readily to use in statistical mechanics. Some applications to the calculation of the equation of state of a gas are included.
This paper studies the problem of a quantum gas of particles whose interparticle potential is attractive enough to support stable bound clusters. The approach to this problem is to employ the time-dependent two-Hilbert-space scattering theory---first developed by Faddeev to treat the three-body problem. From this theory the asymptotic completeness theorem is taken and utilized to give a new definition of the canonical partition function suitable for describing states that are a mixture of free particles and bound clusters. A multispecies cluster expansion is obtained for the grand partition function. The theory of few-particle time delay is employed to determine the values of these cluster integrals. From these solutions are obtained the multidensity virial equation of state, the internal energy, and a density expansion of the chemical mass-action law that gives the effect of interactions between particles and clusters. All of these solutions exhibit the fact that the only aspect of the collision process that has any effect on the macroscopic behavior of the system is the time delay. From another perspective this problem and the solutions obtained represent a specific nontrivial realization of the SSS-matrix description of statistical mechanics put forth by Dashen.