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Physical Review C

Three-boson system with absorptive short range potential

Antonio Eduardo Assis Amorim, Lauro Tomio, T. Frederico

Published Dec 1, 1992• 14 citations
Publisher

Abstract

We study the properties of the three-boson system with absorption, through a short range interaction in the limit where the range reduces to zero. We derive an analytic expression for the three-boson width that relates it to the real part of the three-boson energy, two-boson binding energy, and decay constant. One of the characteristics of this expression is that, in this limit, the ratio between the width and the three-boson binding energy is proportional to the range.

DOI
10.1103/physrevc.46.2224

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We study the properties of the three-boson system with absorption, through a short range interaction in the limit where the range reduces to zero. We derive an analytic expression for the three-boson width that relates it to the real part of the three-boson energy, two-boson binding energy, and decay constant. One of the characteristics of this expression is that, in this limit, the ratio between the width and the three-boson binding energy is proportional to the range.
Universal low-energy properties are studied for three identical bosons confined in two dimensions. The short-range pairwise interaction in the low-energy limit is described by means of the boundary condition model. The wave function is expanded in a set of eigenfunctions on the hypersphere, and the system of hyperradial equations is used to obtain analytical and numerical results. Within the framework of this method, exact analytical expressions are derived for the eigenpotentials and the coupling terms of hyperradial equations. The derivation of the coupling terms is generally applicable to a variety of three-body problems provided the interaction is described by the boundary condition model. The asymptotic form of the total wave function at a small and a large hyperradius $\ensuremath{\rho}$ is studied, and the universal logarithmic dependence $\ensuremath{\sim}{\mathrm{ln}}^{3}\phantom{\rule{0.2em}{0ex}}\ensuremath{\rho}$ in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the (2+1)(2+1)(2+1)-scattering length are calculated.
We estimate the ratio R=g3/g2{R=g}_{3}{/g}_{2}R=g3​/g2​ of the critical coupling constants g2{g}_{2}g2​ and g3{g}_{3}g3​ that are required to achieve binding of two or three bosons, respectively, with a short-range interaction, and examine how this ratio depends on the shape of the potential. Simple monotonous potentials give $R\ensuremath{\simeq}0.8.$ A wide repulsive core pushes this ratio close to R=1.R=1.R=1. On the other hand, for an attractive well protected by an external repulsive barrier, the ratio approaches the rigorous lower bound R=2/3.R=2/3.R=2/3. We also present results for N=4N=4N=4 bosons, sketch the extension to N>4,N>4,N>4, and discuss various consequences.
Two-, three-, and four-boson systems are studied close to the unitary limit using potential models constructed to reproduce the minimal information given by the two-body scattering length aaa and the two-body binding energy or virtual-state energy E2{E}_{2}E2​. The particular path used to reach the unitary limit is given by varying the potential strength. In this way the energy spectrum in the three- and four-boson systems is computed. The lowest-energy states show finite-range effects absorbed in the construction of level functions that can be used to study real systems. Higher-energy levels are free from finite-range effects, therefore the corresponding level functions tend to the zero-range universal function. Using this property, a zero-range equation for the four-boson system is proposed and the four-boson universal function is computed.
A model for the system of three identical bosons with interactions containing a Feshbach resonance is developed. The model assumes short-range interactions, but includes the energy dependence of the interaction due to the finite width of the resonance. The model is applied to the system of three bosons trapped in a harmonic potential. In particular we study the energy in the limit of infinite scattering length. In this limit the energy does not depend on the scattering length, but it does depend on the other parameters of the resonance.
Close upper and lowei bounds are found on the binding energy of a system of N identical bosons interacting in one dimension by an attractive square-well pair potential. In the limit in which the potential has the shape of a δ function the bounds are found to be consistent with McGuire's exact solution. In three dimensions the limit of vanishingly small range, in which the two-particle binding energy is kept constant, gives infinite binding to the N -particle system, in agreement with Thomas's theorem Consideration of other limiting Proc.esses eliminates an apparent contradiction between Thomas and Svartholm.
The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume. Its imaginary part is directly related to the three-body recombination rate in a quantum gas consisting of such particles. We derive an analytical formula of it for weak interactions, and perform its first numerical calculations for bosons with a variable nonzero-range potential. For attractive interactions, we identify several three-body resonances at which the three-body scattering hypervolume becomes divergent or anomalously large.