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Journal of Physics B Atomic Molecular and Optical Physics

Universality of the three-boson system close to a Feshbach resonance

S. Jonsell

Published Mar 24, 2004• 17 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/37/7/069

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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.
We calculate energy levels of two and three bosons trapped in a harmonic oscillator potential with oscillator length aosc{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}aosc​. The atoms are assumed to interact through a short-range potential with a scattering length aaa, and the short-distance behavior of the three-body wave function is characterized by a parameter $\ensuremath{\theta}$. For large positive a/aosca/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}a/aosc​, the energies of states that, in the absence of the trap, correspond to three free atoms approach values independent of aaa and $\ensuremath{\theta}$. For other states, the $\ensuremath{\theta}$ dependence of the energy is strong, but the energy is independent of aaa for $|a/{a}_{\mathrm{o}\mathrm{s}\mathrm{c}}|\ensuremath{\gg}1$.
We consider either 3 spinless bosons or 3 equal mass spin-1/2 fermions, interacting via a short-range potential of infinite scattering length and trapped in an isotropic harmonic potential. For a zero-range model, we obtain analytically the exact spectrum and eigenfunctions: for fermions all the states are universal; for bosons there is a coexistence of decoupled universal and efimovian states. All the universal states, even the bosonic ones, have a tiny 3-body loss rate. For a finite range model, we numerically find for bosons a coupling between zero angular momentum universal and efimovian states; the coupling is so weak that, for realistic values of the interaction range, these bosonic universal states remain long-lived and observable.
We consider a system of three identical bosons near a Feshbach resonance in the universal regime with large scattering length usually described by model-independent zero-range potentials. We employ the adiabatic hyperspherical approximation and derive the rigorous large-distance equation for the adiabatic potential for finite-range interactions. The effective range correction to the zero-range approximation must be supplemented by a new term of the same order. The nonadiabatic term can be decisive. Efimov physics is always confined to the range between effective range and scattering length. The analytical results agree with numerical calculations for realistic potentials.
Three-boson Efimov physics is well known in the bound-state regime, but far less in the three-particle continuum at negative two-particle scattering length where Efimov states evolve into resonances. They are studied solving rigorous three-particle scattering equations for transition operators in the momentum space. The dependence of the three-boson resonance energy and width on the two-boson scattering length is studied with several force models. The universal limit is determined numerically considering highly excited states; simple parametrizations for the resonance energy and width in terms of the scattering length are established. Decreasing the attraction, the resonances rise not much above the threshold but broaden rapidly and become physically unobservable, evolving into subthreshold resonances. Finite-range effects are studied and related to those in the bound-state regime.
We investigate finite-range effects in systems with three identical bosons. We calculate recombination rates and bound state spectra using two different finite-range models that have been used recently to describe the physics of cold atomic gases near Feshbach resonances where the scattering length is large. The models are built on contact potentials which take into account finite-range effects; one is a two-channel model and the other is an effective range expansion model implemented through the boundary condition on the three-body wavefunction when two of the particles are at the same point in space. We compare the results with the results of the ubiquitous single-parameter zero-range model where only the scattering length is taken into account. Both finite-range models predict variations of the well-known geometric scaling factor 22.7 that arises in Efimov physics. The threshold value at negative scattering length for creation of a bound trimer moves to higher or lower values depending on the sign of the effective range compared to the location of the threshold for the single-parameter zero-range model. Large effective ranges, corresponding to narrow resonances, are needed for the reduction to have any noticeable effect.
Four identical spinless bosons with purely attractive two-body short-range interactions and repulsive three-body interactions under external spherically symmetric harmonic confinement are considered. The repulsive three-body potential prevents the formation of deeply bound states with molecular character. The low-energy spectrum with vanishing orbital angular momentum and positive parity for infinitely large two-body sss-wave scattering length is analyzed in detail. Using the three-body contact, states are classified as universal, quasiuniversal, or strongly nonuniversal. Connections with the zero-range interaction model are discussed. The energy spectrum is mapped out as a function of the two-body sss-wave scattering length as,as>0{a}_{s}, {a}_{s}>0as​,as​>0. In the weakly to medium-strongly interacting regime, one of the states approaches the energy obtained for a hard-core interaction model. This state is identified as the energetically lowest-lying ``BEC state.'' Structural properties are also presented.