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Journal of Physics B Atomic Molecular and Optical Physics

The zero-range approximation applied to theN-boson problem

Takaaki Sogo, O Sørensen, A. S. Jensen, D. V. Fedorov

Published Mar 23, 2005• 10 citations
Publisher

Abstract

We use the hyperspherical adiabatic expansion method to formulate the angular Faddeev-like equation for N identical bosons including effects of two-body correlations. Eigenvalues and eigenfunctions determine the effective potentials in the coupled set of radial equations. The zero-range approximation is implemented on the free solutions as a boundary condition obtained by using the effective-range expansion of the two-body phase shift. This results in a new angular eigenvalue equation expressed as a transcendental equation in the index of the Jacobi functions corresponding to solutions for non-interacting particles. The collapse for attractive potentials is avoided by including scattering length, effective range and shape-parameter terms in the effective-range expansion of the phase shift. Correspondingly the hyperradial dependence in the eigenvalue equation is either a polynomial of first, second, or fourth order. Two-body correlations are included. The arithmetic eigenvalue equation allows derivation of analytic properties such as for example simple scaling relations for the particle number dependence. Excellent agreement with finite-range numerical results is obtained. The couplings between adiabatic channels are discussed. Including only the lowest adiabatic potential is shown to be sufficient for most applications on dilute systems.

DOI
10.1088/0953-4075/38/7/021

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We use the hyperspherical adiabatic expansion method to formulate the angular Faddeev-like equation for N identical bosons including effects of two-body correlations. Eigenvalues and eigenfunctions determine the effective potentials in the coupled set of radial equations. The zero-range approximation is implemented on the free solutions as a boundary condition obtained by using the effective-range expansion of the two-body phase shift. This results in a new angular eigenvalue equation expressed as a transcendental equation in the index of the Jacobi functions corresponding to solutions for non-interacting particles. The collapse for attractive potentials is avoided by including scattering length, effective range and shape-parameter terms in the effective-range expansion of the phase shift. Correspondingly the hyperradial dependence in the eigenvalue equation is either a polynomial of first, second, or fourth order. Two-body correlations are included. The arithmetic eigenvalue equation allows derivation of analytic properties such as for example simple scaling relations for the particle number dependence. Excellent agreement with finite-range numerical results is obtained. The couplings between adiabatic channels are discussed. Including only the lowest adiabatic potential is shown to be sufficient for most applications on dilute systems.
We study two-body correlations for N identical bosons by use of the hyperspherical adiabatic-expansion method. We use the zero-range interaction and derive a transcendental equation determining the key ingredient of the hyperradial potential. The necessary renormalization is for both repulsive and attractive interactions achieved with an effective range expansion of the two-body phase shifts. Our solutions including correlations provide the properties of Bose-Einstein condensates exemplified by stability conditions as established by mean-field Gross-Pitaevskii calculations. The many-body Efimov states are unavoidable for large scattering lengths.
We consider the three-boson problem with $\ensuremath{\delta}$-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle off a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired SSS-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum-space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the 1+21+21+2 scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pairwise interactions are short range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.
We investigate systems of identical bosons with the focus on two-body correlations. We use the hyperspherical adiabatic method and a decomposition of the wave function in two-body amplitudes. An analytic parametrization is used for the adiabatic effective radial potential. We discuss the structure of a condensate for arbitrary scattering length. Stability and time scales for various decay processes are estimated. The previously predicted Efimov-like states are found to be very narrow. We discuss the validity conditions and formal connections between the zero- and finite-range mean-field approximations, Faddeev-Yakubovskii formulation, Jastrow ansatz, and the present method. We compare numerical results from present work with mean-field calculations and discuss qualitatively the connection with measurements.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.
We formulate a method to study two-body correlations in a system of N identical bosons interacting via central two-body potentials. We use the adiabatic hyperspherical approach and assume a Faddeev-like decomposition of the wave function. For a fixed hyperradius we derive variationally an optimal integro-differential equation for the hyperangular eigenvalue and wave function. This equation reduces substantially by assuming the interaction range much smaller than the size of the N-body system. At most, one-dimensional integrals then remain. We view a Bose-Einstein condensate pictorially as a structure in the landscape of the potential given as a function of the one-dimensional hyperradial coordinate. The quantum states of the condensate can be located in one of the two potential minima. We derive and discuss properties of the solutions and illustrate with numerical results. The correlations lower the interaction energy substantially. The new multibody Efimov states are solutions independent of details of the two-body potential. We compare with mean-field results and available experimental data.
We investigate systems of identical bosons with the focus on two-body correlations and attractive finite-range potentials. We use a hyperspherical adiabatic method and apply a Faddeev type of decomposition of the wave function. We discuss the structure of a condensate as a function of particle number and scattering length. We establish universal scaling relations for the critical effective radial potentials for distances where the average distance between particle pairs is larger than the interaction range. The correlations in the wave function restore the large-distance mean-field behavior with the correct two-body interaction. We discuss various processes limiting the stability of condensates. With correlations we confirm that macroscopic tunneling dominates when the trap length is about half of the particle number times the scattering length.