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arXiv:cond-mat/0306564 · Journal of Physics B Atomic Molecular and Optical Physics

Structure of boson systems beyond the mean field

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

Published Dec 8, 2003• 16 citations
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Abstract

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.

DOI
10.1088/0953-4075/37/1/006

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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.
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.
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 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 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 formulate a method to study two-body correlations in a condensate of N identical bosons. We use the adiabatic hyperspheric approach and assume a Faddeev-like decomposition of the wave function. We derive for a fixed hyper-radius an integro-differential equation for the angular eigenvalue and wave function. We discuss properties of the solutions and illustrate with numerical results. The correlations lower the interaction energy for N=20N=20N=20 by about a factor of 5.