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arXiv:cond-mat/0403208 · Journal of Physics B Atomic Molecular and Optical Physics

Two-component boson systems with hyperspherical coordinates

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

Published May 8, 2004• 2 citations
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Abstract

The effective potential is computed for two boson systems in one trap as a function of their two individual hyperadii and the distance between their centers. Zero-range interactions are used and only relative s-states are included. Existence and properties of minima are investigated as a function of these three collective coordinates. For sufficiently strong repulsion stable structures are found at a finite distance between the centers. The relative center of masses motion corresponds to the lowest normal mode. The highest normal mode is essentially the breathing mode where the subsystems vibrate by scaling their radii in phase. The intermediate normal mode corresponds to isovector motion where the subsystems vibrate by scaling their radii in opposite phase. Stability conditions are established as substantially more restrictive than in mean-field computations.

DOI
10.1088/0953-4075/37/10/011

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The effective potential is computed for two boson systems in one trap as a function of their two individual hyperadii and the distance between their centers. Zero-range interactions are used and only relative s-states are included. Existence and properties of minima are investigated as a function of these three collective coordinates. For sufficiently strong repulsion stable structures are found at a finite distance between the centers. The relative center of masses motion corresponds to the lowest normal mode. The highest normal mode is essentially the breathing mode where the subsystems vibrate by scaling their radii in phase. The intermediate normal mode corresponds to isovector motion where the subsystems vibrate by scaling their radii in opposite phase. Stability conditions are established as substantially more restrictive than in mean-field computations.
The effective potential is computed for two boson systems in one trap as a function of their two individual hyperadii and the distance between their centers. Zero-range interactions are used and only relative s-states are included. Existence and properties of minima are investigated as a function of these three collective coordinates. For sufficiently strong repulsion stable structures are found at a finite distance between the centers. The relative center of masses motion corresponds to the lowest normal mode. The highest normal mode is essentially the breathing mode where the subsystems vibrate by scaling their radii in phase. The intermediate normal mode corresponds to isovector motion where the subsystems vibrate by scaling their radii in opposite phase. Stability conditions are established as substantially more restrictive than in mean-field computations.
The lowest adiabatic potential expressed in hyperspherical coordinates is estimated for two boson systems in an external harmonic trap. Corresponding conditions for stability are investigated and the related structures are extracted for zero-range interactions. Strong repulsion between non-identical particles leads to two features, respectively, when identical particles attract or repel each other. For repulsion stable structures arise with displaced center of masses. For attraction the mean-field stability region is restricted due to motion of the center of masses.
We investigate two-component boson systems by using their two individual average sizes and the distance between their centers as coordinates. We use zero-range interactions and include only relative sss waves. An effective potential is computed and discussed as a function of particle numbers and interaction strengths. Efimov conditions of two very large scattering lengths can arise if one component is a binary molecule. The corresponding atom-molecule interaction can then exhibit chaotic behavior. We present static and dynamic properties of an atom-molecule two-component system. We discuss the possible atom-molecule oscillations when all atoms are identical. The atomic number is then used as a collective coordinate and the cranking mass is computed. The collective oscillation frequency is estimated and compared with the trap frequency and the three-body recombination rate.
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.