The scattering of a few particles with short-range interactions at low energies
Author(s)
Zhu, Shangguo
Advisor(s)
Tan, Shina
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Abstract
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, which is responsible for the nonuniversal effects in dilute Bose-Einstein condensates. We derive an analytical formula of the three-body scattering hypervolumes for weak interactions. When the interaction supports two-body bound states, the three-body scattering hypervolume gains a negative imaginary part, which is directly related to the rate constant for three-body recombinations. We develop a numerical method to calculate the three-body scattering hypervolumes for some model potentials with variable strengths. For real atoms with van der Waals potential, the three-body scattering hypervolumes are much harder to compute, because the three-body wave function is highly oscillatory at smaller inter-atomic distances. However, they may be extractable from precision data such as the collective-mode frequencies of trapped Bose-Einstein condensates. In many numerical simulations, the system being simulated is put into a large but finite volume, such as a large periodic box of size $L$. To extract the low-energy scattering properties of two particles in infinite space from such simulations, L\"uscher's formula must be used. In the second part of this thesis, we generalize L\"uscher's formula to $d$ spatial dimensions. We obtain its $s$-wave and $p$-wave approximations and the systematic expansions of the energies of the low-lying states. At a $s$-wave resonance in $d\ge4$ dimensions, we identify two low-lying states close to the threshold with nearly opposite energies, $E\sim \pm 1/L^{d/2}$ when $d\ge 5$, or $E \sim \pm 1/L^2 \sqrt{\ln L}$ when $d= 4$. These calculations provide important insights to the physics of three particles at a three-body resonance in finite volumes in two or three dimensions. Three-body resonances are important phenomena not yet completely explored in the experiments on cold atoms.
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Date
2018-11-08
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Dissertation