## Physics Graduate Theses & Dissertations

Spring 1-1-2019

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

John L. Bohn

Eric A. Cornell

Jose P. D' Incao

Victor Gurarie

Carl Lineberger

#### Abstract

In this work, we study the ground state properties of a system of $N$ harmonically trapped bosons of mass $m$ interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained variational (LOCV) method, we are able to semi-quantitatively attain Bose-Einstein condensate ground state energies even for gases with infinite scattering length. In the large particle number limit, our method provides analytical estimates for the energy per particle $E_0/N \approx 2.5 N^{1/3} \hbar \omega$ and two-body contact $C_2/N \approx 16 N^{1/6}\sqrt{m\omega/\hbar}$ for a Bose gas on resonance, where $\omega$ is the trap frequency. Further, by considering only two-body correlations, we note that a sudden quench from small to large scattering lengths leads to out-of-equilibrium resonant BEC. As an alternative, we propose a two-step scheme that involves an intermediate scattering length, between $0$ and $\infty$, which serves to maximize the transfer probability of $N$ bosons in a harmonic trap with frequency $\omega$ to the resonant state. We find that the intermediate scattering length should be $a\approx3.16N^{-2/3}\sqrt{\hbar/(m\omega)}$, and that it produces an optimum transition probability of $1.03N^{-1/6}$.

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