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# VARIATIONAL ADIABATIC HYPERSPHERICAL TREATMENT OF FEW AND MANY PARTICLES

This dissertation works out novel methodologies for approximately computing the adiabatic hyperspherical potential energies of more than 2 particles in terms of the variational principle. The first part uses the root-mean-square radius of a cloud of many atoms as a natural collective hyperradial coordinate, calculating the condensate energy by holding that radius fixed at discrete values. A new step beyond the constant-ansatz "K-Harmonic" approximation (in terms of orbital solutions of the mean-field equation) leads to great improvements in minimization of the ground condensate energy. Careful analysis using linear combinations of orbitals reveals a reduction in macroscopic potential barrier for attractive condensates as a partial description of many-body correlations in these ultracold atomic systems. The second part of this dissertation constructs a new ansatz that explicitly takes two-body correlation into account. Benchmark calculations are performed in comparison with well-known theory of Efimov physics for 3 bosons. New results on the variational potential energies of more than 3 bosons are obtained, revealing characteristic features of a deep minimum and a barrier at small values of hyperradii that are intimately connected to the lowest bound states of N bosons. Comparisons with numerical diagonalization for N = 4 show that the variational potential accounts for the local minimum describing the lowest 4-body bound state but always diabatically converges to the scattering threshold (regardless of the presence of bound trimer thresholds). New results on the asymptotic behavior of unitary Bose gas are obtained for up to 10 particles.

## Funding

### National Science Foundation Grant No. PHY-1912350

## History

## Degree Type

- Doctor of Philosophy

## Department

- Physics and Astronomy

## Campus location

- West Lafayette

## Advisor/Supervisor/Committee Chair

Chris H. Greene## Additional Committee Member 2

Chen-Lung Hung## Additional Committee Member 3

Francis Robicheaux## Additional Committee Member 4

Andrew Weiner## Usage metrics

## Categories

## Keywords

hyperradiushypersphericalBose-Einstein condensateGross-Pitaevskii equationadiabaticEfimovbright solitonvariational principlepseudopotentialBogoliubov excitationsorbitalconfiguration interactionMonte Carloquasirandomscattering lengthFeshbach resonancebosonThomas-FermiSteepest Descent Methodunitary Bose gasAtomic PhysicsAtomic and Molecular PhysicsComputational PhysicsDegenerate Quantum Gases and Atom OpticsQuantum Mechanics

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