Numerical study of magneto-optical traps through a hierarchical tree method

Abstract

We approach the problem of $N$ atoms in a magneto-optical trap through a hierarchical tree method, using an algorithm originally developed by Barnes and Hut (BH) in the astrophysical context. Such an algorithm numerically takes care of the particle-particle interaction by controlling the approximation level in a way that offers more physical fidelity than the mean-field treatment and considerably less time consumption ($\ensuremath{\tau}\ensuremath{\sim}N\phantom{\rule{0.3em}{0ex}}{\mathrm{log}}_{10}\phantom{\rule{0.3em}{0ex}}N$ in the hierarchical BH method, in contrast with the $\ensuremath{\tau}\ensuremath{\sim}{N}^{2}$ and $\ensuremath{\tau}\ensuremath{\sim}{N}^{3∕2}$ dependences found in direct and mean-field approaches, respectively). Our results reproduce the experimentally reported single-ring orbital mode for $N\ensuremath{\lesssim}{10}^{6}$ atoms and also find indication of a double-ring structure for $N\ensuremath{\sim}{10}^{7}$, a situation mimicked by a $N={10}^{6}$ system with enhanced radiative force, in agreement with experimental observations. We stress that this high-density regime is not accessed by direct integration of the equations of motion, due to the enormous computing times required, and is not suitably described through mean-field approaches, due to the rather unphysical enhancement of the particle-particle interactions and the presence of a spurious numerical grid dependence.