Harmonically trapped Bose gases: Weakly-interacting regime and beyond

There exist few exactly solvable problems in physics, but they tend to serve as useful benchmarks and starting points for understanding more complex phenomena. We study dilute harmonically trapped few-boson systems with two-body s-wave interactions and present highly accurate results for the energetics and the condensate fraction. Using the exact two-body wave function, we calculate the condensate fraction by diagonalizing the one-body density matrix as a function of the two-body s-wave scattering length $a_s$. We find that the condensate fraction exhibits an interesting oscillatory behavior. In the weakly-interacting regime, we analytically expand the condensate fraction in terms of $a_s$. For the weakly-interacting three-body system, our numerical results for the energy agree with the analytical predictions of Ref. [1] and, furthermore, allow for the extraction of higher order corrections. In addition, we determine the condensate fraction of weakly- and strongly-interacting three-boson systems. Extensions to larger systems and an interpretation in terms of effective $N$-body interactions are presented.\\[4pt] [1] P. R. Johnson, E. Tiesinga, J. V. Porto and C. J. Williams, New J. Phys 11, 093022 (2009).