Natural orbitals and Bose-Einstein condensates in traps: A diffusion Monte Carlo analysis

We investigate the properties of hard-core bosons in harmonic traps over a wide range of densities. Bose-Einstein condensation is formulated using the one-body density matrix (OBDM) which is equally valid at low and high densities. The OBDM is calculated using diffusion Monte Carlo methods and it is diagonalized to obtain the ''natural'' single-particle orbitals and their occupation, including the condensate fraction. At low boson density, na{sup 3}<10{sup -5}, where n=N/V and a is the hard-core diameter, the condensate is localized at the center of the trap. As na{sup 3} increases, the condensate moves to the edges of the trap. At high density, it is localized at the edges of the trap. At na{sup 3}{<=}10{sup -4}, the Gross-Pitaevskii theory of the condensate describes the whole system within 1%. At na{sup 3}{approx_equal}10{sup -3}, corrections are 3% to the Grass-Pitaevskii energy but 30% to the Bogoliubov prediction of the condensate depletion. At na{sup 3}(greater-or-similar sign)10{sup -2}, mean-field theory fails. At na{sup 3}(greater-or-similar sign)0.1, the bosons behave more like a liquid {sup 4}He droplet than a trapped boson gas.