Exact collective occupancies of the Moshinsky model in two-dimensional geometry

In this paper, we investigate the ground state of N bosonic atoms confined in a two-dimensional isotropic harmonic trap, where the atoms interact via a harmonic potential. We derive an exact diagonal representation of the first-order reduced density matrix in polar coordinates, in which the angular components of the natural orbitals are eigenstates of the angular momentum operator. Furthermore, we present an exact expression for the collective occupancy of the natural orbitals with angular momentum l, quantifying the fraction of particles carrying that angular momentum. The present study explores how the dependence of collective occupancy relies on angular momentum l and the control parameters of the system. Building on these findings, we examine boson fragmentation into components with different l and reveal a unique feature of the system: the natural orbitals contributing to the correlations are uniformly distributed across all significant l components.