Tonks-Girardeau gas in different potential traps by exact diagonalization

Abstract

We computed the second-quantized many-mode Hamiltonian for spinless ultracold bosons trapped in di erent one-dimensional potentials and diagonalized it for a small number of atoms. We observed the transition from the non-interacting gas to the Tonks-Girardeau gas, i.e. the fermionization process, as we increase the strength of the interactions between the atoms. We computed the energy spectrum, the one-body density matrix, the two-body correlations, the momentum distribution, the natural orbitals and the occupations for di erent interaction strengths. We compared the fermionization in di erent potentials with special attention to the largest occupation of the natural orbits, the build up of quantum many-body correlations between di erent parts of the system, and the energy of the ground and excited states.