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Strongly Interacting Quantum Gases in One-Dimensional Traps

Under the second-order degenerate perturbation theory, we show that the physics of $N$ particles with arbitrary spin confined in a one dimensional trap in the strongly interacting regime can be described by super-exchange interaction. An effective spin-chain Hamiltonian (non-translational-invariant Sutherland model) can be constructed from this procedure. For spin-1/2 particles, this model reduces to the non-translational-invariant Heisenberg model, where a transition between Heisenberg anti-ferromagnetic (AFM) and ferromagnetic (FM) states is expected to occur when the interaction strength is tuned from the strongly repulsive to the strongly attractive limit. We show that the FM and the AFM states can be distinguished in two different methods: the first is based on their distinct response to a spin-dependent magnetic gradient, and the second is based on their distinct momentum distribution. We confirm the validity of the spin-chain model by comparison with results obtained from several unbiased techniques