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Two Atoms in a Double Well: An Exact Solution

We propose to experimentally realize an odd parity eigenstate $\left\vert b\right\rangle $ of two atoms in the double well. The occupation probability of this state shows evident dependence on the interaction, distinct from the result of two-mode model adopted in the Heidelberg experiment. The tunneling dynamics of two atoms starting from the $NOON$ state with infinite barrier height can be derived from the exactly solved model of $δ$-barrier split double well based on a Bethe ansatz type hypothesis of the wave functions. We find that the single particle tunneling transfer the probabilities between double occupancy and single occupancy of each well.