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Motional Degrees of Freedom
Negative temperatures have also been achieved in motional degrees of freedom. By using optical lattices], upper bounds were placed on the kinetic energy, interaction energy and potential energy of cold atoms. This was done by tuning the interactions of the atoms from repulsive to attractive using a [[Feshbach resonance and changing the overall harmonic potential from trapping to anti-trapping, thus transforming the Bose-Hubbard Hamiltonian from $$\hat{H} \rightarrow -\hat{H}$$, leading to the evolution of a negative temperature motional state.