Negative Absolute Temperature for Motional Degrees of Freedom

Absolute temperature, the fundamental temperature scale in thermodynamics, is usually bound to be positive. Under special conditions, however, negative temperatures - where high-energy states are more occupied than low-energy states - are also possible. So far, such states have been demonstrated in localized systems with finite, discrete spectra. Here, we were able to prepare a negative temperature state for motional degrees of freedom. By tailoring the Bose-Hubbard Hamiltonian we created an attractively interacting ensemble of ultracold bosons at negative temperature that is stable against collapse for arbitrary atom numbers. The quasi-momentum distribution develops sharp peaks at the upper band edge, revealing thermal equilibrium and bosonic coherence over several lattice sites. Negative temperatures imply negative pressures and open up new parameter regimes for cold atoms, enabling fundamentally new many-body states and counterintuitive effects such as Carnot engines above unity efficiency.