  We show that thermalization of the motion of atoms at negative temperature is
possible in an optical lattice, for conditions that are feasible in current
experiments. We present a method for reversibly inverting the temperature of a
trapped gas. Moreover, a negative-temperature ensemble can be cooled, reducing
abs(T), by evaporation of the lowest-energy particles. This enables the
attainment of the Bose-Einstein condensation phase transition at negative
temperature.
