Editing Negative temperature (section)

=== Motional degrees of freedom ===
Negative temperatures have also been achieved in motional [[Degrees of freedom (physics and chemistry)|degrees of freedom]]. Using an [[optical lattice]], upper bounds were placed on the kinetic energy, interaction energy and potential energy of cold [[potassium-39]] 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 model|Bose–Hubbard Hamiltonian]] from {{math|''Ĥ'' → −''Ĥ''}}. Performing this transformation adiabatically while keeping the atoms in the [[Mott insulator]] regime, it is possible to go from a low entropy positive temperature state to a low entropy negative temperature state. In the negative temperature state, the atoms macroscopically occupy the maximum momentum state of the lattice. The negative temperature ensembles equilibrated and showed long lifetimes in an anti-trapping harmonic potential.<ref name=":0">{{Cite journal | last1 = Braun | first1 = S. | last2 = Ronzheimer | first2 = J. P. | last3 = Schreiber | first3 = M. | last4 = Hodgman | first4 = S. S. | last5 = Rom | first5 = T. | last6 = Bloch | first6 = I. | last7 = Schneider | first7 = U. | url = https://www.mpg.de/research/negative-absolute-temperature | doi = 10.1126/science.1227831 | title = Negative Absolute Temperature for Motional Degrees of Freedom | journal = Science | volume = 339 | issue = 6115 | pages = 52–55 | year = 2013 | pmid =  23288533|arxiv = 1211.0545 |bibcode = 2013Sci...339...52B | s2cid = 8207974 }}</ref>