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Fig. 1 Negative absolute temperature in optical lattices. (A) Sketch of entropy as a function of energy in a canonical ensemble possessing both lower (Emin) and upper (Emax) energy bounds. (Insets) Sample occupation distributions of single-particle states for positive, infinite, and negative temperature, assuming a weakly interacting ensemble. (B) Energy bounds of the three terms of the 2D Bose-Hubbard Hamiltonian: kinetic (Ekin), interaction (Eint), and potential (Epot) energy. (C) Measured momentum distributions (TOF images) for positive (left) and negative (right) temperature states. Both images are averages of about 20 shots; both optical densities (OD) are individually scaled. The contour plots below show the tight-binding dispersion relation; momenta with large occupation are highlighted. The white square in the center indicates the first Brillouin zone.
Fig. 2 Experimental sequence and TOF images. (A) Top to bottom: lattice depth, horizontal trap frequency, and scattering length as a function of time. Blue indicates the sequence for positive, red for negative temperature of the final state. (B) TOF images of the atomic cloud at various times t in the sequence. Blue borders indicate positive, red negative temperatures. The initial picture in a shallow lattice at t = 6.8 ms is taken once for a scattering length of a = 309(5) a0 (top) as in the sequence, and once for a = 33(1) a0 (bottom; OD rescaled by a factor of 0.25), comparable to the final images. All images are averages of about 20 individual shots. See also Fig. 1C.
Fig. 3 Occupation distributions. The occupation of the kinetic energies within the first Brillouin zone is plotted for the final positive (blue) and negative (red) temperature states. Points show experimental data extracted from band-mapped pictures. Solid lines are fits to a noninteracting Bose-Einstein distribution assuming a homogeneous system. (Insets) Top row: Symmetrized positive (left) and negative (right) temperature images of the quasimomentum distribution in the horizontal plane. Bottom row: Fitted distributions for the two cases. All distributions are broadened by the in situ cloud size (9).
Fig. 4 Stability of the positive (blue) and negative (red) temperature states. Main figure: Visibility V = (nb − nr)/(nb + nr) extracted from the atom numbers in the black (nb) and red (nr) boxes (indicated in the TOF images) plotted versus hold time in the final state for various horizontal trap frequencies. Dark red, |ωhor|/2π = 43(1) Hz anti-trapping; medium red, 22(3) Hz anti-trapping; light red, 42(3) Hz trapping; blue, 45(3) Hz trapping. (Inset) Coherence lifetimes τ extracted from exponential fits (solid lines in main figure). The statistical error bars from the fits are smaller than the data points. The color scale of the images is identical to Fig. 2B (see also fig. S3).
