Ultracold atoms in optical lattices have great potential to contribute to a better understanding of some of the most important issues in many-body physics, such as high-temperature superconductivity1. The Hubbard model—a simplified representation of fermions moving on a periodic lattice—is thought to describe the essential details of copper oxide superconductivity2. This model describes many of the features shared by the copper oxides, including an interaction-driven Mott insulating state and an antiferromagnetic (AFM) state. Optical lattices filled with a two-spin-component Fermi gas of ultracold atoms can faithfully realize the Hubbard model with readily tunable parameters, and thus provide a platform for the systematic exploration of its phase diagram3,4. Realization of strongly correlated phases, however, has been hindered by the need to cool the atoms to temperatures as low as the magnetic exchange energy, and also by the lack of reliable thermometry5. Here we demonstrate spin-sensitive Bragg scattering of light to measure AFM spin correlations in a realization of the three-dimensional Hubbard model at temperatures down to 1.4 times that of the AFM phase transition. This temperature regime is beyond the range of validity of a simple high-temperature series expansion, which brings our experiment close to the limit of the capabilities of current numerical techniques, particularly at metallic densities. We reach these low temperatures using a compensated optical lattice technique6, in which the confinement of each lattice beam is compensated by a blue-detuned laser beam. The temperature of the atoms in the lattice is deduced by comparing the light scattering to determinant quantum Monte Carlo simulations7 and numerical linked-cluster expansion8 calculations. Further refinement of the compensated lattice may produce even lower temperatures which, along with light scattering thermometry, would open avenues for producing and characterizing other novel quantum states of matter, such as the pseudogap regime and correlated metallic states of the two-dimensional Hubbard model.
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This work was supported under ARO grant no. W911NF-13-1-0018 with funds from the DARPA OLE programme, NSF, ONR, the Welch Foundation (grant no. C-1133), and an ARO-MURI grant no. W911NF-14-1-003. T.P. acknowledges support from CNPq, FAPERJ, and the INCT on Quantum Information. R.T.S. acknowledges support from the Office of the President of the University of California.
The authors declare no competing financial interests.
Extended data figures and tables
a, Schematic of the compensated optical lattice set-up. Along each axis, the radial confinement of the lattice is compensated with a repulsive compensation beam which is combined with the lattice beam using a dichroic mirror. The compensation beam co-propagates with the lattice beam but is not retroreflected; instead a dichroic mirror before the retro-reflection mirror is used to direct the compensation beam to a beam dump. b, The local value of the lattice depth v (black line; right-hand y axis) is shown as a function of distance from the centre along a body diagonal of the lattice. Owing to the finite extent of the lattice beams, v varies across the density profile of the cloud. The density n, calculated for U0/t0 = 11.1 at T/t0 = 0.60, is shown (blue line; left-hand y axis). c, The inhomogeneity in v results in spatially varying Hubbard parameters t (blue line; left-hand y axis) and U/t (black line; right-hand y axis).
Atom number N which maximizes Sπ as a function of U0/t0. We control N by adjusting the depth of the dimple trap. Using a linear calibration between the depth of the dimple trap and the final atom number, we obtain the value of N corresponding to the data in Fig. 4. The error bars correspond to the s.e.m. of the dimple depths used in at least 40 in situ and 40 time-of-flight realizations of the experiment, corresponding to the data in Fig. 4. The line is a third-order polynomial fit, which is used to interpolate the value of N for numerical calculations shown in Fig. 4.
Measurement of the round-trip T/TF versus hold time th in a compensated lattice with v0 = 7Er and g0 = 3.7Er. The duration of the loading ramps is not included in th. The scattering length is 326a0, which corresponds to U0/t0 = 12.5. Error bars are the s.e.m. of six independent realizations. The temperature in the dimple trap before loading into the lattice is T/TF = 0.04 ± 0.02.
a, Detected counts (from CCD camera) versus th, measured for momentum transfer Q = π for an in situ sample (Iπ0, green circles) and after decay of the Debye–Waller factor (Iπ∞, blue triangles). For longer hold times, the Bragg-scattered intensity Iπ0 decays to match Iπ∞, reflecting the absence of AFM correlations in a sample at higher T. b, The spin structure factor Sπ corresponding to the scattered intensities shown in a. For these measurements the scattering length is 200a0, corresponding to U0/t0 = 7.7 in a 7Er deep lattice. The compensation is g0 = 4.05Er, different from that used for the data in Fig. 4. The increased compensation requires a larger atom number to realize an n ≈ 1 shell in the cloud. The atom number used here is 2.6 × 105 atoms. The duration of the Bragg probe is 2.7 µs for these data. Error bars in a are the s.e.m. of at least 5 measurements for Iπ∞ and at least 10 measurements for Iπ0. Error bars in b are obtained from the s.e.m. of the measured intensities and equation (2).
a, Detected counts versus U0/t0, measured for momentum transfer Q = π for an in situ sample (Iπ0, green circles), and after decay of the Debye–Waller factor (Iπ∞, blue triangles). As U0/t0 increases we use a larger atom number to optimize the Bragg signal. Iπ∞ and Iπ0 both increase with U0/t0 owing to the larger N, but Iπ0 shows an additional enhancement due to the presence of AFM correlations. b, Detected counts versus U0/t0, measured for momentum transfer Q = θ for an in situ sample (Iθ0, green circles), and after decay of the Debye–Waller factor (Iθ∞, blue triangles). For Q= θ most of the dependence for both the in situ and time-of-flight intensities is due to the changing N. Error bars in both a and b are the s.e.m. of at least 40 measurements. The overall count rate is higher for Q = θ owing to the different collection efficiency and gain settings of the CCD camera.
Overall entropy per particle S/(NkB) as a function of U0/t0 for the calculations at various T/t* shown in Fig. 4 (lines are guides to the eye). For the lowest temperatures, S/(NkB) does not vary significantly over the range of U0/t0 covered by the experiment, justifying the treatment at constant T. A value of S/(NkB) ≈ 0.76 is obtained for the temperature determined from the data in Fig. 4.
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Hart, R., Duarte, P., Yang, T. et al. Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms. Nature 519, 211–214 (2015). https://doi.org/10.1038/nature14223
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