Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model

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Science  16 Sep 2016:
Vol. 353, Issue 6305, pp. 1253-1256
DOI: 10.1126/science.aag1430
  • Fig. 1 Experimental technique for measuring spin correlations.

    (A) After loading the atoms into an optical lattice, we use a spin-removal technique to map the spin correlations onto charge correlations, which can then be detected using site-resolved imaging. The two spin states are denoted by green and orange balls. By driving cycling optical transitions for either spin state with the spin-removal beam, we can eject one spin state from the trap. We can then combine charge correlations measured in images where we remove each spin state and where no removal is performed to compute the local spin correlation (24). (B) A typical image where no atoms are removed. (C) A typical image with one of the spins removed. Atoms in doubly occupied sites are removed in both the spin-removal and imaging procedures as a result of light-assisted collisions.

  • Fig. 2 Local observation of density and spin correlations.

    (A to C) Spatial maps and azimuthally averaged profiles (mirrored about Embedded Image and corrected for ellipticity) of the detected density, nearest-neighbor correlator and diagonal next-nearest neighbor correlator for a cold (top) and hot (bottom) cloud. A combined fit determines the temperature T and chemical potential Embedded Image (solid lines). (D) Green symbols show the nearest-neighbor correlator in the center of the cloud for samples prepared at different temperatures. Listed are the values of Embedded Image from fits of a numerical linked-cluster expansion to the radial profile and Embedded Image obtained by comparing the central correlator value to a quantum Monte Carlo calculation at half-filling (solid line) (22). For the coldest data in (D) and (E) (squares), the NLCE theory error is too large for a fit, and we report only the QMC result. (E) An exponential fit to the correlator in the center of the cloud versus d allows us to extract the correlation length for data sets at three different temperatures, giving 0.24(9), 0.39(2), and 0.51(4) sites for decreasing temperature. The asterisk denotes the nearest-neighbor correlator value from the QMC calculation in (D) as Embedded Image. Error bars on Embedded Image and Embedded Image are standard errors after averaging at least 20 sets of combined correlation maps and averaging azimuthally (24). All data shown are at Embedded Image. Horizontal errors in (D) are fit errors.

  • Fig. 3 Thermalization dynamics during lattice loading.

    (A) (Upper) Detected density. (Lower) The nearest- and diagonal next-nearest-neighbor spin correlator. Both are measured at Embedded Image as a function of lattice loading time Embedded Image, where Embedded Image is the radius where Embedded Image is maximized. (B) Computed Embedded Image of simultaneous fits of the density and nearest-neighbor correlator profiles to NLCE data. A value Embedded Image indicates a good fit, consistent with our model, which assumes thermal equilibrium. Embedded Image settles to approximately 1 at a lattice loading time of 20 ms, indicated by the shaded region. (C) Sample profile fits for three different loading times.

  • Fig. 4 Varying the interaction strength.

    Nearest-neighbor correlator at half-filling for varying scattering length. The top y axis gives computed values of Embedded Image for each scattering length (24). The solid lines are isothermal theory curves from the NLCE theory.

Supplementary Materials

  • Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model

    Maxwell F. Parsons, Anton Mazurenko, Christie S. Chiu, Geoffrey Ji, Daniel Greif, Markus Greiner

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