Figure 1

(a) Energy level diagram for Yb. (b) Schematic of the two lattices. Arrows indicate optical polarizations and magnetic field directions with respect to gravity. At the left, the Ramsey pulse sequence is entering: two pulses of duration $t_{1,2}$ separated by dark time $T$ (not to scale). An inset shows a few 2D lattice sites with 0 to 2 atoms per site; two atoms in one of the sites occupy axial motional states $n_1$ and $n_2$. (c) Energy level diagram for two atoms in the rotating frame: three triplet states and one singlet state, with interactions $V$ and $U$, as in Eq. (1).Reuse & Permissions

Figure 2

(a) Collision shift vs excitation fraction, 1D lattice. Blue (red) points show experimental measurements in a vertical (horizontal) lattice with temperature $\mathcal{T}\sim 10\mu \mathrm{K}$ and $⟨\Delta \Omega /\overline{\Omega }{⟩}_{\mathcal{T}}=0.2$. Dashed black line gives an $s$-wave-only fit $⟨U^{eg}{⟩}_{\mathcal{T}}=-2\pi \times 3.0\mathrm{Hz}$) from the mean-field model. Solid red line gives a $p$-wave-only fit with $⟨V^{eg}{⟩}_{\mathcal{T}}=10⟨V^{ee}{⟩}_{\mathcal{T}}=-2\pi \times 2.2\mathrm{Hz}$. Long-dashed green line adds to this a small $s$-wave component ($⟨U^{eg}{⟩}_{\mathcal{T}}=-2\pi \times 1.2\mathrm{Hz}$). (b) Collision shift vs excitation fraction, 2D lattice. Blue (red) points show experimental measurements probing along the vertical (horizontal) direction. Dashed black line is an $s$-wave-only fit with $a_{eg}^{-}\approx -25$ $a_0$ ($a_0$ the Bohr radius); solid red line is a $p$-wave-only fit with $b_{eg}\approx -74$ $a_0$ and $b_{ee}^3=0.1b_{eg}^3$. The long-dashed green line adds to this a small $s$-wave interaction $a_{eg}^{-}=-25$ $a_0$. We emphasize that these are not necessarily accurate determinations of the scattering lengths since their values depend on the spatial distribution of the atoms in the 2D lattice, which is not well characterized. (c) Collision shift vs probe misalignment angle (vertical 1D lattice) for constant excitation fraction 0.12. Using the same parameters as (a), the dashed black line gives an $s$-wave-only fit, solid red line gives a $p$-wave-only fit, and the long-dashed green line has $s$- and $p$-wave terms. In the well-aligned case (0 mrad) there is a residual effective misalignment of $\sim 5\mathrm{mrad}$ due to the imperfect overlap between lattice and probe beams.Reuse & Permissions

Figure 3

(a) Collision shift vs dark time, 1D lattice, for excitation fraction 0.18. Using the same parameters as Fig. 2a, the solid line gives a fit from the mean-field model. (b) Collision shift vs dark time, 2D lattice, for excitation fraction $0.19\pm .03$. The shift crosses zero due to the periodic dependence of the shift on collisional phase, and is a signature of strong interactions. The model calculations (shaded region) use the same parameters as Fig. 2b for an excitation fraction range $0.19\pm .03$. (c) Asymmetric Rabi spectrum, 2D lattice. The solid line is the prediction of the model, using the number of doubly occupied sites as a fitting parameter.Reuse & Permissions