GoKawiilExperimental platform
In our experiment, we prepare a degenerate Fermi gas of 6Li atoms in a balanced mixture of the two lowest hyperfine states, which represent our two spin states. The atomic cloud is loaded into a single plane of a vertical lattice following our previous work41,51, with radial confinement provided by a blue-detuned box potential projected using a digital micromirror device (DMD)40,41.
From there, the atoms are loaded into a two-dimensional square optical lattice in the x–y plane with lattice constants a x,long = 2.28(2) μm and a y = 1.11(1) μm. A DMD pattern is chosen such that a flat central region of approximately 145 sites is surrounded by a low-density reservoir60. The chemical potential of the reservoir, tuned by the light intensity of the DMD, controls the particle density \(\langle \hat{n}\rangle \) at the centre. We realize a state with an average of nearly two particles per lattice site (close to a band insulator) at lattice depths of \({V}_{x}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}=9.0\,{E}_{{\rm{r}}}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}\) and \({V}_{y}=9.3\,{E}_{{\rm{r}}}^{{\rm{s}}{\rm{h}}{\rm{o}}{\rm{r}}{\rm{t}}}\). Dynamics are frozen by ramping the lattice depths to \({V}_{x}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}=35.5\,{E}_{{\rm{r}}}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}\) and \({V}_{y}=45.0\,{E}_{{\rm{r}}}^{{\rm{s}}{\rm{h}}{\rm{o}}{\rm{r}}{\rm{t}}}\), leaving isolated single-wells with mainly two particles per site. Subsequently, we ramp up a second, short-spaced lattice along x (a x,short = a x,long /2) over 25 ms, resulting in isolated, doubly occupied double-wells with total spin S z = 0. In this experiment, the short lattices are generated by laser beams at blue-detuned 532-nm light incident at an angle of about 27°. The long lattice along the x-direction follows the same beam path, except that it is generated with red-detuned 1,064-nm light14.
In all figures, data points were collected in a randomized sequence to prevent systematic bias.
State preparation fidelity
The probability of realizing the desired state in the region of interest is approximately constant within a given dataset and mostly depends on the relative phase drift between the long and short lattices, as well as on the chosen atomic density. It is set largely by the fidelity of preparing an average occupancy close to two atoms of opposite spin per initial lattice site, which ranges from 60% to 85%. Deviations from the target state fall into two categories: (1) empty or singly occupied double-wells, which we remove by post-selection, and (2) double-wells containing three or more atoms, typically with population in higher lattice bands. Because these high-occupancy events can be mistaken for gate errors, we deliberately work at slightly lower atomic densities to suppress them, retaining between 45% and 65% of double-wells in analysis. Recent demonstrations of low-entropy band insulators in optical lattices suggest that considerably higher state preparation fidelity is attainable7,13,61. We note that the state preparation step does not affect the intrinsic performance of the individual gate operations.
Lattice depth calibration
Lattice depth calibration is performed by measuring single-particle oscillations in a double-well, from which we extract the calibration factor by fitting the observed tunnelling rates to theoretical predictions across a range of lattice depths. An initial state consisting of a single particle in a double-well is prepared by adjusting the atom density and tilting the double-well potentials during loading, similar to our previous work14. We then remove the potential offset δ, resulting in a symmetric double-well configuration at lattice depths of \({V}_{x}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}=36.5\,{E}_{{\rm{r}}}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}\) and \(({V}_{x}^{{\rm{s}}{\rm{h}}{\rm{o}}{\rm{r}}{\rm{t}}},{V}_{y})=(56,43)\,{E}_{{\rm{r}}}^{{\rm{s}}{\rm{h}}{\rm{o}}{\rm{r}}{\rm{t}}}\). Quenching the short x lattice depth to a lower value initiates coherent oscillations of the population between the two sites in the double-well. In our analysis, we post-select double-wells containing exactly one atom.
In Extended Data Fig. 1a, we show an example calibration plot in which the calculated calibration curve aligns with the measured tunnelling frequencies with residuals less than 1.5% of \({V}_{x}^{{\rm{short}}}\). The tunnelling frequency of intra-double-well oscillations f t = 2t/h is extracted by fitting a resonant two-level oscillation [1 + cos(2πf t × τ h )]/2 to the population of one of the wells, which is then compared with the frequency expected from a band calculation (see our previous work14 for more details).
To cross-check the lattice depth calibration, we measure spin-exchange oscillation in the U/t ≫ 1 regime (J ≈ 4t2/U), in which virtual doublon-hole excitations are strongly suppressed (Extended Data Fig. 1b). We compare the frequency extracted from the fit to the oscillations (Extended Data Fig. 1b, upper row) with the calculated calibration curve (solid blue line) and find excellent agreement, consistent with the single-particle tunnelling calibration. The initial lattice depths in this case are \(({V}_{x}^{{\rm{s}}{\rm{h}}{\rm{o}}{\rm{r}}{\rm{t}}},{V}_{y})=(56,45)\,{E}_{{\rm{r}}}^{{\rm{s}}{\rm{h}}{\rm{o}}{\rm{r}}{\rm{t}}}\), \({V}_{x}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}=39.5\,{E}_{{\rm{r}}}^{{\rm{l}}{\rm{o}}{\rm{n}}{\rm{g}}}\) and the Feshbach magnetic field is set to 688.2 G to control the on-site interaction strength U through a Feshbach resonance. The long lattice depth \({V}_{x}^{{\rm{long}}}\) is independently calibrated using lattice modulation spectroscopy through band-excitation energies to an accuracy of 5%.
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