Growing extended Laughlin states in a quantum gas microscope: A patchwork construction

Phys. Rev. Res. 6, 013198 (2024) · DOI: 10.1103/physrevresearch.6.013198

License: CC BY 4.0.

Abstract

Fractional quantum Hall states are hard to prepare in quantum simulators because they are gapped, strongly correlated, and grow no faster than the system does. We propose a patchwork construction: prepare small Laughlin patches in a Hofstadter-Bose-Hubbard lattice independently, then couple them adiabatically into one extended state, either by ramping up the hopping between them or by lowering a potential barrier. Using exact diagonalization and matrix-product-state simulations we follow the target-state fidelity, the many-body gap and the entanglement entropy through the ramp for chains of up to five patches and for a large square of four patches, and we show that a fractionally charged quasihole can be pinned and read out from snapshots of the resulting state. The scheme is designed for a quantum gas microscope, where the individual patches and the couplings between them are the operations already available.

Figures

58 panels with data across 18 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF. Page 2 of the PDF.

Fig. 2

Illustrative figure, no extractable data. Shown in the paper PDF. Page 3 of the PDF.

Fig. 3

• panel (a): Hopping protocol: many-body gap $\Delta/J$ of two 4 by 4 patches against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$, for $U/J = \infty$. data.csv
• panel (b): Hopping protocol: many-body gap $\Delta/J$ of two 4 by 4 patches against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$, for $U/J = 8$. The white box marking the zoom of panel (c) is not drawn. data.csv
• panel (c): Zoom of panel (b) onto $0.2 \le \alpha \le 0.4$, hopping protocol: many-body gap $\Delta/J$ of two 4 by 4 patches against $\alpha$ and $J_{\rm coupling}/J$, for $U/J = 8$. data.csv
• panel (d): Hopping protocol: many-body Chern number $C_{\rm Streda} = \partial\bar{n}/\partial\alpha$ of the coupled system against $\alpha$, for $U/J = \infty$, at $J_{\rm coupling}/J$ = 1.0 (solid) and 0.8 (dotted). The dashed line marks 1/2, the value expected for the $\nu = 1/2$ Laughlin state. The grey shading of the printed panel is not shown. data.csv
• panel (e): Hopping protocol: many-body Chern number $C_{\rm Streda} = \partial\bar{n}/\partial\alpha$ of the coupled system against $\alpha$, for $U/J = 8$, at $J_{\rm coupling}/J$ = 1.0 (solid) and 0.8 (dotted). The dashed line marks 1/2, the value expected for the $\nu = 1/2$ Laughlin state. The grey shading and the inset of the printed panel are not shown. data.csv
• panel (f): Zoom of panel (e) onto $0.2 \le \alpha \le 0.4$, hopping protocol: many-body Chern number $C_{\rm Streda} = \partial\bar{n}/\partial\alpha$ of the coupled system against $\alpha$, for $U/J = 8$, at $J_{\rm coupling}/J$ = 1.0 (solid) and 0.8 (dotted). The dashed line marks 1/2, the value expected for the $\nu = 1/2$ Laughlin state. The grey shading of the printed panel is not shown. data.csv

Fig. 4

• panel (1): Hopping protocol: target-state fidelity $\mathcal{F}$ against $t/T$ upon linearly coupling two patches, $U/J = \infty$ (upper panel of the printed figure), for $T/\tau = 1.59$, 3.18, 6.37 and 15.92. The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (2): Hopping protocol: target-state fidelity $\mathcal{F}$ against $t/T$ upon linearly coupling two patches, $U/J = 8$ (lower panel of the printed figure), for $T/\tau = 1.59$, 3.18, 6.37 and 15.92. The final-fidelity labels printed at the right edge are not drawn. data.csv

Fig. 5

• panel (1): Hopping protocol: bulk density $\bar{n}$ against flux per plaquette $\alpha$ for the ground state (filled circles) and for the final state of the ramps with $T/\tau = 1.59$, 3.18, 6.37 and 15.92 (open markers), for $U/J = \infty$. The straight lines are the linear Streda fits, whose slopes $C_{\rm Streda}$ the legend gives. data.csv

Fig. 6

• panel (1): Hopping protocol, upper panel: squared overlap $|\langle \cdot | \Psi(t)\rangle|^2$ against $t/T$ of the time-evolved state with the instantaneous ground state $\langle\Psi_0(J_{\rm coupling})|$ (grey) and with the target state from exact diagonalization (ED) and from TDVP (dotted), for $U/J = \infty$ and $T/\tau = 6.37$. data.csv
• panel (2): Hopping protocol, lower panel: entanglement entropy $S$ between the two patches against $t/T$ for the time-evolved state, $S(t)$ (dotted), and for the instantaneous ground state, $S_0(J_{\rm coupling})$ (grey). The inset of the printed panel is hosted as its own panel. data.csv
• panel (2-inset): Inset of the printed lower panel: the excess entropy $S(t) - S_0$ of the time-evolved state over the instantaneous ground state against $t/T$, with a line at zero. data.csv

Fig. 7

Illustrative figure, no extractable data. Shown in the paper PDF. Page 5 of the PDF.

Fig. 8

• panel (a): Barrier protocol: many-body gap $\Delta/J$ of two 4 by 4 patches against the flux per plaquette $\alpha$ and the barrier height $V_{\rm barrier}/J$, for $U/J = \infty$. data.csv
• panel (b): Barrier protocol: many-body gap $\Delta/J$ of two 4 by 4 patches against the flux per plaquette $\alpha$ and the barrier height $V_{\rm barrier}/J$, for $U/J = 8$. The white box marking the zoom of panel (c) is not drawn. data.csv
• panel (c): Zoom of panel (b) onto $0.2 \le \alpha \le 0.4$, barrier protocol: many-body gap $\Delta/J$ of two 4 by 4 patches against $\alpha$ and $V_{\rm barrier}/J$, for $U/J = 8$. data.csv
• panel (d): Barrier protocol: many-body Chern number $C_{\rm Streda} = \partial\bar{n}/\partial\alpha$ of the target state ($V_{\rm barrier}/J = 0$) against $\alpha$, for $U/J = \infty$. The dashed line marks 1/2, the value expected for the $\nu = 1/2$ Laughlin state. The grey shading of the printed panel is not shown. data.csv
• panel (e): Barrier protocol: many-body Chern number $C_{\rm Streda} = \partial\bar{n}/\partial\alpha$ of the target state ($V_{\rm barrier}/J = 0$) against $\alpha$, for $U/J = 8$. The dashed line marks 1/2, the value expected for the $\nu = 1/2$ Laughlin state. The grey shading and the inset of the printed panel are not shown. data.csv
• panel (f): Zoom of panel (e) onto $0.2 \le \alpha \le 0.4$, barrier protocol: many-body Chern number $C_{\rm Streda} = \partial\bar{n}/\partial\alpha$ of the target state ($V_{\rm barrier}/J = 0$) against $\alpha$, for $U/J = 8$. The dashed line marks 1/2, the value expected for the $\nu = 1/2$ Laughlin state. The grey shading of the printed panel is not shown. data.csv

Fig. 9

• panel (1): Barrier protocol: target-state fidelity $\mathcal{F}$ against $t/T$ upon linearly turning off the potential barrier, $U/J = \infty$ (upper panel of the printed figure), for $T/\tau = 1.59$, 3.18, 6.37 and 15.92. The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (2): Barrier protocol: target-state fidelity $\mathcal{F}$ against $t/T$ upon linearly turning off the potential barrier, $U/J = 8$ (lower panel of the printed figure), for $T/\tau = 1.59$, 3.18, 6.37 and 15.92. The final-fidelity labels printed at the right edge are not drawn. data.csv

Fig. 10

• panel (1): Two-step barrier protocol: target-state fidelity $\mathcal{F}$ against $t/T$, the barrier turned off in two linear steps, $U/J = \infty$ (upper panel), for $T/\tau = 1.59$, 3.18, 6.37 and 15.92. The vertical lines marking the intermediate time $T_1$ and the final-fidelity labels of the printed panel are not drawn. data.csv
• panel (2): Two-step barrier protocol: target-state fidelity $\mathcal{F}$ against $t/T$, the barrier turned off in two linear steps, $U/J = 8$ (lower panel), for $T/\tau = 1.59$, 3.18, 6.37 and 15.92. The vertical lines marking the intermediate time $T_1$ and the final-fidelity labels of the printed panel are not drawn. data.csv

Fig. 11

• panel (1): Final target-state fidelity $\mathcal{F}$ at $t = T$ for the hopping, 1-step barrier and 2-step barrier protocols, one open marker per ramp time, $T/\tau = 1.59$, 3.18, 6.37 and 15.92, for $U/J = \infty$ (left panel of the printed figure). The protocols are evenly spaced here; the print sets the two barrier protocols closer together. data.csv
• panel (2): Final target-state fidelity $\mathcal{F}$ at $t = T$ for the hopping, 1-step barrier and 2-step barrier protocols, one open marker per ramp time, $T/\tau = 1.59$, 3.18, 6.37 and 15.92, for $U/J = 8$ (right panel of the printed figure). The protocols are evenly spaced here; the print sets the two barrier protocols closer together. data.csv

Fig. 12

• panel (a): Super-chain of $N_{\rm patches} = 3$ patches with $N = 6$ particles: many-body gap $\Delta/J$ against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$. data.csv
• panel (b): Super-chain of $N_{\rm patches} = 4$ patches with $N = 8$ particles: many-body gap $\Delta/J$ against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$. data.csv
• panel (c): Super-chain of $N_{\rm patches} = 5$ patches with $N = 10$ particles: many-body gap $\Delta/J$ against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$. data.csv
• panel (d): Super-chain of $N_{\rm patches} = 3$ patches: entanglement entropy $S^{(1)}$ between the first patch and the rest of the system against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$. data.csv
• panel (e): Super-chain of $N_{\rm patches} = 4$ patches: entanglement entropy $S^{(1)}$ between the first patch and the rest of the system against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$. data.csv
• panel (f): Super-chain of $N_{\rm patches} = 5$ patches: entanglement entropy $S^{(1)}$ between the first patch and the rest of the system against the flux per plaquette $\alpha$ and the coupling $J_{\rm coupling}/J$. data.csv

Fig. 13

• panel (1): Super-chain of $N_{\rm patches} = 3$ patches, all couplings ramped linearly at once: target-state fidelity $\mathcal{F}$ against $t/T$ for $T/\tau = 6.37$ (dash-dotted) and 15.92 (dotted). The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (2): Super-chain of $N_{\rm patches} = 4$ patches, all couplings ramped linearly at once: target-state fidelity $\mathcal{F}$ against $t/T$ for $T/\tau = 6.37$ (dash-dotted) and 15.92 (dotted). The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (3): Super-chain of $N_{\rm patches} = 5$ patches, all couplings ramped linearly at once: target-state fidelity $\mathcal{F}$ against $t/T$ for $T/\tau = 6.37$ (dash-dotted) and 15.92 (dotted). The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (4): Super-chain of $N_{\rm patches} = 3$ patches, all couplings ramped linearly at once: entanglement entropy $S^{(1)}$ of the first patch against $t/T$ for $T/\tau = 6.37$ (dash-dotted) and 15.92 (dotted), and for the instantaneous ground state (solid). data.csv
• panel (5): Super-chain of $N_{\rm patches} = 4$ patches, all couplings ramped linearly at once: entanglement entropy $S^{(1)}$ of the first patch against $t/T$ for $T/\tau = 6.37$ (dash-dotted) and 15.92 (dotted), and for the instantaneous ground state (solid). data.csv
• panel (6): Super-chain of $N_{\rm patches} = 5$ patches, all couplings ramped linearly at once: entanglement entropy $S^{(1)}$ of the first patch against $t/T$ for $T/\tau = 6.37$ (dash-dotted) and 15.92 (dotted), and for the instantaneous ground state (solid). data.csv

Fig. 14

• panel (1): Super-chain of four patches grown as two pairs ($t/T < 0.5$) and then joined: target-state fidelity $\mathcal{F}$ against $t/T$ for $T/\tau = 6.37$ and 15.92, with grey lines for all couplings ramped at once. The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (2): Super-chain of four patches grown as two pairs ($t/T < 0.5$) and then joined: entanglement entropy $S^{(1)}$ of the first patch against $t/T$ for $T/\tau = 6.37$ and 15.92. The inset of the printed panel, a sketch of the coupling strengths, and its short line for the target-state entropy are not shown. data.csv

Fig. 15

• panel (1): Super-chain of $N_{\rm patches} = 3$ patches connected one at a time: target-state fidelity $\mathcal{F}$ against $t/T$ for $T/\tau = 6.37$ and 15.92, with grey lines for all couplings ramped at once. The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (2): Super-chain of $N_{\rm patches} = 4$ patches connected one at a time: target-state fidelity $\mathcal{F}$ against $t/T$ for $T/\tau = 6.37$ and 15.92, with grey lines for all couplings ramped at once. The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (3): Super-chain of $N_{\rm patches} = 3$ patches connected one at a time: entanglement entropy $S^{(1)}$ of the first patch against $t/T$ for $T/\tau = 6.37$ and 15.92. The short line the printed panel draws for the target-state entropy is not shown. data.csv
• panel (4): Super-chain of $N_{\rm patches} = 4$ patches connected one at a time: entanglement entropy $S^{(1)}$ of the first patch against $t/T$ for $T/\tau = 6.37$ and 15.92. The short line the printed panel draws for the target-state entropy is not shown. data.csv

Fig. 16

• panel (1): Large square of four coupled 4 by 4 patches: many-body gap $\Delta/J$ against the flux per plaquette $\alpha$ and the barrier height $V_{\rm barrier}/J$, for $U/J = \infty$ (upper left panel of the printed figure). data.csv
• panel (2): Large square of four coupled 4 by 4 patches: many-body gap $\Delta/J$ against the flux per plaquette $\alpha$ and the barrier height $V_{\rm barrier}/J$, for $U/J = 8$ (upper right panel of the printed figure). data.csv
• panel (3): Large square of four coupled 4 by 4 patches: many-body gap $\Delta/J$ against the flux per plaquette $\alpha$ and the barrier height $V_{\rm barrier}/J$, for $U/J = \infty$, zoomed onto low barriers and $0.18 \le \alpha \le 0.42$ (lower left panel of the printed figure). data.csv
• panel (4): Large square of four coupled 4 by 4 patches: many-body gap $\Delta/J$ against the flux per plaquette $\alpha$ and the barrier height $V_{\rm barrier}/J$, for $U/J = 8$, zoomed onto low barriers and $0.18 \le \alpha \le 0.42$ (lower right panel of the printed figure). data.csv

Fig. 17

• panel (1): Large square: target-state fidelity $\mathcal{F}$ against $t/T$ for a linearly decreasing barrier height, $V_{\rm barrier}/J = 1 - t/T$ at $\alpha = 0.25$, for $T/\tau = 6.37$ and 15.92. The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (2): Large square: entanglement entropy $S^{\rm (left)}$ of the left two patches against $t/T$ for a linearly decreasing barrier height, $V_{\rm barrier}/J = 1 - t/T$ at $\alpha = 0.25$, for $T/\tau = 6.37$ and 15.92. data.csv

Fig. 18

• panel (b-1): Large square: density $\langle n\rangle$ over the sites $x$, $y$ of four coupled 4 by 4 patches without a pinning potential, $V_{\rm pin}/J = 0$. The dotted circle of the printed panel is not drawn. data.csv
• panel (b-2): Large square: density $\langle n\rangle$ over the sites $x$, $y$ of four coupled 4 by 4 patches with a cross-shaped pinning potential $V_{\rm pin}/J = 2$ at the centre. The dotted circle of the printed panel is not drawn. data.csv
• panel (c-1): Mean pinned charge $\langle \hat{Q}\rangle$ against $V_{\rm pin}/J$, the white dots of printed panel (c), left, with dashed lines at $-1/2$ and $-1$ and vertical dashed lines at $V_{\rm pin}/J = 0$ and 2. The dots are shown up to $V_{\rm pin}/J = 3$, where the print runs to 10, and the counting-statistics map behind them is not shown. data.csv
• panel (c-2): Full counting statistics of the pinned charge from 2000 snapshots: probability density against $Q$ at $V_{\rm pin}/J = 0$, with a vertical line at $Q = 0$. data.csv
• panel (c-3): Full counting statistics of the pinned charge from 2000 snapshots: probability density against $Q$ at $V_{\rm pin}/J = 2$, with a vertical line at $Q = -1/2$. data.csv

Fig. 19

• panel (a): Large square with a cross-shaped pinning potential $V_{\rm pin}/J = 2$: target-state fidelity $\mathcal{F}$ against $t/T$ for a linearly decreasing barrier height, for $T/\tau = 6.37$ and 15.92. The final-fidelity labels printed at the right edge are not drawn. data.csv
• panel (b): Full counting statistics of the pinned charge $Q$, probability density, for the state prepared by the ramp at $T/\tau = 15.92$ (red bars) and for the target state obtained via DMRG, the ground state, with a vertical line at $Q = -1/2$. The target state, a purple dashed outline in print, is drawn as translucent purple bars. data.csv

Fig. 20

• panel (1): Final target-state fidelity $\mathcal{F}$ at $t = T$ against system size $(L_x \times L_y)$ for the patchwork schemes (open markers) and a scheme coupling chains (small faded filled markers), for $T/\tau = 1.59$, 3.18, 6.37 and 15.92, all at $U/J = \infty$. data.csv

Fig. 21

• panel (a): Final target-state fidelity $\mathcal{F}$ against $N_{\rm patches}$, on log axes, for all patches coupled at once, for $T/\tau = 6.37$ (squares) and 15.92 (triangles), at $U/J = \infty$ and $\alpha = 0.3$. The curves are the exponential fits the legend gives. data.csv
• panel (b): Many-body gap $\Delta/J$ against $N_{\rm patches}$, on log axes: the minimal gap along the preparation path (crosses) and the average gap (circles), at $U/J = \infty$ and $\alpha = 0.3$, with the exponential and power-law fits the legend gives. The hosted legend's power-law exponent differs from the printed one in the second decimal. data.csv

Cite

Felix A. Palm, Joyce Kwan, Brice Bakkali-Hassani, Markus Greiner, Ulrich Schollwöck, Nathan Goldman, Fabian Grusdt. Growing extended Laughlin states in a quantum gas microscope: A patchwork construction. Phys. Rev. Res. 6, 013198 (2024). https://doi.org/10.1103/physrevresearch.6.013198

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