Chiral and bond-ordered phases in a triangular-ladder superconducting-qubit quantum simulator

Abstract

Many-body systems with strong interactions often exhibit macroscopic behavior markedly absent in single-particle or noninteracting limits. Such emergent phenomena are well exemplified in lattice Hubbard models, where the interplay between interactions, geometric frustration, and magnetic flux gives rise to rich physics. Superconducting qubits naturally enable analog quantum simulation of Bose-Hubbard models, while offering tunable parameters, site-resolved control, and rapid experimental repetition rates. Here, we study a superconducting-qubit device that realizes the Bose-Hubbard model on a triangular-ladder lattice. By tuning the magnitude and sign of couplings, we engineer a synthetic magnetic flux to characterize the resulting half-filling ground state for various parameter regimes. We measure observables analogous to current-current correlators and bond kinetic energies, finding signatures consistent with chiral superfluids, Meissner superfluids, and bond-ordered insulators. Our results establish superconducting circuits as a platform for robustly probing quantum phases of matter in frustrated Bose-Hubbard systems, even in strongly correlated and gapless regimes.

Figures (4)

• Figure 1: Realization of the triangular Bose-Hubbard ladder with transmon qubits. (a) False-color image of the 8-qubit triangular ladder with transmon qubits shaded in red and transmon couplers shaded in blue. Readout resonators and flux-bias lines are shaded in white and light blue, respectively. (b) Schematic of the 8-qubit triangular ladder. $J$ and $\tilde{J}_{\parallel}$ denote hoppings along the diagonals (rungs) and parallel edges (legs), respectively. Each plaquette is additionally threaded with a synthetic magnetic flux $\phi$. (c) Full experimental sequence. The four qubits with the highest energy are excited before the adiabatic ramp onto resonance. Arbitrary pairs of spatially adjacent qubits are isolated to characterize the state in two-qubit subspaces. Lattice schematics (bottom) depict the initialized state at half filling, the superfluid state after the ramp, and the isolated state during the measurement protocol.
• Figure 2: Current-current correlations between rungs of the triangular ladder. (a) Schematic of the lattice showing pairs of qubits undergoing beamsplitter rotation in order to perform current measurement. The arrows show the measured correlations with the rung shaded in red. (b) Bloch-sphere representation of the beamsplitter rotation from the current eigenstates $\ket{\pm j}$ to the readout basis. (c) Current measurement trace between qubits $1$ and $2$ as the beamsplitter interaction time is varied. The current value is extracted at the beamsplitter time $t^{\space}_{\text{BS}}=\pi/4J$ shown by the dashed line. The solid line is a sinusoidal fit to the population oscillations. (d) Same as in (c) but for qubits $7$ and $8$. (e). Current correlation between qubit pairs 1-2 and 7-8. Although the current on both rungs vanishes at the beamsplitter time, the current correlation has a nonzero value.
• Figure 3: Chiral and Meissner superfluids on the 8-qubit triangle ladder. (a) Total chiral-current order parameter for various values of $J_{\parallel}/J$. Negative (positive) ratios correspond to a $\pi$-flux ($0$-flux) synthetic magnetic field. The values of $J_{\parallel}/J$ for each plot are (from left to right) $-3.56$, $-2.02$, $-1.22$, $0.98$, $1.96$, and $3.53$. (b) Current correlation matrices for the $\pi$-flux coupling ratios. Rung $j$ links qubits $(j,j+1)$. Experimental data are shown on the lower left and simulation on the upper right of the correlation matrix. (c) Current correlation matrices for the $0$-flux coupling ratios. (d) The same current correlations as in part (b) plotted against distance between the two rungs for the $\pi$-flux coupling ratios. The inset depicts a schematic of the rung currents that produce the positive current-current correlations observed here. (e) The same current correlations as in part (c) plotted against distance between the two rungs for the $0$-flux coupling ratios. The inset depicts a schematic of the rung currents that produce the anticorrelated current-current correlations observed here.
• Figure 4: Bond order insulator in the 8-qubit triangle ladder. (a) Schematic of the bond kinetic energy ordering on the lattice. (b) Pulse diagram depicting the bond kinetic energy measurement. Here, one qubit acquires a phase relative to the other in order to rotate the kinetic energy ($\hat{x}$) axis to the current ($\hat{y}$) axis, before undergoing the beamsplitter interaction to rotate to the measurement ($\hat{z}$) axis. (c) Total bond order parameter for various values of $J_{\parallel}/J$: from left to right, $-3.56$, $-2.02$, $-1.22$, $0.98$, $2.04$, and $2.85$. (d) Bond kinetic energies of each rung for the $\pi$-flux coupling ratios. Experimental data (simulation) is shown on the upper (lower) row of the color plots. The dashed lines indicate the alternating trend in bond kinetic energies seen numerically. (e) Bond kinetic energy of each rung for the $0$-flux coupling ratios. Here, the experimental (red circles) and numerical (dashed lines) results highlight that the bond kinetic energies maintain a uniform sign throughout, without the alternation seen previously.