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

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.

💡 Research Summary

In this work the authors realize a programmable analog quantum simulator of the Bose‑Hubbard model on a triangular‑ladder geometry using eight superconducting transmon qubits and eight flux‑tunable couplers. The ladder consists of two coupled one‑dimensional chains that form triangular plaquettes; hopping along the rungs (J) is fixed at about 6 MHz while hopping along the legs (J∥) can be tuned from +20 MHz down to –17 MHz by adjusting the coupler frequencies. Changing the sign of J∥ implements a synthetic magnetic flux of 0 or π per plaquette, allowing the exploration of flux‑induced frustration. The on‑site interaction U is negative (≈‑186 MHz) due to the transmon anharmonicity, but the authors employ a sign‑flip mapping (–H(ϕ,U)=H(ϕ+π,‑U)) to study the physics of a repulsive model at π flux by preparing the highest excited state of the attractive Hamiltonian and then inverting the sign of the couplings.

The experiment focuses on the half‑filling sector (four photons in eight sites), a regime where kinetic and interaction energies compete most strongly. State preparation proceeds by selectively exciting four qubits, then adiabatically ramping all qubits onto resonance to reach the target ground state. To probe order parameters the authors isolate adjacent qubit pairs, apply a calibrated beamsplitter Hamiltonian HBS=ℏJ(a†j aj+1 + h.c.) for a time tBS=π/4J, and read out the population imbalance. This procedure maps the current operator Jj = iJ(a†j aj+1 – a†j+1 aj) onto a measurable basis, enabling extraction of both the local rung current ⟨Jj⟩ and the current‑current correlator G(i,j)=⟨Ji Jj⟩–⟨Ji⟩⟨Jj⟩.

Measurements reveal that the average rung current vanishes for all flux values, consistent with time‑reversal symmetry, but the current‑current correlator is non‑zero and displays long‑range structure. For J∥/J < 0 (π‑flux) the correlator remains sizable even between the most distant rungs, indicating chiral order that breaks a Z2 parity symmetry. In contrast, for J∥/J > 0 (zero flux) the correlator decays more rapidly, characteristic of a Meissner‑type superfluid with uniform leg currents and no chiral texture. By additionally measuring leg currents through a similar beamsplitter protocol, the authors distinguish Meissner superfluids (uniform leg flow) from chiral superfluids (alternating leg flow).

A further diagnostic is the bond‑kinetic energy operator B = ⟨aj†aj+1 + h.c.⟩. Its spatial pattern reveals a bond‑ordered insulating phase when J∥/J lies in the range roughly –1 to –2: the bond strength alternates between strong and weak on successive bonds, breaking translational symmetry while suppressing both off‑diagonal long‑range order and rung currents. This bond‑ordered phase coexists with a gapped excitation spectrum, as confirmed by exact‑diagonalization and DMRG calculations that match the experimental data.

Overall, the study demonstrates that superconducting‑circuit platforms can simultaneously engineer geometric frustration, synthetic magnetic fields, and strong on‑site interactions, and that site‑resolved measurements of current and bond operators provide direct access to order parameters of exotic many‑body phases. The authors’ beamsplitter‑based measurement technique opens a pathway to probe chiral currents, bond order, and other non‑local observables in larger lattices, paving the way for future investigations of topological states, non‑equilibrium dynamics, and quantum phase transitions in engineered bosonic Hubbard systems.