The 2019 experiment of Schweizer et al. realized a minimal Z2\mathbb{Z}_2 lattice gauge theory (LGT) on a single double well of two-component bosons in an optical lattice, using resonant periodic (Floquet) driving to generate the gauge-invariant matter-gauge coupling. The open problem is to scale this Floquet primitive to extended one- and two-dimensional many-body systems with certified local gauge invariance: realizing exactly one gauge particle per link across many coupled double wells, controlling residual gauge-violating processes (correlated two-particle tunnelling and nearest-neighbour interactions), quantitatively mapping the trade-off between the high-frequency Floquet regime and interaction-enhanced correlated tunnelling, achieving sufficient ensemble homogeneity or single-link addressing, and developing direct measurement and certification of the local Gauss-law operators together with a systematic assessment of the robustness of many-body observables (confinement signatures, phase boundaries, transport) to gauge-variant imperfections. No extended Floquet-based cold-atom Z2\mathbb{Z}_2 LGT experiment has been reported through 2026; the theoretical ingredients for scaling have been demonstrated, gauge-protection and gauge-violation-measurement methodology has advanced substantially, and extended Z2\mathbb{Z}_2 LGTs with dynamical matter have been realized on digital trapped-ion and superconducting platforms, so the Floquet cold-atom realization of such a simulator with certified local gauge invariance remains open.

Background

A lattice gauge theory (LGT) is a discretized quantum field theory in which charged matter lives on lattice sites while gauge (force) fields live on the links between sites. The defining constraint is a local Gauss law: at every site jj a generator G^j\hat{G}_j must commute with the Hamiltonian, so that physical states are eigenstates G^j∣Ψ⟩=gj∣Ψ⟩\hat{G}_j|\Psi\rangle = g_j|\Psi\rangle. For the simplest discrete gauge group, Z2={+1,−1}\mathbb{Z}_2 = \{+1,-1\}, the link is a two-level system (a spin-1/21/2, with Pauli operators τ^z\hat{\tau}^z for the parallel transporter and τ^x\hat{\tau}^x for the electric field) and the Gauss operator is the product of the electric fields on all links touching jj times the matter parity, G^j=(−1)n^j∏i∈⟨i,j⟩τ^⟨i,j⟩x\hat{G}_j = (-1)^{\hat{n}_j}\prod_{i\in\langle i,j\rangle}\hat{\tau}^x_{\langle i,j\rangle}. A Z2\mathbb{Z}_2 LGT with dynamical matter is the simplest setting in which gauge-invariant dynamics, confinement of charges by electric-flux strings, and confinement-deconfinement phase transitions can be studied out of equilibrium, regimes that stress classical simulation methods such as Monte Carlo. Floquet engineering is a technique in which a periodic drive (here resonant modulation at the on-site interaction energy UU of a two-component Bose-Hubbard system) makes the time-averaged effective Hamiltonian contain terms absent in the static system; specifically, photon-assisted tunnelling with Bessel-function amplitudes realizes the gauge-invariant matter-gauge coupling −Jaa^i†τ^⟨i,j⟩za^j-J_a \hat{a}^\dagger_i \hat{\tau}^z_{\langle i,j\rangle}\hat{a}_j on a double well, where aa-atoms (matter) and ff-atoms (gauge particles) occupy the two wells. In 2019 Schweizer et al. demonstrated this primitive on a single double well, observed gauge-invariant charge dynamics, and analysed the symmetry-breaking terms: an inhomogeneous tilt distribution across the atomic cloud, correlated two-particle tunnelling and nearest-neighbour interactions (of order 0.03J0.03J for their parameters), and finite-frequency corrections to the Floquet expansion that define the time window over which the Z2\mathbb{Z}_2 symmetry survives (an optimum near ℏω≈1.01U\hbar\omega \approx 1.01U in their numerics). The experiment did not have direct access to the Gauss-law operators G^j\hat{G}_j and assessed gauge-violating effects only numerically. Feshbach resonances (magnetic-field-tuned scattering lengths) offer a route to larger UU and hence the high-frequency Floquet limit, but simultaneously enhance the correlated tunnelling that itself violates Gauss's law. A companion theoretical work (Barbiero et al.) showed that two-frequency driving can scale the scheme to extended one-dimensional chains, and the required density-dependent Peierls phases were demonstrated experimentally by Görg et al.; nevertheless the cold-atom realization has remained at the building-block level.

Problem Statement

Determine whether the Floquet-engineered double-well primitive of Schweizer et al. (2019) can be scaled into a many-body quantum simulator of Z2\mathbb{Z}_2 lattice gauge theories coupled to dynamical matter, and establish what gauge-theory physics such a simulator can reliably access. This requires, as jointly supporting sub-questions, all of the following, each independently checkable and none sufficient on its own: (1) realize the primitive in extended one-dimensional chains or two-dimensional (e.g. ladder) arrays, which demands preparing exactly one gauge particle per link across many coupled double wells and controlling the residual gauge-violating processes whose relative weight grows with system size, in particular how correlated two-particle tunnelling and nearest-neighbour interactions scale and can be suppressed in extended systems; (2) quantitatively map the parameter trade-off between entering the high-frequency Floquet regime (suppressing finite-frequency gauge-violating corrections) and the concomitant enhancement of correlated tunnelling, identifying optimal operating regimes for many-body implementations; (3) achieve the ensemble homogeneity or single-link control at array scale needed to preserve coherence of array-averaged many-body observables; (4) develop and demonstrate methods for direct measurement and certification of the local Gauss-law operators G^j\hat{G}_j in many-site Floquet-based systems, which the original experiment accessed only indirectly through numerical modelling; and (5) carry out the numerical and experimental studies determining how robust many-body observables of the Z2\mathbb{Z}_2 LGT (confinement signatures, phase boundaries, charge transport) are to the identified gauge-variant imperfections, so as to establish which physical phenomena of Z2\mathbb{Z}_2 lattice gauge theories can be reliably accessed in near-term Floquet-based simulators. The scaling determination and the accessible-physics assessment require a concrete, quantitatively validated realisation scheme with the supporting numerics. Resolution does not require the experiment to have already been performed, but the scheme must be concrete enough to fix its gauge-violation error budget and the observables it makes accessible.

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