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Engineering quantum criticality and dynamics on an analog-digital simulator

Alexandra A. Geim, Nazli Ugur Koyluoglu, Simon J. Evered, Rahul Sahay, Sophie H. Li, Muqing Xu, Dolev Bluvstein, Nik O. Gjonbalaj, Nishad Maskara, Marcin Kalinowski, Tom Manovitz, Ruben Verresen, Susanne F. Yelin, Johannes Feldmeier, Markus Greiner, Vladan Vuletic, Mikhail D. Lukin

arXiv:2602.18555v1quant-ph

TL;DR

The paper addresses how to realize and probe complex out-of-equilibrium many-body dynamics despite limited interaction engineering, measurement flexibility, and error mitigation. It combines analog Rydberg evolution with digital hyperfine-qubit control, coherent mapping, non-destructive readout, and closed-loop optimization. The approach engineers hopping and ring exchange and prepares a non-equilibrium RK-type U(1) quantum spin liquid whose observed signatures agree with theoretical predictions.

  • Problem

    Quantum simulators face challenges in realizing required interactions, measuring arbitrary observables, and mitigating errors in strongly correlated many-body systems.

  • Method

    The experiment coherently transfers states between Rydberg and hyperfine qubits to combine analog many-body evolution, programmable digital control, non-destructive readout, and optimized state preparation.

  • Results

    The approach engineers hopping and ring-exchange dynamics and produces an out-of-equilibrium RK-type U(1) quantum spin liquid with key microscopic signatures matching equilibrium RK predictions.

  • Takeaways & Limitations

    Hybrid analog-digital control provides a route to studying complex quantum many-body systems with programmable observables, loss-based error mitigation, and efficient analog interactions.

  • Takeaways & Limitations

    Engineered Floquet dynamics are sensitive to static detuning offsets, and microstate-dependent van der Waals tails reduce fidelity of number conservation in two dimensions.

Abstract

from arXiv · show

Understanding emergent phenomena in out-of-equilibrium interacting many-body systems is an exciting frontier in physical science. While quantum simulators represent a promising approach to this long-standing problem, in practice it can be challenging to directly realize the required interactions, measure arbitrary observables, and mitigate errors. Here we use coherent mapping between the Rydberg and hyperfine qubits in a neutral atom array simulator to engineer and probe complex quantum dynamics. We combine efficient analog dynamics with fully programmable state preparation and measurement, leverage non-destructive readout for loss information and atomic qubit reuse, and use an atom reservoir for replacing lost atoms. With this analog-digital approach, we first demonstrate dynamical engineering of ring-exchange and particle hopping dynamics via Floquet driving and measure the spectral function of single excitations by evolving initial superposition states. Extending these techniques to a 271-site kagome lattice, we employ closed-loop optimization to target an out-of-equilibrium critical quantum spin liquid of the Rokhsar-Kivelson type. We observe the key features of such a state, including the absence of local order, many-body coherences between nearly equal-amplitude dimer configurations over up to 18 sites, and universal correlations consistent with predictions from field theory. Together, these results pave the way for using dynamical control in analog-digital quantum simulators to study complex quantum many-body systems.

Fast analog-digital quantum simulations

The simulator coherently transfers states between Rydberg and hyperfine qubits, combining analog many-body evolution with programmable digital control. Periodic detuning drives engineer hopping and ring-exchange Hamiltonians, while coherent measurements probe interactions and single-particle spectra.

  • Architecture: Coherent mapping connects analog Rydberg evolution with programmable hyperfine-qubit gates, readout, loss detection, and atom reuse.Non-destructive readout enables loss-based error detection and increases the experimental cycle rate by over an order of magnitude.
  • Floquet Hamiltonian engineering: Periodic global detuning drives perturb a many-body echo to engineer effective Hamiltonians with approximate excitation-number conservation at stroboscopic times.Intermediate dynamics can involve non-number-conserving states even when the stroboscopic evolution approximately conserves excitation number.
  • Floquet Hamiltonian engineering: The Floquet protocol realizes tunable blockaded hopping with a single-particle quantum walk and faster number-conserving dynamics than large static detuning.The effective model includes chemical potential µ, nearest-neighbor hopping J, and next-nearest-neighbor interaction U.
  • Floquet Hamiltonian engineering: Three-body engineering produces six-body ring exchange on hexagonal plaquettes, with hopping and ring-exchange dynamics occurring on the same timescale.Single-excitation and Néel initial states isolate the two-body and six-body processes, respectively.
  • Dynamical probes: Four-basis initialization and Fourier analysis of G(r,t) yield a tight-binding dispersion ω(k)/2π = 0.407(2)−0.281(2) cos ka.The spectral-function measurement also demonstrates phase coherence between hyperfine and Rydberg qubits throughout the evolution.

Non-equilibrium quantum spin liquid

The experiment targets an out-of-equilibrium U(1) quantum spin liquid by mapping a kagome Rydberg array to a honeycomb quantum dimer model. Closed-loop optimization produces highly dimerized states with RK kinetic and potential signatures, while local loss postselection improves fidelity.

  • Target state: The honeycomb quantum dimer model has an RK critical point at v = t, where the predicted U(1) spin liquid is an equal-amplitude, equal-phase superposition of perfect dimer coverings.The kinetic term captures plaquette resonances, while the potential term counts flippable plaquettes.
  • Target state: Rydberg atoms on kagome vertices encode honeycomb dimers, while blockade enforces that each vertex is touched by at most one dimer.The experiment uses an out-of-equilibrium regime because the equilibrium U(1) spin liquid is unstable and finite protocols can produce quantum spin lakes.
  • State preparation: Longer optimized sweeps slow at early times and accelerate before the phase transition, indicating quasi-adiabatic initial evolution followed by faster preparation.The common final sweep rate contrasts with conventional ground-state preparation.
  • State preparation: The slowest sweep yields high dimer population with finite monomer and double-dimer densities, while RK kinetic and potential energies grow with dimer population.The defects arise from excitations and dressing of the low-energy dimer manifold at finite Δ/Ω.
  • State preparation: A closed-loop multi-objective Bayesian optimizer maximizes RK energies and mean Rydberg density while minimizing atom loss.Each run supplies X- and Z-basis snapshots, and the preparation protocol is updated automatically.
  • State preparation: Local loss postselection improves state fidelity and closely reproduces results from postselecting on no loss across the entire lattice.

Signatures of a Rokhsar-Kivelson state

The prepared state exhibits several RK-state signatures: weak local order, nearly uniform dimer-covering probabilities, measurable many-body coherences, and loop behavior consistent with the predicted state.

  • Absence of local order: Boundary light shifts produce uniform populations across kagome sublattices and a nearly uniform nematic-order phase distribution, distinguishing the state from a nematic phase.The angular distribution reveals emergent global rotational symmetry.
  • Dimer-covering statistics: Dimer-covering probabilities are almost flat in isolated subsystems containing up to 27 atoms plus 12 projected bonds.The subsystems are isolated by projecting surrounding bonds onto configurations without dimers.
  • Many-body coherences: Projecting surrounding bonds increases the measured potential-energy expectation by approximately 5×, while kinetic energy remains lower because of measurement errors attributed to mapping-down phase accumulation.The RK prediction is ⟨T⟩ = ⟨V⟩ ≈ 0.29 from classical Monte Carlo simulations.
  • Many-body coherences: Parity oscillations show coherent X6-loop oscillations with a many-body phase equal to 3× the applied single-qubit phase.The observed behavior is consistent with local GHZ-like states on loop perimeters.
  • Loop correlations: Projected X-loop expectations decay by a perimeter law, while the ratio of bare to projected loops decays as the number of dimer configurations grows exponentially with subsystem size.The projected loops are expected to be +1 in the ideal QSL, with decay attributed to virtual magnetic excitations.

Emergent universal correlations

The measured dimer-dimer and dimer-string correlations exhibit finite-size behavior while retaining the universal relationship predicted by field theory. Their decay and normalization support a locally Rokhsar-Kivelson-like state without crystalline order.

  • Dimer correlations: Dimer-dimer correlations decay exponentially in magnitude, show antiferromagnetic sign structure, and plateau at large separations.The plateau suggests weak long-range order induced by open boundaries that break the lattice Z3 symmetry.
  • String correlations: The FM-normalized dimer-string correlator decays toward zero for longer strings, ruling out crystalline order.The normalization mitigates perimeter-law decay from monomers and incoherent errors in closed loops.
  • Universal comparison: Rectification maps both correlators onto vertex-vertex correlators described by the field theory.This provides the basis for comparing their measured relation with the universal prediction.
  • Universal comparison: The logarithmic relation between rectified dimer-dimer and dimer-string correlators agrees with the field-theory ratio of 4.The comparison is made marker-by-marker at each vertex separation.
  • Finite-size behavior: Finite-size effects and monomers explain the observed exponential decay, while larger systems are expected to approach algebraic correlations.The interpretation is consistent with field theory and classical Monte Carlo simulations of the RK state.

Discussion and outlook

The experiments establish an out-of-equilibrium U(1) Rokhsar-Kivelson state whose local properties match equilibrium critical-point predictions despite the state’s equilibrium instability. The authors identify Hamiltonian learning, Floquet engineering, and broader analog-digital control as directions for extending these results.

  • Discussion: The prepared state shows key signatures of a non-equilibrium U(1) Rokhsar-Kivelson state and matches equilibrium critical-point predictions.The state’s microscopic properties remain sensitive to the specific preparation protocol.
  • Discussion: Longer-scale deviations and the interplay of monomer density, system size, and open boundaries remain future research directions.These factors limit how completely the observed state can be compared with ideal predictions.
  • Outlook: Hamiltonian learning could infer an approximate equilibrium parent Hamiltonian, enabling studies of deformations from the ideal Rokhsar-Kivelson model.The proposal uses local measurements to obtain the parent Hamiltonian.
  • Outlook: Floquet Hamiltonian engineering and variational circuits could probe low-energy excitations and support closed-loop optimization of analog and digital controls.The proposed control loop may also incorporate coherent atom rearrangement.
  • Broader significance: Hybrid analog-digital simulation combines efficient analog many-body evolution with digital local control, error detection, and rapid experimental cycles.The approach also supports loss-based error mitigation, qubit reuse, and measurements such as two-time correlations and entanglement witnesses.
  • Related work: Related work realizing U(1) quantum spin liquids with ultracold atoms appeared during completion of this study.

METHODS

The platform coherently transfers atoms between programmable hyperfine and interacting Rydberg qubits to combine digital preparation and measurement with analog evolution. Non-destructive readout detects loss and enables reuse, while reservoir refilling maintains the array during repeated cycles.

  • Experimental system: The experiment uses 87Rb atoms in optical-tweezer arrays to implement both analog and digital evolution.The hyperfine qubit is controlled with Raman excitation, while the Rydberg qubit uses two-photon excitation.
  • Coherent mapping: Coherent mapping transfers quantum states between long-lived hyperfine and interacting Rydberg qubits for hybrid analog-digital control.This architecture is extended to larger two-dimensional arrays and includes non-destructive hyperfine-qubit readout.
  • Control sequence: Programmable local and global single-qubit gates prepare arbitrary local superpositions before analog Rydberg evolution.The sequence transfers population between hyperfine and Rydberg manifolds using fast Raman and Rydberg π-pulses.
  • Control sequence: Local and global gates enable measurements in arbitrary local bases after the analog evolution.A final Rydberg-to-hyperfine transfer maps the state for digital readout.
  • Readout and reuse: Loss-resolved non-destructive readout detects loss while retaining most atoms for reinitialization and reuse.This adds loss information without requiring atoms to be discarded after readout.
  • Atom replacement: Reservoir atoms refill missing sites after readout and evolution, supporting repeated experimental cycles.The reservoir is depleted over time, and analog evolution requires the SLM tweezers to be turned off because the Rydberg state is anti-trapped by 852-nm light.

Details of experiment configuration

The experiment combines programmable atom-array geometries, coherent mapping, loss-resolved readout, and mitigation strategies for population-transfer and phase errors. These measures support measurements in one and two dimensions while exposing limitations from long-range interactions and dephasing.

  • Array architecture: A simulation zone and an 11-by-32-site reservoir zone support programmable geometries and atom replacement after experimental cycles.The reservoir stores atoms used to refill missing sites.
  • Operating regimes: 1D experiments use n = 53, while 2D experiments use n = 70 with lattice spacing a = 6.0 µm.The 2D simulation zone is 62 µm tall and 185 µm wide.
  • Mapping and readout: Coherent mapping transfers states between hyperfine and Rydberg manifolds, but long-range vdW tails reduce population-transfer fidelity, especially in the 2D kagome lattice.The estimated upper bound on Rydberg π-pulse infidelity is ≈1.3% in 2D; these errors are detected through loss-resolved readout.
  • Error sources: Phase errors arise from vdW interaction tails, light shifts, and dephasing during coherent mapping and reduce the fidelity of off-diagonal measurements.For 2D experiments, Doppler shifts and 1013-nm light-shift fluctuations produce a random phase of ∼2π × 0.02 radians, with T*2 ≈1−2 µs.
  • 2D limitations: In 2D, larger vdW tails limit coherent-superposition mapping, while reducing tail energy or reconstructing observables with additional local measurements are proposed improvements.Comparable 1D mapping fidelity requires ≈4× reduced tail energy on the kagome lattice, implying a 25% lattice-spacing increase and slower analog evolution.
  • Loss mitigation: Loss-resolved readout detects true losses and vacancies, enabling postselection whose sample overhead scales with operator weight rather than system size.The mapping protocol also converts some initialization, blockade, and population errors into detectable loss events.

Approach to Floquet Hamiltonian engineering

The experiment uses periodic global-detuning drives to replace the native Rydberg Hamiltonian with programmable Floquet dynamics. A many-body echo supplies the baseline, while perturbations generate hopping, interactions, and spin-flip terms in a constrained manifold.

  • Floquet framework: Periodic global-detuning driving produces an effective Hamiltonian with dynamics qualitatively different from the native Rydberg Hamiltonian.Higher-order corrections can be obtained through a high-frequency Floquet-Magnus expansion.
  • Many-body echo: Equally spaced detuning π-pulses reverse interacting PXP evolution between pulses, creating a many-body echo within the blockade-constrained manifold.The blockade projector removes configurations containing simultaneously excited nearest neighbors.
  • Many-body echo: At stroboscopic times, the ideal echo has teff(nτ) = 0 and Floquet unitary UF = U(τ) = 1, producing periodic revivals.The system undergoes no net PXP evolution after each complete Floquet period.
  • Hamiltonian design: Perturbing the echo makes the time-evolved occupation operator overlap with multibody interactions, whose weights are controlled by the detuning perturbation.The resulting interacting evolution is represented by Uint ≡ exp {−iτHeff}.
  • Hamiltonian design: The engineered Floquet ansatz contains chemical-potential, next-nearest-neighbor interaction, blockaded exchange, and two spin-flip terms.The experiment targets a blockaded XX Hamiltonian through the corresponding parameterized detuning drive.
  • Corrections and limitations: Finite pulse widths renormalize the effective coefficients, while higher-order expansions and vdW tails add operators beyond the idealized PXP description.These corrections become especially relevant for stronger drive parameters and non-negligible interaction tails.

Verification of the effective Hamiltonian

The effective Hamiltonian is tested by fitting measured bitstring distributions from multiple initial states and measurement bases, then independently comparing its parameters with single-particle dispersion data.

  • Verification protocol: The verification protocol combines vacuum, single-excitation, and two-excitation preparations with Z-basis measurements and phase-sensitive superposition measurements.Four superposition states probe dynamics beyond population-only observables.
  • Hamiltonian fitting: Measured bitstring probabilities across configurations are fit by minimizing a multiparameter cost function over J, µ, U, gX, and gY.The fit uses experimentally measured probabilities at times nτ for the prepared states and measurement settings.
  • Verification results: Cmin = 1.34 · 10−4, and the fitted J and µ + 4U agree with values extracted from the single-particle dispersion.The optimal coefficients include J/2π = −0.143 MHz, µ/2π = 0.411 MHz, U/2π = 0.023 MHz, gX/2π = 0.000 MHz, and gY/2π = 0.021 MHz.
  • Many-body spectroscopy: The Green’s-function protocol evolves initial vacuum–single-excitation superpositions, measures site-resolved X-basis expectations, and Fourier transforms in time and space.Taking differences isolates the real and imaginary parts of the target Green’s function before obtaining the density of states.
  • Many-body spectroscopy: For the blockaded hopping model, the imaginary-part peak follows ω(k) = µ + 4U + 2J cos ka.This dispersion provides the comparison used to assess the fitted Hamiltonian parameters.
  • Phase calibration: Relative phases between qubit manifolds are calibrated and compensated before the four phase-sensitive measurements.The procedure includes calibration of Raman and Rydberg π/2-pulses, differential light shifts, and mean-field vdW phases.

Domain wall dynamics under Rydberg blockade

Under blockade, the constrained XX model maps to an integrable XXZ chain, where spin-wave excitations interact and weak spin flips can create domain-wall pairs on the maximally packed Néel state.

  • Spin-wave dynamics: The constrained XX model with gX,Y = 0 maps onto the integrable XXZ Heisenberg chain, whose spin-wave excitations interact.The interaction is evidenced by spin scattering in the reported dynamics.
  • Domain-wall creation: Weak spin-flip terms gX,Y create domain-wall pairs on the maximally packed Néel state |Z2⟩.The passage frames domain-wall dynamics as a consequence of allowing weak spin-flip processes.

Engineering ring-exchange dynamics

The Floquet protocol engineers effective ring-exchange and hopping dynamics from periodically driven Rydberg interactions. Optimized detuning sweeps access a hemidiabatic regime relevant to preparing Rokhsar–Kivelson-type states, while interactions and inhomogeneity limit quantitative control.

  • Engineering the effective dynamics: Floquet driving realizes ring-exchange through engineered higher-body interactions or effective processes generated from three-body interactions.The protocol uses periodic detuning profiles and optimized detuning sweeps to generate the desired effective Hamiltonians.
  • Engineering the effective dynamics: g6 ≈ 2π × 0.22 MHz predicts plaquette oscillations with period π/g6 ≈ 2.3 µs, consistent with experiment.
  • Engineering the effective dynamics: The effective Hamiltonian also contains blockaded hopping with strength J ≈ −2π × 0.23 MHz, comparable to the engineered ring-exchange strength.
  • Limitations: Long-range van der Waals tails and detuning sensitivity reduce number-conservation fidelity when extending the Floquet approach to two dimensions.
  • Preparing the quantum spin liquid: The RK state is an equal-amplitude, equal-phase superposition of dimer configurations whose diagonal correlations map to a classical dimer gas and compact-boson field theory.
  • Preparing the quantum spin liquid: Field theory predicts dimer-dimer correlations scaling as ∼1/x^2 and dimer-string correlations as ∼1/x^1/2, with a logarithmic-exponent ratio of 4 observed in data and simulations.
  • Preparing the quantum spin liquid: Optimized detuning sweeps target a hemidiabatic regime that is adiabatic for monomer excitations but sudden relative to splittings between dimer coverings.

Boundary engineering with local detunings

Programmable optical-tweezer light shifts are applied to boundary atoms to prevent edge-seeded Rydberg excitations and to generate suitable boundaries for the quantum spin liquid.

  • Boundary control: Local light shifts on boundary atoms prevent Rydberg excitations from being seeded at the array edges.
  • Boundary control: For quantum-spin-liquid experiments, boundary detunings are generated using an empirical weighting procedure.

Probability distribution of dimer configurations

The experiment probes local dimer-configuration probabilities using boundary-fixed subsystems and compares them with thermal distributions and field-theory correlators. Small subsystems show nearly uniform dimer populations, while measurement and mapping errors affect coherence estimates.

  • Subsystem populations: The number of allowed dimer configurations grows exponentially with system size, motivating analysis on sufficiently small subsystems.
  • Subsystem populations: Boundary fixing isolates subsystem configurations by postselecting snapshots whose outer connecting bonds have the desired state.
  • Subsystem populations: A thermal model pi ∝ e^−βEi fits the observed configuration probabilities using β as the only free parameter.
  • Correlation functions: Rectified dimer-dimer and dimer-string correlators are formed from endpoint pairs and averaged by radial separation after excluding the boundary layer.
  • Uncertainty estimation: Bootstrap uncertainties use 20 resamples of the full dataset for all correlation functions.
  • Coherence measurement: Mapping down under van der Waals tails reduces the simulated kinetic-to-potential-energy ratio from 0.92 to 0.76 over 100 ns.

Numerical methods

The study combines tensor-network simulations, exact diagonalization, and classical Monte Carlo to model Rydberg dynamics and Rokhsar–Kivelson observables. Worm updates efficiently sample close-packed dimer coverings and allow comparison across topological sectors.

  • Tensor-network simulations: Tensor-network simulations model the 0.8 µs preparation protocol using matrix-product operators implemented in TeNPy.
  • Tensor-network simulations: Two tensor-network models compare nearest-neighbor PXP blockade dynamics with a PXP model including truncated long-range van der Waals tails.
  • Tensor-network simulations: The simulations use strip geometries, boundary detunings, Trotter steps dt = 0.01 Ω^-1, and bond dimensions χ = 256 or χ = 350 depending on the observable.
  • Classical Monte Carlo: Classical Monte Carlo evaluates diagonal and off-diagonal RK observables by uniformly sampling dimer coverings with a worm-update Markov chain.
  • Worm updates: Each worm update temporarily creates two monomers, propagates one through local dimer pivots, and closes into a loop that produces a new covering.
  • Worm updates: The worm terminates when it reconnects to its starting vertex, flipping dimers along a closed loop and restoring close packing.
  • Topological sectors: For toroidal RK systems, four topological sectors are specified by the Z-string eigenvalues of two non-contractible loops; experiment averages over sectors because open boundaries do not define one.
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