Engineering quantum criticality and dynamics on an analog-digital simulator

TL;DR

This work uses coherent mapping between the Rydberg and hyperfine qubits in a neutral atom array simulator to engineer and probe complex quantum dynamics and paves the way for using dynamical control in analog-digital quantum simulators to study complex quantum many-body systems.

Abstract

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.

Figures (16)

• Figure 1: Analog-digital atomic processor for probing many-body quantum matter.a, Analog evolution is performed in the interacting Rydberg qubit manifold and the hyperfine spin qubit is used for digital processing. Non-destructive readout of each atom obtains spin and loss information, after which lost sites are refilled from the reservoir and the spin state is reinitialized. The circuit repeats until the reservoir is depleted. The geometry of the simulation zone is programmed for each experiment, shown here for a 271-site kagome lattice. b, Two $\pi$-pulses are used to map between the qubit manifolds, shown for mapping down from the Rydberg to the hyperfine qubit. c, Rabi oscillations on the Rydberg qubit, measured after mapping down. The contrast is 97.8(1)% and loss is $\approx 1.8\%$ (not shown), dominated by loss when turning off the tweezer traps and rearrangement infidelity as the reservoir depletes. Here and later, all data are locally postselected on no atom loss. d, Ramsey fringe showing coherence after mapping down $\ket{+}$. Atoms are located at next-nearest-neighbor sites for the 1D chain and 2D kagome lattice, emulating a blockaded many-body state. The contrast is reduced in 2D where effects of long-range Rydberg interaction tails are larger. e, Many-body states and dynamics can be engineered and probed using this zoned analog-digital architecture.
• Figure 1: Analog-digital quantum simulations.a, Level diagram showing the relevant atomic transitions in $^{87}$Rb. Raman transitions are used for single-qubit gates, the spin-dependent lattice is used for spin-to-position conversion, and the local SLM is used for local light shifts. The clock qubit $\ket{1} = \ket{F = 2, m_F=0}$ is excited to the lower-energy $m_J=-1/2$ Rydberg state via a two-photon transition. All Rydberg pair states lie higher in energy than the blockaded subspace. We suppress coupling to the nearby $m_J = +1/2$ Rydberg state, which has 3$\times$ smaller matrix element, by increasing our DC magnetic field to 11.3 G. The intermediate state detuning of +1 GHz and $n = 70$ Rydberg state is used for all 2D experiments. For the 1D chain in Figs. 2,3 we instead use $-6.3$ GHz and $n = 53$. b, Schematic experiment timeline. The total loop takes 16.4 ms for 1D experiments and 24 ms for 2D experiments; in 2D, spin-to-position conversion is performed in two groups due to limited laser power for the AOD tweezers, and image processing and rearrangement are slower for the larger lattice. c, Example pulse sequence used for analog-digital quantum simulation. Not all steps are used in all experiments. For the QSL experiments, the first Rydberg $\pi$-pulse is omitted when starting from the all-$\ket{0}$ initial state before mapping up. Gaussian Rydberg $\pi$-pulses are used to minimize coupling to the other $m_J$ state. The Rydberg Rabi frequency is reduced for the analog dynamics via the 420-nm intensity. For 1D experiments at $n=53$ and performed in AOD tweezers, the atom spacing is reconfigured from $a = 6.0\,\mu$m for local Raman rotations to $3.6\,\mu$m for Rydberg dynamics. In 2D, $a = 6.0\,\mu$m is used throughout. d, Characterization of coherent mapping. As in Fig. 1d, single qubits are arranged at next-nearest neighbor sites in 1D and 2D to emulate the vdW tails in a many-body state. Microstate-dependent phase accumulation from vdW tails occurs during the gap for the Raman $\pi$-pulse after the analog dynamics (here, a single-qubit $\pi/2$-pulse). We observe the decay in Ramsey contrast for increasing vdW tail energy, $E_{\mathrm{vdW}}$, by varying the lattice spacing in 1D (180 ns gap time) and also for increasing gap time in 2D ($2\pi \times$670 kHz tail energy at $\sqrt3 a$). Analytical formula includes only phase from vdW tails during the gap time $t$, given as $\cos^N(\varphi/2)$ for $N$ neighboring qubits and $\varphi = E_{\mathrm{vdW}}t$. Numerics simulate the entire measurement where the effect of vdW tails during the Rydberg pulses leads to a further reduction in contrast. Both curves are rescaled by the measured contrast of the hyperfine qubit without mapping and no other sources of error are included in the numerics.
• Figure 2: Hamiltonian engineering via periodic global detuning drives.a, Example detuning profiles per Floquet cycle. Gray profile implements a many-body echo via $\pi$ phase jumps of the Rydberg drive, realized as smooth detuning pulses. Perturbations are added on top of this echo to engineer an effective Hamiltonian (red profile). b, Schematic of a single Rydberg excitation initialized at the center of a 31-site chain, used as the initial state in c-e. c, Dynamics of microstate populations, grouped by Hamming distance from the initial state in the $Z$-basis and computed over the central 19 sites of the chain. Under the many-body echo, the state periodically revives, here twice per Floquet cycle (left). The Floquet drive perturbs the dynamics, illustrated by the modified evolution of the populations, to yield an effective Floquet Hamiltonian in the Rydberg qubit manifold (right). d, At stroboscopic times, the number of spin excitations is approximately conserved. e, Single-spin quantum walk under an effective blockaded XX Hamiltonian. The hopping rate is tuned via the drive parameter $\varepsilon$ (Methods). f, Detuning profile per Floquet cycle used to engineer simultaneous two-body hopping and ring-exchange dynamics, and resulting dynamics on a six-site hexagonal plaquette starting from g, a single spin and h, a Néel state. Data are averaged over 15 plaquettes evolved in parallel. The decay in contrast is primarily attributed to the sensitivity to static detuning offsets (Methods). Explicit parameterizations for a and f and the associated parameters for each plot are given in Methods.
• Figure 2: Loss detection in analog quantum simulations.a, Distribution of losses per shot in QSL experiments. Baseline losses exclude the many-body dynamics and follow a Poisson distribution (solid line). The loss increases when performing analog dynamics and a non-Poissonian tail from correlated losses appears. There are 271 atoms in total. b, Spatial correlations of loss events. The two-point connected correlator follows a dipolar $1/r^3$ decay, suggesting the presence of Rydberg $S$-$P$ interactions resulting from an atom, originally in the $S$ state, undergoing blackbody decay to a nearby $P$ state. These long-range interactions facilitate so-called avalanche events festa_blackbody-radiation-induced_2022goldschmidt_anomalous_2016boulier_spontaneous_2017 in which nearby atoms come into resonance with unwanted transitions and are also lost, explaining the tail in a. A background offset of $\approx 2.5\times10^{-5}$ is subtracted. Data in a,b are postselected on the atom being present in the prior image to remove correlations from mid-circuit rearrangement which are created when the reservoir starts to deplete. c,d Comparison of local and global loss postselection methods. Data are binned according to the number of missing atoms in each snapshot, including both losses and imperfect initial lattice filling. Postselection on no lost qubits within the local operator approximates the expectation value obtained from shots with a vanishing number of missing atoms across the entire 271-site lattice (dark curves). Randomly converting loss to qubit state $\ket{0}$ or $\ket{1}$ in post-processing strongly affects the calculated expectation value (light curves). We illustrate this for the c, three-body dimer populations and d, six-body potential energy from the QSL experiments. Dashed lines show the means of the entire dataset with local postselection. There are 285,948 shots in total and 6 shots have zero missing atoms.
• Figure 3: Dynamical probes of interactions and excitations.a, Two-spin quantum walk under an effective blockaded XX Hamiltonian. b, Taking the difference with the summed density profiles of the two individual spins reveals an effective spin-spin repulsion. For clarity, we subtract the background evolution of the vacuum initial state. c, Protocol for many-body spectroscopy of a single spin excitation on the central site, $r = 0$. In this case, the ancilla in $\ket{+}$ is replaced by preparation of the central site in four bases, $\alpha \in \{\pm X,\pm Y\}$, and classical post-processing to obtain $\mathcal{G}(r,t)$ (Methods). d, The Fourier transform, $\mathcal{G}(k,\omega)$, as a function of momentum, $k$, and frequency, $\omega$, across the 1D Brillouin zone. Left schematic illustrates the spin wave excitation. The imaginary component is the spectral function, whose peak is fit to obtain the single-particle dispersion. From this we extract $(\mu+4U) / 2\pi = 0.407(2)$ MHz and $J/2\pi=-0.140(1)$ MHz, and independently verify these coefficients using Hamiltonian learning of the dynamics of different initial states (Methods). Background evolution, appearing at $k = 0$, is subtracted.