Quantum simulation of the Dicke model in a two-dimensional ion crystal: chaos, quantum thermalization, and revivals

TL;DR

The paper reports a quantum simulation of the Dicke model in a two-dimensional ion crystal of ~100 Be$^{+}$ ions, enabling controlled exploration of non-equilibrium light-matter dynamics. By tuning the detuning and spin-phonon coupling, the authors map regimes to the integrable LMG limit and observe a dynamical phase transition, while entering strongly coupled, non-integrable regimes that exhibit chaos and rapid entanglement growth. They demonstrate two-mode spin-phonon squeezing and long-time coherence via vacuum-Rabi–like collapses and revivals, and quantify entanglement through the Rényi entropy, establishing a scalable platform for studying information scrambling and quantum thermalization in closed many-body systems. The results illuminate how quantum fluctuations drive correlated spin-phonon dynamics and provide a path toward metrological and quantum-information applications in large hybrid quantum systems.

Abstract

Quantum many-body systems driven far from equilibrium can exhibit chaos, entanglement, and non-classical correlations, yet directly observing these phenomena in large, closed quantum systems remains challenging. Here we realize the Dicke model -- a fundamental description of light-matter interactions -- in a two-dimensional crystal of approximately 100 trapped ions. The ions' internal state is optically coupled to the center of mass vibrational mode via an optical spin-dependent force, enabling unitary many-body dynamics beyond the mean-field and few-body limits. In the integrable regime, where the phonons can be adiabatically eliminated, we observe a dynamical phase transition between ferromagnetic to paramagnetic spin phases. In contrast, when the spins and phonons are strongly coupled, we observe clear signatures of non-integrable chaotic dynamics, including erratic phase-space trajectories and the exponential growth of excitations and entanglement quantified by the one-body Rényi entropy. By quenching from an unstable fixed point in the near-integrable regime, quantum noise can generate correlated spin-phonon excitations. Our numerical calculations, in clear agreement with experiment, reveal the generation of two-mode spin-phonon squeezing, 2.6 dB below the standard quantum limit (4.6 dB relative to the initial thermal state), followed by generalized vacuum Rabi collapses and revivals. Our results establish large ion crystals as scalable analog quantum simulators of non-equilibrium light-matter dynamics and provide a controlled platform for experimental studies of information scrambling and entanglement in closed many-body systems.

Figures (13)

• Figure 1: Experimental setup and dynamical regimes. (a) $N\sim 10^2$ beryllium ions (red dots) are confined in a Penning trap, with an axial magnetic field $B=4.46$ T. The Doppler-cooled ions form a 2D crystal, whose axial center-of-mass (COM) motion realizes a high-Q mechanical oscillator with frequency $\omega_Z$ (teal spring), and which may be further cooled by crossed electromagnetically induced transparency (EIT) beams (blue). Phonon excitations of this COM motion collectively couple to the ions' valence electronic spin at rate $g$, induced by a spin-dependent optical dipole force (ODF) from crossed optical beams (yellow) with relative wavevector $\Delta \pmb{k}$ and frequency difference $\mu \equiv \delta + \omega_Z$. The spins can also be globally addressed by microwaves (MW, purple) with Rabi rate $\Omega$. (b) Expected time-averaged magnetization and (c) classical Lyapunov exponent, $\lambda_L$, for spins initially polarized along $-z$, as a function of $\Omega/\delta$ and $\Omega/\chi$, where $\chi\equiv 4g^2/\delta$; inset to (b) shows corresponding MF results. We consider two different experimental parameter sweeps: an "LMG cut" (pink line)---dominated by an effective, integrable LMG spin-spin interaction---which traverses a dynamical phase transition (cyan line) between spin-trapped and spin-untrapped behaviors; and a "chaotic cut" (green line), where the phonons play a more active role and can enable non-integrable chaotic behavior. Colored symbols label parameters considered later in the text, and we also show examples of corresponding classical dynamical trajectories of (d) the spins on the collective Bloch sphere and (e) the COM motion in canonical phase space ($\mathcal{X},\mathcal{P}$). (f) Analogous time-averaged transverse magnetization and experimental sweep ("resonant cut", orange line and symbols) for spins initially polarized along $-x$, which exhibits non-trivial dynamics close to resonance ($\Omega\sim \delta$), driven by quantum fluctuations for an initial bosonic vacuum state. Corresponding MF results in inset exhibit no dynamics. For $\Omega\gg \chi$, the build-up of entanglement manifests in the form of pair production of correlated spin-phonon excitations illustrated in (g). The spin excitations correspond to fully symmetrized spin flips along $x$, which occur together with a phonon excitation. This paired generation leads to exponential growth of mode populations from vacuum or thermal fluctuations and produces nonclassical correlations, including reduced fluctuations in collective quadratures characteristic of two-mode squeezing.
• Figure 2: Dynamical phase transition in the integrable, spin-dominated limit (see pink "LMG cut" in Fig.1 and corresponding pink symbols). (a-c) Representative time traces of $\braket{\hat{S}_z(t)}$ for initial EIT-cooled phonons and $-z$ polarized spins, following a variable time pulse of the ODF beams and MW source to simulate an LMG model (see sketch, top right). We compare against semiclassical (blue-solid) and mean-field (MF) models (lavender-dashed-dot), including relevant experimental details. (d) Magnetization averaged over $3$ ms of evolution for initial EIT (blue, $N\in [86,167]$) and Doppler (orange, $N\in [57,79]$) cooling, compared to semiclassical (blue solid, $\overline{n} = 0.5$; orange-dashed, $\overline{n} = 5.0$) and MF (lavender dashed-dot) results with $N=100$; idealized MF results are also shown (black dotted), averaged over $0-20$ ms. (e) We also present corresponding semiclassical predictions for the time-averaged phonon occupancy, not directly accessible in the experiment. Bloch sphere/phase space schematics next to pink symbols correspond to the parameters and behaviors outlined in Fig. \ref{['fig:Fig1']}b-e. $\Omega/\delta\in [0.12,0.13]$ and $g \in 2\pi \times [0.93,1.00]$ kHz throughout. Error bars denote standard-error of the mean.
• Figure 3: Non-integrable dynamics and chaos in the Dicke model (see green cut and corresponding symbols in Fig.\ref{['fig:Fig1']}). (a-c) Representative spin dynamics for initial EIT (blue) and Doppler (orange) cooling and $-z$ polarized spins, following an analogous sequence to Fig. \ref{['fig:Fig2']} to simulate the Dicke model (see also sketch, top right), compared against semiclassical (blue/orange solid) and MF models (lavender dashed-dot), including relevant experimental details. (d) Magnetization averaged over $3$ ms of evolution, with filled (unfilled) symbols corresponding to results with $N\in [51,108]$ ($N\in [159,178]$). We compare to semiclassical (blue solid, $\overline{n} = 0.5$; orange dashed, $\overline{n} = 5.0$) and MF (lavender dashed-dot) results with $N=100$, (e) and also show the semiclassically predicted time-averaged phonon occupancy. Bloch sphere and phase space diagrams associated with green symbols display typical classical behaviors and corresponding parameters outline in Fig. \ref{['fig:Fig1']}b-e. (f-h) Experimental phase space dynamics corresponding to (a-c) of the mean magnetization (with EIT-cooled initial phonons), with color denoting the elapsed time. $\braket{\hat{S}_y(t)}$ is computed from the time-derivative of smoothed dynamics, see Ref. SM for details. $\Omega/\chi \in [0.41,0.47]$ and $g\in 2\pi\times [1.07,1.15]$ kHz throughout (for unfilled symbols, $g \approx 2\pi \times 0.88$ kHz). Error bars denote standard-error of the mean.
• Figure 4: Fluctuation-driven, unstable fixed-point dynamics in the resonant spin-phonon regime (see orange trace Fig. \ref{['fig:Fig1']}f, and corresponding orange symbols). (a-b) Representative transverse spin dynamics for initial EIT-cooled phonons and $-x$ polarized spins, using a modification of the pulse sequence in Fig. \ref{['fig:Fig2']} to prepare and access the transverse magnetization (see sketch, top). We compare against our semiclassical (blue-solid) and MF models (lavender dashed-dot), including relevant experimental details, as well as analytic results from a large-$N$ expansion (green dotted). (c) We also compare to a classical model accounting for classical correlations from different noise sources, with different levels of added initial thermal noise in the phonons, quantified as multiples of measured occupancy $\overline{n}_{\rm exp}\sim 0.5(2)$. $\Omega/\delta\in [0.89,1.02]$ and $g\in 2\pi\times [0.87,0.91]$ kHz throughout, and $N\in [89,112]$. Error bars denote standard-error of the mean.
• Figure 5: Pair production and entanglement in the resonant spin-phonon regime. (a) Time-averaged transverse magnetization from results in Fig. \ref{['fig:Fig4']}, compared to our semiclassical model (blue solid). Error bars denote standard-error of the mean. (b-c) Representative dynamics of the single-qubit Rényi entropy, compared to our semiclassical model (blue solid). Maximum entropy indicated by black dotted line. (d) Simulated dynamics of composite $\mathcal{V}_{+}$ quadrature (ideal results in gray), showing noise reduction relative to the vacuum state and also the initial EIT-cooled thermal state. We also plot the corresponding minimum achieved variance $\mathcal{V}_+$ versus $\Omega/\chi$ in panel (a); reference value for an uncorrelated vacuum phonon state displayed in red. (e-f) Semiclassical prediction for joint spin-phonon quadrature histograms, simulating $5\times 10^3$ measurements.