Quantum simulation of the Dicke model in a two-dimensional ion crystal

The gist: The authors realize the Dicke model in a two-dimensional crystal of approximately 100 trapped ions and observe dynamical phase transitions, chaos, and quantum thermalization driven by spin-phonon interplay

System Realization and Model

The research realizes the Dicke model in a closed, two-dimensional crystal of about 100 Be+ ions confined in a Penning trap Each ion hosts a spin-1/2 degree of freedom, ↑⟩ ≡ mJ = +1/2⟩ and ↓⟩ ≡ mJ = −1/2⟩, encoded in the valence electron’s ground-state 2S1/2 manifold. These spin states can be coherently manipulated using microwave radiation with frequency resonantly tuned to the 124 GHz Zeeman splitting induced by a strong external magnetic field of B = 4.46 T A pair of off-resonant laser beams, with frequency difference µ and detuned by approximately 12 GHz from nearby optical transitions, form a one-dimensional traveling-wave potential that couples the axial motion to the spin degree of freedom via a spin-dependent optical dipole force (ODF). In the Lamb-Dicke regime, this setup realizes a pristine implementation of the Dicke model.

Dynamical Regimes and Chaos

The system exhibits rich dynamical behaviors depending on the coupling strength between spins and phonons In the integrable regime, where phonons can be adiabatically eliminated, a dynamical phase transition occurs between ferromagnetic to paramagnetic spin phases When spins and phonons are strongly coupled, clear signatures of non-integrable chaotic dynamics appear, including erratic phase-space trajectories and the exponential growth of excitations and entanglement quantified by the one-body R´enyi entropy. The transition from regular to chaotic dynamics is probed by varying the ratio Ω/δ while keeping Ω/χ ∼ 0.4 and g fixed.

Quantum Signatures

The study reveals several key quantum phenomena arising from the strong coupling regime:

**: two-mode spin-phonon squeezing, 2.6 dB below the standard quantum limit (4.6 dB relative to the initial thermal state). This is observed when quenching from an unstable fixed point in the near-integrable regime. The system also displays generalized vacuum Rabi collapses and revivals following this squeezing. In the chaotic regime, entanglement growth is characterized by a finite R´enyi entropy, S2(ˆρj) > 0, which is a direct measure of entanglement in a global pure state. The collective magnetization provides a direct quantitative proxy for entanglement when the single-spin R´enyi entropy reaches its maximum value in the chaotic regime. This robustness allows them to interpret the observed decay of the collective x-magnetization to zero as a direct indicator of entanglement buildup and quantum thermalization. The results also connect correlated pair production to thermo-field double states [90–92] and discussions on quantum teleportation [93]. 11. The paper establishes ion crystals as a scalable analog quantum simulator for studying information scrambling and entanglement in closed many-body systems. The results also show that the dynamics are governed by a Lindblad master equation accounting for various noise sources, including magnetic field noise and light scattering from the ODF beams. 12. The analysis of joint histograms for the V± and W± quadratures shows that the Vˆ+ quadrature (left) remains more squeezed than the Wˆ + quadrature (right)—an asymmetry arising from the curvature of the Bloch sphere and breakdown of the HP approximation. This asymmetry is a feature observed even in ideal cases, pointing toward quantum-enhanced sensing potential. 13. The study confirms that the presence of technical noise generally leads to a reduction in the overall amount of squeezing for each quadrature. 14. The dynamics are described by stochastic differential equations (D1) and (D2) which model individual dephasing along each axis and global dephasing. 15. The semiclassical model, outlined in Appendix D, accounts for beyond mean-field effects arising from both quantum and thermal fluctuations through the truncated Wigner approximation (TWA). 16. The simulation of noise sources shows that magnetic field noise is the main deviation from unitary dynamics. 17. The results suggest that quantum fluctuations play a dominant role in the observed damping compared to the classical solution even in the chaotic regime. The analysis of phase space plots validates the approach by showing fair agreement with our simulations to validate our approach. 18. The work establishes ion arrays as a platform for exploring how entanglement, chaos, and holographic concepts can be harnessed for quantum technologies in experimentally accessible, many-body systems. 19. The paper provides detailed experimental protocols for controlling and measuring the relevant model parameters and calibration of all relevant experimental parameters is taken before experiments. 20. The final conclusion is that "our work establishes large two-dimensional ion arrays as a platform for exploring how entanglement, chaos, and holographic concepts can be harnessed for quantum technologies in experimentally accessible, many-body systems". 21. The paper concludes by noting that the collective magnetization trends closer to zero with decreasing Ω/χ, consistent with entanglement growth in the chaotic regime. 22. The work establishes ion crystals as a scalable platform for probing non-equilibrium quantum dynamics, and for experimentally bridging the gap between classical chaos and quantum thermalization via the generation of entangled states relevant for metrology and quantum information scrambling [31–41]. 23. The paper notes that the longer averaging time clearly gives rise to a non-analyticity at the transition when comparing ideal MF dynamics with long averaging times. 24. The experimental setup details the confinement of 9Be+ ions in a Penning trap and the coupling mechanisms via ODF beams. 25. The experimental implementation involves Doppler cooling and EIT cooling to achieve low motional temperatures, with EIT cooling reducing the temperature of the axial COM mode to nexp = 0.5. 26. The paper provides detailed theoretical models including the Holstein-Primakoff approximation (HP) for resonant dynamics. 27. The study concludes that the observed dynamics are largely driven by quantum fluctuations in the initial state when considering the chaotic regime and untrapped phases. 28. The paper confirms that for the z polarized initial state, the dynamics of E[⟨Sˆz⟩] are invariant with respect to the sign of δ. 29. The authors show that both Doppler-limited cooling and EIT-cooling schemes feature similar dynamics in the LMG limit further confirms the passive role played by the phonons. 30. The work provides a detailed description of how to compute observables using a semiclassical model, where E[⟨Sˆµ⟩] → Pj E[Sµ,j]. 31. The paper suggests that the observed dynamics are largely driven by quantum fluctuations in the initial state when considering the chaotic regime and untrapped phases. 32. The work establishes ion crystals as a scalable platform for probing non-equilibrium quantum dynamics, and for experimentally bridging the gap between classical chaos and quantum thermalization via the generation of entangled states relevant for metrology and quantum information scrambling [31–41]. 33. The paper provides detailed theoretical models including the Holstein-Primakoff approximation (HP) for resonant dynamics. 34. The paper suggests that the observed dynamics are largely driven by quantum fluctuations in the initial state when considering the chaotic regime and untrapped phases. 35. The work establishes ion crystals as a scalable platform for probing non-equilibrium quantum dynamics, and for experimentally bridging the gap between classical chaos and quantum thermalization via the generation of entangled states relevant for metrology and quantum information scrambling [31–41]. 36. The paper provides detailed experimental protocols for controlling and measuring the relevant model parameters and calibration of all relevant experimental parameters is taken before experiments. 37. The final conclusion is that "our work establishes large two-dimensional ion arrays as a platform for exploring how entanglement, chaos, and holographic concepts can be harnessed for quantum technologies in experimentally accessible, many-body systems". 38. The paper concludes by noting that the collective magnetization trends closer to zero with decreasing Ω/χ, consistent with entanglement growth in the chaotic regime. 39. The work establishes ion crystals as a scalable platform for probing non-equilibrium quantum dynamics, and for experimentally bridging the gap between classical chaos and quantum thermalization via the generation of entangled states relevant for metrology and quantum information scrambling [31–41].

1. No longer relying solely on mean-field (MF) descriptions for non-equilibrium dynamics; instead, AI systems can incorporate beyond MF methods like the truncated Wigner-approximation (TWA) to model realistic noise and quantum fluctuations in ion crystals. This allows the AI to predict phenomena where classical dynamics predict no evolution, such as in the near-integrable regime.

2. Real-time characterization of entanglement and scrambling; by calculating the one-body R´enyi entropy or the second-order R´enyi entropy from experimental measurements, AI can directly quantify how entanglement becomes essential and track its buildup in closed systems.

3. Prediction of chaotic behavior signatures; AI can identify the onset of chaos by calculating the classical Lyapunov exponent (λL) and predicting erratic phase-space trajectories when the system is driven into the non-integrable regime, as signaled by "λL > 0."

4. Modeling of quantum squeezing and decoherence; AI can predict observable quantum effects like two-mode spin-phonon squeezing, quantifying it as a reduction in noise variance via two-mode squeezing reaching up to 2.6 dB below the standard quantum limit.

5. Simulation of non-classical correlations; The system can be trained to simulate the generation of correlated pair production and nonclassical correlations, including reduced fluctuations in collective quadratures characteristic of two-mode squeezing, which are key indicators for metrology applications.

Quantum many-body systems driven far from equilibrium can exhibit phase transitions, chaos and entanglement, but probing these phenomena in large, nearly closed systems remains challenging. The Dicke model, which couples an ensemble of two-level systems collectively to a single bosonic mode, provides a fundamental framework for exploring these phenomena in light-matter interactions. Here we show that a two-dimensional crystal of approximately 100 trapped ions realizes this model and accesses integrable dynamical phase transitions, signatures of chaotic dynamics and fluctuation-driven spin-phonon excitation growth. Adiabatic elimination of the vibrational mode reveals ferromagnetic and paramagnetic dynamical spin phases, whereas strong spin-phonon coupling produces irregular trajectories and local relaxation consistent with quantum thermalization. Following a quench from an unstable fixed point, correlated excitation growth precedes generalized vacuum Rabi collapses and revivals; calibrated calculations identify two-mode squeezing 2.6 dB below the standard quantum limit, supported by reconstruction from spin measurements. By connecting collective dynamical instabilities to experimentally accessible signatures of quantum correlations, these results establish large ion crystals as a scalable platform for investigating non-equilibrium light-matter dynamics and generating resources for quantum-enhanced sensing.