Probing the Crossover between Dynamical Phases with Local Correlations in a Rydberg Atom Array

TL;DR

The paper tackles the challenge of detecting non-equilibrium quantum critical dynamics by focusing on local connected correlations $C(\mathbf{r})$ in a driven Rydberg-atom Ising-like system. Using a second-order Magnus expansion, the authors derive closed-form analytic expressions for these local correlations under a detuning ramp and show a smooth AF-to-FM crossover controlled by the relation $U_c(\delta)$, with results agreeing with exact numerics and displaying robustness to finite-size effects. They demonstrate that higher-order ME terms capture extended-path contributions, reveal universal behavior of $C^{(n)}_R$ across lattice geometries, and identify finite-size thresholds for the universal form. The findings establish local correlations as practical, scalable probes for non-equilibrium critical dynamics in programmable quantum simulators, enabling experimental observation of dynamical phase behavior without global signatures like the Loschmidt echo. Overall, the work provides analytic insight, numerical validation, and a clear experimental pathway for mapping dynamical phase crossovers in Rydberg atom arrays.

Abstract

The experimental detection of non-equilibrium quantum criticality remains a challenge, as traditional signatures like dynamical quantum phase transitions rely on hard-to-measure global properties. Here, we demonstrate that local connected correlation functions provide a superior, practical means to directly probe the dynamics of magnetic order in a quenched Rydberg atom array. Using a Magnus expansion formalism, we derive analytic expressions for these correlations that capture a smooth crossover from antiferromagnetic to ferromagnetic dominance. Our analytic results, which reveal the critical parameter relationship $U_{c}(δ)$, are validated against exact numerical simulations and exhibit robustness to finite-size effects. By shifting the focus from global singularities to local correlations, our protocol establishes a direct and feasible path to observe the rich critical dynamics in scalable quantum simulators.

Figures (6)

• Figure 1: Temporal density distribution $|f(n)|^{2}$ for 6 sites with $U/h=3 \, \rm MHz$ at $t=0 \, \mu s$ (a1), $t=0.3 \, \mu s$ (a2), $t=0.5 \, \mu s$ (a3), $t=0.6 \, \mu s$ (a4), $t=0.75 \, \mu s$ (a5). (b) The quench protocol, showing the time-dependent parameters $\delta(t)$ and $\Omega(t)$ (right vertical axis), and the resulting dynamics of the nearest-neighbor correlation function $C_{R=1}$ (left vertical axis).
• Figure 2: The crossover is mapped as a function of interaction and detuning, with case studies at $\Omega/(2\pi) = 1 \, \rm MHz$ (a) and $\Omega/(2\pi) = 2 \, \rm MHz$ (b) upon quench completion. Comparison between the analytical expansion terms $C_{R=1}^{(n)}$ (colored dotted lines) and numerical results (black dotted lines) for the nearest-neighbor correlation function.
• Figure 3: Evolution of the long-range correlation function with detuning at the end of the quench. The next-nearest-neighbor $C_{R=2}^{(n)}$ (a) and next-next-nearest-neighbor $C_{R=3}^{(n)}$ (b) correlation functions are investigated at the fixed interaction strength $U/h=3\,\rm MHz$ and Rabi frequency $\Omega/2\pi = 2\, \rm MHz$. The analytical results from the first-order ($\tilde{C}_{R}^{(n)}$) and second-order ($C_{R}^{(n)}$) ME are compared against numerical simulations.
• Figure 4: Analytic expressions for the $n$-th order approximation of the nearest-neighbor correlation function, $C_{R=1}^{(n)}$, are identical for the three lattice geometries shown (1D chain, ring, bifurcate), provided the system size $N$ meets a minimum threshold. Dots indicate the lattice can be extended arbitrarily. The required minimum size increases with the order $n$: for $C_{R=1}^{(1)}$, $N \geq 2$ suffices for all geometries; for $C_{R=1}^{(2)}$, $N \geq 4$ (chain), $N \geq 3$ (ring), and $N \geq 6$ (bifurcate).
• Figure 5: Adiabaticity estimation in a two-site model. The adiabatic coefficient of the system is expressed as $\eta(T) = \frac{\hbar |\langle m | \Delta H | n \rangle|}{T |E_m - E_n|^{2}}$. With the interaction strength fixed at $U/h = 3\,\text{MHz}$, we evaluate how the adiabatic coefficient varies with the final detuning $\delta_{f}$ and the Rabi frequency $\Omega$.