Algebraic law of local correlations in a driven Rydberg atomic system

TL;DR

The paper addresses how antiferromagnetic correlations build up in driven, nonuniform Rydberg lattices during quenches through an Ising-like phase transition. It combines a physically accurate effective two-level Rydberg model with a second-order Magnus expansion to derive analytical expressions for local correlations that are then validated by full Schrödinger-evolution simulations. A key finding is a universal path superposition principle: the AF correlation magnitude at Manhattan distance $R$ is the algebraic sum of contributions from all shortest paths, robust to lattice geometry and quench protocol variations. This principle provides a practical framework for understanding correlation propagation in quantum simulators and suggests generalizations to other strongly correlated systems and driving schemes.

Abstract

Understanding the mechanism behind the buildup of inner correlations is crucial for studying nonequilibrium dynamics in complex, strongly interacting many-body systems. Here we investigate both analytically and numerically the buildup of antiferromagnetic (AF) correlations in a dynamically tuned Ising model with various geometries, realized in a Rydberg atomic system. Through second-order Magnus expansion (ME), we demonstrate quantitative agreement with numerical simulations for diverse configurations including $2 \times n$ lattice and cyclic lattice with a star. We find that the AF correlation magnitude at fixed Manhattan distance obeys a universal superposition principle: It corresponds to the algebraic sum of contributions from all shortest paths. This superposition law remains robust against variations in path equivalence, lattice geometries, and quench protocols, establishing a new paradigm for correlation propagation in quantum simulators.

Figures (9)

• Figure 1: (a) Schematic description of the lattice, and atoms are excited from the ground state $|g\rangle$ to the Rydberg state $|r\rangle$ through a two-photon process. (b) The protocol of Rabi frequency $\Omega(t)$ and detunning $\delta(t)$.
• Figure 2: Superposition principle validation for the correlations in square lattice. Analytic verification of the next-nearest-neighbor correlation superposition with parameters referred to the experiment Lienhard2018: $T = 0.5 \mu s$, $U_{1}/h = 2.8\, \rm MHz$, $U_{2}/h = 1.4\, \rm MHz$, $\Omega/(2\pi) = 0.8\, \rm MHz$, $\delta_{0}/(2\pi) = -6\, \rm MHz$, and $\delta_{f}/(2\pi) \in [1,5]\, \rm MHz$. The near-perfect overlap between $C_{11}$ (blue solid line) and $2 \times C_{20}$ (red dashed line) confirms path contribution additivity.
• Figure 3: The buildup of antiferromagnetic correlation on $2 \times n$ lattice. Schematics of $2 \times n$ Rydberg array (a) and equivalent 1D chain capturing single-path contributions to $C_{11, \rm square}$ (b). The nearest-neighbor correlation $C_{R=1}$ (c) and the next-nearest-neighbor correlations $C_{R=2}$ (d) as the function of $\delta_{f}$. The results of numerically solving Schrödinger equation for Hamiltonian (\ref{['eq1']}) for $2 \times n$ lattice (yellow circles; green triangles) are compared with the analytic ones on the local lattice geometries (blue solid; red dashed).
• Figure 4: The buildup of antiferromagnetic correlation on cyclic lattice with a star. Schematics of cyclic lattice with a star (a) and equivalent 1D chains according to single-path contributions in the correlation $C_{20, \rm cyclic}$ (b). The next-nearest-neighbor correlations $C_{20, \rm cyclic}$ (c) and $C_{20, \rm star}$ (d) as the function of $\delta_{f}$. The results of numerically solving Schrödinger equation for Hamiltonian (\ref{['eq1']}) for cyclic lattice with a star (yellow circles; green triangles) are compared with the analytic ones on the local lattice geometries (blue solid; red dashed).
• Figure 5: The buildup of antiferromagnetic correlation on hexagonal and octagonal lattices. Schematic descriptions of hexagon lattice (a), octagon lattice (b) and corresponding single-path contribution. The third-nearest-neighbor correlation $C_{30,\rm hexagon}$ (c) and the fourth-nearest-neighbor correlation $C_{40,\rm octagon}$ (d) as the function of $\delta_{f}$. The yellow dotted lines with circle and the green dotted lines with triangle are numerical results.