Scalable Spin Squeezing in Power-Law Interacting XXZ Models with Disorder

Abstract

While spin squeezing has been traditionally considered in all-to-all interacting models, recent works have shown that spin squeezing can occur in systems with power-law interactions, leading to direct testing in Rydberg atoms, trapped ions, ultracold atoms and nitrogen vacancy (NV) centers in diamond. For the latter, Wu. et al. Nature 646 (2025) demonstrated that spin squeezing is heavily affected by positional disorder, reducing any capacity for a practical squeezing advantage, which requires scalability with the system size. In this Letter we explore the robustness of spin-squeezing in two-dimensional lattices with a fraction of unoccupied lattice sites. Using semi-classical modeling, we demonstrate the existence of scalable squeezing in power-law interacting XXZ models up to a disorder threshold, above which squeezing is not scalable. We produce a phase diagram for scalable squeezing, and explain its absence in the aforementioned NV experiment. Our work illustrates the maximum disorder allowed for realizing scalable spin squeezing in a host of quantum simulators, highlights a regime with substantial tolerance to disorder, and identifies controlled defect creation as a promising route for scalable squeezing in solid-state systems.

Figures (4)

• Figure 1: Squeezing parameter $\xi^2$ (blue) and magnetization $M_{xy}$ (red) vs time for different $N$ values (faded lines) in the case of $\Delta = -1$. The results correspond to vacancy probability (a) $p = 0$, (b) $p = 0.5$, (c) $p=0.75$, (d) $p = 0.85$. Darkness of lines scales with $\sqrt{N}$ over a system size range of approximately $N \in \{10^2, 10^4 \}$. Results correspond to an average over 10 disorder realizations with 6400 dTWA samples for each system.
• Figure 2: Distribution of effective interaction strengths $P(J^{\rm eff})$ for varying vacancy probability (a) $p=0.1$, (b) $p = 0.5$, (c) $p = 0.9$. (d)-(f) Corresponding spatial positions of spins (dark points) in the 2D plane for a single disorder realization. Results in (a)-(c) are obtained from lattices with $N\sim \mathcal{O}(10^3)-\mathcal{O}(10^4)$, and are averaged over 25 disorder realizations.
• Figure 3: (a) Optimal squeezing parameter $\xi_{\rm opt}^2$ vs $N$ for a range of vacancy probabilities $p$, where the colors range from $p=0$ to $p=0.85$ (legend) and we have set $\Delta = -1$. The dotted line gives the scaling for OAT: $\xi_{\rm opt}^2\sim N^{-2/3}$. Inset: $\nu$ (circles) vs $p$ and $\alpha$ vs $p$ (triangles), extracted from fits to the data in panels (a) and (b) respectively (dashed lines). Error bars indicate uncertainty of the fit. (b) Late-time magnetization $\overline{M}_{xy}$ vs system size $N$ for different $p$ values [same legend as (a)]. Data in (a)[(b)] is obtained from an average over 10 (25) disorder realizations, with 6400 (1024) dTWA samples for each system.
• Figure 4: (a) Scaling exponent $\alpha$ (color) as a function of vacancy fraction $p$ and anisotropy $\Delta$, where $\overline{M}_{xy} \sim N^{-\alpha}$. (b) Squeezing exponent $\nu$ (color) as a function of vacancy fraction $p$ and anisotropy $\Delta$, where $\xi^2 \sim N^{-\nu}.$ Blue dots correspond to estimates of the phase boundary from $\alpha$ [data in (a)], while red triangles correspond to the estimate from $\nu$ [data in (b)]. Results in (a) [(b)] correspond to an average of 25 disorder realizations and 1024 (12800) dTWA samples for each system. Error bars include sampling uncertainty and discreteness of simulated $p$-grid SM. Color data in the y-direction is linearly interpolated. For $\Delta =0$ and $\Delta = 0.5$, red triangles represent a lower bound of $p_c =0.85$ since we see scalable squeezing ($\nu >0$) for the entire $p-$grid. Hatched region in (a) indicates simulations where the magnetization has not fully converged to the steady-state, in which data is taken for the largest times available.