Quantum Saturation of Magnetoelectric Coupling in Fe${}_3$O${}_4$ Nanoparticles

We report magnetoelectric coupling in nanoparticle assemblies that persists to temperatures over 300 times lower than in previous studies. The ME response saturates at low temperature, revealing a quantum plateau of the coupling. A field dependence analysis shows a crossover from quadratic to linear behavior, captured by a phenomenological expansion $C(B,T)\simeq C_0(T)+a_1(T)B+a_2(T)B^2$. The magnitude of the extracted quadratic coefficient, $\mathrm{\mid }{a}_2(T)\mathrm{\mid }$, follows a power law $\mathrm{\mid }{a}_2(T)\mathrm{\mid }\sim T^{-\alpha }$ with $\alpha \approx 1.15$, indicating proximity to a quantum critical regime. The observed saturation reflects a new intrinsic energy scale, distinct from finite-size or extrinsic effects. These results establish nanoparticle assemblies as a new platform for studying quantum magnetoelectric phenomena.