Anomalous Mean-Squared Displacement in Quantum Active Matter from a Wigner Phase-Space Framework

Abstract

Active matter is driven out of equilibrium by a local influx of energy. While classical active matter has been extensively studied, the extension of active matter concepts to quantum systems has been explored far less. In this work we develop a quantum description based on the Wigner function for a quantum system influenced by a classical noise. For the model considered here, the hybrid dynamics reduces to an Ornstein--Uhlenbeck process, enabling us to compute the quantum mean-squared displacement (MSD) using established techniques from classical active matter. We analytically derive the time dependence of the MSD and clarify the conditions under which the characteristic scaling with time MSD $t^6$ emerges, namely the regime of long persistence time and large active noise strength. We also show that, for certain parameter and initial conditions, the MSD can exhibit an even steeper scaling regime MSD $t^7$. In addition, explicit expressions are derived that precisely predict the onset times of $t^6$ and $t^7$ scaling behaviors. Finally, we examine the robustness of these behaviors against quantum fluctuations of the initial state.