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

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 full quantum description based on the Wigner function. By introducing a hybrid Wigner master equation that incorporates classical active motion and quantum degrees of freedom, we 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 $\mathrm{MSD}\sim t^{6}$ emerges. We further show that, for certain parameter and initial conditions, the MSD can exhibit an even steeper scaling regime $\mathrm{MSD}\sim t^{7}$, and we examine the robustness of these behaviors against quantum fluctuations of the initial state.