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arXiv:2604.22600

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

Sangyun Lee, Yehor Tuchkov, Alexander P. Antonov, Benno Liebchen, Hartmut Löwen, Giovanna Morigi, Michael te Vrugt

cond-mat.softquant-phSubmitted Apr 24, 2026• Announced Apr 27, 2026• 0 citations
v1 · Apr 24, 2026v2 · Jul 3, 2026v3 · Sep 15, 2026
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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 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.

Comments
24 pages, 9 figures

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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 MSD∼t6\mathrm{MSD}\sim t^{6}MSD∼t6 emerges. We further show that, for certain parameter and initial conditions, the MSD can exhibit an even steeper scaling regime MSD∼t7\mathrm{MSD}\sim t^{7}MSD∼t7, and we examine the robustness of these behaviors against quantum fluctuations of the initial state.
Active matter is characterized by a constant influx and dissipation of energy that gives rise to directed motion. Dissipation requires interactions with an external environment, such that extending the paradigm of active matter to a quantum framework requires an appropriate description of this environment. In this work, we consider a driven quantum particle undergoing noise and dissipation, with external driving exhibiting characteristics of classical activity. We model the non-unitary dynamics with time-local master equations and analyze the particle motion at different time scales for different forms of the master equations, satisfying different criteria. We systematically compare predictions on the dynamics of particle trajectories and thereby we uncover how the particle motion evolves under the interplay of quantum effects, dissipation, and active-like dynamics. These results are essential for guiding possible experiments aimed at realizing quantum analogues of classical active systems.
We introduce a framework for engineering active quantum matter that involves mimicking the role of self-propulsion through an external trapping potential that is moving along imposed trajectories traced by classical active dynamics. This approach in the presence of dissipation not only recovers essential dynamical behavior of classical activity, including the ballistic to diffusive crossover of its mean-square displacement, but also reveals additional features of activity that are of quantum origin. These quantum-active features are revealed in nondissipative systems, and manifest as novel exponents of the mean-square displacement at short timescales.
Suspended carbon nanotubes provide a useful platform for studying controlled nanomechanical motion in regimes where environmental fluctuations, memory effects, and quantum coherence can all influence the dynamics. We model the fundamental flexural mode of a carbon nanotube driven by an atomic force microscope as a harmonic quantum Brownian oscillator subject to a prescribed classical force and coupled linearly to a thermal environment. Starting from the generalized Langevin equation, we formulate the reduced dynamics directly in phase space, where the Wigner function evolves through a Gaussian propagator determined by dissipative linear transport, a deterministic force-induced displacement, and bath-induced fluctuations. For a Drude memory kernel, the retarded response is expressed through the three roots of a characteristic cubic equation. In the classical-noise limit, the three independent entries of the finite-time covariance matrix reduce to explicit finite sums over these response poles, eliminating numerical double-time covariance integration. We extend the construction to the full quantum fluctuation spectrum, obtaining a thermal frequency series representation that retains zero-point and low-temperature fluctuations. For an initial two-packet superposition, we examine interference attenuation and the integrated negative part of the Wigner function, which provides a phase space measure of nonclassicality. These results provide an explicit analytical framework for finite-memory quantum Brownian motion of bound oscillators and a tractable phase space description of driven carbon nanotube dynamics relevant to future quantum state preparation and quantum tomography experiments.
Quantum fluctuations are inherent in open quantum systems and they affect not only the statistical properties of the initial state but also the time evolution of the system. Using a generic minimal model, we show that quantum noise smooths the transition between a stationary and a dynamical phase corresponding to a limit cycle (LC) in the semiclassical mean-field approximation of a generic open quantum system. Employing truncated Wigner approximation, we show that the inherent quantum noise pushes the system to exhibit signatures of LCs for interaction strengths lower than the critical value predicted by the standard mean-field theory, suggesting a noise-induced emergence of temporal ordering. Our work demonstrates that the apparent crossover-like behavior between stationary phases brought by finite-size effects from quantum fluctuations also apply to transitions involving dynamical phases. To demonstrate this on a specific physical system, we consider a transversely pumped atom-cavity setup, wherein LCs have been observed and identified as continuous time crystals. We compare the oscillation frequencies of the LCs in the one-dimensional (1D) and two-dimensional regimes, and find that the frequencies have larger shot-to-shot fluctuations in 1D. This has an important consequence in the effectiveness of entrainment of LCs for a periodically driven pump intensity or light-matter coupling strength.
Quantum diffusion is a major topic in condensed-matter physics, and the Caldeira-Leggett model has been one of the most successful approaches to study this phenomenon. Here, we generalize this model by coupling the bath to the system through a Liouville fractional derivative. The Liouville fractional Langevin equation is then derived in the classical regime, without imposing a non-Ohmic macroscopic spectral function for the bath. By investigating the short- and long-time behavior of the mean squared displacement, we show that this model is able to describe a large variety of anomalous diffusion. Indeed, we find ballistic, sub-ballistic, and super-ballistic behavior for short times, whereas for long times, we find saturation and sub- and super-diffusion.
The work derives the quantum evolution in a fluctuating vacuum by introducing the related (dark) mass density noise into the Madelung quantum hydrodynamic model. The paper shows that the classical dynamics can spontaneously emerge on the cosmological scale allowing the realization of the classical system-environment super system. The work shows that the dark matter-induced noise is not spatially white and owns a well defined correlation function with the intrinsic vacuum physical length given by the De Broglie one. The resulting model, in the case of microscopic systems, whose dimension is much smaller than the De Broglie length, leads to the Langevin-Schrodinger equation whose friction coefficient is not constant. The derivation puts in evidence the range of application of the Langevin-Schrodinger equation and the approximations inherent to its foundation. The work shows that the classical physics can be achieved in a description whose length scale is much bigger both than the De Broglie length and the quantum potential range of interaction. The model shows that the quantum-to-classical transition is not possible in linear systems, and defines the long-distance characteristics as well as the range of interaction of the non-local quantum potential in order to have a coarse-grained large-scale classical phase. The theory also shows that the process of measurement (by a large-scale classical observer) satisfies the minimum uncertainty conditions if interactions and information do not travel faster than the light speed, reconciling the quantum entanglement with the relativistic macroscopic locality.