Time-Dependent Perturbations Induce Diffusion in All States of the Harper Model

The behaviour of quantum particles in patterned landscapes, such as those described by the Harper model, typically falls into distinct categories, remaining localized, spreading diffusively, or travelling ballistically. Hiroaki Yamada from Yamada Physics Research Laboratory and Kensuke Ikeda from Ritsumeikan University, along with their colleagues, demonstrate that even these fundamentally different behaviours can be altered by carefully applied time-varying forces. Their research reveals that introducing harmonic perturbations, composed of multiple, unrelated frequencies, drives all states within the Harper model towards diffusive behaviour, regardless of their initial characteristics. This finding is significant because it suggests a universal mechanism for controlling quantum transport, potentially offering new avenues for manipulating the flow of energy and information in complex systems.

The research details how the Harper model, a mathematical representation of particle movement in a periodic potential, responds to the introduction of time-dependent harmonic perturbations. The model exhibits three distinct states, localized, diffusive, and ballistic, depending on the strength of the underlying potential. Researchers demonstrate that applying perturbations composed of multiple incommensurate frequencies drives the system from any of these initial states towards diffusive behavior as the perturbation strength increases. The team meticulously mapped the transitions between these states, creating a diagram that illustrates the parameter space governing these changes.

Disorder, Quasiperiodicity, and Dynamical Localization

This document provides an overview of research concerning Anderson localization, quasiperiodic systems, such as the Aubry-Andre and Harper models, and dynamical localization. It explores the fundamental concepts and significance of these phenomena, detailing how disorder and periodic variations influence the behavior of quantum particles. The document highlights the importance of understanding these effects for both fundamental physics and potential technological applications. Anderson localization describes the phenomenon where disorder within a material prevents electrons from moving freely, leading to a loss of conductivity.

Quasiperiodic systems, unlike traditional crystals, exhibit order without perfect repetition, described by irrational numbers and possessing unique electronic properties. Dynamical localization occurs when a time-dependent perturbation induces localization even without static disorder, a more subtle effect often observed in driven systems. Mobility edges, sharp boundaries in the energy spectrum of quasiperiodic systems, define regions of localized and delocalized states, crucially influencing transport properties. The Aubry-Andre and Harper models serve as foundational frameworks for studying localization in quasiperiodic potentials, exhibiting well-defined transitions between localized and delocalized states.

The kicked rotor provides a model system for investigating dynamical localization and chaotic behavior. Researchers commonly employ tight-binding models to describe the electronic structure of materials in conjunction with quasiperiodic potentials, often focusing on one-dimensional systems where localization is guaranteed with any amount of disorder. The document covers a broad range of research topics, including transitions between localized and extended states, analysis of the energy spectrum to identify key features, and studies of how wave packets propagate in quasiperiodic potentials. Researchers investigate driven systems and dynamical localization, explore the effects of non-Hermitian Hamiltonians, and consider the impact of interactions between particles.

Experimental efforts focus on creating and studying quasiperiodic potentials in various physical systems, such as optical lattices and atomic gases, with potential applications in quantum biology, materials science, and fundamental physics. Researchers investigate critical phenomena and universality, exploring the universal features of localization transitions and their dependence on system parameters. They also study anomalous diffusion, deviations from normal diffusion due to localization and disorder. The document implicitly suggests contributions from researchers in condensed matter physics, quantum optics, atomic physics, nonlinear dynamics, and mathematical physics, demonstrating the breadth and depth of research in this active field.

Complexity Drives Transition to Diffusive Motion

Researchers have investigated the behavior of particles within a specifically designed model system, revealing a surprising transition between distinct states of motion when subjected to external disturbances. The model, initially capable of supporting both localized states, where particles remain confined, and ballistic motion, where particles travel freely, demonstrates a shift to diffusive behavior, characterized by random wandering, when exposed to time-varying forces. This transition occurs regardless of the initial state of the particle, whether it was initially localized or moving ballistically. The research demonstrates that the key to triggering this transition is the complexity of the applied disturbance.

Applying simple, single-frequency disturbances does not induce diffusion, but introducing disturbances composed of multiple, carefully chosen frequencies, at least three, reliably drives the system towards diffusive behavior. This suggests that a certain level of “disorder” in the driving force is necessary to overcome the inherent tendencies towards localization or free movement. The team quantified this change in motion by tracking the mean square displacement of the particles, finding that it transitions from remaining constant (localized), to growing proportionally to the square of time (ballistic), to growing proportionally to time (diffusive). Interestingly, the strength of the disturbance is not the sole determinant of the transition.

Researchers observed that the transition is not simply a matter of exceeding a certain threshold of energy input. Instead, the number of frequencies composing the disturbance plays a critical role, indicating a more nuanced interplay between the system and the external force. Furthermore, the team found that the diffusion constant, a measure of how quickly particles spread out, initially increases with the strength of the disturbance, but then decreases at higher strengths, suggesting a complex relationship between the driving force and the resulting diffusive behavior. The team’s analysis extends to the critical point where the system transitions between localization and ballistic motion, revealing the possibility of a new type of transition.

They observed that the diffusion index, a measure of how the mean square displacement changes over time, can exhibit unusual behavior at this point, potentially indicating a transition between different types of diffusion. This suggests that the system is not simply moving from one diffusive state to another, but rather undergoing a more fundamental change in its diffusive properties. These findings offer new insights into the dynamics of complex systems and could have implications for understanding transport phenomena in various physical contexts.

Perturbations Induce Diffusion Across All States

The study reveals a nuanced understanding of diffusion, showing that the diffusion constant initially increases with perturbation strength but eventually decreases. Furthermore, the research suggests a transition between different types of diffusion occurring near the boundaries between localized/ballistic and diffusive phases, hinting at complex behavior at these critical points. The authors acknowledge that their calculations are performed with specific system sizes and numerical methods, which may introduce limitations, and they suggest that further investigation into the anomalous diffusion observed at the transition points would be valuable. Future work could explore the behavior of the system with different parameters or investigate the impact of varying the number and frequencies of the applied perturbations.