Percolation to Anderson Localization: Study Reveals Smooth Crossover in One-Dimensional Speckle Potentials

The behaviour of particles moving through disordered environments presents a long-standing challenge in physics, often described by either a percolation transition, where particles become trapped, or Anderson localization, where they become entirely immobile. Margaux Vrech from Institut Langevin, alongside Jan Major and Dominique Delande from Laboratoire Kastler Brossel, and their colleagues, now demonstrate a crucial connection between these two seemingly distinct phenomena. Their research investigates how particles propagate in a specific type of disordered environment, a one-dimensional ‘speckle’ potential, revealing a smooth transition between percolation-like trapping and Anderson localization. This work is significant because it explains how the characteristics of the disorder itself influence particle behaviour, predicting a unique bimodal transmission distribution not typically observed in one-dimensional systems and offering a refined understanding of transport in complex media.

Wave Transport in Disordered Atomic Systems

This research delves into the physics of disordered systems, exploring how waves propagate through random environments, including those created with ultracold atoms. Scientists investigate phenomena like Anderson localization and percolation, aiming to understand transitions between different wave behaviours, ballistic, diffusive, and localized. The work combines theoretical analysis, computer simulations, and experimental observations with cold atoms to provide a comprehensive understanding of wave propagation in complex media. Central to this research is Anderson localization, where waves become trapped in disordered potentials, suppressing diffusion.

The team draws parallels between Anderson localization and percolation theory, suggesting the transition between localized and extended states can be understood as a type of percolation process dependent on the connectivity of the disordered environment. The study focuses on systems with random disorder, such as laser speckle patterns or random potentials created with optical lattices, recognizing that the type and strength of this disorder are crucial to wave behaviour. Experiments with ultracold atoms, trapped in optical lattices, provide a highly controllable platform for studying these effects. The research also connects to random matrix theory, used to understand the statistical properties of wave functions in disordered systems.

Scientists explore different regimes of wave transport, including ballistic propagation, diffusive scattering, and localized trapping. They employ semiclassical approximations to analyze wave behaviour and investigate the quantum boomerang effect, where wave packets exhibit unusual reflection in random potentials. This work contributes to our understanding of fundamental physics and has implications for materials science, photonics, and quantum information, potentially aiding the design of materials with specific optical or electronic properties and the development of new quantum technologies.

Disordered Potential Simulates Quantum to Classical Transition

Scientists have investigated the transition between classical and quantum particle transport in disordered systems, focusing on how particles propagate through a one-dimensional “red speckle” potential, a type of random energy landscape. They employed both computer simulations and theoretical calculations to examine particle behaviour as the system shifts from fully classical to increasingly quantum-dominated regimes. A central technique was the transfer-matrix method, a computational approach used to model particle propagation by discretizing the fundamental equation of quantum mechanics onto a lattice. To ensure accuracy, the simulations used a lattice spacing carefully chosen to be smaller than both the disorder correlation length and the particle’s wavelength, guaranteeing a faithful representation of the continuous random potential.

The team generated the red-speckle potential by combining a complex random field with a specific mathematical function, creating a realistic energy landscape with controlled statistical properties. Complementing the simulations, the researchers developed a semi-classical theoretical framework to predict particle behaviour, calculating the localization length, a measure of confinement. They found that standard theoretical descriptions, based on uncorrelated disorder, break down in the red-speckle potential due to its correlated and non-Gaussian nature. Instead, the team predicted and confirmed the emergence of a bimodal transmission distribution, an unusual phenomenon in one dimension.

Quantum and Classical Crossover in Localization

Scientists have achieved a detailed understanding of particle propagation in disordered systems, revealing a surprising connection between classical and quantum behaviours. The research focuses on how particles move through a specifically designed “red speckle” potential, a type of random energy landscape, and demonstrates a crossover between two previously distinct regimes of particle transport. Traditionally, particles in random potentials are thought to either become trapped in clusters or be localized by quantum effects. This work shows that these two behaviours connect in a non-trivial way. The team measured the localization length, a key indicator of how confined a particle is, as they varied the strength of quantum effects.

They discovered that as the system transitions from the classical limit, the behaviour characteristic of classical trapping smoothly connects to a continuous increase in the localization length. This connection was confirmed through both computer simulations and a semi-classical theoretical approach. The simulations, employing a transfer-matrix method, tracked particle propagation through the red speckle potential, revealing that the localization length does not converge to expected classical values, but instead vanishes as quantum effects diminish. Further analysis revealed a bimodal transmission distribution, suggesting that particles are not simply scattered randomly, but exhibit a distinct pattern of either being transmitted or blocked.

Disorder Correlates to Localization Length Transition

This work investigates the transition between classical percolation and Anderson localization, examining particle propagation in a one-dimensional disordered potential. Researchers demonstrate that as the system moves away from purely classical behaviour, the behaviour characteristic of classical percolation gradually transforms into a continuous increase in the localization length, which describes how confined a particle becomes. This crossover is governed by an effective Planck constant, reflecting the relationship between particle wavelength and the scale of the disorder. The study reveals that the standard theoretical framework for Anderson localization, typically applied to uncorrelated disorder, breaks down in this regime due to the correlated and non-Gaussian nature of the disordered potential. Instead, a semi-classical approach successfully describes the localization length and predicts the emergence of a bimodal transmission distribution, a behaviour not usually observed in one-dimensional systems with uncorrelated disorder. The researchers performed extensive transfer-matrix simulations, calculating the average transmission through the disordered potential for a range of system sizes and quantum strengths, accurately reproducing the behaviour predicted by the semi-classical theory.