We explore numerically, analytically, and experimentally the relationship between quasi-normal modes (QNMs) and transmission resonance (TR) peaks in the transmission spectrum of one-dimensional (1D) and quasi-1D open disordered systems. It is shown that for weak disorder there exist two types of the eigenstates: ordinary QNMs which are associated with a TR, and hidden QNMs which do not exhibit peaks in transmission or within the sample. The distinctive feature of the hidden modes is that unlike ordinary ones, their lifetimes remain constant in a wide range of the strength of disorder. In this range, the averaged ratio of the number of transmission peaks $N_{\rm res}$ to the number of QNMs $N_{\rm mod}$, $N_{\rm res}/N_{\rm mod}$, is insensitive to the type and degree of disorder and is close to the value $\sqrt{2/5}$, which we derive analytically in the weak-scattering approximation. The physical nature of the hidden modes is illustrated in simple examples with a few scatterers. The analogy between ordinary and hidden QNMs and the segregation of superradiant states and trapped modes is discussed. When the coupling to the environment is tuned by an external edge reflectors, the superradiace transition is reproduced. Hidden modes have been also found in microwave measurements in quasi-1D open disordered samples. The microwave measurements and modal analysis of transmission in the crossover to localization in quasi-1D systems give a ratio of $N_{\rm res}/N_{\rm mod}$ close to $\sqrt{2/5}$. In diffusive quasi-1D samples, however, $N_{\rm res}/N_{\rm mod}$ falls as the effective number of transmission eigenchannels $M$ increases. Once $N_{\rm mod}$ is divided by $M$, however, the ratio $N_{\rm res}/N_{\rm mod}$ is close to the ratio found in 1D.
The local distribution of exciton levels in disordered cyanine-dye-based molecular nano-aggregates has been elucidated using fluorescence line narrowing spectroscopy. The observation of a Wigner-Dyson-type level spacing distribution provides direct evidence of the existence of level repulsion of strongly overlapping states in the molecular wires, which is important for the understanding of the level statistics, and therefore the `functional properties, of a large variety of nano-confined systems.
Starting from first principles, we formulate a theory of wave packet propagation in a nonlinear, disordered medium of any dimension, through the derivation of a Fokker-Planck transport equation. Our theory is based on a diagrammatic expansion of the wave packet's density, and is supported by a heuristic picture that involves a Boltzmann equation with an effective, external potential. Our approach also confirms results obtained in previous work for two-dimensional, nonlinear disordered media.
The conventional single-particle Monte Carlo simulation of charge transport in disordered media is based on the truncated density of localized states (DOLS) which benefits from very short time execution. Although this model successfully clarifies the properties of electron transport in moderately disordered media, it overestimates the electron diffusion coefficient for strongly disordered media. The origin of this deviation is discussed in terms of zero-temperature approximation in the truncated DOLS and the ignorance of spatial occupation of localized states. Here, based on the multiple-trapp…
The aging effects of a 'diffusing glass', a single elastic string diffusing in a two-dimensional disordered media assisted by thermal noise are studied by Monte Carlo (MC) simulations. We find for the first time convincing numerical evidence of non-trivial aging effects both in the linear response against transverse force and the associated correlation function. Our results retain some important predictions of the conventional dynamical mean-field theory (MFT) but also reveal remarkable differences from it.
We study the kinetic roughening of a driven domain wall between spin-up and spin-down domains for a model with non-conserved order parameter and quenched disorder. To understand the scaling behavior of this interface we construct an equation of motion and study it theoretically.
The propagation of classical waves in the presence of a disordered medium is studied. We consider wave pulses containing a broad range of frequencies in terms of the configurationally averaged Green function of the system. Damped oscillations in the time-dependent response trailing behind the direct arrival of the pulse (coda) are predicted, the periods of which are governed by the density of scatterers.
We provide some analytical tests of the density of states estimation from the "localization landscape" approach of Ref. [Phys. Rev. Lett. 116, 056602 (2016)]. We consider two different solvable models for which we obtain the distribution of the landscape function and argue that the precise spectral singularities are not reproduced by the estimation of the landscape approach.
We analyze mechanisms and regimes of wave packet spreading in nonlinear disordered media. We predict that wave packets can spread in two regimes of strong and weak chaos. We discuss resonance probabilities, nonlinear diffusion equations, and predict a dynamical crossover from strong to weak chaos. The crossover is controlled by the ratio of nonlinear frequency shifts and the average eigenvalue spacing of eigenstates of the linear equations within one localization volume. We consider generalized models in higher lattice dimensions and obtain critical values for the nonlinearity power, the dimen…
A theoretical analysis of the statistical distributions of the reflected intensities from random media is presented. We use random matrix theory to analytically deduce the probability densities in the localization regime. Numerical calculations of the coupling to backward modes in surface corrugated waveguides are also put forward for comparison. Interestingly, the speckle distributions are found to be independent of the transport regime. Despite the scattering being highly non-isotropic, the predicted probability densities reproduce accurately the numerical results.
We address the issue of whether amplification, like absorption, suppresses wave transmission at large gain, as has been claimed in previous studies of wave propagation in active random media. A closer examination reveals that the paradoxical symmetry between absorption and amplification is an artifact of unphysical solutions from the time-independent wave equation. Solutions from the time-dependent equation demonstrate clearly that when gain is above the threshold, the amplitude of both the transmitted and the reflected wave actually increases with time, apparently without bound. The implicati…
The polaron to bipolaron transition in disordered media under the influence of a broadband ac electric field is suggested: ac conductivity vs. frequency measurements in disordered media with inhomogeneous disorder induce spatio-temporal fluctuations of the density of polarons yielding polaron to bipolaron transformation. The external field results in the transition, alternatively to an increasing doping level. The assertion is confirmed by performing successive complex impedance measurements in disordered media. A systematic increase of the real part of the ac conductivity in the frequency dom…
Optical scattering strength of fractal optical disordered media with varying fractal dimension is reported. The diffusion limited aggregation (DLA) technique is used to generate fractal samples in 2D and 3D, and fractal dimensions are calculated using the box-counting method. The degree of structural disorder of these samples are calculated using their light localization strength, using the inverse participation ratio (IPR) analyses of the optical eigenfunctions. Results show non-monotonous behavior of the disorder-induced scattering strength with the fractal dimension, attributed to the compe…
We study the fluid flow through disordered porous media by numerically solving the complete set of the Navier-Stokes equations in a two dimensional lattice with a spatially random distribution of solid obstacles (plaquettes). We simulate viscous and non-viscous flow through these idealized pore spaces to determine the origin of the deviations from the classical Darcy's law behavior. Due to the non-linear contribution of inertia to the transport of momentum at the pore scale, we observe a typical departure from Darcy's law at sufficiently high Reynolds numbers. Moreover, we show that the classi…
The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Fina…
We present a detailed, microscopic transport theory for light in strongly scattering disordered systems whose constituent materials exhibit linear absorption or gain. Starting from Maxwell's equations, we derive general expressions for transport quantities such as energy transport velocity, transport mean free path, diffusion coefficient, and absorption/gain length. The approach is based on a fully vectorial treatment of the generalized kinetic equation and utilizes an exact Ward identity (WI). While for loss- and gainless media the WI reflects local energy conservation, the effects of absorpt…
We show that the common local Monte Carlo rules used to simulate the motion of driven flux lines in disordered media cannot capture the interplay between elasticity and disorder which lies at the heart of these systems. We therefore discuss a class of generalized Monte Carlo algorithms where an arbitrary number of line elements may move at the same time. We prove that all these dynamical rules have the same value of the critical force and possess phase spaces made up of a single ergodic component. A variant Monte Carlo algorithm allows to compute the critical force of a sample in a single pass…
We present a diagrammatic theory for coherent backscattering from disordered dilute media in the nonlinear regime. The approach is non-perturbative in the strength of the nonlinearity. We show that the coherent backscattering enhancement factor is strongly affected by the nonlinearity, and corroborate these results by numerical simulations. Our theory can be applied to several physical scenarios like scattering of light in nonlinear Kerr media, or propagation of matter waves in disordered potentials.
Correlated, or extended, impurities play an important role in the transport properties of dirty metals. Here, we examine, in the framework of a tight-binding lattice, the transmission of a single electron through an array of correlated impurities. In particular we show that particles transmit through an impurity array in identical fashion, regardless of the direction of transversal. The demonstration of this fact is straightforward in the continuum limit, but requires a detailed proof for the discrete lattice. We also briefly demonstrate and discuss the time evolution of these scattering state…
We show that the entanglement of an initially frequency-entangled photon pair propagating through a disordered scattering medium can be read off in transmission from the coincidence counting rate. The very same quantity also encodes the Thouless time of the medium.
We study the time delay of reflected and transmitted waves in 1D disordered media with high transmission. Highly transparent and translucent random media are found in nature or can be synthetically produced. We perform numerical simulations of microwaves propagating in disordered waveguides to show that reflection amplitudes are described by complex Gaussian random variables with the remarkable consequence that the time-delay statistics in reflection of 1D disordered media are described as in random media in the diffusive regime. For transmitted waves, we show numerically that the time delay i…
The backscattering line shape is analytically predicted for thick disordered medium films where, remarkably, the medium configuration is periodic along the direction perpendicular to the incident light. A blunt triangular peak is found to emerge on the sharp top. The phenomenon roots in the coexistence of quasi-1D localization and 2D extended states.
A free massive scalar field in inhomogeneous random media is investigated. The coefficients of the Klein-Gordon equation are taken to be random functions of the spatial coordinates. The case of an annealed-like disordered medium, modeled by centered stationary and Gaussian processes, is analyzed. After performing the averages over the random functions, we obtain the two-point causal Green's function of the model up to one-loop. The disordered scalar quantum field theory becomes qualitatively similar to a $λϕ^{4}$ self-interacting theory with a frequency-dependent coupling.
We investigate the transport of a passive tracer in a two-dimensional stratified random medium with flow parallel and perpendicular to the strata. Assuming a Gaussian random flow with a Gaussian correlation function, it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterising the tracer transport, but also to investigate the self-averaging properties of these quantities. We give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time. As it turns out, the sample to sample fluctuation…
We review our recent theoretical results on multiple scattering of waves in nonlinear disordered media. In particular, we discuss angular correlations of scattered waves, coherent backscattering cone, and the new phenomenon of temporal instability of multiply scattered waves in disordered media with sufficiently strong nonlinearity. Instability threshlod is derived, dynamics of speckle pattern beyond the threshold is discussed.
Understanding light-matter interaction in disordered photonic media allows people to manipulate light scattering and achieve exciting applications using seemingly scrambled media. As the concentration of scattering particles rises, they are inclined to step into near fields of each other in deep subwavelength scale. The fundamental physics involving the interplay between disorder and near-field interaction (NFI) is still not fully understood. We theoretically examine the role of NFI by analyzing the underlying multiple scattering mechanism. We find NFI leads to a stronger collective behavior i…
We consider pairs of non-interacting quantum particles transmitted through a disordered medium, with emphasis on the role of their quantum statistics. It is shown that particle-number correlations measured in transmission are strikingly sensitive to the quantum nature of the particles when they undergo Anderson localisation, due to bosonic bunching and fermionic anti-bunching in the scattering channels of the medium. The case of distinguishable particles is also discussed.
Spatial field correlation functions represent a key quantity for the description of mesoscopic phenomena in disordered media and the optical characterization of complex materials. Yet many aspects related to the vector nature of light waves have not been investigated so far. We study theoretically the polarization and coherence properties of electromagnetic waves produced by a dipole source in a three-dimensional uncorrelated disordered medium. The spatial field correlation matrix is calculated analytically using a multiple scattering theory for polarized light. This allows us to provide a for…
While there have been important theoretical advances in understanding the universality classes of interfaces moving in porous media, the developed tools cannot be directly applied to experiments. Here we introduce a method that can identify the universality class from snapshots of the interface profile. We test the method on discrete models whose universality class is well known, and use it to identify the universality class of interfaces obtained in experiments on fluid flow in porous media.
We study propagation of stationary waves in disordered non-linear media described by the non-linear Schroedinger equation and show that for given boundary conditions and a given coherent wave incident on a sample the number of solutions of the equation increase exponentially with the sample size. We also discuss the ballistic case, the sensitivity of the solutions to a change of external parameters, the similarity of this problem to the problem of spin glasses and time-dependent solutions.
We demonstrate order of magnitude coherent control of total transmission of light through random media by shaping the wavefront of the input light. To understand how the finite illumination area on a wide slab affects the maximum values of total transmission, we develop a model based on random matrix theory that reveals the role of long-range correlations. Its predictions are confirmed by numerical simulations and provide physical insight into the experimental results.
Folklore has, that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks Folklore has, that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks (SAWs) on percolation clusters and their critical exponent $ν_{\text{SAW}}$, with SAW implicitly referring to \emph{average} SAW. Hitherto, static averaging has been commonly used, e.g. in numerical simulations, to determine what the \emph{average} SAW is. We assert that only kinetic, rather than …
The conventional theory of burning works well in the case of uniform media where all system parameters are spatially independent. We develop a theory of burning in disordered media. In this case, rare regions (hot spots) where the burning process is more effective than on average may control the heat propagation in an explosive sample. We show that most predictions of the theory of burning are quite different from the conventional case. In particular, we show that a system of randomly distributed hot spots exhibits a dynamic phase transition, which is similar to the percolation transition. Dep…
Interfaces moving in a disordered medium exhibit stochastic velocity fluctuations obeying universal scaling relations related to the presence or absence of conservation laws. For fluid invasion of porous media, we show that the fluctuations of the velocity are governed by a geometry-dependent length scale arising from fluid conservation. This result is compared to the statistics resulting from a non-equilibrium (depinning) transition between a moving interface and a stationary, pinned one.
We review the random matrix theory describing elastic scattering through zero-dimensional ballistic cavities (having chaotic classical dynamics) and quasi-one dimensional disordered systems. In zero dimension, general symmetry considerations (flux conservation and time reversal symmetry) are only considered, while the combination law of scatterers put in series is taken into account in quasi-one dimension. Originally developed for calculating the distribution of the electrical conductance of mesoscopic systems, this theory naturally reveals the universal behaviors characterizing elastic scatte…
Wave localization is ubiquitous in disordered media -- from amorphous materials, where soft-mode localization is closely related to materials failure, to semi-conductors, where Anderson localization leads to metal-insulator transition. Our main understanding, though, is based on discrete models. Here, we provide a continuum perspective on the wave localization in two-phase disordered elastic media by studying the scalar wave equation with heterogeneous modulus and/or density. At low frequencies, soft modes arise as a result of disordered elastic modulus, which can also be predicted by the loca…
These notes contain a rapid overview of the methods and results obtained in the field of propagation of waves in disordered media. The case of Schrödinger and Helmholtz equations are considered that describe respectively electrons in metals and scalar electromagnetic waves. The assumptions on the nature of disorder are discussed and perturbation methods in the weak disorder limit are presented. A central quantity, namely the probability of quantum diffusion is defined and calculated in the same limit. It is then shown that several relevant physical quantities are related to the return probabil…
We introduce a wavefront shaping protocol for focusing inside disordered media based on a generalization of the established Wigner-Smith time-delay operator. The key ingredient for our approach is the scattering (or transmission) matrix of the medium and its derivative with respect to the position of the target one aims to focus on. A specifc experimental realization in the microwave regime is presented showing that the eigenstates of a corresponding operator are sorted by their focusing strength - ranging from strongly focusing on the designated target to completely bypassing it. Our protocol…
Temporal fluctuations of the speckle pattern formed upon backscattering of a laser beam from an interface between gold and nonlinear polymer film have been observed as a function of optical power. The instability can be explained by coupling of laser light to surface plasmons and other guided modes, which experience multiple scattering while propagating in the film along the interface. The speckle pattern produced in this process is extremely sensitive to fluctuations of the scattering potential near the interface.
We develop a model based on a multiple scattering theory to describe the diffusion of polarized light in disordered media exhibiting short-range structural correlations. Starting from exact expressions of the average field and the field spatial correlation function, we derive a radiative transfer equation for the polarization-resolved specific intensity that is valid for weak disorder and we solve it analytically in the diffusion limit. A decomposition of the specific intensity in terms of polarization eigenmodes reveals how structural correlations, represented via the standard anisotropic sca…
Light scattering in a magnetic medium with uncorrelated inclusions is theoretically studied in the approximation of ladder diagram. Correlation between polarizations of electromagnetic waves that are produced by infinitely-distant dipole source is considered. Here white noise disorder model with Gaussian distribution is taken into account. In such a medium the magneto-optical interaction leads to correlation between perpendicular light polarizations. Spatial field correlation matrix with nonzero nondiagonal elements is obtained in the first order on gyration.
In these lectures, given at the NATO ASI at Windsor (2001), applications of the replicas nonlinear sigma model to disordered systems are reviewed. A particular attention is given to two sets of issues. First, obtaining non-perturbative results in the replica limit is discussed, using as examples (i) an oscillatory behaviour of the two-level correlation function and (ii) long-tail asymptotes of different mesoscopic distributions. Second, a new variant of the sigma model for interacting electrons in disordered normal and superconducting systems is presented, with demonstrating how to reduce it, …
Synchronized oscillations are of critical functional importance in many biological systems. We show that such oscillations can arise without centralized coordination in a disordered system of electrically coupled excitable and passive cells. Increasing the coupling strength results in waves that lead to coherent periodic activity, exhibiting cluster, local and global synchronization under different conditions. Our results may explain the self-organized transition in a pregnant uterus from transient, localized activity initially to system-wide coherent excitations just before delivery.
Recent advances in microscopy techniques make it possible to study the growth, dynamics, and response of complex biophysical systems at single-cell resolution, from bacterial communities to tissues and organoids. In contrast to ordered crystals, it is less obvious how one can reliably distinguish two amorphous yet structurally different cellular materials. Here, we introduce a topological earth mover's (TEM) distance between disordered structures that compares local graph neighborhoods of the microscopic cell-centroid networks. Leveraging structural information contained in the neighborhood mo…
We study diffusion-limited pair annihilation $A+A\to 0$ on one-dimensional lattices with inhomogeneous nearest neighbour hopping in the limit of infinite reaction rate. We obtain a simple exact expression for the particle concentration $ρ_k(t)$ of the many-particle system in terms of the conditional probabilities $P(m;t|l;0)$ for a single random walker in a dual medium. For some disordered systems with an initially randomly filled lattice this leads asymptotically to $\bar{ρ(t)}=\bar{P(0;2t|0;0)}$ for the disorder-averaged particle density. We also obtain interesting exact relations for single…
Equilibrium distribution of interacting ionic particles in a charged disordered background is studied using the nonlinear Poisson-Boltzmann equation. For an arbitrarily given realization of the disorder, an exact solution of the equation is obtained in one dimension using a mapping of the nonlinear Poisson-Boltzmann equation to a self-consistent Schrodinger equation. The resulting density profile shows that the ions are delocalized, despite what the equivalent Schrodinger formulation in one dimension would suggest. It is shown that the ions are not distributed so as to locally neutralize the b…
We investigate the driven states of a two-dimensional crystal whose ground state can be tuned through a square-triangular transition. The depinning of such a system from a quenched random background potential occurs via a complex sequence of dynamical states, which include plastic flow states, hexatics, dynamically stabilized triangle and square phases and intermediate regimes of phase coexistence. These results are relevant to transport experiments in the mixed phase of several superconductors which exhibit such structural transitions as well as to driven colloidal systems whose interactions …
The space- and temperature-dependent electron distribution $n(\mathbf r,T)$ is essential for the theoretical description of the opto-electronic properties of disordered semiconductors. We present two powerful techniques to access $n(\mathbf r,T)$ without solving the Schrödinger equation. First, we derive the density for non-degenerate electrons by applying the Hamiltonian recursively to random wave functions (RWF). Second, we obtain a temperature-dependent effective potential from the application of a universal low-pass filter (ULF) to the random potential acting on the charge carriers in diso…
Random walk models in one-dimensional disordered media with an oscillatory input current are investigated theoretically as generic models of the boundary perturbation experiment. It is shown that the complex admittance obtained in the experiment is not self-averaging when the jump rates $w_i$ are random variables with the power-law distribution $ρ(w_i)\sim {w_i}^{α-1} (0 < α\leq 1)$. More precisely, the frequency-dependence of the disorder-averaged admittance $<χ>$ disagrees with that of the admittance $χ$ of any sample. It implies that the Cole-Cole plot of $<χ>$ shows a different shape from …
The statistical properties of the phases of several modes nonlinearly coupled in a random system are investigated by means of a Hamiltonian model with disordered couplings. The regime in which the modes have a stationary distribution of their energies and the phases are coupled is studied for arbitrary degrees of randomness and energy. The complexity versus temperature and strength of nonlinearity is calculated. A phase diagram is derived in terms of the stored energy and amount of disorder. Implications in random lasing, nonlinear wave propagation and finite temperature Bose-Einstein condensa…
In this work we present a method for generating random matrices describing electromagnetic scattering from disordered media containing dielectric particles with prescribed single particle scattering characteristics. Resulting scattering matrices automatically satisfy the physical constraints of unitarity, reciprocity and time reversal, whilst also incorporating the polarization properties of electromagnetic waves and scattering anisotropy. Our technique therefore enables statistical study of a variety of polarization phenomena, including depolarization rates and polarization-dependent scatteri…
The inference of Markov models from data on stochastic dynamical trajectories over the large time-window $T$ is revisited via the Large Deviations at Level 2.5 for the time-empirical density and the time-empirical flows. The goal is to obtain the large deviations properties for the probability distribution of the inferred Markov parameters in order to characterize their possible fluctuations around the true Markov parameters for large $T$. The explicit rate functions are given for several settings, namely discrete-time Markov chains, continuous-time Markov jump processes, and diffusion process…
After reviewing the basics of the cavity method in classical systems, we show how its quantum version, with some appropriate approximation scheme, can be used to study a system of spins with random ferromagnetic interactions and a random transverse field. The quantum cavity equations describing the ferromagnetic-paramagnetic phase transition can be transformed into the well-known problem of a classical directed polymer in a random medium. The glass transition of this polymer problem translates ino the existence of a `Griffith phase' close to the quantum phase transition of the quantum spin pro…
We theoretically and numerically investigate the capability of disordered media to enhance the optical path length in dielectric slabs and augment their light absorption efficiency due to scattering. We first perform a series of Monte Carlo simulations of random walks to determine the path length distribution in weakly to strongly (single to multiple) scattering, non-absorbing dielectric slabs under normally incident light and derive analytical expressions for the path length enhancement in these two limits. Quite interestingly, while multiple scattering is expected to produce long optical pat…
Partial summations of perturbation expansions of the directed polymer in disordered media (DPRM) enables one to represent the latter as skeleton expansions in powers of the effective coupling constant $Δ(t)$, which corresponds to the binding state between two replicas in the replica field theory of DPRM, and is equivalent to the binding state of a quantum particle in an external $δ$% -potential. The strong coupling phase is characterized by the exponential dependence of $Δ(t)$ on $t$, $Δ(t)\sim \exp (p_{c}t)$ with $% p_{c} $ being the binding energy of the particle. For dimensions $d>2$ the st…
