One-dimensional Random Medium Research Articles

Delocalization of a non-Hermitian quantum walk on random media in one dimension

We first review the localization–delocalization transition of a non-Hermitian random tight-binding Anderson model, called the Hatano–Nelson model. We then report a new result for a non-Hermitian extension of a discrete-time quantum walk on a one-dimensional random medium; we numerically find a delocalization transition similar to one of the Hatano–Nelson model. As a common feature to both models, at the transition point, an eigenvector gets delocalized and at the same time the corresponding energy eigenvalue (for the latter quantum-walk model, the imaginary unit times the phase of the eigenvalue of the time-evolution operator) becomes complex. One of the unique properties of the present non-Hermitian quantum walk is that the localization length of all eigenvectors is the same, and thereby all eigenstates simultaneously undergo the delocalization transition and all energy eigenvalues become complex at the same time when we turn up a non-Hermitian parameter.

Monte Carlo study of multiple scattering of steady-state waves in a one-dimensional random multi-layer medium

An analytical solution of the one-dimensional wave propagation is proposed to predict the steady-state SH wave behaviour in a one-dimensional random multi-layer medium with both periodic and non-periodic structures. A finite element model sandwiched by non-reflective boundary is also proposed and used to verify the analytical solution. Monte Carlo studies are conducted to demonstrate the influence of randomness of segment length on the wave propagation behaviour and the band structure. A macroscopic elastic model with structural damping is proposed to describe the homogenized steady-state wave behaviour in the multi-layer medium.

Moderate deviations for diffusion in time dependent random media

The position x(t) of a particle diffusing in a one-dimensional uncorrelated and time dependent random medium is simply Gaussian distributed in the typical direction, i.e. along the ray x = v 0 t, where v 0 is the average drift. However, it has been found that it exhibits at large time sample to sample fluctuations characteristic of the Kardar–Parisi–Zhang (KPZ) universality class when observed in an atypical direction, i.e. along the ray x = v t with v ≠ v 0. Here we show, from exact solutions, that in the moderate deviation regime x − v 0 t ∝ t 3/4 these fluctuations are precisely described by the finite time KPZ equation, which thus describes the crossover between the Gaussian typical regime and the KPZ fixed point regime for the large deviations. This confirms heuristic arguments given in []. These exact results include the discrete model known as the Beta random walk in a time dependent random environment, and a continuum diffusion. They predict the behavior of the maximum of a large number of independent walkers, which should be easier to observe (e.g. in experiments) in this moderate deviations regime.

Investigation of Anderson localization in disordered heterostructures irradiated by a Gaussian beam

The propagation of a Gaussian beam through a one-dimensional disordered media is studied. By employing the transfer matrix method, the localization length as a function of frequency is calculated for different values of transverse coordinate r. It is demonstrated that the localization length significantly depends on r in different frequency ranges. This result is in contrast to those reported for a plane wave incident on disordered structures in which the localization length is transversely constant. For some frequency regions, the peak of localization length is red-shifted and becomes smaller with increasing the transverse coordinate. At some frequencies, the system is in the localized state for particular values of r, while at other r values the system is in the extend regime at the same frequencies. It is observed that the quality of localization at each frequency depends on r. To quantify the localization behavior of the whole Gaussian beam, a modified localization length is defined in terms of the input and output powers of the Gaussian beam where the dependence of Anderson localization on the transverse coordinate is considered. It is suggested that this modified localization length is used in experiments performed for study of wave propagation in one-dimensional random media under illumination of laser beams.

S-wave impedance measurements of the uppermost material in surface ground layers: Vertical load excitation on a circular disk

S-wave impedance is one of the most effective parameters used to study the ground motion amplification of soil deposits. We propose a new approach to measure the S-wave impedance of the uppermost material in surface ground layers. First, a circular disk is set on the ground surface, and it is vertically loaded by sinusoidal wave excitation. When the time series of the loading velocity is synchronized with the reaction force, the ratio of the reaction force to the loading velocity is proportional to the S-wave impedance. We then estimate the proportionality coefficient from numerical experiments and check its accuracy. The measurement error is estimated to be within 1% for the homogeneous half-space case. We also discuss the applicability of this new approach and its limitations on the basis of numerical experiments for inhomogeneous media: a two-layered medium and a one-dimensional (1-D) random medium. The proposed approach is effective for both cases if we select the appropriate circular disk size.

Delocalization of Acoustic Waves in a One-Dimensional Random Dimer Media

The propagation of classical waves in one-dimensional random media is examined in presence of short-range correlation in disorder. A classical analogous of the Kronig-Penney model is proposed by means a chain of repeated sub-systems, each of them constituted by a mass connected to a rigid foundation by a spring. The masses are related to each other by a string submitted to uniform tension. The nature of the modes is investigated by using different transfer matrix formalisms. It is shown that in presence of short-range correlation in the medium which corresponds to the RD model- the localization-delocalization transition occurs at a resonance frequency . The divergence of near is studied, and the critical exponent that characterizes the power-law behavior of near is estimated. Moreover an exact analytical study is carried out for the delocalization properties of the waves in the RD media. In particular, we predict the resonance frequency at which the waves can propagate in the entire chain. The transmission properties of the system are numerically studied using a statistical procedure yielding various physical magnitudes such the transmission coefficient, the localization length and critical exponents. In particular, it is shown that the presence of correlation in disorder restores a large number of extended Bloch-like modes in contradiction with the general conclusion of the localization phenomenon in one-dimensional systems with correlated disorder.

Phase of the transmission coefficient of waves in one-dimensional random media

We study the behavior of the phase of the transmission coefficient t in random media by using the invariant imbedding method of wave propagation. We calculate the disorder average of t/t* for waves propagating in one-dimensional random media with uncorrelated Gaussian disorder in a numerically exact manner. We find that the analytical formula derived by Mello et al. [Phys. Rev. Lett. 67, 342 (1991)] is valid only after a major modification and only in the limit where the so-called random phase approximation is valid. We find that the disorder average of t/t* converges to a finite complex number in the large-size limit. We discuss the implications of our results for the probability distribution of the phase of the transmission coefficient.

Effective fractional acoustic wave equations in one-dimensional random multiscale media

This paper considers multiple scattering of waves propagating in a non-lossy one-dimensional random medium with short- or long-range correlations. Using stochastic homogenization theory it is possible to show that pulse propagation is described by an effective deterministic fractional wave equation, which corresponds to an effective medium with a frequency-dependent attenuation that obeys a power law with an exponent between 0 and 2. The exponent is related to the Hurst parameter of the medium, which is a characteristic parameter of the correlation properties of the fluctuations of the random medium. Moreover the frequency-dependent attenuation is associated with a special frequency-dependent phase, which ensures that causality and Kramers-Kronig relations are satisfied. In the time domain the effective wave equation has the form of a linear integro-differential equation with a fractional derivative.

Stochastic differential equation approach for waves in a random medium

We present a mathematical approach that simplifies the theoretical treatment of electromagnetic localization in random media and leads to closed-form analytical solutions. Starting with the assumption that the dielectric permittivity of the medium has delta-correlated spatial fluctuations, and using Ito's lemma, we derive a linear stochastic differential equation for a one-dimensional random medium. The equation leads to localized wave solutions. The localized wave solutions have a localization length that scales as L approximately omega(-2) for low frequencies whereas in the high-frequency regime this length behaves as L approximately omega(-2/3) .

Influence of gain localization in one-dimensional random media

Based on the transfer matrix method, we studied the effects of gain distribution on the performance of light localization in one-dimensional completely random systems. Due to the scattering of the pump light, the gain can also be localized. Compared with homogenous gain distribution, the localized gain can induce light confinement more efficiently. The overlapping between the laser modes and the localized gain brings on the high-Q modes that are amplified preferentially, and reduces the lasing threshold. The threshold can be hundreds times lowered in some special cases. With increase of the random system size, the localization position goes deeper into the media first, and then saturates when the system is thicker than 550 layers. For a given kind of random laser systems, an optimal system size exists, for which the mode intensity is significantly enhanced. The influence of the system structure on the lasing intensity and localization position is also discussed.

Pulse Propagation in Random Media with Long-Range Correlation

This paper analyzes wave propagation in a one-dimensional random medium with long-range correlations. The asymptotic regime where the fluctuations of the medium parameters are small and the propagation distance is large is studied. In this regime pulse propagation is characterized by a random time shift described in terms of a fractional Brownian motion and a deterministic spreading described by a pseudodifferential operator. This operator is characterized by a frequency-dependent attenuation that obeys a power law with an exponent ranging from 1 to 2 that is related to the power decay rate of the autocorrelation function of the fluctuations of the medium parameters. This frequency-dependent attenuation is associated with a frequency-dependent phase, which ensures causality of the filter that realizes the approximation. A discussion is provided showing that the mean-field theory cannot capture the correct attenuation rate; this is because it also averages the random time delay. Numerical results are given...

ANALYSIS OF PULSE PROPAGATION THROUGH A ONE-DIMENSIONAL RANDOM MEDIUM USING COMPLEX MARTINGALES

We show that the propagation of pulses in a one-dimensional random medium can be characterized using a new complex martingale representation of the transmission coefficient. This representation holds in the frequency domain and in a certain asymptotic limit where diffusion approximations can be used.

Quantum walks on a random environment

Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength $W$, where a quantum to classical crossover at time ${t}_{c}\ensuremath{\propto}{W}^{\ensuremath{-}2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution, we relate our results to previously studied models of decoherence for quantum walks, and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.

Scaling properties of delay times in one-dimensional random media

The scaling properties of the inverse moments of Wigner delay times are investigated in finite one-dimensional (1D) random media with one channel attached to the boundary of the sample. We find that they follow a simple scaling law which is independent of the microscopic details of the random potential. Our theoretical considerations are confirmed numerically for systems as diverse as 1D disordered wires and optical lattices to microwave waveguides with correlated scatterers.

Analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in one-dimensional random media

Consider N points randomly distributed along a line segment of unitary length. A walker explores this disordered medium, moving according to a partially self-avoiding deterministic walk. The walker, with memory mu , leaves from the leftmost point and moves, at each discrete time step, to the nearest point that has not been visited in the preceding mu steps. Using open boundary conditions, we have calculated analytically the probability P{N}(mu)=(1-2{-mu}){N-mu-1} that all N points are visited, with N>>mu>>1 . This approximated expression for P{N}(mu) is reasonable even for small N and mu values, as validated by Monte Carlo simulations. We show the existence of a critical memory mu{1}=lnNln2 . For mu<mu{1}-e(2ln2) , the walker gets trapped in cycles and does not fully explore the system. For mu>mu{1}+e(2ln2) , the walker explores the whole system. Since the intermediate region increases as lnN and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker need not have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order log{2}N .

Correlation of spontaneous emission in a one-dimensional random medium with Anderson localization

Under Anderson localization, two types of correlation of spontaneous emission in one-dimensional random media are investigated: i.e., time-domain field correlation ${C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau})$ and energy-spectrum correlation ${C}_{\ensuremath{\omega}}^{P}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\Delta}\ensuremath{\omega})$. The results show that the spatial correlation length of ${C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau}=0)$ is unrelated to the localization length; however, the increase of the correlation length of ${\mathrm{max}}_{\ensuremath{\tau}}\ensuremath{\mid}{C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau})\ensuremath{\mid}$ with the localization length is sensitive and monotonous. In particular, we find that the fields at the different locations keep on exchanging with each other in a certain fixed speed by investigation of time-domain field correlation. The speed is almost not affected by the random strength and almost equal to the group velocity of the corresponding periodic structure. Therefore the localized mode is really a dynamic equilibrium state though the energy of the localized mode is localized. In addition, because it is not convenient to characterize the localization length by ${C}_{t}^{E}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\tau})$, another correlation---energy-spectrum correlation ${C}_{\ensuremath{\omega}}^{P}({\mathbf{r}}_{1},{\mathbf{r}}_{2},\ensuremath{\Delta}\ensuremath{\omega})$---is proposed. By investigation of the energy-spectrum correlation for $\ensuremath{\Delta}\ensuremath{\omega}=0$, we obtain that there is an approximately linear relation between the spatial correlation length of energy-spectrum and the localization length. Obviously, in the aspect of characterizing the localization length, the energy-spectrum correlation is more convenient than the time-domain field correlation.

Time reversal detection in one-dimensional random media

In this paper, we study time reversal in the reflection of an acoustic wave in a one-dimensional random medium with an embedded reflector. The main result is that time reversal allows us to find the location of a reflector even in the absence of a coherent reflection. We carry out the analysis in the asymptotic regime of separation of scales, where the probing pulse is large compared to the medium inhomogeneities but small relative to the depth of the reflector. In the limit, the quantity of interest, namely the density of the time reversal refocusing kernel, is given as a solution to a deterministic system of transport equations, which is solved by using Monte Carlo simulations.

Enhanced nonradiative relaxation and photoluminescence quenching in random, doped nanocrystalline powders

Nonradiative relaxation and photoluminescence quenching in nanocrystalline powders doped with rare-earth elements are of interest in optical bistability, random laser, and other optoelectronic applications. Here, the luminescence quenching of a one-dimensional random medium made of multilayer nanoparticles (Y2O3) doped with rare-earth elements (Yb3+) is analyzed by considering the transport, transition, and interaction of the fundamental energy carriers. The nonradiative decay and luminescence quenching in random media are enhanced compared to single crystals, due to multiple scattering, enhanced absorption, and low thermal conductivity. The coherent wave treatment is used to calculate the photon absorption, allowing for field enhancement and photon localization. The luminescent and thermal emission is considered as incoherent. The size-dependent absorption coefficient and penetration depth are observed. The nonradiative decay is identified as a multiphonon relaxation process, and is found to be enhanced compared to bulk materials. The luminescence quenching and nonlinear thermal emission, occurring with increasing irradiation intensity, are predicted.

Effect of eigenmodes on the optical transmission through one-dimensional random media

Using the transfer matrix method as well as numerical solutions to the one-dimensional wave equation derived from the Maxwell equations, we study the transition from a photonic crystal with a finite size to a random dielectric medium. We examine the validity of the Kronig-Penney model for a finite size crystal and analyze the spatial structure of the eigenmodes as the crystal is made more irregular.

Amplification in one-dimensional random active medium near the lasing threshold

We studied the transmittance of a random amplifying medium near the lasing threshold by using the convergence criterion proposed by Nam and Zhang [Phys. Rev. B66 73101, 2002] that allows separating the physical solutions of the time-independent Maxwell equations from the unphysical ones and describing critical size Lc of a random system in statistical terms. We found that the dependence of the critical gain ∈″c (at which the lasing threshold occurs) as a function of number of layers is configuration-dependent and thus the lasing condition for random media is described by means of the probability of finding of physical solutions and evaluated by averaging over the ensemble of random configurations. By employing this approach we inspect the validity of the two-parameter scaling model by Zhang [Phys. Rev. B52 7960, 1995], according to which the behavior of the random system with gain is described by relation 1/ξ = 1/ξ0 + 1/lg, where ξ and ξ0 are localization length with and without gain, respectively, and lg = 2/ω∈″, is gain length, where ∈″ is imaginary part of the dielectric constant that represents gain. We show that the range of validity of this relation depends on the ratio of both lengths and also affects the slope of the ln Λc(q) (where Λc ≡ Lc/ξ0 is normalized critical size and q−1 ≡ lg/ξ0 is dimensionless gain length). We extend the study of the relation for the critical size Lc by inspecting the dependence of the slope of the ln Λc(q) on the strength of the randomness. We interpret its behavior in terms of the statistical properties of the localized states. Namely, by studying of the variance of the Lyapunov exponent λ (the inverse of the localization length ξ0) we have found that the slope of the ln Λc(q)) reflects the transition between two different regimes of localization with Anderson and Lifshits-like behavior that is known to be indicated by peak in var(λ). We discuss the generalization of two-parameter scaling model by implementing revisited single parameter scaling (SPS) theory by Deych et al. [Phys. Rev. Lett.84 2678, 2000] which allows describing non-SPS regime in terms of a new scale ls.