We develop an analytical theory for diffusive random lasers by coupling the transport theory of the disordered medium to the semiclassical laser rate equations, accounting for (coherent) stimulated and (incoherent) spontaneous emission. From the causality of wave propagation in an amplifying, diffusive medium we derive a novel length scale which we identify with the average mode radius of the lasing quasi-modes. We show further that loss at the surface of the laser-active medium is crucial in order to stabilize a stationary lasing state. The solution of the transport theory of random lasers for a layer geometry with appropriate surface boundary conditions yields the spatial profile of the light intensity and of the population inversion. The dependence of the intensity correlation length on the pump rate is in qualitative agreement with experimental and numerical findings.
We study the crossover between the diffusive and quasi-ballistic regimes of random lasers. In particular, we compare incoherent models based on the diffusion equation and the radiative transfer equation (RTE), which neglect all wave effects, with a coherent wave model for the random laser threshold. We show that both the incoherent and the coherent models predict qualitatively similar thresholds, with a smooth transition from a diffuse to a quasi-ballistic regime. The shape of the intensity distribution in the sample as predicted by the RTE model at threshold is also in good agreement with the…
We study the quantum transport through networks of diffusive wires connected to reservoirs in the Landauer-Büttiker formalism. The elements of the conductance matrix are computed by the diagrammatic method. We recover the combination of classical resistances and obtain the weak localization corrections. For arbitrary networks, we show how the cooperon must be properly weighted over the different wires. Its nonlocality is clearly analyzed. We predict a new geometrical effect that may change the sign of the weak localization correction in multiterminal geometries.
The purpose of this work is to understand the fundamental connection between structural correlations and light localization in three-dimensional (3D) open scattering systems of finite size. We numerically investigate the transport of vector electromagnetic waves scattered by resonant electric dipoles spatially arranged in 3D space by stealthy hyperuniform disordered point patterns. Three-dimensional stealthy hyperuniform disordered systems (3D-SHDS) are engineered with different structural correlation properties determined by their degree of stealthiness $χ$. Such fine control of exotic states…
We study a weakly disordered 2D electron gas with two bands and a spectral node within the weak-localization approach and compare its results with those of Gaussian fluctuations around the self-consistent Born approximation. The appearance of diffusive modes depends on the type of disorder. In particular, we find for a random gap a diffusive mode only from ladder contributions, whereas for a random scalar potential the diffusive mode is created by ladder and by maximally crossed contributions. The ladder (maximally crossed) contributions correspond to fermionic (bosonic) Gaussian fluctuations.…
The full distribution of the conductance $P(G)$ in quasi-one-dimensional wires with rough surfaces is analyzed from the diffusive to the localization regime. In the crossover region, where the statistics is dominated by only one or two eigenchannels, the numerically obtained P(G) is found to be independent of the details of the system with the average conductance <G> as the only scaling parameter. For <G> < e^2/h, P(G) is given by an essentially ``one-sided'' log-normal distribution. In contrast, for e^2/h < <G> <= 2e^2/h, the shape of P(G) remarkable agrees with those predicted by random matr…
In this paper, we discuss the transport phenomena of electromagnetic waves in a two-dimensional random system which is composed of arrays of electrical dipoles, following the model presented earlier by Erdogan, et al. (J. Opt. Soc. Am. B {\bf 10}, 391 (1993)). A set of self-consistent equations is presented, accounting for the multiple scattering in the system, and is then solved numerically. A strong localization regime is discovered in the frequency domain. The transport properties within, near the edge of and nearly outside the localization regime are investigated for different parameters s…
We present a numerical study of electromagnetic wave transport in disordered quasi-one-dimensional waveguides at terahertz frequencies. Finite element method calculations of terahertz wave propagation within LiNbO$_{3}$ waveguides with randomly arranged air-filled circular scatterers exhibit an onset of Anderson localization at experimentally accessible length scales. Results for the average transmission as a function of waveguide length and scatterer density demonstrate a clear crossover from diffusive to localized transport regime. In addition, we find that transmission fluctuations grow dra…
Self-induced transparency pulses propagating in a random medium embedded in a two-level system can transfer energy to localized Anderson states. This allows the onset of two-level laser-like action.
Quantum phase transitions of three-dimensional (3D) Weyl semimetals (WSMs) subject to uncorrelated on-site disorder are investigated through quantum conductance calculations and finite-size scaling of localization length. Contrary to previous claims that a direct transition from a WSM to a diffusive metal (DM) occurs, an intermediate phase of Chern insulator (CI) between the two distinct metallic phases should exist due to internode scattering that is comparable to intranode scattering. The critical exponent of localization length is $ν\simeq 1.3$ for both the WSM-CI and CI-DM transitions, in …
We investigate the statistics of single-mode delay times of waves reflected from a disordered waveguide in the presence of wave localization. The distribution of delay times is qualitatively different from the distribution in the diffusive regime, and sensitive to coherent backscattering: The probability to find small delay times is enhanced by a factor close to sqrt(2) for reflection angles near the angle of incidence. This dynamic effect of coherent backscattering disappears in the diffusive regime.
We study high-temperature magnetization transport in a many-body spin-1/2 chain with on-site quasiperiodic potential governed by the Fibonacci rule. In the absence of interactions it is known that the system is critical with the transport described by a continuously varying dynamical exponent (from ballistic to localized) as a function of the on-site potential strength. Upon introducing weak interactions, we find that an anomalous noninteracting dynamical exponent becomes diffusive for any potential strength. This is borne out by a boundary-driven Lindblad dynamics as well as unitary dynamics,…
This work presents results of ab-initio simulations of continuous wave transport in disordered absorbing waveguides. Wave interference effects cause deviations from diffusive picture of wave transport and make the diffusion coefficient position- and absorption-dependent. As a consequence, the true limit of a zero diffusion coefficient is never reached in an absorbing random medium of infinite size, instead, the diffusion coefficient saturates at some finite constant value. Transition to this absorption-limited diffusion exhibits a universality which can be captured within the framework of the …
In this paper we propose that phonons, free diffusive electrons, and diffusion in two-and three--dimensional disordered lattice problems belong to a different class of localization than bonded electrons. This is manifested by three effects that can be observed numerically. First, there are extended states even at two dimensions, whereas there are no extended states in the usual electronic models. Second, the correlation length does not diverge at the mobility edges in three dimensions, and finally, the participation ratio of the extended states, decays to zero at this edge. This indicates zero…
We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of long-range interactions decaying as power-law $V_{ij}/(r_i-r_j)^α$ with distance and having random coefficients $V_{ij}$. We demonstrate that MBL survives even for $α<1$ and is preceded by a broad non-ergodic sub-diffusive phase. Starting from parameters at which the short-range interacting system shows infinite temperature MBL phase, turning on random power-law interactions results in many-body mobility edges in the spectrum with a larger fraction …
The conductance of a quantum wire with off-diagonal disorder that preserves a sublattice symmetry (the random hopping problem with chiral symmetry) is considered. Transport at the band center is anomalous relative to the standard problem of Anderson localization both in the diffusive and localized regimes. In the diffusive regime, there is no weak-localization correction to the conductance and universal conductance fluctuations are twice as large as in the standard cases. Exponential localization occurs only for an even number of transmission channels in which case the localization length does…
Exact calculations of the transmittance of surface corrugated optical waveguides are presented. The elastic scattering of diffuse light or other electromagnetic waves from a rough surface induces a diffusive transport along the waveguide axis. As the length of the corrugated part of the waveguide increases, a transition from the diffusive to the localized regime is observed. This involves an analogy with electron conduction in nanowires, and hence, a concept analogous to that of ``resistance'' can be introduced. We show an oscillatory behavior of both the elastic mean free path and the localiz…
Diffusive transport is among the most common phenomena in nature [1]. However, as predicted by Anderson [2], diffusion may break down due to interference. This transition from diffusive transport to localization of waves should occur for any type of classical or quantum wave in any media as long as the wavelength becomes comparable to the transport mean free path $\ell^*$ [3]. The signatures of localization and those of absorption, or bound states, can however be similar, such that an unequivocal proof of the existence of wave localization in disordered bulk materials is still lacking. Here we…
Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we propose a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model and show that it exhibits a MBL transition. The model on a bipartite lattice has $N$ Majorana fermions with SYK interactions on each site of the $A$ sublattice and $M$ free Majorana fermions on each site the of $B$ sublattice, where $N$ and $M$ are large and finite. For $r$$\equiv$$M/N\!<\!r_c$=1, it describes a diffusive metal exhibiting maximal chaos. Remarkably, its diffusive constant $D$ vanishes [$D$$\propto$$ (r_c-r)^{1/…
We study quantum phase coherence and weak localization (WL) in disordered metals with restricted back-scattering and phenomenologically formulate a large class of unconventional transport mechanisms as modified diffusion processes not captured by the Boltzmann picture. Inspired by conductivity measurements in ferromagnetic films and semiconductors where anomalous power law corrections have been observed, we constrain memory dependent, self avoidance effects onto the quantum enhanced back-scattered trajectories, drastically altering the effect of weak localization in two dimensions (2D). Scale …
We study the quantum transport in multiterminal networks of quasi-one-dimensional diffusive wires. When calculating the weak localization correction to the conductances, we show that the Cooperon must be properly weighted over each wire. This can even change the sign of the weak localization correction in certain geometries.
We report on anomalous light transport in the strong scattering regime. Using low-coherence interferometry, we measure the reflection matrix of titanium dioxide powders, revealing crucial features of strong optical scattering which can not be observed with transmission measurements: (i) a subdiffusive regime of transport at early times of flight that is a direct consequence of predominant recurrent scattering loops, and (ii) a crossover to a conventional, but extremely slow, diffusive regime at long times. These observations support previous predictions that near-field coupling between scatter…
The spatial extension and complexity of the eigenfunctions of an open finite-size two-dimensional (2D) random system are systematically studied for a random collection of systems ranging from weakly scattering to localized. The eigenfunctions are obtained by introducing gain in the medium and pumping just above threshold. All lasing modes are found to correspond to quasimodes of the passive system, for all regimes of propagation. We demonstrate the existence of multipeaked quasimodes or necklace states in 2D at the transition from localized to diffusive, resulting from the coupling of localize…
We calculate two-point energy level correlation function in weakly disorderd metallic grain with taking account of localization corrections to the universal random matrix result. Using supersymmetric nonlinear sigma model and exactly integrating out spatially homogeneous modes, we derive the expression valid for arbitrary energy differences from quantum to diffusive regime for the system with broken time reversal symmetry. Our result coincides with the one obtained by Andreev and Altshuler [Phys. Rev. Lett. 72, 902 (1995)] where homogeneous modes are perturbatively treated.
Statistical fluctuations of the light emitted from amplifying random media are studied theoretically and numerically. The characteristic scales of the diffusive motion of light lead to Gaussian or power-law (Levy) distributed fluctuations depending on external control parameters. In the Levy regime, the output pulse is highly irregular leading to huge deviations from a mean--field description. Monte Carlo simulations of a simplified model which includes the population of the medium, demonstrate the two statistical regimes and provide a comparison with dynamical rate equations. Different statis…
In this work, we study long-time wave transport in correlated and uncorrelated disordered 2D arrays. When a separation of dimensions is applied to the model, we find that the predicted 1D random dimer phenomenology also appears in so-called pseudo-2D arrays. Therefore, a threshold behavior is observed in terms of the effective size for eigenmodes, as well as in long-time dynam- ics. For this threshold behavior to be observed a minimum system size is required, what is very important when considering a possible experimental realization. For the long-time evolution, we find that for short-range c…
Many complex systems exhibit hydrodynamic (or macroscopic) behavior at large scales characterized by few variables such as the particle number density, temperature and pressure obeying a set of hydrodynamic (or macroscopic) equations. Does the hydrodynamic description exist also for waves in complex open media? This is a long-standing fundamental problem in studies on wave localization. Practically, if it does exist, owing to its simplicity macroscopic equations can be mastered far more easily than sophisticated microscopic theories of wave localization especially for experimentalists. The pur…
Unlike conventional lasers, diffusive random lasers (DRLs) have no resonator to trap light and no high-Q resonances to support lasing. Due to this lack of sharp resonances the DRL has presented a challenge to conventional laser theory. We present a theory able to treat the DRL rigorously, and provide results on the lasing spectra, internal fields and output intensities of DRLs. Typically DRLs are highly multimode lasers, emitting light at a number of wavelengths. We show that the modal interactions through the gain medium in such lasers are extremely strong and lead to a uniformly spaced frequ…
We compute the distribution of the decay rates (also referred to as residues) of the eigenstates of a disordered slab from a numerical model. From the results of the numerical simulations, we are able to find simple analytical formulae that describe those results well. This is possible for samples both in the diffusive and in the localised regime. As example of a possible application, we investigate the lasing threshold of random lasers.
We investigate coherent matter wave transport in isotropic 3D speckle potentials by using the Bethe-Salper equation and the self-consistent theory of localization. This model constitutes an efficient tool to properly evaluate corrections to Boltzmann diffusion by taking into consideration quantum interference terms between the multiple-scattering paths. We calculate analytically and numerically the static current density, the density of states, the dipolar contribution and the reduced diffusion coefficient. Our results reveal that quantum corrections to diffusive transport, known as weak local…
Results of detailed investigations of the conductivity and Hall effect in gated single quantum well GaAs/InGaAs/GaAs heterostructures with two-dimensional electron gas are presented. A successive analysis of the data has shown that the conductivity is diffusive for $k_F l=25-2$ and behaves like diffusive one for $k_F l=2-0.5$ down to the temperature T=0.4 K. It has been therewith found that the quantum corrections are not small at low temperature when $k_F l\simeq 1$. They are close in magnitude to the Drude conductivity so that the conductivity $σ$ becomes significantly less than $e^{2}/h$ (t…
We define a simple model of conformal field theory in random space-time environments, which we refer to as stochastic conformal field theory. This model accounts for the effects of dilute random impurities in strongly interacting critical many-body systems. On one hand, surprisingly, although impurities are separated by macroscopic distances, we find that the infinite-time steady state is factorized on microscopic lengths, a signature of the emergence of localization. The stationary state also displays vanishing energy current and strong uncorrelated spatial fluctuations of local observables. …
We study the propagation of waves in quasi-one-dimensional finite periodic systems whose classical (ray) dynamics is diffusive. By considering a random matrix model for a chain of $L$ identical chaotic cavities, we show that its average conductance as a function of $L$ displays an ohmic behavior even though the system has no disorder. This behavior, with an average conductance decay $N/L$, where $N$ is the number of propagating modes in the leads that connect the cavities, holds for $1\ll L \lesssim \sqrt{N}.$ After this regime, the average conductance saturates at a value of ${\mathcal O}(\sq…
Localization in one-dimensional disordered or quasiperiodic non-interacting systems in presence of power-law hopping is very different from localization in short-ranged systems. Power-law hopping leads to algebraic localization as opposed to exponential localization in short-ranged systems. Exponential localization is synonymous with insulating behavior in the thermodynamic limit. Here we show that the same is not true for algebraic localization. We show, on general grounds, that depending on the strength of the algebraic decay, the algebraically localized states can be actually either conduct…
A mixture of light and heavy atoms is considered. We study the kinetics of the light atoms, scattered by the heavy ones, the latter undergoing slow diffusive motion. In three-dimensional space we claim the existence of a crossover region (in energy), which separates the states of the light atoms with fast diffusion and the states with slow diffusion; the latter is determined by the dephasing time. For the two dimensional case we have a transition between weak localization, observed when the dephasing length is less than the localization length (calculated for static scatterers), and strong loc…
We study the transport and localization properties of scalar vibrations on a lattice with random bond strength by means of the transfer matrix method. This model has been recently suggested as a means to investigate the vibrations and heat conduction properties of structural glasses. In three dimensions we find a very rich phase diagram. The delocalization transition is split, so that between the localized and diffusive phases which have been identified in the Anderson problem, we observe a phase with anomalous, sub-exponential localization. For low frequencies, we find a strongly conducting p…
Stark-Many body localization in interacting infinite dimensional systems
School of Chemistry, Tel Aviv University（特拉维夫大学化学学院）；Department of Physics, University of Michigan（密歇根大学物理系）；Department of Physics, Ben-Gurion University of the Negev（内盖夫本古里安大学物理系）；The Raymond and Beverley Sackler Center for Computational Molecular and Materials Science, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利萨克勒计算分子与材料科学中心）
In the absence of confinement localization of waves takes place due to randomness or nonlinearity and relies on their phase coherence. We quantitatively probe the sensitivity of localized wave packets to random phase fluctuations and confirm the necessity of phase coherence for localization. Decoherence resulting from a dynamical random environment leads to diffusive spreading and destroys linear and nonlinear localization. We find that maximal spreading is achieved for optimal phase fluctuation characteristics which is a consequence of the competition between diffusion due to decoherence and …
Several aspects of interplay between Anderson localization and coherent amplification/absorption, and aspect of mirrorless laser for a laser-active (amplifying) disordered dielectric medium have been addressed. We have calculated the statistics of the reflection coefficient and the associated phase for a light wave reflected from a coherently amplifying/absorbing one-dimensional disordered dielectric medium for different lengths, different disorder strengths and different "amplification"/"absorption" parameters of the disordered sample. Problems with modeling coherent amplification/absorption …
We study the propagation and scattering of electromagnetic waves by random arrays of dipolar cylinders in a uniform medium. A set of self-consistent equations, incorporating all orders of multiple scattering of the electromagnetic waves, is derived from first principles and then solved numerically for electromagnetic fields. For certain ranges of frequencies, spatially localized electromagnetic waves appear in such a simple but realistic disordered system. Dependence of localization on the frequency, radiation damping, and filling factor is shown. The spatial behavior of the total, coherent an…
We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model [1]. We also present the exact solution of a semiclassical model of conductors and i…
We report on the measurement of phase coherence length in a high mobility two-dimensional electron gas patterned in two different geometries, a wire and a ring. The phase coherence length is extracted both from the weak localization correction in long wires and from the amplitude of the Aharonov-Bohm oscillations in a single ring, in a low temperature regime when decoherence is dominated by electronic interactions. We show that these two measurements lead to different phase coherence lengths, namely $L_Φ^\mathrm{wire}\propto T^{-1/3}$ and $L_Φ^\mathrm{ring}\propto T^{-1/2}$. This difference re…
Optomechanical arrays are a promising future platform for studies of transport, many-body dynamics, quantum control and topological effects in systems of coupled photon and phonon modes. We introduce disordered optomechanical arrays, focusing on features of Anderson localization of hybrid photon-phonon excitations. It turns out that these represent a unique disordered system, where basic parameters can be easily controlled by varying the frequency and the amplitude of an external laser field. We show that the two-species setting leads to a non-trivial frequency dependence of the localization l…
We report on a numerical study of quantum transport in disordered two dimensional graphene and graphene nanoribbons. By using the Kubo and the Landauer approaches, transport length scales in the diffusive (mean free path, charge mobilities) and localized regimes (localization lengths) are computed, assuming a short range disorder (Anderson-type). In agreement with localization scaling theory, the electronic systems are found to undergo a conventional Anderson localization in the zero temperature limit. Localization lengths in weakly disordered ribbons are found to differ by two orders of magni…
Diffusive random walks feature the surprising property that the average length of all possible random trajectories that enter and exit a finite domain is determined solely by the domain boundary. Changes in the diffusion constant or the mean-free path, that characterize the diffusion process, leave the mean path length unchanged. Here we demonstrate experimentally that this result can be transferred to the scattering of waves, even when wave interference leads to marked deviations from a diffusion process. Using a versatile microwave setup, we establish the mean path length invariance for the …
We study the thermodynamic and dynamic behaviors of twist-induced denaturation bubbles in a long, stretched random sequence of DNA. The small bubbles associated with weak twist are delocalized. Above a threshold torque, the bubbles of several tens of bases or larger become preferentially localized to \AT-rich segments. In the localized regime, the bubbles exhibit ``aging'' and move around sub-diffusively with continuously varying dynamic exponents. These properties are derived using results of large-deviation theory together with scaling arguments, and are verified by Monte-Carlo simulations.
