We develop a general dressed-state framework for computing fluorescence spectra, probe absorption spectra, and photon-photon correlations of light scattered by ensembles of N_at two-level atoms with arbitrary J_g → J_e transitions driven by intense coherent fields. The approach employs a full vectorial treatment of the electromagnetic field, handles any atomic geometries, illumination directions, and polarizations, and yields optical observables as explicit sums of Lorentzian lines whose positions, widths, and weights are directly tied to the eigenvalues and eigenvectors of the Lindbladian. The framework is implemented in an open-source Python package and benchmarked against exact single- and two-atom calculations. We identify geometries in which the full vectorial description is essential, and the scalar approximation fails qualitatively. Applying the method to pairs of atoms with a J_g=0→ J_e=1 transition, we show that elastic and inelastic scattered intensities collapse onto universal master curves controlled by a single collective saturation parameter built from the dominant superradiant mode, across several orders of magnitude in drive strength and interatomic distance. We identify collective phenomena that require a description beyond this single-mode picture. Extending the analysis to atoms with ground-state degeneracy, we find that most collective features carry over, while two qualitatively new effects emerge: an incoherent spontaneous Raman channel that modifies the scaling of inelastic emission, and a slow timescale in the time-delayed correlations g^(2)(τ) governed by the competition between Raman scattering and subradiant decay, controlled by a single dimensionless parameter. These results provide both physical insight and practical computational tools for engineering collective optical responses in few-atom systems such as optical tweezer arrays.

We perform a theoretical study of defect states within the photonic band gap of three-dimensional diamond crystals composed of point scatterers and doped with substitutional defects. The defects introduce localized states inside the photonic band gap, whose existence conditions and eigenfrequencies are expressed in terms of the on-site Green's function of the ideal defect-free crystal. Off-site Green's functions are also calculated as function of distance and are shown to vanish within approximately two unit cells. Finite-size effects are analyzed by comparing the results obtained in the infinite-crystal limit with numerical simulations based on the coupled-dipole method. The latter not only reproduce the eigenfrequencies of the defect states within the band gap, but also provide their lifetimes originating from the finite crystal size. The lifetimes of the defect states increase exponentially with crystal size, becoming very long for large crystals. In addition to defect states in the three-dimensional photonic band gap, the defects also give rise to strongly detuned states outside the gap, which decouple from the spectrum of the ideal defect-free crystal.

A prominent phenomenon in contemporary condensed matter physics is Anderson localization – suppression of wave propagation in disordered systems as a result of interference effects. Despite being observed with various types of waves over the years, all prior attempts to reach Anderson localization of light in three-dimensional systems have been hampered by experimental artifacts. Here, we report an unambiguous experimental proof of three-dimensional Anderson localization of microwaves in disordered metal aggregates. By studying samples with different metal volume fractions, we show a clear difference between diffusive and localized behaviors, and the latter is confirmed by a scaling analysis of transmitted beam width in excellent agreement with theoretical and numerical results. Our demonstration opens avenues for both fundamental studies and practical applications of this extraordinary phenomenon.

The concept of random walk, in which particles or waves undergo multiple collisions with the microscopic constituents of a surrounding medium, is central to understanding diffusive transport across many research areas. However, this paradigm may break down in complex systems, where quantum interference and memory effects render the particle propagation anomalous, often fostering localization. Here we report on the observation of such anomalous dynamics in a minimal setting: an ultracold mass-imbalanced mixture of two fermionic gases in three dimensions. We release light impurities into a gas of heavier atoms and follow their evolution across different collisional regimes. Under strong interspecies interactions, by lowering the temperature we unveil a crossover from normal diffusion to subdiffusion. Simultaneously, a localized fraction of the light gas emerges, displaying no discernible dynamics over hundreds of collisions. Our findings, incompatible with the conventional Fermi-liquid picture, are instead captured by a model of an atom propagating through a (quasi-)static disordered landscape of point-like scatterers. These results highlight the key role of quantum interference in our mixture, which emerges as a versatile platform for exploring disorder-free localization phenomena.

Common belief, confirmed by existing experiments, is that arbitrarily weak disorder should lead to spatial localization of eigenmodes of scalar wave equations when wave propagation is two-dimensional (2D). We predict that contrary to this belief, a localization-delocalization transition can take place for light scattered by two-level atoms placed at random positions in the middle plane of a parallel-plate 2D waveguide fed by its fundamental transverse-magnetic (TM) mode (electric field polarized perpendicular to the waveguide and atomic planes). This transition, driven by near-field dipole-dipole interactions between atoms, occurs upon increasing the areal number density of atoms beyond some critical value. A finite-size scaling analysis of the transition yields an estimate of its critical exponent ν = 1.4 ± 0.2.

We study Anderson transition for light in three dimensions by performing large-scale simulations of electromagnetic wave transport in disordered ensembles of perfect-electric-conducting spatially overlapping spheres. A mobility edge that separates diffusive transport and Anderson localization is identified, revealing a sharp transition from diffusion to localization for light. Critical behavior in the vicinity of the mobility edge is well described by a single-parameter scaling law. The critical exponent is found to be consistent with the value known for the Anderson transition of the orthogonal universality class. Statistical distribution of total transmission at the mobility edge is described without any fit parameter by the diagrammatic perturbation theory originally developed for scalar wave diffusion, but notable deviation from the theory is found when Anderson localization sets in.

Topologically nontrivial band structure of a material may give rise to special states that are confined to the material's boundary and protected against disorder and scattering. Quantum spin Hall effect (QSHE) is a paradigmatic example of phenomenon in which such states appear in the presence of time-reversal symmetry in two dimensions. Whereas the spatial structure of these helical edge states has been largely studied, their dynamic properties are much less understood. We design a microwave experiment mimicking QSHE and explore the spatiotemporal dynamics of unidirectional transport of optical angular momentum (or pseudospin) by edge states. Pseudospin-polarized signal propagation is shown to be immune to scattering by defects introduced along the edge. Its velocity is 2 to 3 orders of magnitude slower than the speed of light in the free space, which may have important consequences for practical applications of topological edge states in modern optical and quantum-information technologies.

A solid transparent medium with randomly positioned, immobile impurity atoms is a promising candidate for observation of Anderson localization of light in three dimensions. It can have low losses and allows for mitigation of the detrimental effect of longitudinal optical fields by an external magnetic field, but it has its own issues: thermal oscillations of atoms around their equilibrium positions and inhomogeneous broadening of atomic spectral lines due to random local electric fields. Our calculations suggest that these complications should not impede observation of Anderson localization of light in such materials provided that a sufficiently high number densities of impurities can be reached. The thermal oscillations hardly affect light propagation whereas the inhomogeneous broadening can be compensated for by increasing the number density of impurities.

We present a thorough study of the complexity of optical localized modes in two-dimensional disordered photonic crystals. Direct experimental measurements of complexity were made using an interferometric setup that allowed for extraction of phases and, hence, complex-valued wavefunctions. The comparison of experimental and theoretical results allows us to propose a metric for Anderson localization based on the average value and statistical distribution of complexity. Being an alternative to other known criteria of localization, the proposed metric exploits the openness of the disordered medium and provides a quantitative characterization of the degree of localization allowing for determining the localization length.

For a Hamiltonian H containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions un(r) obeying Hun(r) = un-1(r) with u0(r) = 1. For n -* oc, 1/vn(r) = un-1(r)/un(r) converges to the lowest eigenenergy E1 of H whereas uc(r) yields the corresponding wave function psi 1(r). For large but finite n, vn(r) can be approximated by a piecewise constant function vn(r) v(m) yields progressively improving estimations of eigenenergies Em = 1/vn(m) of locally fundamental eigenstates psi m(r) cx un(r) in spatial domains Omega m. These general results are illustrated by a number of examples in one dimension: box potential, sequence of randomly placed infinite potential barriers, smooth and spatially uncorrelated random potentials, quasiperodic potential, as well as for the uncorrelated random potential in two dimensions.

We study Anderson transition for light in three dimensions by performing large-scale ab-initio simulations of electromagnetic wave transport in disordered ensembles of conducting spheres. A mobility edge that separates diffusive transport and Anderson localization is identified, revealing a sharp transition from diffusion to localization for light. Critical behavior in the vicinity of the mobility edge is well described by a single-parameter scaling law. The critical exponent is found to be consistent with the value known for the Anderson transition of the orthogonal universality class. Statistical distribution of total transmission at the mobility edge is described without any fit parameter by the diagrammatic perturbation theory originally developed for scalar wave diffusion, but notable deviation from the theory is found when Anderson localization sets in.

lattice of two-level atoms coupled by the in-plane electromagnetic field may exhibit band gaps that can be opened either by applying an external magnetic field or by breaking the symmetry between the two triangular sublattices of which the honeycomb one is a superposition. We establish the conditions of band gap opening, compute the width of the gap, and characterize its topological property by a topological index (Chern number). The topological nature of the band gap leads to inversion of the population imbalance between the two triangular sublattices for modes with frequencies near band edges. It also prohibits a transition to the trivial limit of infinitely spaced, noninteracting atoms without closing the spectral gap. Surrounding the lattice by a Fabry-Pérot cavity with small intermirror spacing d < \pi/k_0d<π/k0, where k_0k0 is the free-space wave number at the atomic resonance frequency, renders the system Hermitian by suppressing the leakage of energy out of the atomic plane without modifying its topological properties. In contrast, a larger dd allows for propagating optical modes that are built up due to reflections at the cavity mirrors and have frequencies inside the band gap of the free-standing lattice, thus closing the latter.

Disorder in atomic positions can induce a topologically nontrivial phase—topological Anderson insulator (TAI)—for transverse electric optical quasimodes of a two-dimensional honeycomb lattice of immobile atoms. TAI requires both time-reversal and inversion symmetries to be broken to similar extents. It is characterized by a nonzero topological invariant, a reduced density of states and spatially localized quasimodes in the bulk, as well as propagating edge states. A transition from TAI to the topological insulator (TI) phase can take place at a constant value of the topological invariant, showing that TAI and TI represent the same topological phase.

Anderson localization is a halt of diffusive wave propagation in disordered systems. Despite extensive studies over the past 40 years, Anderson localization of light in three dimensions has remained elusive, leading to the question of its very existence. Recent advances have enabled finite-difference time-domain calculations to be sped up by orders of magnitude, allowing us to conduct brute-force numerical simulations of light transport in fully disordered three-dimensional systems with unprecedented dimension and refractive index difference. We show numerically three-dimensional localization of vector electromagnetic waves in random aggregates of overlapping metallic spheres, in sharp contrast to the absence of localization for dielectric spheres with a refractive index up to 10 in air. Our work opens a wide range of avenues in both fundamental research related to Anderson localization and potential applications using three-dimensional localized light. Whether Anderson localization of light can be achieved in three dimensions has remained an open question. Numerical calculations now show that it is possible with a random arrangement of metallic spheres, but not with dielectric ones.

This publisher's note contains a correction to Opt. Lett.47, 1439 (2022)10.1364/OL.449084.

A honeycomb lattice of atoms coupled by light exhibits topological bandgaps and protected edge modes when subjected to an external magnetic field, even when atomic positions are partially randomized. Moreover, disorder in atomic positions can be at the origin of topological properties. Finally, topological nature of a bandgap influences on the localization length of spatially localized modes that arise inside the bandgap due to disorder.

We present a laser-speckle imaging technique, termed Echo speckle imaging (ESI), that quantifies the local dynamics in biological tissue and soft materials with a noise level around or below 10% of the measured signal without affecting the spatial resolution. We achieve this through an unconventional speckle beam illumination that creates changing, statistically independent illumination conditions and substantially increases the measurement accuracy. Control experiments for dynamically homogeneous and heterogeneous soft materials and tissue phantoms illustrate the performance of the method. We show that this approach enables us to precision-monitor purely dynamic heterogeneities in turbid soft media with a lateral resolution of 100 µm and better.

We explore the interplay of disorder and topological phenomena in honeycomb lattices of atoms coupled by the electromagnetic field. On the one hand, disorder can trigger transitions between distinct topological phases and drive the lattice into the topological Anderson insulator state. On the other hand, the nontrivial topology of the photonic band structure suppresses Anderson localization of modes that disorder introduces inside the band gap of the ideal lattice. Furthermore, we discover that disorder can both open a topological pseudogap in the spectrum of an otherwise topologically trivial system and introduce spatially localized modes inside it.

Frequency-dependent intensity correlation function measurements can be employed to determine the optical turbidity of solid disordered dielectrics. Here we demonstrate a speckle frequency correlation experiment with a focused beam and using an area detector. We show how to apply frequency correlation measurements to optically thin solid samples with the aim of determining the light diffusion coefficient and transport mean free path ℓ*. To give a practical example, we extract the optical transport mean free path of PTFE (Teflon) slabs, with a thickness of L = 0.4-3.5 mm, covering optical densities L/ℓ* ∼ 4-15.