Diffuse field cross-correlations: Scattering theory and electromagnetic experiments.

The passive estimation of impulse responses from ambient noise correlations arouses increasing interest in seismology, acoustics, optics, and electromagnetism. Assuming the equipartition of the noise field, the cross-correlation function measured with noninvasive receiving probes converges towards the difference of the causal and anticausal Green's functions. Here, we consider the case when the receiving field probes are antennas which are well coupled to a complex medium-a scenario of practical relevance in electromagnetism. We propose a general approach based on the scattering matrix formalism to explore the convergence of the cross-correlation function. The analytically derived theoretical results for chaotic systems are confirmed in microwave measurements within a mode-stirred reverberation chamber. This study provides fundamental insight into the Green's function retrieval technique and paves the way for a new technique to characterize electromagnetic antennas.

Wavefront dislocations reveal the topology of quasi-1D photonic insulators.

Phase singularities appear ubiquitously in wavefields, regardless of the wave equation. Such topological defects can lead to wavefront dislocations, as observed in a humongous number of classical wave experiments. Phase singularities of wave functions are also at the heart of the topological classification of the gapped phases of matter. Despite identical singular features, topological insulators and topological defects in waves remain two distinct fields. Realising 1D microwave insulators, we experimentally observe a wavefront dislocation - a 2D phase singularity - in the local density of states when the systems undergo a topological phase transition. We show theoretically that the change in the number of interference fringes at the transition reveals the topological index that characterises the band topology in the insulator.

Experimental Tuning of Transport Regimes in Hyperuniform Disordered Photonic Materials.

We present wave transport experiments in hyperuniform disordered arrays of cylinders with high dielectric permittivity. Using microwaves, we show that the same material can display transparency, photon diffusion, Anderson localization, or a full band gap, depending on the frequency ν of the electromagnetic wave. Interestingly, we find a second weaker band gap, which appears to be related to the second peak of the structure factor. Our results emphasize the importance of spatial correlations on different length scales for the formation of photonic band gaps.

Non-Hermitian CT-Symmetric Spectral Protection of Nonlinear Defect Modes.

We investigate, using a microwave platform consisting of a non-Hermitian Su-Schrieffer-Heeger array of coupled dielectric resonators, the interplay of a lossy nonlinearity and CT symmetry in the formation of defect modes. The measurements agree with the theory which predicts that, up to moderate pumping, the defect mode is an eigenstate of the CT-symmetric operator and retains its frequency at the center of the gap. At higher pumping values, the system undergoes a self-induced explicit CT-symmetry violation which removes the spectral topological protection and alters the shape of the defect mode.

Observation of supersymmetric pseudo-Landau levels in strained microwave graphene.

Using an array of coupled microwave resonators arranged in a deformed honeycomb lattice, we experimentally observe the formation of pseudo-Landau levels in the whole crossover from vanishing to large pseudomagnetic field strengths. This result is achieved by utilising an adaptable setup in a geometry that is compatible with the pseudo-Landau levels at all field strengths. The adopted approach enables us to observe the fully formed flat-band pseudo-Landau levels spectrally as sharp peaks in the photonic density of states and image the associated wavefunctions spatially, where we provide clear evidence for a characteristic nodal structure reflecting the previously elusive supersymmetry in the underlying low-energy theory. In particular, we resolve the full sublattice polarisation of the anomalous 0th pseudo-Landau level, which reveals a deep connection to zigzag edge states in the unstrained case.

Anisotropic nonlinear refractive index measurement of a photorefractive crystal via spatial self-phase modulation.

We show that the refractive index modification photoinduced in a biased nonlinear photorefractive crystal can be accurately measured and controlled by means of a background incoherent illumination and an external electric field. The proposed easy-to-implement method is based on the far-field measurement of the diffraction patterns of a laser beam propagating through a self-defocusing medium undergoing spatial self-phase modulation. For various experimental conditions, both saturation intensity and maximum refractive index modification have been measured. We also clearly evidence and characterise the anisotropic nonlinear response of the crystal in the stationary regime.

Fluctuations in an established transmission in the presence of a complex environment.

In various situations where wave transport is preeminent, like in wireless communication, a strong established transmission is present in a complex scattering environment. We develop a nonperturbative approach to describe emerging fluctuations which combines a transmitting channel and a chaotic background in a unified effective Hamiltonian. Modeling such a background by random matrix theory, we derive exact results for both transmission and reflection distributions at arbitrary absorption that is typically present in real systems. Remarkably, in such a complex scattering situation, the transport is governed by only two parameters: an absorption rate and the ratio of the so-called spreading width to the natural width of the transmission line. In particular, we find that the established transmission disappears sharply when this ratio exceeds unity. The approach exemplifies the role of the chaotic background in dephasing the deterministic scattering.

Microwave experiments simulating quantum search and directed transport in artificial graphene.

A series of quantum search algorithms have been proposed recently providing an algebraic speedup compared to classical search algorithms from N to √N, where N is the number of items in the search space. In particular, devising searches on regular lattices has become popular in extending Grover's original algorithm to spatial searching. Working in a tight-binding setup, it could be demonstrated, theoretically, that a search is possible in the physically relevant dimensions 2 and 3 if the lattice spectrum possesses Dirac points. We present here a proof of principle experiment implementing wave search algorithms and directed wave transport in a graphene lattice arrangement. The idea is based on bringing localized search states into resonance with an extended lattice state in an energy region of low spectral density-namely, at or near the Dirac point. The experiment is implemented using classical waves in a microwave setup containing weakly coupled dielectric resonators placed in a honeycomb arrangement, i.e., artificial graphene. Furthermore, we investigate the scaling behavior experimentally using linear chains.

Selective enhancement of topologically induced interface states in a dielectric resonator chain.

The recent realization of topological phases in insulators and superconductors has advanced the search for robust quantum technologies. The prospect to implement the underlying topological features controllably has given incentive to explore optical platforms for analogous realizations. Here we realize a topologically induced defect state in a chain of dielectric microwave resonators and show that the functionality of the system can be enhanced by supplementing topological protection with non-hermitian symmetries that do not have an electronic counterpart. We draw on a characteristic topological feature of the defect state, namely, that it breaks a sublattice symmetry. This isolates the state from losses that respect parity-time symmetry, which enhances its visibility relative to all other states both in the frequency and in the time domain. This mode selection mechanism naturally carries over to a wide range of topological and parity-time symmetric optical platforms, including couplers, rectifiers and lasers.

Topological transition of Dirac points in a microwave experiment.

By means of a microwave tight-binding analogue experiment of a graphenelike lattice, we observe a topological transition between a phase with a pointlike band gap characteristic of massless Dirac fermions and a gapped phase. By applying a controlled anisotropy on the structure, we investigate the transition directly via density of states measurements. The wave function associated with each eigenvalue is mapped and reveals new states at the Dirac point, localized on the armchair edges. We find that with increasing anisotropy, these new states are more and more localized at the edges.

Statistics of resonance states in a weakly open chaotic cavity with continuously distributed losses.

In this Rapid Communication, we demonstrate that a non-Hermitian random matrix description can account for both spectral and spatial statistics of resonance states in a weakly open chaotic wave system with continuously distributed losses. More specifically, the statistics of resonance states in an open two-dimensional chaotic microwave cavity are investigated by solving the Maxwell equations with lossy boundaries subject to Ohmic dissipation. We successfully compare the statistics of its complex-valued resonance states and associated widths with analytical predictions based on a non-Hermitian effective Hamiltonian model defined by a finite number of fictitious open channels.

Avoided-level-crossing statistics in open chaotic billiards.

We investigate a two-level model with a large number of open decay channels in order to describe avoided level crossing statistics in open chaotic billiards. This model allows us to describe the fundamental changes in the probability distribution of the avoided level crossings compared with the closed case. Explicit expressions are derived for systems with preserved and broken time-reversal symmetry. We find that the decay process induces a modification at small spacings of the probability distribution of the avoided level crossings due to an attraction of the resonances. The theoretical predictions are in complete agreement with the recent experimental results of Dietz [Phys. Rev. E 73, 035201 (2006)].

Statistics of resonance states in open chaotic systems: a perturbative approach.

We investigate the statistical properties of the complexness parameter which characterizes uniquely complexness (nonorthogonality) of resonance eigenstates of open chaotic systems. Specifying to the regime of weakly overlapping resonances, we apply the random matrix theory to the effective Hamiltonian formalism and derive analytically the probability distribution of the complexness parameter for two statistical ensembles describing the systems invariant under time reversal. For those with rigid spectra, we consider a Hamiltonian characterized by a picket-fence spectrum without spectral fluctuations. Then, in the more realistic case of a Hamiltonian described by the Gaussian orthogonal ensemble, we reveal and discuss the role of spectral fluctuations.

Controlled excitation of scar modes in passive and active multimode chaotic fiber.

A multimode optical fiber with a D-shaped cross section is an experimental paradigm of a wave system with chaotic ray dynamics. We show that seldom but usable modes, called scar modes, localized along some particular direction of the geometric trajectories, can be selectively excited. We report numerical simulations that demonstrate the importance of the so-called self-focal point in the scar mode selection process. We use a localized illumination in a passive fiber, or a localized gain in a ytterbium-doped fiber, located in the vicinity of this special point to control scar mode selection.

Cosine law at the atomic scale: toward realistic simulations of Knudsen diffusion.

We propose to reconsider the diffusion of atoms in the Knudsen regime in terms of a complex dynamical reflection process. By means of molecular dynamics simulations, we emphasize the asymptotic nature of the cosine law of reflection at the atomic scale, and carefully analyze the resulting strong correlations in the reflection events. A dynamical interpretation of the accommodation coefficient associated with the slip at the wall interface is also proposed. Finally, we show that the first two moments of the stochastic process of reflection depend nonuniformly on the incident angle.

Selective amplification of scars in a chaotic optical fiber.

In this Letter we propose an original mechanism to select scar modes through coherent gain amplification in a multimode D-shaped fiber. More precisely, we demonstrate the selective amplification of scar modes by positioning a gain region in the vicinity of the self-focal point of the shortest periodic orbit in the transverse motion.

Localized modes in a finite-size open disordered microwave cavity.

We present measurements of the spatial intensity distribution of localized modes in a two-dimensional open microwave cavity randomly filled with cylindrical dielectric scatterers. We show that each of these modes displays a range of localization lengths, and we successfully relate the largest value to the measured leakage rate at the boundary. These results constitute unambiguous signatures of the existence of strongly localized electromagnetic modes in two-dimensional open random media.

Diffractive orbits in the length spectrum of a two-dimensional microwave cavity with a small scatterer.

In a two-dimensional rectangular microwave cavity dressed with one pointlike scatterer, a semiclassical approach is used to analyze the spectrum in terms of periodic orbits and diffractive orbits. We show, both numerically and experimentally, how the latter can be accounted for in the so-called length spectrum, which is retrieved from two-point correlations of a finite-range frequency spectrum. Beyond its fundamental interest, this first experimental evidence of the role played by diffractive orbits in the spectrum of an actual cavity, can be the first step towards a technique to detect and track small defects in wave cavities.

Complete S matrix in a microwave cavity at room temperature.

We experimentally study the widths of resonances in a two-dimensional microwave cavity at room temperature. By developing a model for the coupling antennas, we are able to discriminate their contribution from those of Ohmic losses to the broadening of resonances. Concerning Ohmic losses, we experimentally put to evidence two mechanisms: damping along propagation and absorption at the contour, the latter being responsible for variations of widths from mode to mode due to its dependence on the spatial distribution of the field at the contour. A theory, based on an S -matrix formalism, is given for these variations. It is successfully validated through measurements of several hundreds of resonances in a rectangular cavity.

Speckle statistics in a chaotic multimode fiber.

Wave chaos is devoted to the study of wave motion when the geometrical limit of rays is chaotic. Imprints of ray chaos may be found either in spectral and spatial properties of modes or in spatio-temporal evolution of wave packets. In this paper, we present a thorough experimental and theoretical analysis of field statistics for light propagating in a multimode fiber with a noncircular cross section. This optical fiber serves as a powerful tool to image waves in a system where light rays exhibit a chaotic dynamics. We show that, in the speckle regime, the experimentally measured statistical properties of intensity patterns are well accounted for by a "random Gaussian" hypothesis. A comparison is also made in the case of regular ray motion by using a circular optical fiber. Possible extensions and applications of the tools and concepts of wave chaos are mentioned in modern communication technology.