Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality

Nonorthogonal Eigenvectors, Fluctuation-Dissipation Relations, and Entropy Production

Celebrated fluctuation-dissipation theorem (FDT) linking the response function to time dependent correlations of observables measured in the reference unperturbed state is one of the central results in equilibrium statistical mechanics. In this Letter we discuss an extension of the standard FDT to the case when multidimensional matrix representing transition probabilities is strictly non-normal. This feature dramatically modifies the dynamics, by incorporating the effect of eigenvector nonorthogonality via the associated overlap matrix of Chalker-Mehlig type. In particular, the rate of entropy production per unit time is strongly enhanced by that matrix. We suggest, that this mechanism has an impact on the studies of collective phenomena in neural matrix models, leading, via transient behavior, to such phenomena as synchronization and emergence of the memory. We also expect, that the described mechanism generating the entropy production is generic for wide class of phenomena, where dynamics is driven by non-normal operators. For the case of driving by a large Ginibre matrix the entropy production rate is evaluated analytically, as well as for the Rajan-Abbott model for neural networks.

Quantum Chaos and Level Dynamics

We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, such as level avoided-crossing distributions, level slope distributions, or level curvature distributions. We show both the aspects of universality in these distributions and their limitations. We concentrate in some detail on measures imported from the quantum information approach such as the fidelity susceptibility, and more generally, geometric tensor matrix elements. The possible open problems are suggested.

Scattering and transport properties of the three classical Wigner-Dyson ensembles at the Anderson transition

An extensive numerical analysis of the scattering and transport properties of the power-law banded random matrix model (PBRM) at criticality in the presence of orthogonal, unitary, and symplectic symmetries is presented. Our results show a good agreement with existing analytical expressions in the metallic regime and with heuristic relations widely used in studies of the PBRM model in the presence of orthogonal and unitary symmetries. Moreover, our results confirm that the multifractal behavior of disordered systems at criticality can be probed by measuring scattering and transport properties, which is of paramount importance from the experimental point of view. Thus, a full picture of the scattering and transport properties of the PBRM model at criticality corresponding to the three classical Wigner-Dyson ensembles is provided.

Chaotic scattering with localized losses: S-matrix zeros and reflection time difference for systems with broken time-reversal invariance

Motivated by recent studies of the phenomenon of coherent perfect absorption, we develop the random matrix theory framework for understanding statistics of the zeros of the (subunitary) scattering matrices in the complex energy plane, as well as of the recently introduced reflection time difference (RTD). The latter plays the same role for S-matrix zeros as the Wigner time delay does for its poles. For systems with broken time-reversal invariance, we derive the n-point correlation functions of the zeros in a closed determinantal form, and we study various asymptotics and special cases of the associated kernel. The time-correlation function of the RTD is then evaluated and compared with numerical simulations. This allows us to identify a cubic tail in the distribution of RTD, which we conjecture to be a superuniversal characteristic valid for all symmetry classes. We also discuss two methods for possible extraction of S-matrix zeros from scattering data by harmonic inversion.

Evanescent Wave Approximation for Non-Hermitian Hamiltonians

The counterpart of the rotating wave approximation for non-Hermitian Hamiltonians is considered, which allows for the derivation of a suitable effective Hamiltonian for systems with some states undergoing decay. In the limit of very high decay rates, on the basis of this effective description we can predict the occurrence of a quantum Zeno dynamics, which is interpreted as the removal of some coupling terms and the vanishing of an operatorial pseudo-Lamb shift.

Necklace-State-Mediated Anomalous Enhancement of Transport in Anderson-Localized non-Hermitian Hybrid Systems

Non-Hermiticity is known to manifest interesting modifications in the transport properties of complex systems. We report an intriguing regime of transport of hybrid quasiparticles in a non-Hermitian setting. We calculate the probability of transport, quantified by the Thouless conductance, of hybrid plasmons under varying degrees of disorder. With increasing disorder, we initially observe an expected decrease in average transmission, followed by an anomalous rise at localizing disorder. The behavior originates from the confluence of hybridization and non-Hermiticity, in which the former realizes the aggregation of eigenvalues migrating under disorder, while the latter enables energy transfer between the eigenmodes. We find that the enhanced transmission is mediated by quasiparticle hopping over various Anderson-localized states within the so-formed necklace states. We note that, in this scenario, all configurations exhibit the formation of necklace states and enhanced transport, unlike the conventionally known behavior of necklace states which only occurs in rare configurations.

Wave scattering properties of multiple weakly coupled complex systems

The statistics of the scattering of waves inside single ray-chaotic enclosures have been successfully described by the random coupling model (RCM). We expand the RCM to systems consisting of multiple complex ray-chaotic enclosures with various coupling scenarios. The statistical properties of the model-generated quantities are tested against measured data of electrically large multicavity systems of various designs. The statistics of model-generated transimpedance and induced voltages on a load impedance agree well with the experimental results. The RCM coupled chaotic enclosure model is general and can be applied to other physical systems, including coupled quantum dots, disordered nanowires, and short-wavelength electromagnetic and acoustic propagation through rooms in buildings, aircraft, and ships.

Fidelity susceptibility in Gaussian random ensembles

The fidelity susceptibility measures the sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here, we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system sizes. The results are verified by a comparison with numerical data.

Statistical properties of chaotic microcavities in small and large opening cases

We study the crossover behavior of statistical properties of eigenvalues in a chaotic microcavity with different refractive indices. The level spacing distributions change from Wigner to Poisson distributions, as the refractive index of a microcavity decreases. We propose a non-Hermitian matrix model with random elements describing the spectral properties of the chaotic microcavity, which exhibits the crossover behaviors as the opening strength increases.

Selectively exciting quasi-normal modes in open disordered systems

Transmission through disordered samples can be controlled by illuminating a sample with waveforms corresponding to the eigenchannels of the transmission matrix (TM). But can the TM be exploited to selectively excite quasi-normal modes and so control the spatial profile and dwell time inside the medium? We show in microwave and numerical studies that spectra of the TM can be analyzed into modal transmission matrices of rank unity. This makes it possible to enhance the energy within a sample by a factor equal to the number of channels. Limits to modal selectivity arise, however, from correlation in the speckle patterns of neighboring modes. In accord with an effective Hamiltonian model, the degree of modal speckle correlation grows with increasing modal spectral overlap and non-orthogonality of the modes of non-Hermitian systems. This is observed when the coupling of a sample to its surroundings increases, as in the crossover from localized to diffusive waves.

The authors present a study of the modal contributions to the transmission matrix in a scattering medium. They show that the incident wave form can be manipulated to control the net energy deposited in the sample, as well as the energy deposited in a selected quasi-normal mode.

Resonance Eigenfunction Hypothesis for Chaotic Systems

A hypothesis about the average phase-space distribution of resonance eigenfunctions in chaotic systems with escape through an opening is proposed. Eigenfunctions with decay rate γ are described by a classical measure that (i) is conditionally invariant with classical decay rate γ and (ii) is uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle. This explains the localization of fast-decaying resonance eigenfunctions classically. It is found to occur in the phase-space region having the largest distance to the chaotic saddle. We discuss the dependence on the decay rate γ and the semiclassical limit. The hypothesis is numerically demonstrated for the standard map.

Focusing inside Disordered Media with the Generalized Wigner-Smith Operator

We introduce a wave front 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 specific 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 works without optimization or phase conjugation and we expect it to be particularly attractive for optical imaging in disordered media.

Eigenvector statistics of the product of Ginibre matrices

We develop a method to calculate left-right eigenvector correlations of the product of m independent N×N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O=1+(m/2)(N-1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N→∞ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

Microwave experiments in the realm of fidelity

In this review article, we will demonstrate the power of microwave experiments in the realm of fidelity also known as Loschmidt echoes. As the determination of the fidelity itself is experimentally tedious and error prone, we will introduce the scattering fidelity which under the conditions of chaotic systems and weak coupling approaches the fidelity itself. The main ingredient in fidelity investigations is the type and strength of a perturbation. The perturbations presented here will be both global and local boundary perturbations, as well as local perturber movements but also the change of coupling to the environment. All these perturbations will produce their own fidelity decay as a function of the perturbation strength, which will be discussed in this article.

Localization in chaotic systems with a single-channel opening

We introduce a single-channel opening in a random Hamiltonian and a quantized chaotic map: localization on the opening occurs as a sensible deviation of the wave-function statistics from the predictions of random matrix theory, even in the semiclassical limit. Increasing the coupling to the open channel in the quantum model, we observe a similar picture to resonance trapping, made of a few fast-decaying states, whose left (right) eigenfunctions are entirely localized on the (preimage of the) opening, and plentiful long-lived states, whose probability density is instead suppressed at the opening. For the latter, we derive and test a linear relation between the wave-function intensities and the decay rates, similar to the Breit-Wigner law. We then analyze the statistics of the eigenfunctions of the corresponding (discretized) classical propagator, finding a similar behavior to the quantum system only in the weak-coupling regime.

Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations

The change of resonance widths in an open system under a perturbation of its interior has been recently introduced by Fyodorov and Savin [Phys. Rev. Lett. 108, 184101 (2012)] as a sensitive indicator of the nonorthogonality of resonance states. We experimentally study universal statistics of this quantity in weakly open two-dimensional microwave cavities and reverberation chambers realizing scalar and electromagnetic vector fields, respectively. We consider global as well as local perturbations, and also extend the theory to treat the latter case. The influence of the perturbation type on the width shift distribution is more pronounced for many-channel systems. We compare the theory to experimental results for one and two attached antennas and to numerical simulations with higher channel numbers, observing a good agreement in all cases.

Dysonian dynamics of the Ginibre ensemble

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a nontrivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of a new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the nonanalytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics that we observe for the Ginibre ensemble is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

Experimental observation of the spectral gap in microwave n-disk systems

Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmonic inversion the distribution of the imaginary parts of the resonances is determined. With increasing opening of the systems, a spectral gap is observed for thick as well as for thin repellers and for the latter case it is compared with the known topological pressure bounds. The maxima of the distributions are found to coincide for a large range of the distance to radius parameter with half of the classical escape rate. This confirms theoretical predictions based on rigorous mathematical analysis for the spectral gap and on numerical experiments for the maxima of the distributions.