arXiv subjects

Dmitry V. Savin

Publications and source records attributed to Dmitry V. Savin.

16 recordsLinked to original sources

arXiv2026

Random Matrix Theory for Chaotic Wave Scattering and Transport

We review random matrix approaches to chaotic wave scattering and transport in open systems. Starting from the effective non-Hermitian Hamiltonian formulation, we discuss the scattering matrix, reaction matrix, time delays, and complex resonances as complementary probes of open chaotic dynamics. We emphasize universal statistics governed by symmetry, openness, and channel coupling. Topics include the maximum-entropy description of fixed-energy scattering and its applications to quantum transport, energy correlations, resonance and eigenfunction statistics, and selected wave-chaotic phenomena induced by finite absorption. The focus throughout is on non-perturbative methods and universal structures underlying open quantum and wave chaotic systems.

nlin.CD↗

arXiv2025

Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices

Non-Hermitian random matrices with statistical spectral characteristics beyond the standard Ginibre ensembles have recently emerged in the description of dissipative quantum many-body systems as well as in non-ergodic wave transport in complex media. We investigate the class AI$^\dag$ of complex symmetric random matrices, for which available analytic results remain scarce. Using a recently proposed framework by one of the authors, we analyze this class for Gaussian entries and derive an explicit, closed-form expression for the joint distribution of a complex eigenvalue and its right eigenvector for arbitrary matrix size $N\ge 2$ in the entire complex plane. From this, we obtain the distribution of the eigenvector non-orthogonality overlap and the mean eigenvalue density, both for finite $N$ and in the large-$N$ limit. Notably, at the spectral edge both the eigenvalue density and eigenvector statistics exhibits a limiting behavior that differs from the Ginibre universtality class. This behavior is expected to be universal, as further supported by numerical evidence for Bernoulli random matrices.

math-ph↗

arXiv2022

Complex Dynamics of the Implicit Maps Derived from Iteration of Newton and Euler Method

Special exotic class of dynamical systems~ -- the implicit maps~ -- is considered. Such maps, particularly, can appear as a result of using of implicit and semi-implicit iterative numerical methods. In the present work we propose the generalization of the well-known Newton-Cayley problem. Newtonian Julia set is a fractal boundary on the complex plane, which divides areas of convergence to different roots of cubic nonlinear complex equation when it is solved with explicit Newton method. We consider similar problem for the relaxed, or damped, Newton method, and obtain the implicit map, which is non-invertible both time-forward and time-backward. It is also possible to obtain the same map in the process of solving of certain nonlinear differential equation via semi-implicit Euler method. The nontrivial phenomena, appearing in such implicit maps, can be considered, however, not only as numerical artifacts, but also independently. From the point of view of theoretical nonlinear dynamics they seem to be very interesting object for investigation. Earlier it was shown that implicit maps can combine properties of dissipative non-invertible and Hamiltonian systems. In the present paper strange invariant sets and mixed dynamics of the obtained implicit map are analyzed.

nlin.CD↗

arXiv2021

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 non-invasive receiving probes converges towards the difference of the causal and anti-causal 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 new fundamental insights into the Green's function retrieval technique and paves the way for a new technique to characterize electromagnetic antennas.

cond-mat.dis-nn↗

arXiv2020

Implementing Non-Universal Features with a Random Matrix Theory Approach: Application to Space-to-Configuration Multiplexing

We consider the efficiency of multiplexing spatially encoded information across random configurations of a metasurface-programmable chaotic cavity in the microwave domain. The distribution of the effective rank of the channel matrix is studied to quantify the channel diversity and to assess a specific system's performance. System-specific features such as unstirred field components give rise to nontrivial inter-channel correlations and need to be properly accounted for in modelling based on random matrix theory. To address this challenge, we propose a two-step hybrid approach. Based on an ensemble of experimentally measured scattering matrices for different random metasurface configurations, we first learn a system-specific pair of coupling matrix and unstirred contribution to the Hamiltonian, and then add an appropriately weighted stirred contribution. We verify that our method is capable of reproducing the experimentally found distribution of the effective rank with good accuracy. The approach can also be applied to other wave phenomena in complex media.

physics.app-ph↗

arXiv2020

Statistics of a simple transmission mode on a lossy chaotic background

Scattering on a resonance state coupled to a complicated background is a typical problem for mesoscopic quantum many-body systems as well as for wave propagation in the presence of a complex environment. On average, such a simple mode acquires an effective damping, the so-called "spreading" width, due to mixing with the background states. Modeling the latter by random matrix theory and employing the strength function formalism, we derive the joint distribution of the reflection and total transmission at arbitrary absorption in the background. The distribution is found to possess a remarkable symmetry between its reflection and transmission sectors, which is controlled by the ratio of the spreading to escape width. This in turn results in a symmetry relation between the marginal densities, despite the absence of the flux conservation at finite absorption. As an application, we study the statistics of total losses in the system at arbitrary coupling to the background.

cond-mat.mes-hall↗

arXiv2019

Wigner-Smith time-delay matrix in chaotic cavities with non-ideal contacts

We consider wave propagation in a complex structure coupled to a finite number $N$ of scattering channels, such as chaotic cavities or quantum dots with external leads. Temporal aspects of the scattering process are analysed through the concept of time delays, related to the energy (or frequency) derivative of the scattering matrix $\mathcal{S}$. We develop a random matrix approach to study the statistical properties of the symmetrised Wigner-Smith time-delay matrix $\mathcal{Q}_s=-\mathrm{i}\hbar\,\mathcal{S}^{-1/2}\big(\partial_\varepsilon\mathcal{S}\big)\,\mathcal{S}^{-1/2}$, and obtain the joint distribution of $\mathcal{S}$ and $\mathcal{Q}_s$ for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) $0<T\leq1$. We derive two representations of the distribution of $\mathcal{Q}_s$ in terms of matrix integrals specified by the Dyson symmetry index $β=1,2,4$ (the general case of unequally coupled channels is also discussed). We apply this to the Wigner time delay $τ_\mathrm{W}=(1/N)\,\mathrm{tr}\big\{\mathcal{Q}_s\big\}$, which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution $\mathscr{P}_{N,β}(τ)$ of the Wigner time delay in the weak coupling limit $NT\ll1$ and identify three different asymptotic regimes.

math-ph↗

arXiv2018

Envelope and phase distribution of a resonance transmission through a complex environment

A transmission amplitude is considered for quantum or wave transport mediated by a single resonance coupled to the background of many chaotic states. Such a model provides a useful approach to quantify fluctuations in an established signal induced by a complex environment. Applying random matrix theory to the problem, we derive an exact result for the joint distribution of the transmission intensity (envelope) and the transmission phase at arbitrary coupling to the background with finite absorption. The intensity and phase are distributed within a certain region, revealing essential correlations even at strong absorption. In the latter limit, we obtain a simple asymptotic expression that provides a uniformly good approximation of the exact distribution within its whole support, thus going beyond the Rician distribution often used for such purposes. Exact results are also derived for the marginal distribution of the phase, including its limiting forms at weak and strong absorption.

cond-mat.mes-hall↗

arXiv2017

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 novel 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 non-perturbative 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.

cond-mat.mes-hall↗

arXiv2015

Resonance width distribution in RMT: Weak coupling regime beyond Porter-Thomas

We employ the random matrix theory (RMT) framework to revisit the distribution of resonance widths in quantum chaotic systems weakly coupled to the continuum via a finite number M of open channels. In contrast to the standard first-order perturbation theory treatment we do not a priory assume the resonance widths being small compared to the mean level spacing. We show that to the leading order in weak coupling the perturbative $χ^2_M$ distribution of the resonance widths (in particular, the Porter-Thomas distribution at M=1) should be corrected by a factor related to a certain average of the ratio of square roots of the characteristic polynomial ("spectral determinant") of the underlying RMT Hamiltonian. A simple single-channel expression is obtained that properly approximates the width distribution also at large resonance overlap, where the Porter-Thomas result is no longer applicable.

cond-mat.mes-hall↗

arXiv2014

Efficient semiclassical approach for time delays

The Wigner time delay, defined by the energy derivative of the total scattering phase shift, is an important spectral measure of an open quantum system characterising the duration of the scattering event. It is related to the trace of the Wigner-Smith matrix Q that also encodes other time-delay characteristics. For chaotic cavities, these exhibit universal fluctuations that are commonly described within random matrix theory. Here, we develop a new semiclassical approach to the time-delay matrix which is formulated in terms of the classical trajectories that connect the exterior and interior regions of the system. This approach is superior to previous treatments because it avoids the energy derivative. We demonstrate the method's efficiency by going beyond previous work in studying the time-delay statistics for chaotic cavities with perfectly connected leads. In particular, the universality for moment generating functions of the proper time-delays (eigenvalues of Q) is established up to third order in the inverse number of scattering channels for systems with and without time-reversal symmetry. Semiclassical results are then obtained for a further two orders. We also show the equivalence of random matrix and semiclassical results for the second moments and for the variance of the Wigner time delay at any channel number.

nlin.CD↗

arXiv2004

Statistics of Impedance, Local Density of States, and Reflection in Quantum Chaotic Systems with Absorption

We are interested in finding the joint distribution function of the real and imaginary parts of the local Green function for a system with chaotic internal wave scattering and a uniform energy loss (absorption). For a microwave cavity attached to a single-mode antenna the same quantity has a meaning of the complex cavity impedance. Using the random matrix approach, we relate its statistics to that of the reflection coefficient and scattering phase and provide exact distributions for systems with beta=2 and beta=4 symmetry class. In the case of beta=1 we provide an interpolation formula which incorporates all known limiting cases and fits excellently available experimental data as well as diverse numeric tests.

cond-mat.mes-hall↗

arXiv2001

Distribution of proper delay times in quantum chaotic scattering: A crossover from ideal to weak coupling

The probability distribution of the proper delay times during scattering on a chaotic system is derived in the framework of the random matrix approach and the supersymmetry method. The result obtained is valid for an arbitrary number of scattering channels as well as arbitrary coupling to the energy continuum. The case of statistically equivalent channels is studied in detail. In particular, the semiclassical limit of infinite number of weak channels is paid appreciable attention.

cond-mat.dis-nn↗

arXiv2001

Reducing nonideal to ideal coupling in random matrix description of chaotic scattering: Application to the time-delay problem

We write explicitly a transformation of the scattering phases reducing the problem of quantum chaotic scattering for systems with M statistically equivalent channels at nonideal coupling to that for ideal coupling. Unfolding the phases by their local density leads to universality of their local fluctuations for large M. A relation between the partial time delays and diagonal matrix elements of the Wigner-Smith matrix is revealed for ideal coupling. This helped us in deriving the joint probability distribution of partial time delays and the distribution of the Wigner time delay.

cond-mat.dis-nn↗

arXiv1997

Quantum Versus Classical Decay Laws in Open Chaotic Systems

We study analytically the time evolution in decaying chaotic systems and discuss in detail the hierarchy of characteristic time scales that appeared in the quasiclassical region. There exist two quantum time scales: the Heisenberg time t_H and the time t_q=t_H/\sqrt{κT} (with κ>> 1 and T being the degree of resonance overlapping and the transmission coefficient respectively) associated with the decay. If t_q < t_H the quantum deviation from the classical decay law starts at the time t_q and are due to the openness of the system. Under the opposite condition quantum effects in intrinsic evolution begin to influence the decay at the time t_H. In this case we establish the connection between quantities which describe the time evolution in an open system and their closed counterparts.

cond-mat.mes-hall↗

arXiv1997

Parametric Correlations of Phase Shifts and Statistics of Time Delays in Quantum Chaotic Scattering: Crossover between Unitary and Orthogonal Symmetries

We analyse universal statistical properties of phase shifts and time delays for open chaotic systems in the crossover regime of partly broken time-reversal invariance. In particular, we find that the distribution of the time delay shows $τ^{-3/2}$ behavior for weakly open systems of any symmetry.

cond-mat.mes-hall↗