Probing nonorthogonality of eigenfunctions and its impact on transport through open systems

Davy, Matthieu; Genack, Azriel Z.

doi:10.1103/physrevresearch.1.033026

Cited by 16 publications

(7 citation statements)

References 60 publications

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“…This was experimentally verified [31]. Remarkably, the exact statistics (1.6) appeared very recently in an experiment from [15] for microscopic separation of eigenvalues, suggesting some universality of this formula. Unfortunately, many of the models considered in the physics literature are perturbative, and most of the examined statistics are limited to expectations.…”

Section: Introductionsupporting

confidence: 61%

The distribution of overlaps between eigenvectors of Ginibre matrices

Bourgade

Dubach

2019

Probab. Theory Relat. Fields

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We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove:(i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables;(ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues;(iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale;(v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.

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Section: Introductionsupporting

confidence: 61%

The distribution of overlaps between eigenvectors of Ginibre matrices

Bourgade

Dubach

2019

Probab. Theory Relat. Fields

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“…Another steady source of interest in the statistics of eigenvector overlaps is due to its role in chaotic wave scattering. In that context O 1 (z) and O 2 (z 1 , z 2 ) have been studied for a few special models different from Ginibre (both theoretically [19,26,27] and very recently experimentally [10,11]) and in the associated models of random lasing [38,39]. In the scattering context all eigenvalues are necessarily complex, and the lasing threshold is associated with the eigenvalue with the smallest imaginary part.…”

Section: Gin2mentioning

confidence: 99%

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Fyodorov

Tarnowski

2020

Ann. Henri Poincaré

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We study the distribution of the eigenvalue condition numbers $$\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}$$ κ i = ( l i ∗ l i ) ( r i ∗ r i ) associated with real eigenvalues $$\lambda _i$$ λ i of partially asymmetric $$N\times N$$ N × N random matrices from the real Elliptic Gaussian ensemble. The large values of $$\kappa _i$$ κ i signal the non-orthogonality of the (bi-orthogonal) set of left $${\mathbf{l}}_i$$ l i and right $${\mathbf{r}}_i$$ r i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) $${{\mathcal {P}}}_N(z,t)$$ P N ( z , t ) of $$t=\kappa _i^2-1$$ t = κ i 2 - 1 and $$\lambda _i$$ λ i taking value z, and investigate its several scaling regimes in the limit $$N\rightarrow \infty $$ N → ∞ . When the degree of asymmetry is fixed as $$N\rightarrow \infty $$ N → ∞ , the number of real eigenvalues is $$\mathcal {O}(\sqrt{N})$$ O ( N ) , and in the bulk of the real spectrum $$t_i=\mathcal {O}(N)$$ t i = O ( N ) , while on approaching the spectral edges the non-orthogonality is weaker: $$t_i=\mathcal {O}(\sqrt{N})$$ t i = O ( N ) . In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as $$N\rightarrow \infty $$ N → ∞ . In such a regime eigenvectors are weakly non-orthogonal, $$t=\mathcal {O}(1)$$ t = O ( 1 ) , and we derive the associated JDF, finding that the characteristic tail $${{\mathcal {P}}}(z,t)\sim t^{-2}$$ P ( z , t ) ∼ t - 2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

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“…When there is a step in the potential or dielectric constant demarcating the sample from the surroundings in which the potential or dielectric constant is uniform so that waves do not scatter back into the sample, modes form a complete biorthogonal set. Departures from orthogonality grow as spectrally overlap of modes increases and neighboring modes become correlated [9,10,38]. However, the field within the medium can still be expressed as a superposition of modes [9,14].…”

Section: Introductionmentioning

confidence: 99%

Wave excitation and dynamics in non-Hermitian disordered systems

Huang

Kang

Genack

2022

Phys. Rev. Research

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Dynamic and steady state aspects of wave propagation are deeply connected in lossless open systems in which the scattering matrix is unitary. There is then an equivalence among the energy excited within the medium through all channels, the Wigner time delay, which is the sum of dwell times in all channels coupled to the medium, and the density of states. But these equivalences fall away in the presence of material loss or gain. In this paper, we use microwave measurements, numerical simulations, and theoretical analysis to discover the changing relationships among the internal field, transmission, transmission time, dwell time, total excited energy, and the density of states in with loss and gain, and their dependence upon dimensionality and spectral overlap. The field, including the contribution of the still coherent incident wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. The total energy excited is equal to the dwell time. When modes are spectrally isolated, the energy is proportional to the sum of products of the modal contribution to the density of states and the ratio of the linewidth without and with absorption. When modes overlap, however, the total energy is further reduced by loss and enhanced by gain. In 1D, the average transmission time is independent of loss, gain, and scattering strength. In higher dimensions, however, the average transmission time falls with loss, scattering strength and channel number, and increases with gain. It is the sum over Lorentzian functions associated with the zeros as well as the poles of the transmission matrix. This endows transmission zeros with robust topological characteristics: in unitary media, zeros occur either singly on the real axis or as conjugate pairs in the complex frequency plane. The transmission zeros are moved down or up in the complex plane by an amount equal to the internal rate of decay or growth of the field. In weakly absorbing media, the spectrum of the transmission time of the lowest transmission eigenchannel is the sum of Lorentzians due to transmission zeros plus a background due to far-off-resonance poles. As the scattering strength increases, conjugate pairs of zeros are brought to the real axis and converted to two single zeros which are constrained to move on the real axis until they combine and leave the real axis as a conjugate pair. The average density of transmission zeros in the complex plane is found from the fall of the average transmission time with absorption as zeros are swept into the lower half of the complex frequency plane. As a sample is deformed, two single zeros and a conjugate pair of zeros may interconvert on the real axis with a square root singularity in the sensitivity of the spacing between transmission zeros to structural change. Thus, the disposition of poles and zeros in the complex frequency plane provides a framework for understanding and controlling wave propagation in non-Hermitian systems.

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Probing nonorthogonality of eigenfunctions and its impact on transport through open systems

Cited by 16 publications

References 60 publications

The distribution of overlaps between eigenvectors of Ginibre matrices

The distribution of overlaps between eigenvectors of Ginibre matrices

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Wave excitation and dynamics in non-Hermitian disordered systems

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