An official website of the United States government
Abstract The diffusion model is used to calculate both the time-averaged flow of particles in stochastic media and the propagation of waves averaged over ensembles of disordered static configurations. For classical waves exciting static disordered samples, such as a layer of paint or a tissue sample, the flux transmitted through the sample may be dramatically enhanced or suppressed relative to predictions of diffusion theory when the sample is excited by a waveform corresponding to a transmission eigenchannel. Even so, it is widely assumed that the velocity of waves is irretrievably randomized in scattering media. Here we demonstrate in microwave measurements and numerical simulations that the statistics of velocity of different transmission eigenchannels are distinct and remains so on all length scales and are identical on the incident and output surfaces. The interplay between eigenchannel velocities and transmission eigenvalues determines the energy density within the medium, the diffusion coefficient, and the dynamics of propagation. The diffusion coefficient and all scattering parameters, including the scattering mean free path, oscillate with the width of the sample as the number and shape of the propagating channels in the medium change.
more »
« less
Ohm’s law lost and regained: observation and impact of transmission and velocity zeros
Abstract The quantum conductance and its classical wave analogue, the transmittance, are given by the sum of the eigenvalues of the transmission matrix. However, neither measurements nor theoretical analysis of the transmission eigenchannels have been carried out to explain the dips in conductance found in simulations as new channels are introduced. Here, we measure the microwave transmission matrices of random waveguides and find the spectra of all transmission eigenvalues, even at dips in the lowest transmission eigenchannel that are orders of magnitude below the noise in the transmission matrix. Transmission vanishes both at topological transmission zeros, where the energy density at the sample output vanishes, and at crossovers to new channels, where the longitudinal velocity vanishes. Zeros of transmission pull down all the transmission eigenvalues and thereby produce dips in the transmittance. These dips and the ability to probe the characteristics of even the lowest transmission eigenchannel are due to correlation among the eigenvalues. The precise tracking of dips in the conductance by peaks in the density of states points to a further correlation between zeros and poles of the transmission matrix. The conductance approaches Ohm’s law as the sample width increases in accord with the correspondence principle.
more »
« less
- Award ID(s):
- 2211646
- PAR ID:
- 10558805
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 15
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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 fundamental wave properties with loss and gain, and their dependence upon dimensionality and spectral overlap. We begin with the demonstrations that the transmission time in random 1D media is equal to the density of states even in the presence of ultrastrong absorption and that its ensemble average is independent of the strengths of scattering and absorption. In contrast, the Wigner time becomes imaginary in the presence of loss, with real and imaginary parts that fall with absorption. In multichannel media, the transmission time remains equal to the density of states and is independent of the scattering strength in unitary systems but falls with absorption to a degree that increases with the strengths of absorption and scattering, and the number of channels coupled to the medium. We show that the relationships between key propagation variables in non-Hermitian systems can be understood in terms of the singularities of the phase of the determinant of the transmission matrix. The poles of the transmission matrix are the same as those of the scattering matrix, but the transmission zeros are fundamentally different. Whereas the zeros of the scattering matrix are the complex conjugates of the poles, the transmission zeros are topological: in unitary systems they occur only singly on the real axis or as conjugate pairs. We follow the evolution and statistics of zeros in the complex plane as random samples are deformed. The sensitivity of the spacing of zeros in the complex plane with deformation of the sample has a square-root singularity at a zero point at which two single zeros and a complex pair interconvert. The transmission time is a sum of Lorentzian functions associated with poles and zeros. The sum over poles is the density of states with an average that is independent of scattering and dissipation. But the sum over zeros changes with loss, gain, scattering strength and the number of channels in ways that make it possible to control ultranarrow spectral features in transmission and transmission time. We show that 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 energy excited may be expressed in terms of the resonances of the medium and is equal to the dwell time even in the presence of loss or gain.more » « less
-
Understanding vanishing transmission in Fano resonances in quantum systems and metamaterials and perfect and ultralow transmission in disordered media has advanced the knowledge and applications of wave interactions. Here, we use analytic theory and numerical simulations to understand and control the transmission and transmission time in complex systems by deforming a medium and adjusting the level of gain or loss. Unlike the zeros of the scattering matrix, the position and motion of the zeros of the determinant of the transmission matrix (TM) in the complex plane of frequency and field decay rate have robust topological properties. In systems without loss or gain, the transmission zeros appear either singly on the real axis or as conjugate pairs in the complex plane. As the structure is modified, two single zeros and a complex conjugate pair of zeros may interconvert when they meet at a square root singularity in the rate of change of the distance between the transmission zeros in the complex plane with sample deformation. The transmission time is the spectral derivative of the argument of the determinant of the TM. It is a sum over Lorentzian functions associated with the resonances of the medium, which is the density of states, and with the zeros of the TM. Transmission vanishes, and the transmission time diverges as zeros are brought near the real axis. Monitoring the transmission and transmission time when two zeros are close may open new possibilities for ultrasensitive detection.more » « less
-
Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $$\{\lambda _i\}$$ and eigenvectors $$\{\boldsymbol{u}_i\}$$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $$n$$ to obtain empirical eigenvalues $$\{\tilde{\lambda }_i\}$$ and eigenvectors $$\{\tilde{\boldsymbol{u}}_i\}$$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $$\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$$ while leveraging the assumption that the true covariance matrix having size $$p$$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $$p\to \infty $$ to a spectral density $$\rho (\lambda )$$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $$p$$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $$\lambda $$ and approximate the distribution of the expected square error $$r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$$ across the matrix ensemble for all $$\boldsymbol{u}_i$$ associated with $$\lambda _i=\lambda $$. We find, for example, that for sufficiently large matrix size $$p$$ and sample size $n> p$, the probability density of $$r$$ scales as $1/nr^2$. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if $$r$$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.more » « less
-
Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least 67.9% of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery’s theorem and show how to apply it to prove the following result on simple zeros: If all the zeros ρ=β+iγ of the Riemann zeta-function such that T3/8<γ≤T satisfy ∣∣β−1/2∣∣<1/(2logT), then, as T tends to infinity, at least 61.7% of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where β=1/2. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.more » « less
