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Register a new userOn the Kramers-Kronig transform with logarithmic kernel for the reflection phase in the Drude model
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J.-M. Andre
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We use the Kramers-Kronig transform (KKT) with logarithmic kernel to obtain the reflection phase and, subsequently, the complex refractive index of a bulk mirror from reflectance. However, there remains some confusion regarding the formulation for this analysis. Assuming the damped Drude model for the dielectric constant and the oblique incidence case, we calculate the additional terms: phase at zero frequency and Blashke factor and we propose a reformulated KKT within this model. Absolute reflectance in the s-polarization case of a gold film is measured between 40 and 350 eV for various glancing angles using synchrotron radiation and its complex refractive index is deduced using the reformulated KKT that we propose. The results are discussed with respect to the data available in the literature.
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The ultimate precision in any measurement is dictated by the physical process implementing the observation. The methods of quantum metrology have now succeeded in establishing bounds on the achievable precision for phase measurements over noisy channels. In particular, they demonstrate how the Heisenberg scaling of the precision can not be attained in these conditions. Here we discuss how the ultimate bound in presence of loss has a physical motivation in the Kramers-Kronig relations and we show how they link the precision on the phase estimation to that on the loss parameter.
A Kramers-Kronig receiver with a continuous wave tone added digitally at the transmitter is combined with a digital resolution enhancer to limit the increase in transmitter quantization noise. Performance increase is demonstrated, as well as the ability to reduce the number of bits in the digital-to-analog converter.
For tunable control of asymmetric light reflection, we propose a Rydberg atomic system of the optical response varying in space induced by the long-range position-dependent Rydberg dipole-dipole interaction either in the type of self-van der Waals dipole-dipole interaction or the cross F{o}rster-like dipole-dipole exchange interaction. In such a one-dimensional system consisting of a control atomic driven upon the Rydberg state and a homogeneous target atomic ensemble, the non-localized action from the control atom on the target atoms gradually decreases with the distance between the control and target atoms. Our scheme yields a nonlinear correspondence from a finite spectra range to a finite spatial range of susceptibility via the nonlinear characteristics of Rydberg interaction relative to the position. Therefore, the asymmetric reflection can be induced via the spatial modulation on the target ensemble. In particular, the reflection from one direction can be completely suppressed when the absorption and dispersion parts of the susceptibility are modulated to satisfy the spatial Kramers-Kronig relation in an infinite spectral range. The opposite reflection exhibits a band of a small nonzero reflectivity due to the realistic restriction of the cold atomic density of a relatively small value. Thus, via trapping the target atoms in the optical lattice for the Bragg scattering, we enhance the nonzero reflection obviously and retain the directional reflectionlessness.
Kernel methods are a highly effective and widely used collection of modern machine learning algorithms. A fundamental limitation of virtually all such methods are computations involving the kernel matrix that naively scale quadratically (e.g., constructing the kernel matrix and matrix-vector multiplication) or cubically (solving linear systems) with the size of the data set $N.$ We propose the Fast Kernel Transform (FKT), a general algorithm to compute matrix-vector multiplications (MVMs) for datasets in moderate dimensions with quasilinear complexity. Typically, analytically grounded fast multiplication methods require specialized development for specific kernels. In contrast, our scheme is based on auto-differentiation and automated symbolic computations that leverage the analytical structure of the underlying kernel. This allows the FKT to be easily applied to a broad class of kernels, including Gaussian, Matern, and Rational Quadratic covariance functions and physically motivated Greens functions, including those of the Laplace and Helmholtz equations. Furthermore, the FKT maintains a high, quantifiable, and controllable level of accuracy -- properties that many acceleration methods lack. We illustrate the efficacy and versatility of the FKT by providing timing and accuracy benchmarks and by applying it to scale the stochastic neighborhood embedding (t-SNE) and Gaussian processes to large real-world data sets.
The macroscopic electric permittivity of a given medium may depend on frequency, but this frequency dependence cannot be arbitrary, its real and imaginary parts are related by the well-known Kramers-Kronig relations. Here, we show that an analogous paradigm applies to the macroscopic electric conductivity. If the causality principle is taken into account, there exists Kramers-Kronig relations for conductivity, which are mathematically equivalent to the Hilbert transform. These relations impose strong constraints that models of heterogeneous media should satisfy to have a physically plausible frequency dependence of the conductivity and permittivity. We illustrate these relations and constraints by a few examples of known physical media. These extended relations constitute important constraints to test the consistency of past and future experimental measurements of the electric properties of heterogeneous media.
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