Multiple Core-Hole Coherence in X-Ray Four-Wave-Mixing Spectroscopies

Correlation-function expressions are derived for the coherent nonlinear response of molecules to three resonant ultrafast pulses in the x-ray regime. The ability to create two-core-hole states with controlled attosecond timing in four-wave-mixing and pump probe techniques should open up new windows into the response of valence electrons, which are not available from incoherent x-ray Raman and fluorescence techniques. Closed expressions for the necessary four-point correlation functions are derived for the electron-boson model by using the second order cumulant expansion to describe the fluctuating potentials. The information obtained from multidimensional nonlinear techniques could be used to test and refine this model, and establish an anharmonic oscillator picture for electronic excitations

Wavelet Electrodynamics I

A new representation for solutions of Maxwell's equations is derived. Instead of being expanded in plane waves, the solutions are given as linear superpositions of spherical wavelets dynamically adapted to the Maxwell field and well-localized in space at the initial time. The wavelet representation of a solution is analogous to its Fourier representation, but has the advantage of being local. It is closely related to the relativistic coherent-state representations for the Klein-Gordon and Dirac fields developed in earlier work.Comment: 8 Pages in Plain Te

Carrier mode selective working point and side band imbalance in LIGO I

In gravitational wave interferometers, the input laser beam is phase modulated to generate radio-frequency side bands that are used to lock the cavities. The mechanism is the following: the frequency of the side bands and the carrier is chosen in such a way that their response to small changes of the longitudinal degrees of freedom is different. This difference is therefore monitored and it serves as an error signal for controlling the optical cavity lengths, as they are linearly related to the set of observed phases between carrier and side bands. Among the others, one longitudinal degree of freedom is optimally sensitive to the space-time distortions propagating through the cosmos, as predicted by the general theory of relativity. The observation of the astrophysical signal relies on the measurement of that specific degree of freedom. The entire problem is more complex when the transverse degrees of freedom are taken into account, because the relative phase between the fields also depends on their overlap. In order to establish an unambiguous relation between length changes and phase measurements, there must be one circulating optical mode and the only difference between carrier and side bands must be their amplitude. We will show that the variability of the transverse degrees of freedom and their different actions on carrier and side band fields puts a severe limit on this assumption. Unless the system is made of perfect and perfectly matched optical cavities, it is never governed by one unique coherent state and any adjustment of the optical lengths results from a compromise between the lengths that are optimal for the carrier field and the side band ones. Such a compromise alters the correspondence between error signals and cavity lengths, calculated in the one-dimensional treatment. We assess the strength of this effect and relate it to the sensitivity of the instrument (which relies on the reconstruction of that correspondence) in realistic circumstances

Slepian functions and their use in signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla

Data-adaptive harmonic spectra and multilayer Stuart-Landau models

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency, and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey furthermore a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled---provided the decay of temporal correlations is sufficiently well-resolved---within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise, are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.Comment: 26 pages, double columns; 15 figure