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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Factorized time correlation diagram analysis of paired causal systems excited by twin stochastic driving functions

Daniel P. Biebighauser, Daniel B. Turner, Darin J. Ulness

Published Jan 25, 2002• 7 citations
Publisher

Abstract

This work examines the properties and mathematical structure of factorized time correlation (FTC) diagram analysis in a general context. The goal is to extract general principles and analytic behavior that are not tied to any particular phenomenon in physics. It is hoped that this will provide a basis for expanded use of FTC diagram analysis beyond its current employment in the study of noisy light-based nonlinear optical spectroscopy. Furthermore, the concept of indirect correlation in a two-channel system driven by twin stationary circular Gaussian stochastic inputs is defined and discussed both analytically and through FTC diagram analysis.

DOI
10.1103/physreve.65.026142

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This work examines the properties and mathematical structure of factorized time correlation (FTC) diagram analysis in a general context. The goal is to extract general principles and analytic behavior that are not tied to any particular phenomenon in physics. It is hoped that this will provide a basis for expanded use of FTC diagram analysis beyond its current employment in the study of noisy light-based nonlinear optical spectroscopy. Furthermore, the concept of indirect correlation in a two-channel system driven by twin stationary circular Gaussian stochastic inputs is defined and discussed both analytically and through FTC diagram analysis.
Correlations are ubiquitous in nature and their principled study is of paramount importance in scientific development. The seminal contributions from John Bell offer a framework for analyzing the correlations between the components of quantum mechanical systems and have instigated an experimental tradition which has recently culminated with the Nobel Prize in Physics (2022). In physics, Bell's framework allows the demonstration of the non-classical nature of quantum systems just from the analysis of the observed correlation patterns. Bell's ideas need not be restricted to physics. Our contribution is to show an example of a Bell approach, based on the insight that correlations can be broken down into a part due to common, ostensibly significant causes, and a part due to noise. We employ data from finance (price changes of securities) as an example to demonstrate our approach, highlighting several general applications: first, we demonstrate a new measure of association, informed by the assumed causal relationship between variables. Second, our framework can lead to streamlined Bell-type tests of widely employed models of association, which are in principle applicable to any discipline. In the area of finance, such models of association are Factor Models and the bivariate Gaussian model. Overall, we show that Bell's approach and the models we consider are applicable as general statistical techniques, without any domain specificity. We hope that our work will pave the way for extending our general understanding for how the structure of associations can be analyzed.
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Using the recently developed phase-space techniques for the treatment of quantum-mechanical problems, we set up a procedure for calculating multitime-correlation functions in terms of the joint distribution functions. The correlation functions are then expressed simply as integrals over the associated phase space. Explicit expressions are given for these joint distribution functions, in terms of Green's functions of the ccc-number equations of motion for the phase-space equivalent of the density operator. Using these joint distribution functions, an exact regression theorem is rederived, and the connection with the multitime correspondence between classical and quantum stochastic processes is discussed.
Degenerate four-wave mixing using broadband, non-transform-limited (incoherent) laser light having a Lorentzian spectral density is treated analytically for a Bloch two-level system. We consider the configuration in which an incoherent beam from a Lorentzian source is split into two beams having distinct k vectors, k and ${\mathit{k}}^{\ensuremath{'}}$. The twin beam along ${\mathit{k}}^{\ensuremath{'}}$ has been subjected to a controllable delay, \ensuremath{\tau}, relative to k. These twin beams, focused into the sample of interest, generate new fourth fields, the signal at ks{\mathit{k}}_{\mathit{s}}ks​=2k-${\mathit{k}}^{\ensuremath{'}}$ being examined here as a function of \ensuremath{\tau}. A set of factorized time correlation (FTC) diagrams is introduced to organize the calculation and to yield both analytic expressions as well as physical insight. Not only is the signal not symmetric in \ensuremath{\tau} but the signal can peak at \ensuremath{\tau}\ensuremath{\ne}0 even in the absence of inhomogeneous broadening. Finally, the physical insights and the use of the FTC diagrams are extended to rationalize several other incoherent light spectroscopies. \textcopyright{} 1996 The American Physical Society.