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IEEE Statistical Signal Processing Workshop (SSP)

The Wigner distribution for ordinary linear differential equations and wave equations

Lorenzo Galleani, Leon W. Cohen

Published Nov 7, 2002• 3 citations
Publisher

Abstract

A new method is presented to study systems governed by ordinary linear differential equations and partial differential equations whose solutions are waves. We show that one can obtain a differential equation for the Wigner distribution of the solution of a dynamical equation of evolution. As an example we derive in a new way the equation governing the Wigner distribution for the Schrodinger equation. We also consider differential equations where the forcing terms are random processes.

DOI
10.1109/ssap.2000.870193

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A new method is presented to study systems governed by ordinary linear differential equations and partial differential equations whose solutions are waves. We show that one can obtain a differential equation for the Wigner distribution of the solution of a dynamical equation of evolution. As an example we derive in a new way the equation governing the Wigner distribution for the Schrodinger equation. We also consider differential equations where the forcing terms are random processes.
We present a general procedure for obtaining equations of motion for the Wigner distribution of functions that are governed by ordinary and partial differential equations. For the case of fields we show that in general one must consider Wigner distribution of the four variables, position, momentum, time and frequency. We also show that in general one cannot write an equation of motion for position and momentum however it can be done in some cases, the Schrodinger equation being one such case. Our method leads to an equation of motion for the Schrodinger equation with time dependent potentials in contrast to the result obtained by Wigner and Moyal which was for time independent potentials.
We use the Wigner distribution to study systems subjected to random forces. We define the instantaneous spectrum as the ensemble average of the Wigner distribution, and we write the differential equation whose solution gives us the time-varying spectrum of the state variable. We consider the cases of both constant and time-varying coefficients. We apply the method to study the instantaneous spectrum of a harmonic oscillator driven by Gaussian noise, with both constant and time-varying coefficients. In the latter case our method clearly reveals the nonstationarity of the power spectrum and we confirm our result by numerical simulations.
We have previously presented a method to write equations for the Wigner distribution corresponding to the solution of a linear differential equation. We extend this method to arbitrary time-frequency distributions, the Short Time Fourier Transform, and the wavelet transform. Also, we apply the method to stochastic signals and address a number of applications including Brownian motion. We obtain the Wigner distribution for Brownian motion by solving the Langevin equation and also by using our method. We obtain exact solutions.
A phase space description of Schrödinger dynamics is provided in terms of a quantum kinetic formalism relying on the introduction of an appropriate extension of the well-known Wigner transform, also accounting for time delocalizations. This “space-time Wigner distribution,” built up in the framework of two-time correlation functions, is shown to be governed by a non-Markovian, integro-differential equation of convolution type. Its utility in investigating long time dynamics of quantum systems is also discussed and illustrated with some examples.
We present a novel new way-called Schrödingerization-to simulate general (quantum and nonquantum) systems of linear ordinary and partial differential equations (PDEs) via quantum simulation. We introduce a new transform, referred to as the warped phase transformation, where any linear-including nonautonamous-system of ordinary or partial differential equation can be recast into a system of Schrödinger's equations, in real time, in a straightforward way. This approach is not only applicable to PDEs for classical problems but is also useful for quantum problems, including the preparation of quantum ground states and Gibbs thermal states, the simulation of quantum states in random media in the semiclassical limit, simulation of Schrödinger's equation in a bounded domain with artificial boundary conditions, and other non-Hermitian physics. This formulation is versatile enough to be applicable in a simple way to both digital quantum simulation as well as to analog quantum simulation, and using either qubits or continuous-variable quantum systems (qumodes).