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Physical Review A

Bayesian analysis of nonlinear fluctuating-measurement systems

Masahiro Agu

Published Oct 1, 1981• 0 citations
Publisher

Abstract

Bayesian analysis of the physical-measurement process is formulated with respect to the physical quantity being subject to the master equation. Here nonlinear dynamics of the measurement system and its own fluctuations are taken into account with the use of a nonlinear Langevin equation. The coupled nonlinear equations of the most probable path and of the variance of the fluctuations around the path are derived with the results of the measurement taken into account. The theory is applied to the measurement in which the field is detected with the use of a cavity resonator or laser amplifier. The minimum uncertainty in the estimate of the most probable path of the field is determined as a function of the variance of the fluctuations of the measurement system itself.

DOI
10.1103/physreva.24.2174

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Bayesian analysis of the physical-measurement process is formulated with respect to the physical quantity being subject to the master equation. Here nonlinear dynamics of the measurement system and its own fluctuations are taken into account with the use of a nonlinear Langevin equation. The coupled nonlinear equations of the most probable path and of the variance of the fluctuations around the path are derived with the results of the measurement taken into account. The theory is applied to the measurement in which the field is detected with the use of a cavity resonator or laser amplifier. The minimum uncertainty in the estimate of the most probable path of the field is determined as a function of the variance of the fluctuations of the measurement system itself.
A Bayesian analysis of measurement process is formulated with respect to the physical quantity being subject to the master equation. The dynamical equation of the a posteriori probability distribution of the physical quantity is derived with the use of the system size expansion. The coupled nonlinear equations of the most probable path with the results of measurements taken into account and of the variance of the fluctuations around the path are derived, which are the extensions of those of the Kalman's filter in the linear control theory to the nonlinear physical system on the basis of a physically reasonable assumption, the "local nature of transition." The formula of the entropy reduction and the HHH theorem which are associated with the measurement process are also derived.
The Langevin-equation (1) for the fluctuating light-field E in a nonlinear semiconductor resonator has been derived in Ref.[1]: All quantities are dimensionless, εi being proportional to the amplitude of the incoming field, κ is a damping constant, ω" is the ratio of imaginary and real part of the dielectric function and Δ is proportional to the detuning. The additive term f(τ) is the classical noise source representing a Gauss-Markov process. The staionary properties of eq.(1) have been investigated in Ref.[1] and the details of optical bistable operation has been determined.
A detailed quantum-mechanical description of an electron in a Penning trap is developed in which nonlinear relativistic effects as well as stochastic effects arising from radiative damping and the continuous measurement of the electronic motion are taken into account. A master equation is derived that describes the influence of these environmental effects on the bistable dynamics of the electron. Manifestations of measurement-induced fluctuations on the bistable dynamics of individual continuous measurement processes are investigated with the help of the quantum state diffusion model.
The field in the micromaser cavity, the atoms which both pump and probe the micromaser, and the detectors that determine the outgoing states of the atoms are treated as a single system in order to derive a nonlinear master equation for the field in the micromaser cavity which takes the effect of undetected atoms into account. Analytic solutions for this master equation are found for both phase-insensitive and phase-sensitive detection. These solutions can be used to explicitly show the back-action effects that measurement has on the spectrum of the micromaser. In addition, these solutions are used to find analytic expressions for the mean number of successive detector clicks of the same kind as well as atom-atom and photon-photon correlation functions. In this way, the analytic solutions are linked to experimentally accessible data.
In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schrödinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described by a stochastic equation for the state vector. We present a method of deriving explicit evolution operators for linear quantum trajectories, and apply this to a number of physical examples of varying mathematical complexity. In the Heisenberg picture evolution resulting from continuous observation may be described by quantum Langevin equations. We use this method to examine the noise spectrum that results from a continuous observation of the position of a moving mirror, and examine the possibility of detecting the noise resulting from the quantum back-action of the measurement. In addition to the work on continuous measurement theory, we also consider the problem of reconstructing the state of a quantum system from a set of measurements. We present a scheme for determining the state of a single cavity mode from the photon statistics measured both before and after an interaction with one or two two-level atoms.
This study summarizes the theory of laser-amplifier noise based on the master equation, the quantum-mechanical Langevin equation, and the rate equation. The three approaches are used differently depending on the application. Laser-amplifier noise in the direct-detection receiver is described by the master equation. Meanwhile, the quantum-mechanical Langevin equation is most effective for analyzing noise appearing in coherent optical communication systems. The rate equation approach provides a clear understanding of the intensity-noise characteristics of laser oscillators, including the amplitude squeezing.