Optimal determination of the equilibrium displacement of a damped harmonic oscillator in the presence of thermal noise

Abstract

From the title, one might assume that issues raised in this article must have been settled long ago. In a sense this is true. This may be one of those cases of failed communication between mathematicians and physicists, experimental physicists in particular. In the Introduction we show that a common approach to this linear, single-parameter estimation problem converges to the minimum variance, unbiased estimate only when analyzing measurements of pendulum motion over intervals much shorter than a single oscillation period (<1∕Q periods) where Q is the oscillator quality factor, or much longer (>Q periods). In the practical regime for long-period torsion oscillators used in gravitation studies (between 1∕Q and Q periods), the estimator variance may exceed the optimal value by a factor of order Q. Moreover, we have found no discussion in the experimental literature of the optimal estimation procedure appropriate to this measurement interval. In this article, using a matched filter technique, we derive the minimum variance, unbiased estimator for the equilibrium displacement of a damped harmonic oscillator in thermal equilibrium when interactions with the thermal bath are the leading source of noise. We compare the variance in this optimal estimator with the variance in other, commonly used estimators in the presence of thermal noise and white noise. We also compare the variance in these estimators for a mixture of thermal and white noise. This result has implications for experimental design, for the collection and analysis of data, and for the extension of this method to the nonlinear, multiparameter case.