Fluctuations of the retarded van der Waals force

Abstract

The retarded van der Waals force between a polarizable particle and a perfectly conducting plate is reexamined. The expression for this force given by Casimir and Polder represents a mean force, but there are large fluctuations around this mean value on short time scales that are of the same order of magnitude as the mean force itself. However, these fluctuations occur on time scales which are typically of the order of the light travel time between the atom and the plate. As a consequence, they will not be observed in an experiment that measures the force averaged over a much longer time. In the large time limit, the magnitude of the mean-squared velocity of a test particle due to this fluctuating van der Waals force approaches a constant, and is similar to a Brownian motion of a test particle in a thermal bath with an effective temperature. However, the fluctuations are not isotropic in this case, and the shift in the mean-square velocity components can even be negative. We interpret this negative shift to correspond to a reduction in the velocity spread of a wave packet. The force fluctuations discussed in this paper are a special case of the more general problem of stress tensor fluctuations. These are of interest in a variety of areas of physics, including gravity theory. Thus the effects of van der Waals force fluctuations serve as a useful model for better understanding quantum effects in gravity theory.