Superresolving optical ruler based on spatial mode demultiplexing for systems evolving under Brownian motion

The development of superresolution techniques, i.e., allowing for efficient resolution below the Rayleigh limit, became one of the important branches in contemporary optics and metrology. Recent findings show that perfect spatial mode demultiplexing (SPADE) into Hermite-Gauss modes followed by photon counting enables one to reach the quantum limit of precision in the task of estimation of separation between two weak stationary sources in the sub-Rayleigh regime. In order to check the limitations of the method, various imperfections such as misalignment or crosstalk between the modes were considered. Possible applications of the method in microscopy call for the adaptive measurement scheme, as the position of the measured system can evolve in time, causing non-negligible misalignment. In this paper, we examine the impact of Brownian motion of the center of the system of two weak incoherent sources of arbitrary relative brightness on adaptive SPADE measurement precision limits. The analysis is carried out using Fisher information, from which the limit of precision can be obtained by Cram\'er-Rao bound. As a result, we find that Rayleigh's curse is present in such a scenario; however, SPADE measurement can outperform perfect direct imaging. What is more, a suitable adjustment of the measurement time between alignments allows measurement with near-optimal precision.