Camera-based single-particle tracking enables quantitative determination of transport properties and provides nanoscale information about material characteristics such as viscosity and elasticity. However, static localization noise and the blurring of a particle's position over camera integration times introduce artifacts into measurement results even for a particle executing simple diffusion. Common data analysis methods based on the mean-square displacement do not properly account for these effects. In this paper, we analyze the statistics of tracking data for freely diffusing particles in realistic experimental scenarios. We derive a convenient and asymptotically optimal maximum likelihood estimator for the diffusion coefficient and for the magnitude of localization noise together with the corresponding Fisher information, which bounds the performance of all unbiased estimators. We find that the effect of varying the illumination profile during the camera integration time is quantified by a motion blur coefficient, R . We also find that a double-pulse illumination sequence maximizes the information content in some common experimental scenarios. Our results provide a rigorous theoretical framework and practical experimental recipe for achieving optimal performance in camera-based single-particle tracking.
Using spontaneous parametric down-conversion, we produce polarization-entangled states of two photons and characterize them using two-photon tomography to measure the density matrix. A controllable decoherence is imposed on the states by passing the photons through thick, adjustable birefringent elements. When the system is subject to collective decoherence, one particular entangled state is seen to be decoherence-free, as predicted by theory. Such decoherence-free systems may have an important role for the future of quantum computation and information processing.
Single-particle tracking is increasingly used to extract quantitative parameters on single molecules and their environment, while advances in spatial and temporal resolution of tracking techniques inspire new questions and avenues of investigation. Correspondingly, sophisticated analytical methods are constantly developed to obtain more refined information from measured trajectories. Here we point out some fundamental limitations of these approaches due to the finite length of trajectories, the presence of localization error, and motion blur, focusing on the simplest motion regime of free diffusion in an isotropic medium (Brownian motion). We show that two recently proposed algorithms approach the theoretical limit of diffusion coefficient uncertainty. We discuss the practical performance of the algorithms as well as some important implications of these results for single-particle tracking.
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