Quantum Limits in Scanning Transmission Electron Microscopy

We analyze the quantum limits of four-dimensional scanning transmission electron microscopy (4D-STEM). In estimating the moduli and phases of the Fourier coefficients of the sample's electrostatic potential, we find that 4D-STEM does not permit the quantum limit of precision. In particular, 4D-STEM can attain about half of the available quantum Fisher information. Preclusion of the quantum limit is the result of detection in the diffraction plane, and thus applies to all 4D-STEM techniques. Near-optimum information transfer is achieved by a delocalized speckled probe. We compare with Zernike phase-contrast imaging, which can attain the quantum limit for all spatial frequencies admitted by the optical system, though at lower spatial resolution. Our conclusions also apply to analogous imaging techniques employing other forms of coherent scalar radiation.