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The eigenstate thermalization hypothesis (ETH) explains why nonintegrable quantum many-body systems thermalize internally if the Hamiltonian lacks symmetries. If the Hamiltonian conserves one quantity ("charge"), the ETH implies thermalization within a charge sector-in a microcanonical subspace. But quantum systems can have charges that fail to commute with each other and so share no eigenbasis; microcanonical subspaces may not exist. Furthermore, the Hamiltonian will have degeneracies, so the ETH need not imply thermalization. We adapt the ETH to noncommuting charges by positing a non-Abelian ETH and invoking the approximate microcanonical subspace introduced in quantum thermodynamics. Illustrating with SU(2) symmetry, we apply the non-Abelian ETH in calculating local operators' time-averaged and thermal expectation values. In many cases, we prove, the time average thermalizes. However, we find cases in which, under a physically reasonable assumption, the time average converges to the thermal average unusually slowly as a function of the global-system size. This work extends the ETH, a cornerstone of many-body physics, to noncommuting charges, recently a subject of intense activity in quantum thermodynamics.

Maintaining an in-focus image over long time scales is an essential and non-trivial task for a variety of microscopic imaging applications. Here, we present an autofocusing method that is inexpensive, fast, and robust. It requires only the addition of one or a few off-axis LEDs to a conventional transmitted light microscope. Defocus distance can be estimated and corrected based on a single image under this LED illumination using a neural network that is small enough to be trained on a desktop CPU in a few hours. In this work, we detail the procedure for generating data and training such a network, explore practical limits, and describe relevant design principles governing the illumination source and network architecture.