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Quantum gas microscopes have revolutionized quantum simulations withultracold atoms, allowing to measure local observables and snapshots of quantumstates. However, measurements so far were mostly carried out in the occupationbasis. Here, we demonstrate how all kinetic operators, such as kinetic energyor current operators, can be measured and manipulated with single bondresolution. Beyond simple expectation values of these observables, thesingle-shot measurements allow to access full counting statistics and complexcorrelation functions. Our work paves the way for the implementation ofefficient quantum state tomography and hybrid quantum computing protocols foritinerant particles on a lattice. In addition, we demonstrate how site-resolvedprogrammable potentials enable a spatially-selective, parallel readout indifferent bases as well as the engineering of arbitrary initial states.

Local readout and control of current and kinetic energy operators in optical lattices