  We propose and analyze continuous measurements of atom number and atomic
currents using dispersive probing in an optical cavity. For an atom-number
measurement in a closed system, we relate both the detection noise and the
heating rate due to measurement back-action to Tan's contact, and identify an
emergent universal quantum non-demolition (QND) regime in the good-cavity
limit. We then show that such a continuous QND measurement of atom number
serves as a quantum-limited current transducer in a two-terminal setup. We
derive a universal bound on the precision of current measurement, which results
from a tradeoff between detection noise and back-action of the atomic current
measurement. Our results apply regardless of the strength of interaction or the
state of matter and set fundamental bounds on future precision measurements of
transport properties in cold-atom quantum simulators.
