  Nuclear magnetic resonance (NMR) spectroscopy is a powerful technique for
analyzing the structure and function of molecules, and for performing
three-dimensional imaging of the spin density. At the heart of NMR
spectrometers is the detection of electromagnetic radiation, in the form of a
free induction decay (FID) signal, generated by nuclei precessing around an
applied magnetic field. While conventional NMR requires signals from 1e12 or
more nuclei, recent advances in sensitive magnetometry have dramatically
lowered this number to a level where few or even individual nuclear spins can
be detected. It is natural to ask whether continuous FID detection can still be
applied at the single spin level, or whether quantum back-action modifies or
even suppresses the NMR response. Here we report on tracking of single nuclear
spin precession using periodic weak measurements. Our experimental system
consists of carbon-13 nuclear spins in diamond that are weakly interacting with
the electronic spin of a nearby nitrogen-vacancy center, acting as an optically
readable meter qubit. We observe and minimize two important effects of quantum
back-action: measurement-induced decoherence and frequency synchronization with
the sampling clock. We use periodic weak measurements to demonstrate sensitive,
high-resolution NMR spectroscopy of multiple nuclear spins with a priori
unknown frequencies. Our method may provide the optimum route for performing
single-molecule NMR at atomic resolution.
