  Magnetic resonance imaging (MRI) is a powerful technique for investigating
the microscopic properties and dynamics of physical systems. In this work we
demonstrate state-sensitive MRI of ultracold atoms in an optical lattice.
Single-shot spatial resolution is 120 nm, well below the lattice spacing, and
number sensitivity is +/-2.4 for 150 atoms on a single site, well below
Poissonian atom-number fluctuations. We achieve this by combining
high-spatial-resolution control over the atomic spin using an atom chip,
together with nearly quantum-limited spin measurement, obtained by dispersively
coupling the atoms to light in a high-finesse optical cavity. The MRI is
minimally disruptive of the atoms' internal state, preserving the magnetisation
of the gas for subsequent experiments. Using this technique, we observe the
nonequilibrium transport dynamics of the atoms among individual lattice sites.
We see the atom cloud initially expand ballistically, followed by the onset of
interaction-inhibited transport.
