  Quantum control and measurement are two sides of the same coin. To affect a
dynamical map, well-designed time-dependent control fields must be applied to
the system of interest. To read out the quantum state, information about the
system must be transferred to a probe field. We study a particular example of
this dual action in the context of quantum control and measurement of atomic
spins through the light-shift interaction with an off-resonant optical probe.
By introducing an irreducible tensor decomposition, we identify the coupling of
the Stokes vector of the light field with moments of the atomic spin state.
This shows how polarization spectroscopy can be used for continuous weak
measurement of atomic observables that evolve as a function of time.
Simultaneously, the state-dependent light shift induced by the probe field can
drive nonlinear dynamics of the spin, and can be used to generate arbitrary
unitary transformations on the atoms. We revisit the derivation of the master
equation in order to give a unified description of spin dynamics in the
presence of both nonlinear dynamics and photon scattering. Based on this
formalism, we review applications to quantum control, including the design of
state-to-state mappings, and quantum-state reconstruction via continuous weak
measurement on a dynamically controlled ensemble.
