  This dissertation studies the statistics and modeling of a quantum system
probed by a coherent laser field. We focus on an ensemble of qubits
dispersively coupled to a traveling wave light field. The first research topic
explores the quantum measurement statistics of a quasi-monochromatic laser
probe. We identify the shortest timescale that successive measurements
approximately commute. Our model predicts that for a probe in the near
infrared, noncommuting measurement effects are apparent for subpicosecond
times.
  The second dissertation topic attempts to find an approximation to a
conditional master equation, which maps identical product states to identical
product states. Through a technique known as projection filtering, we find such
a equation for an ensemble of qubits experiencing a diffusive measurement of a
collective angular momentum projection, and global rotations. We then test the
quality of the approximation through numerical simulations. In the presence of
strong randomized rotations, the approximation reproduces the exact expectation
values to within 95%.
  The final topic applies the projection filter to the problem of state
reconstruction. We find an initial state estimate based on a single continuous
measurement of an identically prepared atomic ensemble. Given the ability to
make a continuous collective measurement and simultaneously applying time
varying controls, it is possible to find an accurate estimate given based upon
a single measurement realization. Here we explore the fundamental limits of
this protocol by studying an idealized model for pure qubits, which is limited
only by measurement backaction. Using the exact dynamics to produce simulated
measurements, we then numerically search for a maximum likelihood estimate
based on the approximate expression. Our estimation technique nearly achieves
an average fidelity bound set by an optimum POVM.
