  Quantum state reconstruction based on weak continuous measurement has the
advantage of being fast, accurate, and almost non-perturbative. In this work we
present a pedagogical review of the protocol proposed by Silberfarb et al., PRL
95 030402 (2005), whereby an ensemble of identically prepared systems is
collectively probed and controlled in a time-dependent manner so as to create
an informationally complete continuous measurement record. The measurement
history is then inverted to determine the state at the initial time through a
maximum-likelihood estimate. The general formalism is applied to the case of
reconstruction of the quantum state encoded in the magnetic sublevels of a
large-spin alkali atom, 133Cs. We detail two different protocols for control.
Using magnetic interactions and a quadratic ac-Stark shift, we can reconstruct
a chosen hyperfine manifold F, e.g., the 7-dimensional F=3 manifold in the
electronic-ground state of Cs. We review the procedure as implemented in
experiments (Smith et al., PRL 97 180403 (2006)). We extend the protocol to the
more ambitious case of reconstruction of states in the full 16-dimensional
electronic-ground subspace (F=3 \oplus F=4), controlled by microwaves and
radio-frequency magnetic fields. We give detailed derivations of all physical
interactions, approximations, numerical methods, and fitting procedures,
tailored to the realistic experimental setting. For the case of light-shift and
magnetic control, reconstruction fidelities of \sim 0.95 have been achieved,
limited primarily by inhomogeneities in the light shift. For the case of
microwave/RF-control we simulate fidelity >0.97, limited primarily by
signal-to-noise.

Quantum Physics