  We propose to apply the Back and Forth Nudging (BFN) method used for
geophysical data assimilations to estimate the initial state of a quantum
system. We consider a cloud of atoms interacting with a magnetic field while a
single observable is being continuously measured over time using homodyne
detection. The BFN method relies on designing an observer forward and backwards
in time. The state of the BFN observer is continuously updated by the measured
data and tends to converge to the systems state. The proposed estimator seems
to be globally asymptotically convergent when the system is observable. A
detailed convergence proof and simulations are given in the 2-level case. A
discussion on the extension of the algorithm to the multilevel case is also
presented.
