Back and forth nudging for quantum state estimation by continuous weak measurement

We propose to apply the Back and Forth Nudging (BFN) method used for geophysical data assimilations [1] 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 system's 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. An extension of the algorithm to the multilevel case is also presented.