  We investigate the determination of a Hamiltonian parameter in a quantum
system undergoing continuous measurement. We demonstrate a computationally
rapid yet statistically optimal method to estimate an unknown and possibly
time-dependent parameter, where we maximize the likelihood of the observed
stochastic readout. By dealing directly with the raw measurement record rather
than the quantum state trajectories, the estimation can be performed while the
data is being acquired, permitting continuous tracking of the parameter during
slow drifts in real time. Furthermore, we incorporate realistic nonidealities,
such as decoherence processes and measurement inefficiency. As an example, we
focus on estimating the value of the Rabi frequency of a continuously measured
qubit, and compare maximum likelihood estimation to a simpler fast Fourier
transform. Using this example, we discuss how the quality of the estimation
depends on both the strength and duration of the measurement; we also discuss
the trade-off between the accuracy of the estimate and the sensitivity to drift
as the estimation duration is varied.
