  We study a large ensemble of nuclear spins interacting with a single electron
spin in a quantum dot under optical excitation and photon detection. When a
pair of applied laser fields satisfy two-photon resonance between the two
ground electronic spin states, detection of light scattering from the
intermediate exciton state acts as a weak quantum measurement of the effective
magnetic (Overhauser) field due to the nuclear spins. If the spin were driven
into a coherent population trapping state where no light scattering takes
place, then the nuclear state would be projected into an eigenstate of the
Overhauser field operator and electron decoherence due to nuclear spins would
be suppressed: we show that this limit can be approached by adapting the laser
frequencies when a photon is detected. We use a Lindblad equation to describe
the time evolution of the driven system under photon emission and detection.
Numerically, we find an increase of the electron coherence time from 5 ns to
500 ns after a preparation time of 10 microseconds.
