  Coherent population trapping is a well-known quantum phenomenon in a driven
$\Lambda$ system, with many applications across quantum optics. However, when a
stochastic bath is present in addition to vacuum noise, the observed trapping
is no longer perfect. Here we derive a time-convolutionless master equation
describing the equilibration of the $\Lambda$ system in the presence of
additional temporally correlated classical noise, with an unknown decay
parameter. Our simulations show a one-to-one correspondence between the decay
parameter and the depth of the characteristic dip in the photoluminescence
spectrum, thereby enabling the unknown parameter to be estimated from the
observed spectra. We apply our analysis to the problem of qubit state
initialization in a $\Lambda$ system via dark states and show how the
stochastic bath affects the fidelity of such initialization as a function of
the desired dark-state amplitudes. We show that an optimum choice of Rabi
frequencies is possible.
