= Solution
\b[Excited-state decay is suppressed in the ideal dark-state limit.] The dark state has zero amplitude on $|2\rangle$, so a spontaneous-emission jump of the form $\sqrt\Gamma|j\rangle\langle2|$ annihilates it. There is consequently no excited-state <level population> to decay in perfect adiabatic following. Finite speed introduces bright-state admixture and a small excited <level population>; its integrated loss is of order $\Gamma\int P_2(t)dt$. Thus robustness is conditional on good adiabaticity, not immunity to arbitrarily large decay at fixed pulse strength and duration. In a strongly overdamped regime the relevant relaxation gap can also limit following.
\b[Moderate envelope perturbations are tolerated] because this transfer is fixed by the endpoints of the dark-state path, not by an accurately calibrated resonant <pulse area>. Common amplitude changes do not change the angle, although reducing them too far closes the useful gap. Independent smooth changes still work if they preserve the counter-intuitive order, the endpoint ratios, resonance and a sufficiently large gap relative to $|\dot\theta|$. A wrong sequence, missing overlap, rapid fluctuations or unstable relative optical phase can spoil the transfer.
\b[Ground-state <decoherence> is not suppressed by the dark-state mechanism.] During the overlap, the dark <orthogonal projector> has <quantum coherence> $\rho_{13}=-\sin\theta\cos\theta$. Dephasing of the $1$–$3$ transition removes precisely this relative-phase information, even if $P_2=0$ initially. For example, with $L=\sqrt{\gamma/2}(|1\rangle\langle1|-|3\rangle\langle3|)$ in a <Lindblad equation>, its dissipator gives $\dot\rho_{13}|_{\rm deph}=-\gamma\rho_{13}$. It therefore drives a nontrivial dark superposition away from its pure <orthogonal projector>. The transfer should be slow enough for adiabaticity but fast compared with the relevant ground-state <quantum coherence> lifetime. Ideal protection from excited-state <spontaneous emission> does not protect this ground-state <quantum coherence>.
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