Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 60 / 5 / c / Solution

Excited-state decay is suppressed in the ideal dark-state limit. The dark state has zero amplitude on $|2⟩$, so a spontaneous-emission jump of the form $\sqrt{\Gamma }|j⟩⟨2|$ 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.

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.

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 $L=\sqrt{\gamma /2}(|1⟩⟨1|-|3⟩⟨3|)$ in a Lindblad equation, its dissipator gives ${\dot{\rho }}_{13}{|}_{\mathrm{deph}}=-\gamma {\rho }_{13}$. 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.