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Start by turning on ${\Omega }_{12}$ first, while ${\Omega }_{23}=0$. The dark state is then exactly $|3⟩$, which is uncoupled even while the first pulse rises. Next turn on ${\Omega }_{23}$ while ${\Omega }_{12}$ remains appreciable, and then turn off ${\Omega }_{12}$ while ${\Omega }_{23}$ remains on. The ratio ${\Omega }_{23}/{\Omega }_{12}$ rises from zero to infinity, soThe final minus sign is an irrelevant global phase for level population transfer. This is the counter-intuitive sequence of stimulated Raman adiabatic passage: the laser coupling the target state to the excited state precedes the laser coupling the initial state to the excited state.

The quantum adiabatic theorem requires the dark state to rotate slowly compared with its separation from the bright states. Here $\dot{D}=-\dot{\theta }B$, so its coupling to either bright eigenstate has magnitude $|\dot{\theta }|/\sqrt{2}$. In the printed convention the entries ${\Omega }_{jk}=A_{jk}{d}_{jk}/(2\hslash )$ are angular-frequency couplings, and the condition isIf one instead uses an energy-valued Quantum Hamiltonian, its spectral gap is $\hslash \Omega$ and the same condition is $\hslash |\dot{\theta }|\ll \hslash \Omega$. Maintain nonzero overlap and gap while the angle changes. The pulses can vanish at the beginning and end, where the dark state has stopped rotating and is an exactly decoupled basis state.

An explicit smooth sequence has ${\Omega }_{12}={\Omega }_0{\sin }^2(\pi t/(6T))$ on $0<t<6T$ and zero otherwise, while ${\Omega }_{23}={\Omega }_0{\sin }^2(\pi (t-2T)/(6T))$ on $2T<t<8T$ and zero otherwise. Choose ${\Omega }_{0}T\gg 1$. The first pulse comes earlier, both overlap during the rotation, and the second persists after the first has ended. The following original sketch shows the envelopes and instantaneous dark-state level populations.

The adiabatic dark state of a driven Hamiltonian has exactly zero component on the lossy intermediate level throughout the ideal passage. It therefore suppresses spontaneous emission, whereas the two-pulse route explicitly transfers the population into $|2⟩$ before moving it to $|3⟩$. For an intermediate-level decay rate $\Gamma$, the leading loss probability is controlled by $\Gamma \int P_2(t)dt$. In stimulated Raman adiabatic passage, nonadiabatic leakage makes $P_2$ of order $(\hslash \dot{\theta }/\Omega {)}^2$, rather than the order-one intermediate population of the sequential swaps. Transfer is also less sensitive to precise pulse areas, provided the ordering, gap and slowness requirements hold. Making the process slower indefinitely is not always beneficial if the outer states also decohere; their coherence must survive the total passage duration.

For real nonnegative envelopes, put $\Omega =\sqrt{{\Omega }_1^2+{\Omega }_2^2}$ and choose a continuous $\theta =atan2({\Omega }_1,{\Omega }_2)$. The state $|D⟩=\cos \theta |1⟩-\sin \theta |3⟩$ is dark because its two amplitudes feeding $|2⟩$ cancel: ${\Omega }_1\cos \theta -{\Omega }_2\sin \theta =0$. The orthogonal bright combination $|B⟩=\sin \theta |1⟩+\cos \theta |3⟩$ couples to $|2⟩$ with strength $\Omega$. Thus the other eigenstates are $(|B⟩\pm |2⟩)/\sqrt{2}$ with energies $\pm \Omega$.

Start with ${\Omega }_1=0$ and ${\Omega }_2\ne 0$, so $\theta =0$ and $|D⟩=|1⟩$. Slowly increase ${\Omega }_1/{\Omega }_2$ to infinity while maintaining a nonzero gap; end with ${\Omega }_2=0$ and ${\Omega }_1\ne 0$, so $|D⟩=-|3⟩$. This target-to-intermediate pulse first is the counterintuitive ordering of stimulated Raman adiabatic passage. Its quantitative slowness condition isIt follows from $|⟨\pm |\dot{D}⟩|=|\dot{\theta }|/\sqrt{2}$ and the gap $\Omega$. Avoid switching both envelopes off while $\theta$ is changing; switch off together only after the transfer angle has become stationary. The dark eigenvalue is zero, so there is no dynamical phase; with this real eigenvector $⟨D|\dot{D}⟩=0$ as well. The ideal adiabatic quantum control therefore transfers $|1⟩$ to $-|3⟩$, physically the same target ray.

For a chosen orthonormal basis, the off-diagonal entries of a density operator encode quantum coherence between distinct basis states. A diagonal density operator is an incoherent mixture in that basis. This property is basis-dependent: diagonalizing the operator does not imply that it lacked coherence in the measurement or energy basis of interest. Relative phases affect these entries, while a global wavefunction phase cancels from the density operator. Dephasing damps the entries and can spoil stimulated Raman adiabatic passage by removing the ground-state interference that maintains the dark state.