Extended Data Fig. 2: Dark-state physics in two-photon Rydberg gates.

a, In the far-detuned limit, the three-level system can be effectively described as a two-level system with a ‘dark’ state |D⟩ and a ‘bright’ state |B⟩. The population in the excited state |E⟩ is suppressed by a factor ∝ Δ⁻⁴ and does not participate in system dynamics. b, The Hamiltonian of the bare atomic system and the effective two-level system, in which α is the (time-dependent) ratio between the blue and red Rabi frequencies and \(\mathop{\alpha }\limits^{.}\) is its time derivative. c, Intermediate-state population during the parameterized time-optimal gate in the dark and bright configurations together with their Bloch-sphere trajectories in the {|D⟩, |B⟩} basis. d, Intermediate-state population during the smooth-amplitude gate and its Bloch-sphere trajectory in the instantaneous {|D⟩, |B⟩} basis. The simulation parameters correspond to those mentioned in Extended Data Fig. 3a and Extended Data Table 1.