Figure 2

Controlled-phase (cz) gate protocol. (a) Two global Rydberg pulses of length τ and detuning Δ drive Bloch sphere rotations around two different axes due to a laser phase change ξ between pulses. (b) As a result of the evolution, each basis state returns to itself with an accumulated dynamical phase. | 00 ⟩ is uncoupled and, therefore, accumulates no phase. | 01 ⟩ and | 10 ⟩ are equivalent by symmetry ( ϕ 01 = ϕ 10 ), while | 11 ⟩ accumulates phase ϕ 11 . The cz gate is realized for ϕ 11 = 2 ϕ 01 − π . (c) The dynamics of the | 01 ⟩ and | 11 ⟩ states can be understood in terms of two-level systems with the same detuning Δ but different effective Rabi frequencies. The pulse length τ is chosen such that the | 11 ⟩ system undergoes a complete detuned Rabi cycle during the first pulse, while the | 01 ⟩ system undergoes an incomplete oscillation. The laser phase ξ is chosen such that the second pulse drives around a different axis to close the trajectory for the | 01 ⟩ system, while driving a second complete cycle for the | 11 ⟩ system. (d) The dynamical phases ϕ 01 and ϕ 11 are determined by the shaded area enclosed by the Bloch sphere trajectory and vary from 2 π to 0 as a function of Δ , corresponding to increasingly shallow trajectories. Insets show family of trajectories for different detunings. Choosing Δ ≈ 0.377 Ω realizes the cz gate.