Fig. 3

Geometric phase. a The system is initialized in the black state. A π/2 pulse on the first transition with zero phase creates a coherent superposition of the black and the red state. While no pulse is applied to the black state, a sequence of two π pulses is sent on the 2nd transition. The phase difference between these two pulses being Θ, a geometric phase equal to Θ is accumulated. Finally, a second π/2 pulse is sent on the first transition with a phase φ. This creates \(\cos ^2\frac{{{\mathbf{\Theta }} + \varphi }}{2}\) interference between the black and the red states, revealed by the probability P_{|−3/2〉, |−1/2〉} map as the function of φ and Θ. b The system is initialized in the black state. A π/2 pulse on the first transition with zero phase creates a coherent superposition of the black and the red state. While no pulse is applied to the black state, a sequence of two π pulses is sent on the 2nd and the 3rd transition, as shown in Fig. 2b. The phase difference between these two pulses being Θ, a geometric phase equal to 2Θ is accumulated. Finally, a second π/2 pulse is sent on the first transition with a phase φ. It creates \({\mathrm{cos}}^2\frac{{2{\mathbf{\Theta }} + \varphi }}{2}\) interference between the black and the red states, revealed by the probability P_{|−3/2〉, |−1/2〉} map as the function of φ and Θ. c The system is initialized in the red state. A π/2 pulse on the second transition with a zero phase creates a coherent superposition of the red and the green states. Simultaneously, a sequence of two π pulses is sent on the 1st and the 3rd transition, as shown in Fig. 2c The phase difference between these two pulses being Θ₁ and Θ₃, respectively. As a consequence geometric phases equal to Θ₁ and Θ₃ are accumulated on the red and on the green states, respectively. Finally, a second π/2 pulse on the first transition with a zero phase creates \({\mathrm{cos}}^2\frac{{{\mathbf{\Theta }}_1 + {\mathbf{\Theta }}_3}}{2}\) interference between the red and the green states, revealed by the probability P_{|−1/2〉, |1/2〉} map as the function of Θ₁ and Θ₃. This interference pattern involves geometric phases only