Fig. 2: Closed loops that lead to QHE and QR.

a Counterclockwise loop for the QR starting from \(\left|{\psi }_{-}\right\rangle\). b Clockwise loop for the QHE starting from \(\left|{\psi }_{+}\right\rangle\). Evolution of the system’s state along the trajectory of the closed loop is characterized by the fidelity \(\langle {\psi }_{-}|\rho (t)\left|{\psi }_{-}\right\rangle\) in (a) and the fidelity \(\langle {\psi }_{+}|\rho (t)\left|{\psi }_{+}\right\rangle\) in (b). The circles and error bars denote, respectively, the average and standard deviations of 10,000 measurements. The solid curves connecting the dots are obtained by simulating master equations. c Blue and red solid curves represent, respectively, the time evolution of the network in the counterclockwise and clockwise loops, where only three of the strokes associated with doing work are depicted. Durations of the five strokes are T₁ = T₅ = 6 μs, T₃ = 12 μs, and T₂ = T₄ = 150 μs. Other parameters are Ω/2π = 120 kHz, Δ_min/2π = −400 kHz, Δ_max/2π = 400 kHz, γ_min ≈ 0 kHz, and γ_max ≈ 1.45 MHz