Fig. 5

Two-dimensional rephasing spectra \(S^{(3)}(\omega _3, \omega _1)\) for a system of two anyonic oscillators as a function of statistical angle \(\theta\) (vertical axis) and environmental noise correlation \(\xi\) (horizontal axis). Each panel shows the real part of the rephasing response function computed at fixed values of \(\theta \in \{0, \pi /2, \pi \}\) and \(\xi \in \{-1, -1/2, 0, +1/2, +1\}\). The contours highlight the formation of long-lived coherence pathways as a function of fractional statistics and bath correlation. Leftmost panels (red) correspond to purely absorptive responses in the absence of coherence. As \(\xi\) approaches ±1, mode protection emerges—enhanced or suppressed depending on the match between normal mode symmetry and noise structure. Exchange statistics (\(\theta \ne 0\)) break the symmetry between protected and unprotected modes, leading to asymmetric line shapes and phase-sensitive decoherence. Simulation parameters: oscillator frequencies \(\omega _1 = \omega _2 = 1\), inter-oscillator coupling \(J = 0.1\), temperature \(T = 0.1\), and bare dissipation rate \(\gamma = 0.1\).