Fig. 2: Eigenvalue analysis of time-delayed \({{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}\) system.
From: Theoretical and experimental characterization of non-Markovian anti-parity-time systems
Eigenvalues λ = u + iv of Liouvillian Lτ occur at the intersections of F(u, v) = 0 (blue dot–dash) and G(u, v) = 0 (red solid) contours, shown here for κτ = 1.5 and Δω/κ = 1. Properties of G(u, v) = 0 contours and their intersections with the two axes are analytically determined by the Lambert W function38,39. At small detuning, the hyperbolic F(u, v) = 0 contour always intersects the u > 0 axis and gives the central dome that survives in the Markovian limit. At large Δω, intersections of the G = 0 and F = 0 contours on the vertical axis (u = 0) give an infinite sequence of Uτ(Δω) > 0 ↔ Uτ(Δω) < 0 transitions that manifest as sideband oscillations seen in Fig. 1b–e.
