Fig. 1: Non-Markovian \({{{{{{{\mathcal{APT}}}}}}}}\)-symmetric system.
From: Theoretical and experimental characterization of non-Markovian anti-parity-time systems
a Two modes E1,2(t) evolve with opposite phases ± Δωt in frame rotating with frequency ω0. Due to finite speed of light, each mode at time t (filled circles) couples to the other at an earlier time t − τ (open circles). This non-Markovian coupling κ is shown along the (shaded) past light cones. This model, described by Eq. (1), is experimentally realized with two semiconductor lasers with bidirectional, time-delayed feedback; see Fig. 4. b Amplification rate Uτ shows sideband oscillations with a constant width (SOW) (solid black traces). Results for \({U}_{\exp }\) show that the SOW is halved (dashed blue traces). U > 0 region (pink) denotes amplifying modes, while U < 0 region (violet) denotes decaying modes. Inset: in the Markovian limit τ = 0, the \({{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}\) transition from U > 0 to U = 0 occurs at Δω = κ. c Steady-state intensity I1(Δω) obtained from four, coupled, nonlinear rate equations shows sideband oscillations whose constant width is halved when ω1 is varied (\({L}_{\exp }\); dashed blue traces) instead of varying Δω while keeping ω0 constant (Lτ; solid black traces). Despite obvious similarities, explicit mapping from U(Δω) to the steady-state I1,2(Δω) is unknown. Inset: At τ = 0, a central dome at small detuning changes into a flat intensity profile for Δω ≥ κ. d Exemplary traces of experimentally measured intensity I1(Δω) obtained by sweeping ω1 at τ = 0.75 ns (blue) and τ = 1.3 ns (red) show that observed SOW is reduced with increasing τ. Their features are consistent with our model and full laser dynamics simulations. e Exemplary traces of intensity I1(Δω) obtained at κ=1.1 GHz (blue) and κ = 1.9 GHz (red) show that the observed SOW is insensitive to the coupling κ. The central dome in (b–e) at small Δω is present in the Markovian limit (τ = 0) and signals the standard \({{{{{{{\mathcal{A}}}}}}}}{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}\)-transition. We analytically determine the behavior of the key non-Markovian signature SOWn(κ, τ) for Lτ and \({L}_{\exp }\).
