Fig. 3: Dynamics of spin-boson model with truncated Dyck paths.
From: Unified framework for open quantum dynamics with memory
a1, b1, c1 Magnetization (〈σz(t)〉) dynamics predicted using \({{{\bf{{{{\mathcal{K}}}}}}}}\) constructed via Dyck diagrams with increasing truncation orders (from light to darker colors) compared to exact results (see Supplementary Note 6). a2, b2, c2 Bath spectral densities extracted through the Dyck diagrammatic method with increasing truncation order (from white to black colors) compared to exact spectral densities (dashed), see Supplementary Note 3F for more details. These results come from numerically exact trajectories, initiated from linearly independent initial states \({\rho }_{1}(0)=\frac{1}{2}({{{\bf{1}}}}+{\sigma }_{z}),\,{\rho }_{2}(0)=\frac{1}{2}({{{\bf{1}}}}-{\sigma }_{z}),\,{\rho }_{3}(0)=\frac{1}{2}({{{\bf{1}}}}+{\sigma }_{x}),\,{\rho }_{4}(0)=\frac{1}{2}({{{\bf{1}}}}+{\sigma }_{x}+{\sigma }_{y}+{\sigma }_{z})\). Parameters used are: Δ = 1 (other parameters are expressed relative to Δ), ϵ =  0, β  =  5, Δt  =  0.1 (a1, b1, c1) or Δt  = 0.05 (a2, b2, c2), ωc  =  7.5, and ξ  =  0.1 and s  =  1 (a1 and a2), ξ  =  0.5 and s =  1 (b1 and b2), or ξ  = 0.5 and s =  0.5 (c1 and c2).
