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The population evolution of single excitation is studied in the Aubry-Andr\'{e}- Harper (AAH) model coupled to a $d (=1,2,3)$-dimensional simplelattices bath with a focus on the effect of localization in the system and thedimensionality of bath. By performing a precise evaluation of time-independentSchr\"{o}dinger equation, the reduced energy levels of the system can bedetermined. It is found that the reduce energy levels show significantrelevance for the bath dimensions. Subsequently, the time evolution ofexcitation is studied in both the system and bath. It is found that excitationin the system can decay super-exponentially when $d=1$ or exponentially when$d=2,3$. Regarding the finite nature of bath, the spreading of excitation inthe lattices bath is also studied. We find that, depending on the dimensions ofbath and the initial state, the spreading of excitation in the bath isdiffusive or behaves localization.

Open dynamics in the Aubry-Andr\'{e}-Harper model coupled to a finite bath: the influence of localization in the system and dimensionality of bath