Steady-state relations for a two-level system locally and relatively-strongly coupled to a generic many-body quantum chaotic environment

We study the long-time average of the reduced density matrix (RDM) of a two-level system as the central system, which is locally coupled to a generic many-body quantum chaotic system as the environment, under an overall Schr{o}dinger evolution. The system-environment interaction has a generic form with dissipation. It is shown that, in addition to the exact relations due to unit trace and hermiticity, an approximate relation exists among elements of the averaged RDM computed in the eigenbasis of the central systems Hamiltonian in some interaction regimes. In particular, an explicit expression of the relation is derived for relatively strong interactions, whose strength is above the mean level spacing of the environment, meanwhile, remains small compared with the central systems level spacing. Numerical simulations performed in a model with the environment as a defect Ising chain confirm the analytical predictions.