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We study the long-time average of the reduced density matrix (RDM) of atwo-level system as the central system, which is locally coupled to a genericmany-body quantum chaotic system as the environment, under an overallSchr\"{o}dinger evolution. The system-environment interaction has a genericform with dissipation. It is shown that, in addition to the exact relations dueto unit trace and hermiticity, an approximate relation exists among elements ofthe averaged RDM computed in the eigenbasis of the central system's Hamiltonianin some interaction regimes. In particular, an explicit expression of therelation is derived for relatively strong interactions, whose strength is abovethe mean level spacing of the environment, meanwhile, remains small comparedwith the central system's level spacing. Numerical simulations performed in amodel with the environment as a defect Ising chain confirm the analyticalpredictions.

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