  Many recent advancements in quantum computing leverage strong drives on
nonlinear systems for state preparation, signal amplification, or gate
operation. However, the interplay within such strongly driven system introduces
multi-scale dynamics that affects the long-time behavior of the system in
non-trivial ways that are very difficult to model. Therefore, the analysis of
these systems often relies on effective Hamiltonian models that introduce
additional nonlinear processes which approximate the long-time dynamics so that
highly oscillatory terms may be ignored. However, the removal of such high
frequency transitions can only be performed rigorously within a systematic
framework of time-coarse graining, which is a fundamentally irreversible
operation. This implies that standard approaches with unitary effective models
cannot accurately capture the long-time behavior of strongly driven nonlinear
quantum systems in general. We introduce a systematic perturbation theory for
obtaining the complete non-unitary effective model of the time-coarse grained
(TCG) dynamics of a driven quantum system to any order in the coupling
strengths. We derive a closed-form analytical formula for both unitary and
non-unitary contributions, in the form of an effective Hamiltonian and
non-unitary (pseudo-)dissipators. Remarkably, even though the effective theory
presumes unitary time evolution at the microscopic level, the time-coarse
grained dynamics is found to follow a non-unitary time evolution in general.
This occurs even when there is no open heat reservoir for the system to become
entangled with or dissipate into. We demonstrate the effectiveness of the new
method using several typical models of driven nonlinear systems in
superconducting circuits, and show that it generalizes and improves on existing
methods by providing more accurate results and explaining phenomena that have
not been accounted for.
