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The information of quantum pathways can be extracted in the framework of theHamiltonian-encoding and Observable-decoding method. For closed quantumsystems, only off-diagonal elements of the Hamiltonian in the Hilbert space isrequired to be encoded to obtain the desired transitions. For open quantumsystems, environment-related terms will appear in the diagonal elements of theHamiltonian in the Liouville space. Therefore, diagonal encodings have to beperformed to differentiate different pathways, which will lead to self-to-selftransitions and inconsistency of pathway amplitudes with Dyson expansion. Inthis work, a well-designed transformation is proposed to avoid thecounter-intuitive transitions and the inconsistency, with or without controlfields. A three-level open quantum system is employed for illustration, andnumerical simulations show that the method are consistent with Dyson expansion.

Self-to-self transitions in open quantum systems: the origin and solutions