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Solving realistic quantum systems coupled to an environment is a challengingtask. Here we develop a hierarchical functional derivative (HFD) approach forefficiently solving the non-Markovian quantum trajectories of an open quantumsystem embedded in a bosonic bath. An explicit expression for arbitrary orderHFD equation is derived systematically. Moreover, it is found that for ananalytically solvable model, this hierarchical equation naturally terminates ata given order and thus becomes exactly solvable. This HFD approach provides asystematic method to study the non-Markovian quantum dynamics of an open systemcoupled to a bosonic environment.

Higher-order solutions to non-Markovian quantum dynamics via hierarchical functional derivative