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An open quantum system refers to a system that is further coupled to a bathsystem consisting of surrounding radiation fields, atoms, molecules, orproteins. The bath system is typically modeled by an infinite number ofharmonic oscillators. This system-bath model can describe the time-irreversibledynamics through which the system evolves toward a thermal equilibrium state atfinite temperature. In nuclear magnetic resonance and atomic spectroscopy,dynamics can be studied easily by using simple quantum master equations underthe assumption that the system-bath interaction is weak (perturbativeapproximation) and the bath fluctuations are very fast (Markovianapproximation). However, such approximations cannot be applied in chemicalphysics and biochemical physics problems, where environmental materials arecomplex and strongly coupled with environments. The hierarchical equations ofmotion (HEOM) can describe numerically "exact" dynamics of a reduced systemunder nonperturbative and non-Markovian system--bath interactions, which hasbeen verified on the basis of exact analytical solutions (non-Markovian tests)with any desired numerical accuracy. The HEOM theory has been used to treatsystems of practical interest, in particular to account for various linear andnonlinear spectra in molecular and solid state materials, to evaluate chargeand exciton transfer rates in biological systems, to simulate resonanttunneling and quantum ratchet processes in nanodevices, and to explore quantumentanglement states in quantum information theories. This article, presents anoverview of the HEOM theory, focusing on its theoretical background andapplications, to help further the development of the study of open quantumdynamics.

Perspective: Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM)