Onsets of hierarchy truncation and self-consistent Born approximation with quantum mechanics prescriptions invariance

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

An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.

Fokker-Planck quantum master equation for mixed quantum-semiclassical dynamics

We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.