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

Abstract

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 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.

Effect of quantum resonances on local temperature in nonequilibrium open systems

“…To investigate the properties of the impurity, the hierarchical equations of motion (HEOM) approach [65][66][67][68][69][70] is employed, which takes the reduced density matrix of the system ρ and a hierarchical set of auxiliary density operators as the basic variables. The HEOM theory is, in principle, formally exact, and its numerical outcomes are guaranteed to be quantitatively accurate if the results converge with respect to the truncation tier of the hierarchy.…”

Quantum-Classical Simulation of Molecular Motors Driven Only by Light

“…The calculation details are presented in the Supplementary Information Section 3. Propagation methods accounting explicitly for thermalization 25,26,26,27 may improve the efficiency estimate in the future. To estimate the time at which the complete rotation occurs for a part of the wavefunction the probability density at 40 • is monitored as a function of time in Fig.…”

Quantendynamik offener Systeme, Trendbericht Theoretische Chemie 2020, Teil 3

“…Beide Methodenklassen wurden systematisch zur Behandlung komplexer Systeme weiterentwickelt. [12][13][14][15][16] Eine Übersicht liefern die Literaturstellen 17,18) .…”