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 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.
Highlights
Time irreversibility is not a problem to be solved, but a reality to be experienced
In particular in the quantum case, realization of time irreversibility is difficult because the fundamental kinetic equations, including the Schrödinger equation and the Dirac equation, ensure that the dynamics are reversible in time
This system–bath model can describe the time irreversibility of the dynamics toward the thermal equilibrium state in which the energy supplied by fluctuations and the energy lost through dissipation are balanced, while the bath temperature does not change because its heat capacity is infinite
Summary
Time irreversibility is not a problem to be solved, but a reality to be experienced. This is true for the physical, chemical, and biological phenomena that we encounter throughout our lives. The variational approaches can be used to treat nonlinear system–bath coupling, anharmonic bath modes,[60] and a variety of Hamiltonians, for example, the Holstein Hamiltonian;[61–63] because the bath is described as a finite number of oscillators, the number of bath modes must be increased until convergence is realized to obtain the accurate results This implies that the study of long-time behavior using these approaches requires an intensive computational effort, whereas a reduced equation of motion approach requires a numerical effort that scales only linearly with the simulation time. The reduced hierarchical equations of motion (HEOM) theory is a method that can describe the dynamics of a system with a nonperturbative and non-Markovian system–bath interaction at finite temperature, even under strong time-dependent perturbations.[42–44,86–92] In this formalism, the effects of higher-order nonMarkovian system–bath interactions are mapped into the hierarchical elements of the reduced density matrix.
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