The canonically consistent quantum master equation (CCQME) method to treat system-bath dynamics is used to describe intramolecular proton transfer in the thioacetylacetone molecule (TAA, C5H8OS), modeled as an N-level quantum system coupled to a solvent. The solvent is represented as a harmonic bath (a continuum of oscillators) characterized by an Ohmic-Drude spectral density. We benchmark the secularized population dynamics and steady-state populations predicted by CCQME against numerically exact hierarchical equations of motion (HEOM) theory and compare it to the corresponding secularized Redfield results. Our results reveal that Redfield dynamics deviates increasingly from the HEOM reference as the system-bath coupling strength grows. In contrast, for not-too-strong couplings, the secularized CCQME population dynamics remains consistent with HEOM over an extended system-bath coupling range, and approaches the second-order mean-force Gibbs state. A complementary non-secular calculation shows that retaining population-coherence coupling reveals limitations of the second-order treatment for coherence-sensitive observables.
Comments
24 pages, 9 figures; revised version following peer review, with clarifications and additional discussion. Main results and conclusions unchanged
The canonically consistent quantum master equation (CCQME) method to treat system-bath dynamics is used to describe intramolecular proton transfer in the thioacetylacetone molecule (TAA, C5H8OS), modeled as an N-level quantum system coupled to a solvent. The solvent is represented as a harmonic bath (a continuum of oscillators) characterized by an Ohmic-Drude spectral density. We benchmark the secularized population dynamics and steady-state populations predicted by CCQME against numerically exact hierarchical equations of motion (HEOM) theory and compare it to the corresponding secularized Redfield results. Our results reveal that Redfield dynamics deviates increasingly from the HEOM reference as the system-bath coupling strength grows. In contrast, for not-too-strong couplings, the secularized CCQME population dynamics remains consistent with HEOM over an extended system-bath coupling range, and approaches the second-order mean-force Gibbs state. A complementary non-secular calculation shows that retaining population-coherence coupling reveals limitations of the second-order treatment for coherence-sensitive observables.
We put forth a new class of quantum master equations that correctly reproduce the asymptotic state of an open quantum system beyond the infinitesimally weak system-bath coupling limit. Our method is based on incorporating the knowledge of the reduced steady state into its dynamics. The correction not only steers the reduced system toward a correct steady state but also improves the accuracy of the dynamics, thereby refining the archetypal Born-Markov weak-coupling second-order master equations. In case of equilibrium, we use the exact mean-force Gibbs state to correct the Redfield quantum master equation. By benchmarking it with the exact solution of the damped harmonic oscillator, we show that our method also helps correct the long-standing issue of positivity violation, albeit without complete positivity. Our method of a canonically consistent quantum master equation opens a new perspective in the theory of open quantum systems leading to a reduced density matrix accurate beyond the commonly used Redfield and Lindblad equations, while retaining the same conceptual and numerical complexity.
A central challenge in the theory of open quantum systems is the development of theoretical frameworks capable of accurately describing large, strongly interacting quantum many-body systems in the regime of strong system-bath interactions. In this work, we take a step toward this goal by formulating a polaron-transformed version of the canonically consistent quantum master equation (CCQME) [T. Becker \emph{et~al.}, Phys. Rev. Lett. \textbf{129}, 200403 (2022)]. The CCQME extends beyond standard weak coupling approaches while retaining the same numerical complexity as conventional quantum master equations, thereby enabling the treatment of large quantum systems. The polaron transformation further enhances the accessible system-bath interaction strengths, allowing us to move from moderate to ultra-strong interaction regimes. We present a unified and transparent derivation of these two approaches and combine them to obtain the polar-transformed CCQME (PT-CCQME). Applying our method to the paradigmatic spin boson model, we find excellent agreement with numerically exact time-evolving matrix product operator (TEMPO) simulations. Finally, we predict an initial-state-independent \emph{slowing} down of thermalization in the strong-coupling regime of the spin-boson model.
The simulation of electronically nonadiabatic dynamics and spectroscopy remains as a central challenge in computational chemistry. Amongst the reduced dynamic methods, various forms of the perturbative quantum master equation (QME) including Fermi's golden rule (FGR), Redfield equation, and Marcus theory enjoy widespread use for their efficiency in the perturbative regime, and provides important physical intuition for the underlying dynamics. In the first chapter of this dissertation, the accuracy of various QMEs that treat off-diagonal electronic coupling perturbatively (OD-QME) will be examined on a spin-boson model. OD-QME works in the interaction picture, limiting its application to population dynamics. A novel way of retrieving Schrodinger picture coherences from interaction picture dynamics will be introduced, and the resulting dynamics will be compared against the numerically exact multi-configuration time dependent Hartree method. It will be shown that OD-QME can generate accurate population and coherence in appropriate parameter ranges, and that the time-nonlocal QME is more accurate than time-local QME, especially if the QME uses a population only projection. In the second chapter, the aforementioned method of calculating electronic coherence via OD-QME is applied to the simulation of absorption spectra. The result will be compared against the numerically exact hierarchical equations of motion, as well as the Redfield time-local QME which is widely applied in the context of spectroscopy. It will be shown that OD-QME exhibits comparable accuracy to Redfield QME over a wide parameter range. Both QMEs fail to predict the spectra accurately in the case of a high-frequency bath at a low temperature, and this error is likely to be linked to the quantum behavior of the bath. In the third chapter, the FGR rates of electronically nonadiabatic reactions enhanced by a microcavity is calculated. It will be shown that the cavity-enhanced reaction rates can be evaluated from the cavity-free FGR correlation function with minimum additional effort, and this method will be applied to a model CT reaction. While a high-frequency cavity enhances reactions in the Marcus inverted region selectively, a low-frequency cavity enhances reactions in all regions uniformly. In the fourth chapter, the FGR charge transfer rate constants for a molecular triad system solvated in explicit solvent is calculated via linear semiclassical (LSC) approach. A Marcus-like expression can be derived after applying a hierarchy of approximations. The validity of such approximations will be confirmed for the triad model. It will also be shown that the charge transfer rate depends deeply on the conformation of the molecule, the fluctuation between which is the rate-limiting step of the charge transfer.
The Redfield equation describes the dynamics of a quantum system weakly coupled to one or more reservoirs and is widely used in theory of open quantum system. However, the assumption of weak system-reservoir coupling is often not fully adequate and higher-order corrections to the Redfield equation with respect to the system-bath coupling is required. Here we propose a general method of derivation of higher-order corrections to the Redfield quantum master equation based on the hierarchical equations of motion (HEOM). Also we derive conditions of validity of the Redfield equation as well as the additional secular approximation for it.
A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.
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.