A Road Map to Various Pathways for Calculating the Memory Kernel of the Generalized Quantum Master Equation

Abstract

The generalized quantum master equation (GQME) provides a powerful framework for simulating electronic energy, charge, and coherence transfer dynamics in molecular systems. Within this framework, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density matrix is fully captured by a memory kernel superoperator. However, the actual memory kernel depends on the choice of projection operator and is therefore not unique. Furthermore, calculating the memory kernel can be done in multiple ways that use different forms of projection-free inputs. Although the electronic dynamics is invariant to those choices when quantum-mechanically exact projection-free inputs are used, this is not the case when they are obtained via more feasible semiclassical or mixed quantum-classical approximate methods. Furthermore, the accuracy and numerical stability of the resulting electronic dynamics has been observed to be sensitive to the above-mentioned choices when approximate methods are used to calculate the projection-free inputs. In this article, we provide a systematic road map to 30 possible pathways for calculating the memory kernel and highlight how they are related as well as the ways in which they differ. We also compare the performance of different pathways in the context of the spin-boson benchmark model, with the projection-free inputs obtained via a mapping Hamiltonian linearized semiclassical method. In this case, we find that expressing the memory kernel with an exponential operator where the projection operator precedes the Liouvillian yields the most accurate and most numerically stable results.

Perturbation theory under the truncated Wigner approximation reveals how system-environment entanglement formation drives quantum decoherence

“…For the case where the environment is harmonic and where the system state dependent harmonic potentials differ at most by a relative shift of the potential minima, we have demonstrated how one can utilize the described framework to derive expressions for the N th order perturbative contribution to operator expectation values in the combined system-environment Hilbert space, even including polynomial bath operator dependence in the perturbative operator. In contrast to dynamical methods based on projection operator techniques [25,27,28,53,56] or influence functionals [30,57,58], our framework retains full dynamical information for the system and its environment while also avoiding common assumptions about the initial environmental density.…”

Compact and complete description of non-Markovian dynamics

“…In such cases, calculating the memory kernels instead of the full projected dynamics can greatly improve computational efficiency. Recently, this has inspired many successful approaches [42][43][44][45][46][47][48][49][50][51]. Nevertheless, while this approach provides a complete computational framework to predict non-Markovian dynamics, extracting physical insight from these memory kernels is challenging and often requires expert knowledge.…”

“…In particular, we develop a simple, accurate, and efficient means to exploit the advantages of TCL-GMEs to provide a highly compact and complete representation of the non-Markovian dynamics governing projected observables and thereby reduce the computational cost and extend the applicability of both classical and quantum dynamics [57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73] approaches. We show that the TCL approach requires as little reference dynamics to construct as the TC schemewhich has already been demonstrated to provide a compact means to encode non-Markovian dynamics of various systems [6,16,42,44,[46][47][48][49][50][51][74][75][76][77][78][79][80][81][82][83][84][85][86]-while entirely avoiding the complexities of the time-nonlocal convolution over the memory kernel and its construction. Admittedly, as suggested above, it has long been appreciated [1] that a time-local description is not guaranteed to exist for all problems due to the appearance of mathematical divergences in the time-local generator.…”

A comparative study of different machine learning methods for dissipative quantum dynamics

“…Methods based on Nakajima-Zwanzig generalized quantum master equation (GQME) [20][21][22][23][24][25][26] including transfer tensor method (TTM), [27][28][29][30][31] allow to obtain long-time dynamics of the reduced density matrix of a quantum system at a significantly lower computational cost coma) Electronic mail: dral@xmu.edu.cn b) Electronic mail: akanane@udel.edu pared to numerically exact methods, provided the GQME memory kernels are available. However, in general, it is difficult to numerically calculate the exact memory kernel for GQME.…”