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Journal of Molecular Structure THEOCHEM

The non-Markovian master equation for stochastically perturbed systems: effect on spectral lineshape

Gautam Gangopadhyay, Deb Shankar Ray

Published Jan 1, 1996• 0 citations
Publisher
DOI
10.1016/0166-1280(95)04301-2

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Non-Markovian effects in open quantum systems are central to understanding spectral lineshape. Here, we quantify the non-Markovianity associated with both overdamped and underdamped vibrations in terms of information flow between the bath and the system and compare this with the broadening and ellipticity of two-dimensional spectra. Using the Breuer Laine Piilo (BLP) measure, we link the well-known stochastic models for spectral lineshape with modern quantum information theory. Specifically, we study the effect of non-Markovianity in a system in contact with underdamped vibrations and examine the differences observed on increasing the damping to the overdamped limit. The open quantum system dynamics are evolved using the hierarchical equations of motion, efficiently terminated with a Markovian cutoff, where separate hierarchies are derived for the underdamped and overdamped environments. It is shown that the BLP measure is quantitatively correlated with the ellipticity of two-dimensional spectra and memory effects are more pronounced in underdamped environments, due to the long-lived feedback of information between the system and its bath, compared to overdamped environments. Environmental signatures in spectral lineshapes emerge as a result of information flow from the bath back into the system.
Based on the stochastic unravelling of the reduced density operator in the Feynman path integral formalism for an open quantum system in touch with harmonic environments, a new non-Markovian stochastic Schrödinger equation (NMSSE) has been established that allows for the systematic perturbation expansion in the system-bath coupling to arbitrary order. This NMSSE can be transformed in a facile manner into the other two NMSSEs, i.e., non-Markovian quantum state diffusion and time-dependent wavepacket diffusion method. Benchmarked by numerically exact results, we have conducted a comparative study of the proposed method in its lowest order approximation, with perturbative quantum master equations in the symmetric spin-boson model and the realistic Fenna-Matthews-Olson complex. It is found that our method outperforms the second-order time-convolutionless quantum master equation in the whole parameter regime and even far better than the fourth-order in the slow bath and high temperature cases. Besides, the method is applicable on an equal footing for any kind of spectral density function and is expected to be a powerful tool to explore the quantum dynamics of large-scale systems, benefiting from the wavefunction framework and the time-local appearance within a single stochastic trajectory.
Positivity preservation is naturally guaranteed in exact non-Markovian master equations for open quantum system dynamics. However, in many approximated non-Markovian master equations, the positivity of the reduced density matrix is not guaranteed. In this paper we provide a general class of time-local, perturbative, and positivity-preserving non-Markovian master equations generated from stochastic Schr\"odinger equations, particularly quantum-state-diffusion equations. Our method has an expanded range of applicability for accommodating a variety of non-Markovian environments. We show the positivity-preserving master equation for a three-level system coupled to a dissipative bosonic environment as a particular example to exemplify our general approach. We illustrate the numerical simulations with an analysis explaining why the previous approximated non-Markovian master equations cannot guarantee positivity. Our work provides a consistent master equation for studying the non-Markovian dynamics in ultrafast quantum processes and strong-coupling systems.