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The Journal of Chemical Physics

Closures of the functional expansion hierarchy in the non-Markovian quantum state diffusion approach

Gerhard Ritschel, Walter T. Strunz, Alexander Eisfeld

Published Aug 14, 2017• 9 citations
Publisher

Abstract

To find a practical scheme to numerically solve the non-Markovian Quantum State Diffusion equation (NMQSD), one often uses a functional expansion of the functional derivative that appears in the general NMQSD equation. This expansion leads to a hierarchy of coupled operators. It turned out that if one takes only the zeroth order term into account, one has a very efficient method that agrees remarkably well with the exact results for many cases of interest. We denote this approach as zeroth order functional expansion (ZOFE). In the present work, we investigate two extensions of ZOFE. Firstly, we investigate how the hierarchy converges when taking higher orders into account (which, however, leads to a fast increase in numerical size). Secondly, we demonstrate that by using a terminator that approximates the higher order contributions, one can obtain significant improvement, at hardly any additional computational cost. We carry out our investigations for the case of absorption spectra of molecular aggregates.

DOI
10.1063/1.4986251

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To find a practical scheme to numerically solve the non-Markovian Quantum State Diffusion equation (NMQSD), one often uses a functional expansion of the functional derivative that appears in the general NMQSD equation. This expansion leads to a hierarchy of coupled operators. It turned out that if one takes only the zeroth order term into account, one has a very efficient method that agrees remarkably well with the exact results for many cases of interest. We denote this approach as zeroth order functional expansion (ZOFE). In the present work, we investigate two extensions of ZOFE. Firstly, we investigate how the hierarchy converges when taking higher orders into account (which, however, leads to a fast increase in numerical size). Secondly, we demonstrate that by using a terminator that approximates the higher order contributions, one can obtain significant improvement, at hardly any additional computational cost. We carry out our investigations for the case of absorption spectra of molecular aggregates.
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The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.
Non-Markovian quantum state diffusion (NMQSD) is a non-relativistic but otherwise exact theory which expresses the reduced density matrix of an arbitrary subsystem, interacting linearly with an uncoupled harmonic oscillator bath, as an average of diadics formed from state vectors which obey stochastic variational-differential equations. The vacuum radiation field can be represented as such an oscillator bath, and so this model is in widespread use in quantum optics. Prior to the development of NMQSD, exact subsystem solutions could only be obtained in a few special cases (e.g. spin-1/2, harmonic oscillator). Unfortunately, it has not yet been possible to obtain exact solutions to new problems using NMQSD due to the difficulty of solving the variational-differential equations. Here we show that these equations can be transformed into a pair of coupled nonlinear integrodifferential equations. We develop exact numerical methods for the integrodifferential equations and show that solutions can be readily obtained to good accuracy for quite general subsystems. We exactly solve various examples including tunneling in a double well representing molecular isomerization or racemization, suppression of fluorescence from a two-level atom in a band gap, and intermittent fluorescence from a driven three level system representing electronic states of singly ionized magnesium.
Based on the path integral approach, we derive a new realization of the exact non-Markovian stochastic Schrödinger equation (SSE). The main difference from the previous non-Markovian quantum state diffusion (NMQSD) method is that the complex Gaussian stochastic process used for the forward propagation of the wave function is correlated, which may be used to reduce the amplitude of the non-Markovian memory term at high temperatures. The new SSE is then written into the recently developed hierarchy of pure states scheme, in a form that is more closely related to the hierarchical equation of motion approach. Numerical simulations are then performed to demonstrate the efficiency of the new method.
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.
Non-Markovian Quantum State Diffusion: Application to Quantum Jumps in 24^{24}24Mg+^++
Non-Markovian quantum state diffusion (NMQSD) is an exact method for calculating the reduced density matrix of an arbitrary subsystem interacting linearly with the radiation field. Applications of the theory have however been few due to the intractable nature of the variational-differential NMQSD evolution equation. Recently, we argued that the variational-differential equation can be rewritten as an integrodifferential equation which can be readily solved numerically. This manuscript provides an explicit derivation of the modified equations. Applications to intermittent fluorescence in 24^{24}24Mg+^++ are discussed in detail. Earlier speculations that quantum jumps occur on all time scales are verified on a picosecond timescale. We show that a plot of the probability density of the signal vs signal strength shows the two characteristic peaks associated with the bright and dark manifolds, and that the ratio of the areas under the peaks is 16 as observed experimentally. We also show that the shape of this distribution is sensitive to bath memory, but has a mathematical form common to both the Markovian and non-Markovian cases.