Spectral Densities, Structured Noise and Ensemble Averaging within Open Quantum Dynamics

Abstract

Although recent advances in simulating open quantum systems have lead to significant progress, the applicability of numerically exact methods is still restricted to rather small systems. Hence, more approximate methods remain relevant due to their computational efficiency, enabling simulations of larger systems over extended timescales. In this study, we present advances for one such method, namely the Numerical Integration of Schrödinger Equation (NISE). Firstly, we introduce a modified ensemble-averaging procedure that improves the long-time behavior of the thermalized variant of the NISE scheme, termed Thermalized NISE. Secondly, we demonstrate how to use the NISE in conjunction with (highly) structured spectral densities by utilizing a noise generating algorithm for arbitrary structured noise. This algorithm also serves as a tool for establishing best practices in determining spectral densities from excited state calculations along molecular dynamics or quantum mechanics/molecular mechanics trajectories. Finally, we assess the ability of the NISE approach to calculate absorption spectra and demonstrate the utility of the proposed modifications by determining population dynamics.

Figures (13)

• Figure 1: Long-time population dynamics of an 8-site FMO Hamiltonian calculated with a) HEOM b) original TNISE c) new averaging TNISE without interpolation and d) new averaging TNISE with interpolation. The populations of sites 3-8 are summed together for clarity. Furthermore, the thermal distributions according to the right-hand side of inequality \ref{['eq:averagingissue']} are shown as dashed lines an according to the left-hand side as dotted lines and the short-time dynamics of the first 0.2 ps are shown as an inset. 10,000 realizations were used for the different TNISE results.
• Figure 2: The graph presents a running average of the mean squared error (MSE) along a 1 ps-long trajectory with 1000 realizations, compared to the HEOM results, as a function of the reorganization energy $\lambda$ for two-site systems with varying parameters. The data represents a composite of 10,000 systems, where the parameters were randomly picked from the ranges listed in Tab. \ref{['table:sampleSytemParameters']}. Each point represents a moving average over 250 systems to smooth out individual discrepancies and to highlight trends more effectively.
• Figure 3: Example population dynamics of two-site systems with the NISE trajectories averaged over 10,000 realizations. The parameters can be found in Tab. \ref{['table:sampleSytemParameters']}. Panel a) shows a case where the new averaging outperforms the original averaging, while in panel b) the opposite is true. In panel c) an example is delineated in which the lifetime with the empirical factor of five from Eq. \ref{['eq:empirical_factor']} is too high, while panel d) shows the same system with the factor set to one.
• Figure 4: Reconstruction of the experimental spectral density of the FMO complex raet07amait20a by using a 100 ps-long trajectory with a 10 fs time step and 100,000 realizations.
• Figure 5: Autocorrelation function of the experimental spectral density of the FMO complex (see Fig. \ref{['fig:realizations_fullSD']}). The left panel shows the theoretical autocorrelation function calculated by inverting Eq. \ref{['eq:jw']} compared to the autocorrelation functions obtained from Eq. \ref{['eq:autocorrelation_from_noise']} for a 1 ns and a 100 ps-long trajectory. The right panel again shows the autocorrelation obtained from the 100 ps-long trajectory, but in this panel together with a 500 fs moving average over the absolute value of the same autocorrelation function.