Stochastic Schrödinger equation (SSE) methods provide a powerful approach for simulating open quantum system dynamics by unraveling the influence functional into an ensemble average of pure-state trajectories. However, long-time simulations using SSE methods are hindered by the inherent growth of ensemble variance. An alternative strategy is to extract essential memory kernels or transfer tensors from short-time dynamics and extrapolate to arbitrarily long times, as in the transfer tensor method (TTM). Conventional TTM implementations, however, require dynamical maps constructed from an informationally complete set of N 2 initial states─where N is the system dimension─limiting their applicability to small-scale problems. Here, we show that the SSE naturally provides an overcomplete set of Kraus operators that captures the essential information needed to construct dynamical maps for TTM. This connection enables the extension of TTM to high-dimensional systems using results obtained from a single converged SSE simulation with a sufficient number of stochastic realizations. We validate our approach with two representative SSE methods applied to a symmetric spin-boson model and a 24-site Fenna–Matthews–Olson (FMO) complex. The results demonstrate that SSE-TTM offers an accurate and computationally efficient strategy for simulating long-time quantum dynamics in relatively large-scale systems.
Stochastic Schrödinger equation (SSE) methods provide a powerful approach for simulating open quantum system dynamics by unraveling the influence functional into an ensemble average of pure-state trajectories. However, long-time simulations using SSE methods are hindered by the inherent growth of ensemble variance. An alternative strategy is to extract essential memory kernels or transfer tensors from short-time dynamics and extrapolate to arbitrarily long times, as in the transfer tensor method (TTM). Conventional TTM implementations, however, require dynamical maps constructed from an informationally complete set of N 2 initial states─where N is the system dimension─limiting their applicability to small-scale problems. Here, we show that the SSE naturally provides an overcomplete set of Kraus operators that captures the essential information needed to construct dynamical maps for TTM. This connection enables the extension of TTM to high-dimensional systems using results obtained from a single converged SSE simulation with a sufficient number of stochastic realizations. We validate our approach with two representative SSE methods applied to a symmetric spin-boson model and a 24-site Fenna–Matthews–Olson (FMO) complex. The results demonstrate that SSE-TTM offers an accurate and computationally efficient strategy for simulating long-time quantum dynamics in relatively large-scale systems.
The initial stages of the evolution of an open quantum system encode the key information of its underlying dynamical correlations, which in turn can predict the trajectory at later stages. We propose a general approach based on non-Markovian dynamical maps to extract this information from the initial trajectories and compress it into non-Markovian transfer tensors. Assuming time-translational invariance, the tensors can be used to accurately and efficiently propagate the state of the system to arbitrarily long time scales. The non-Markovian transfer tensor method (TTM) demonstrates the coherent-to-incoherent transition as a function of the strength of quantum dissipation and predicts the noncanonical equilibrium distribution due to the system-bath entanglement. TTM is equivalent to solving the Nakajima-Zwanzig equation and, therefore, can be used to reconstruct the dynamical operators (the system Hamiltonian and memory kernel) from quantum trajectories obtained in simulations or experiments. The concept underlying the approach can be generalized to physical observables with the goal of learning and manipulating the trajectories of an open quantum 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.
Accurate simulations of open quantum system dynamics is a long standing issue in the field of chemical physics. Exact methods exist, but are costly, while perturbative methods are limited in their applicability. Recently a new black-box type method, called transfer tensor method (TTM), was proposed [J. Cerrillo and J. Cao, Phys. Rev. Lett. 112, 110401 (2014)]. It allows one to accurately simulate long time dynamics with a numerical cost of solving a time-convolution master equation, provided many initial system evolution trajectories are obtained from some exact method beforehand. The possible time-savings thus strongly depend on the ratio of total versus initial evolution lengths. In this work, we investigate the parameter regimes where an application of TTM would be most beneficial in terms of computational time. We identify several promising parameter regimes. Although some of them correspond to cases when perturbative theories could be expected to perform well, we find that the accuracy of such approaches depends on system parameters in a more complex way than it is commonly thought. We propose that the TTM should be applied whenever system evolution is expected to be long and accuracy of perturbative methods cannot be ensured or in cases when the system under consideration does not correspond to any single perturbative regime.
The non-Markovian stochastic Schrödinger equation (NMSSE) offers a promising approach for open quantum simulations owing to its low scaling complexity and suitability for parallel computing. However, its application at low temperatures faces significant convergence challenges. While short-time evolution converges quickly, long-time evolution requires a much larger number of stochastic trajectories, leading to high computational costs. To this end, we propose a scheme that utilizes neural networks to extract effective information from long-time simulations of the non-Markovian stochastic Schrödinger equation, fine-tunes the networks using converged short-time data, and predicts long-term system evolution, eliminating the influence of stochastic oscillations. By integrating convolutional neural networks (CNNs) and long short-term memory recurrent neural networks (LSTMs), along with the iterative attentional feature fusion (iAFF) technique, this approach significantly reduces the number of trajectories required for long-time simulations, particularly at low temperatures, thus substantially lowering computational costs and improving convergence. To demonstrate our approach, we investigated the dynamics of the spin-boson model and the Fenna-Matthews-Olson (FMO) complex across a range of parameter variations.
The stochastic Schrödinger equation (SSE) provides a trajectory-level route to simulate the dynamics of open quantum systems with applications ranging from molecular processes to quantum technologies. We study a colored-noise extension of the SSE based on an Ornstein-Uhlenbeck (OU) noise drive and benchmark its ensemble-averaged dynamics against the standard white-noise SSE and against a fluctuating OU random Hamiltonian. When the environment exhibits a finite correlation time, averaging over pure-state trajectories yields master equations that are generally open-form and not of Lindblad type, yet remain positive by construction. By considering the differential of the OU process, we define an effective correlated noise, whose properties we analyze and use to formulate an SSE unraveling of its associated open-form quantum master equation. We show that the averaged dissipator separates into a Lindblad contribution stemming from the white-noise component and additional correlation terms arising from the fluctuations of the OU Hamiltonian. To obtain a practical closed description and physical intuition, we introduce a Redfield-inspired perturbative closure for these correlation terms, providing an effective master equation for the colored SSE. For a two-level system, the resulting dynamics exhibit long-lived coherences, nontrivial stationary (including oscillatory) states, and multi-timescale relaxation, rationalized through the components of a time-dependent Redfield tensor.
The Non-Markovian Stochastic Schrodinger Equation (NMSSE) offers a promising approach for open quantum simulations, especially in large systems, owing to its low scaling complexity and suitability for parallel computing. However, its application at low temperatures faces significant convergence challenges. While short-time evolution converges quickly, long-time evolution requires a much larger number of stochastic trajectories, leading to high computational costs. To this end,we propose a scheme that combines neural network techniques with simulations of the non-Markovian stochastic Schrodinger equation. By integrating convolutional neural networks (CNNs) and long short-term memory recurrent neural networks (LSTMs),along with the iterative attentional feature fusion (iAFF) technique, this approach significantly reduces the number of trajectories required for long-time simulations, particularly at low temperatures, thereby substantially lowering computational costs and improving convergence. To demonstrate our approach, we investigated the dynamics of the spin-boson model and the Fenna-Matthews-Olson (FMO) complex across a range of parameter variations.