The non-Markovian dynamics of open quantum systems is still a challenging task, particularly in the nonperturbative regime at low temperatures. While the stochastic Liouville--von Neumann equation (SLN) provides a formally exact tool to tackle this problem for both discrete and continuous degrees of freedom, its performance deteriorates for long times due to an inherently nonunitary propagator. Here we present a scheme that combines the SLN with projector operator techniques based on finite dephasing times, gaining substantial improvements in terms of memory storage and statistics. The approach allows for systematic convergence and is applicable in regions of parameter space where perturbative methods fail, up to the long-time domain. Findings are applied to the coherent and incoherent quantum dynamics of two- and three-level systems. In the long-time domain sequential and superexchange transfer rates are extracted and compared to perturbative predictions.
The non-Markovian dynamics of open quantum systems is still a challenging task, particularly in the nonperturbative regime at low temperatures. While the stochastic Liouville--von Neumann equation (SLN) provides a formally exact tool to tackle this problem for both discrete and continuous degrees of freedom, its performance deteriorates for long times due to an inherently nonunitary propagator. Here we present a scheme that combines the SLN with projector operator techniques based on finite dephasing times, gaining substantial improvements in terms of memory storage and statistics. The approach allows for systematic convergence and is applicable in regions of parameter space where perturbative methods fail, up to the long-time domain. Findings are applied to the coherent and incoherent quantum dynamics of two- and three-level systems. In the long-time domain sequential and superexchange transfer rates are extracted and compared to perturbative predictions.
This cumulative thesis presents new advances in the field of non-Markovian dynamics of open quantum systems based on the formally exact and non-perturbative stochastic Liouville-von Neumann (SLN) framework. More specifically, it presents a general platform to dynamically describe the complete, out-of-equilibrium operation of quantum thermal devices in all temperature and damping ranges and for arbitrary driving frequencies or strengths. Such an approach is urgently required due to the current experimental progress with realizations based on trapped ions, NV-centers in diamond, or superconducting circuits. By providing a comprehensive picture of a quantum Otto engine, the relevance of the thermal coupling work is revealed, an ingredient in the energy balance which has been basically neglected so far and which becomes increasingly relevant in the quantum regime. The thesis further solves crucial issues in dealing with numerical complexity that is inherent to the probabilistic SLN approach in the paradigmatic case of discrete few-level systems in contact with an Ohmic environment. It provides numerically exact benchmark data which is needed to generalize a related, numerically efficient technique for heat transport across open quantum chains to nonlinear systems that show rectification.
The stochastic Liouville-von Neumann (SLN) equation describes the dynamics of an open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. The interaction with the environment is represented by complex colored noises which drive the system, and whose correlation functions are set by the properties of the environment. We present a number of schemes capable of generating colored noises of this kind that are built on a noise amplitude reduction procedure [Imai et al., Chem. Phys. 446, 134 (2015)CMPHC20301-010410.1016/j.chemphys.2014.11.014], including two analytically optimized schemes. In doing so, we pay close attention to the properties of the correlation functions in Fourier space, which we derive in full. For some schemes the method of Wiener filtering for deconvolutions leads to the realization that weakening causality in one of the noise correlation functions improves numerical convergence considerably, allowing us to introduce a well-controlled method for doing so. We compare the ability of these schemes, along with an alternative optimized scheme [Schmitz and Stockburger, Eur. Phys. J.: Spec. Top. 227, 1929 (2019)1951-635510.1140/epjst/e2018-800094-y], to reduce the growth in the mean and variance of the trace of the reduced density matrix, and their ability to extend the region in which the dynamics is stable and well converged for a range of temperatures. By numerically optimizing an additional noise scaling freedom, we identify the scheme which performs best for the parameters used, improving convergence by orders of magnitude and increasing the time accessible by simulation.
Phase-space methods are powerful tools for the treatment of Markovian open quantum systems: they map the reduced dynamics of a system S, in interaction with an environment E, exactly onto Langevin equations for c-number stochastic variables, as opposed to Heisenberg-Langevin equations for operators. Langevin equations provide analytical insight in key regimes and excel at handling strong nonlinearities and couplings, where other methods often falter. Extending phase-space methods to non-Markovian dynamics, however, has remained a long-standing challenge. Here we address this gap by applying phase-space representations to the full S+E system; integrating out the environmental degrees of freedom then yields a general Langevin framework for S that incorporates both deterministic and stochastic contributions from E. Normally ordered representations, such as the Glauber-Sudarshan P representation and its positive variant due to Drummond and Gardiner, lead to Langevin equations in which (i) non-Markovian effects emerge exclusively in the deterministic terms, via a memory kernel, and (ii) noise contributions vanish when E is initially in the vacuum state. To demonstrate the power of this framework, we address the paradigmatic problem of delayed coherent feedback, in which the system is driven by its own past state, and study its impact on the laser linewidth: we recover the narrowing observed well above threshold and predict an enhanced narrowing just above it. Crucially, the number of stochastic variables scales linearly with the system size, making the framework suitable for problems ranging from a few degrees of freedom to genuinely many-body systems. This opens the way to the systematic study of non-Markovian driven-dissipative quantum systems using the same analytical and numerical tools that have long made phase-space methods so successful in the Markovian regime.
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.
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.
The time-convolutionless (TCL) projection operator technique allows a systematic analysis of the non-Markovian quantum dynamics of open systems. We introduce a class of projection superoperators which project the states of the total system onto certain correlated system-environment states. It is shown that the application of the TCL technique to this class of correlated superoperators enables the non-perturbative treatment of the dynamics of system-environment models for which the standard approach fails in any finite order of the coupling strength. We demonstrate further that the correlated superoperators correspond to the idea of a best guess of conditional quantum expectations which is determined by a suitable Hilbert space average. The general approach is illustrated by means of the model of a spin which interacts through randomly distributed couplings with a finite reservoir consisting of two energy bands. Extensive numerical simulations of the full Schroedinger equation of the model reveal the power and efficiency of the method.