We extend the Nakajima-Zwanzig projection operator technique to the determination of multitime correlation functions of open quantum systems. The correlation functions are expressed in terms of certain multitime homogeneous and inhomogeneous memory kernels for which suitable equations of motion are derived. We show that under the condition of finite memory times, these equations can be used to determine the memory kernels by employing an exact stochastic unraveling of the full system-environment dynamics. The approach thus allows us to combine exact stochastic methods, feasible for short times, with long-time master equation simulations. The applicability of the method is demonstrated by numerical simulations of two-dimensional spectra for a donor-acceptor model, and by comparison of the results with those obtained from the reduced hierarchy equations of motion. We further show that the formalism is also applicable to the time evolution of a periodically driven two-level system initially in equilibrium with its environment.
We extend the Nakajima-Zwanzig projection operator technique to the determination of multitime correlation functions of open quantum systems. The correlation functions are expressed in terms of certain multitime homogeneous and inhomogeneous memory kernels for which suitable equations of motion are derived. We show that under the condition of finite memory times, these equations can be used to determine the memory kernels by employing an exact stochastic unraveling of the full system-environment dynamics. The approach thus allows us to combine exact stochastic methods, feasible for short times, with long-time master equation simulations. The applicability of the method is demonstrated by numerical simulations of two-dimensional spectra for a donor-acceptor model, and by comparison of the results with those obtained from the reduced hierarchy equations of motion. We further show that the formalism is also applicable to the time evolution of a periodically driven two-level system initially in equilibrium with its environment.
Memory effects in open quantum dynamics are often incorporated in the equation of motion through a superoperator known as the memory kernel, which encodes how past states affect future dynamics. However, the usual prescription for determining the memory kernel requires information about the underlying system-environment dynamics. Here, by deriving the transfer tensor method from first principles, we show how a memory kernel master equation, for any quantum process, can be entirely expressed in terms of a family of completely positive dynamical maps. These can be reconstructed through quantum process tomography on the system alone, either experimentally or numerically, and the resulting equation of motion is equivalent to a generalised Nakajima-Zwanzig equation. For experimental settings, we give a full prescription for the reconstruction procedure, rendering the memory kernel operational. When simulation of an open system is the goal, we show how our procedure yields a considerable advantage for numerically calculating dynamics, even when the system is arbitrarily periodically (or transiently) driven or initially correlated with its environment. Namely, we show that the long time dynamics can be efficiently obtained from a set of reconstructed maps over a much shorter time.
Efficient simulations of the dynamics of open systems is of wide importance for quantum science and technology. Here, we introduce a generalization of the transfer-tensor, or discrete-time memory kernel, formalism to multitime measurement scenarios. The transfer-tensor method sets out to compute the state of an open few-body quantum system at long times, given that only short-time system trajectories are available. Here, we show that the transfer-tensor method can be extended to processes which include multiple interrogations (e.g., measurements) of the open system dynamics as it evolves, allowing us to propagate high-order short-time correlation functions to later times, without further recourse to the underlying system-environment evolution. Our approach exploits the process-tensor description of open quantum processes to represent and propagate the dynamics in terms of an object from which any multitime correlation can be extracted. As an illustration of the utility of the method, we study the buildup of system-environment correlations in the paradigmatic spin-boson model and compute steady-state emission spectra, taking fully into account system-environment correlations present in the steady state.
We present a projection-based, stability-preserving methodology for computing time correlation functions in open quantum systems governed by generalized quantum master equations with non-Markovian effects. Building upon the memory kernel coupling theory framework, our approach transforms the memory kernel hierarchy into a system of coupled linear differential equations through Mori-Zwanzig projection, followed by spectral projection onto stable eigenmodes to ensure numerical stability. By systematically eliminating unstable modes while preserving the physically relevant dynamics, our method guaranties long-time convergence without introducing artificial damping or ad hoc modifications. The theoretical framework maintains mathematical rigor through orthogonal projection operators and spectral decomposition. Benchmark calculations on the spin-boson model show excellent agreement with exact hierarchical equations of motion results while achieving significant computational efficiency. This approach provides a versatile and reliable framework for simulating non-Markovian dynamics in complex systems.
A projection operator is introduced, which exactly transforms the inhomogeneous Nakajima--Zwanzig generalized master equation for the relevant part of a system +bath statistical operator, containing the inhomogeneous irrelevant term comprising the initial corrrelations, into the homogeneous equation accounting for initial correlations in the kernel governing its evolution. No "molecular chaos"-like approximation has been used. The obtained equation is equivalent to completely closed (homogeneous) equation for the statistical operator of a system of interest interacting with a bath. In the Born approximation (weak system-bath interaction) this equation can be presented as the time-local Redfield-like equation with additional terms caused by initial correlations. As an application, a quantum oscillator, interacting with a Boson field and driven from a Gibbs initial equilibrium system+bath state by an external force, is considered. All terms determining the oscillator evolution over time are explicitly calculated at all timescales. It is shown, how the initial correlations influence the evolution process. It is also demonstrated, that at the large timescale this influence vanishes, and the evolution equation for the quantum oscillator statistical operator acquires the Lindblad form..
We show that the exact master equation incorporating initial correlations for open quantum systems, within the Nakajima-Zwanzig operator-projection method, is a homegenous master equation for the reduced density matrix. We also derive explicitly the exact master equation for a large class of bosonic and fermionic open quantum systems incorporating initial correlations, the resulting master equation is homogenous. We find that the effects of the initial correlations can be fully embedded into the fluctuation dynamics through the exact homogenous master equation. Also a generalized nonequilibrium fluctuation-dissipation theorem incorporating the initial correlations is obtained.
A novel approach to accounting for the influence of initial system-bath correlations on the dynamics of an open quantum system, based on the conventional projection operator technique, is suggested. To avoid the difficulties of treating the initial correlations, the conventional Nakajima-Zwanzig inhomogeneous generalized master equations (GMEs) for a system's reduced statistical operator and correlation function are exactly converted into the homogeneous GMEs (HGMEs), which take into account the initial correlations in the kernel governing the evolution of these HGMEs. In the second order (Born) approximation in the system-bath interaction, the obtained HGMEs are local in time and valid at all timescales. They are further specialized for a realistic equilibrium Gibbs initial (at t=t0) system+bath state (for a system reduced statistical operator an external force at t>t0 is applied) and then for a bath of oscillators (Boson field). As an example, the evolution of a selected quantum oscillator (a localized mode) interacting with a Boson field (Fano-like model) is considered at different timescales. It is shown explicitly how the initial correlations influence the oscillator evolution process. In particular, it is shown that the equilibrium system's correlation function acquires at the large timescale the additional constant phase factor conditioned by survived initial system-bath correlations.