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 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 2D-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 define an ensemble of projection operators, each of which has an exact associated Nakajima-Zwanzig master equation for quantum open system evolution. A mean field approximation for the memory kernels is introduced that yields, for an optimally chosen projection operator, a completely determined inhomogeneous master equation. Previous proofs of positivity and equilibration are extended to these new inhomogeneous non-Markovian master equations. We study a nitrogen vacancy center in diamond interacting with 13C impurities to illustrate the method.
There is current interest in dynamical description of dif- ferent decompositions of a quantum system into subsystems. We investi- gate usefulness of the Nakajima-Zwanzig projection method in this context. Particularly, we are interested in simultaneous description of dynamics of open systems pertaining to dierent system-environment splits (decompo- sitions). We find that the Nakajima-Zwanzig and related projection meth- ods are system-environment split specific, that every system-environment split requires specific projector, and that projector adapted to a split nei- ther provides information…
We consider the exact reduced dynamics of a two-level system coupled to a bosonic reservoir, further obtaining the exact time-convolutionless and Nakajima-Zwanzig non-Markovian equations of motion. The considered system includes the damped and undamped Jaynes-Cummings model. The result is obtained by exploiting an expression of quantum maps in terms of matrices and a simple relation between the time evolution map and time-convolutionless generator as well as Nakajima-Zwanzig memory kernel. This non-perturbative treatment shows that each operator contribution in Lindblad form appearing in the e…
We develop a new operator algebraic formulation of the Nakajima-Mori-Zwanzig (NMZ) method of projections. The new theory is built upon rigorous mathematical foundations, and it can be applied to both classical and quantum systems. We show that a duality principle between the NMZ formulation in the space of observables and in the state space can be established, analogous to the Heisenberg and Schrödinger pictures in quantum mechanics. Based on this duality we prove that, under natural assumptions, the projection operators appearing in the NMZ equation must be conditional expectations. The propo…
The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath $\varrho_{B}(t)$ and the system $\varrho_{S}(t)$ are written down. A formal solution of the equation for $\varrho_{B}(t)$, having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in $\varrho_{S}(t)$,…
Driven two-level system is a useful model to describe many quantum objects, particularly in quantum information processing. However, the exact master equation for such a system is barely explored. Making use of the Feynman-Vernon influence functional theory, we derive an exact non-Markovian master equation for the driven two-level system and show the lost feature in the perturbative treatment for this system. The perturbative treatment leads to the time-convolutionless (TCL) and the Nakajima-Zwanzig (NZ) master equations. So to this end, we derive the time-convolutionless (TCL) and the Nakajim…
We discuss the derivation of master equations in the presence of initial correlations with the reservoir. In van Hove's limit, the total system behaves as if it started from a factorized initial condition. A proper choice of Nakajima-Zwanzig's projection operator is crucial and the reservoir should be endowed with the mixing property.
There exist two canonical approaches to describe open quantum systems by a time-evolution equation: the Nakajima-Zwanzig quantum master equation, featuring a time-nonlocal memory kernel $\mathcal{K}$, and the time-convolutionless equation with a time-local generator $\mathcal{G}$. These key quantities have recently been shown to be connected by an exact fixed-point relation [Phys. Rev. X 11, 021041 (2021)]. Here we show that this implies a recursive relation between their perturbative expansions, allowing a series for the kernel $\mathcal{K}$ to be translated directly into a corresponding seri…
The Nakajima-Zwanzig(N-Z) equation is a powerful tool to analyze the open quantum system and non-Markovian behavior. In this paper, we modify the N-Z equation with assignment maps. This approach extends the application to those situations in which the bipartite state of the system and environment contains non-negligible discord initial and/or during the dynamics. We applied the equation in quantum lattice model and bound the influence of inhomogeneous terms, which figure out how the dimension of the environment impact the non-Markovian behavior.
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 nonequilibriu…
Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. Employing the exact solution for the model we determine analytical expressions for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows a detailed investigation and comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, we find that the structure of widely used phenomenological master equations with memory kernel may be incompatible with a …
A new post-Markovian quantum master equation is derived, that includes bath memory effects via a phenomenologically introduced memory kernel k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t) is presented, in addition to a prescription for the experimental determin…
We relate the memory kernel in the Nakajima-Zwanzig-Mori time-convolution approach to the reduced system propagator which is often used to obtain the kernel in the Tokuyama-Mori time-convolutionless approach. The connection provides a robust and simple formalism to compute the memory kernel for a generalized system-bath model circumventing the need to compute high order system-bath observables. We illustrate this for a model system with electron-electron and electron-phonon couplings, driven away from equilibrium.
We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed p…
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 p…
A Fractional Calculus Framework for Open Quantum Dynamics: From Liouville to Lindblad to Memory Kernels
Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory（太平洋西北国家实验室物理与计算科学部）；Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室理论部）
We consider the problem of the measurement of a system occurring during a finite time interval, while environmentally-induced noise decreases the system-state coherence. We assume a Markovian measuring device and, therefore, use a Lindbladian description for the measurement dynamics. For studying the case of noise produced by a non-Markovian environment, whose definition does not include the measuring apparatus, we use the Redfield approach to the interaction between system and environment. In the present hybrid theory, to trace out the environmental degrees of freedom, we introduce an analyti…
The perturbative master equation (Bloch-Redfield) is extensively used to study dissipative quantum mechanics - particularly for qubits - despite the 25 year old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom, and cast its perturbative master equation (derived from a perturbative treatment of Nakajima-Zwanzig or Schoeller-Schon equations) in the form of a Lindblad master equation. We find that the equation's parameters are time-dependent. This time-dependence is rarely accounted …
Master equations are commonly used to describe time evolution of open systems. We introduce a general method for calculating a Markovian solution of the Nakajima-Zwanzig generalized master equation. We do so for a time dependent transport of interacting electrons through a complex nano scale system in a photon cavity. The central system, described by 120 many-body states in a Fock space, is weakly coupled to the external leads. The very diverse relaxation times of the open system, reflecting radiative or non-radiative transitions, require information about the time evolution through many order…
We study the properties of the Nakajima-Zwanzig memory kernel for a qubit immersed in a many-body localized (i.e., disordered and interacting) bath. We argue that the memory kernel decays as a power law in both the localized and ergodic regimes, and show how this can be leveraged to extract $t\to\infty$ populations for the qubit from finite time ($J t \leq 10^2$) data in the thermalizing phase. This allows us to quantify how the long-time values of the populations approach the expected thermalized state as the bath approaches the thermodynamic limit. This approach should provide a good complem…
We analyze the influence of noise for qubits implemented using a triple quantum dot spin system. We give a detailed description of the physical realization and develop error models for the dominant external noise sources. We use a Davies master equation approach to describe their influence on the qubit. The triple dot system contains two meaningful realizations of a qubit: We consider a subspace and a subsystem of the full Hilbert space to implement the qubit. We test the robustness of these two implementations with respect to the qubit stability. When performing the noise analysis, we extract…
We describe a method for simulating exciton dynamics in protein-pigment complexes, including effects from charge transfer as well as fluorescence. The method combines the hierarchical equations of motion, which are used to describe quantum dynamics of excitons, and the Nakajima-Zwanzig quantum master equation, which is used to describe slower charge transfer processes. We study the charge transfer quenching in light harvesting complex II, a protein postulated to control non-photochemcial quenching in many plant species. Using our hybrid approach, we find good agreement between our calculation …
We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovi…
Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively. Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel $\mathcal{K}$, whereas the second achieves the same using a time-local generator $\mathcal{G}$. Here we show that the two are connected by a simple yet general fixed-point relation: $\mathcal{G} = \hat{\mathcal{K}}[\mathcal{G}]$. This allows one to extract nontrivial relations between the two completely d…
A symmetry-protected pseudo-Hermitian phase of quantum memory-kernel generators
State Key Laboratory of Bioinspired Interfacial Materials Science, Institute of Functional Nano & Soft Materials (FUNSOM), Soochow University（苏州大学功能纳米与软材料（FUNSOM）研究院仿生界面材料科学国家重点实验室）
We study the reduced time-evolution of open quantum systems by combining quantum-information and statistical field theory. Inspired by prior work [EPL 102, 60001 (2013) and Phys. Rev. Lett. 111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling. This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage defines the dissipative timescales of the system --before having integrated out the environm…
