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Projection Operator Techniques

Heinz‐Peter Breuer, Francesco Petruccione

Published Jan 25, 2007• 18 citations
Publisher

Abstract

Abstract This chapter is devoted to the foundations and to the basic mathematical structure of the non-Markovian quantum dynamics of open systems. It gives a survey of the Nakajima–Zwanzig projection operator methods with the help of which one derives so-called generalized master equations for the reduced system dynamics. In the non-Markovian regime, these master equations involve a retarded memory kernel, i.e. a time-convolution integral taken over the history of the reduced system. The chapter also describes an alternative method of particular relevance in many applications, which is based on a time-local quantum master equation, and which is known as the time-convolutionless projection operator method. This method serves as a starting point for a systematic expansion in terms of ordered cumulants and forms the basis for a non-Markovian stochastic unravelling of the reduced system dynamics.

DOI
10.1093/acprof:oso/9780199213900.003.09

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Abstract This chapter is devoted to the foundations and to the basic mathematical structure of the non-Markovian quantum dynamics of open systems. It gives a survey of the Nakajima–Zwanzig projection operator methods with the help of which one derives so-called generalized master equations for the reduced system dynamics. In the non-Markovian regime, these master equations involve a retarded memory kernel, i.e. a time-convolution integral taken over the history of the reduced system. The chapter also describes an alternative method of particular relevance in many applications, which is based on a time-local quantum master equation, and which is known as the time-convolutionless projection operator method. This method serves as a starting point for a systematic expansion in terms of ordered cumulants and forms the basis for a non-Markovian stochastic unravelling of the reduced system dynamics.
A generalization of the stochastic wave-function method to quantum master equations which are not in Lindblad form is developed. The proposed stochastic unraveling is based on a description of the reduced system in a doubled Hilbert space and it is shown that this method is capable of simulating quantum master equations with negative transition rates. Non-Markovian effects in the reduced systems dynamics can be treated within this approach by employing the time-convolutionless projection operator technique. This ansatz yields a systematic perturbative expansion of the reduced systems dynamics in the coupling strength. Several examples such as the damped Jaynes-Cummings model and the spontaneous decay of a two-level system into a photonic band gap are discussed. The power as well as the limitations of the method are demonstrated.
Abstract This chapter applies the time-convolutionless projection operator technique to some typical systems featuring strong non-Markovian behaviour. The first example models the decay of a two-level system into a reservoir with arbitrary spectral density. For this problem the expansion of the master equation is carried out to all orders in the system-reservoir coupling. As specific cases the damped Jaynes–Cummings model and the spontaneous decay into a photonic band gap are studied. The model is used to exemplify the breakdown of the expansion in the strong coupling regime, and to illustrate the emergence of completely positive master equations which are not in Lindblad form and involve negative decay rates. The chapter also treats the non-Markovian dynamics in quantum Brownian motion and in the spin-boson model, and illustrates the stochastic unravelling of non-Markovian quantum processes.
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.
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 ϱB(t)\varrho_{B}(t)ϱB​(t) and the system ϱS(t)\varrho_{S}(t)ϱS​(t) are written down. A formal solution of the equation for ϱB(t)\varrho_{B}(t)ϱB​(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 ϱS(t)\varrho_{S}(t)ϱS​(t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of ϱB(t)\varrho_{B}(t)ϱB​(t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.
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 system considered 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 the time-convolutionless generator as well as the Nakajima-Zwanzig memory kernel. This nonperturbative treatment shows that each operator contribution in Lindblad form appearing in the exact time-convolutionless master equation is multiplied by a different time-dependent function. Similarly, in the Nakajima-Zwanzig master equation each such contribution is convoluted with a different memory kernel. It appears that, depending on the state of the environment, the operator structures of the two sets of equations of motion can exhibit important differences.
We present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima—Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.