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Non-Markovian Dynamics in Physical Systems

Heinz‐Peter Breuer, Francesco Petruccione

Published Jan 25, 2007• 2 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780199213900.003.10

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Journal of Physics A Mathematical and Theoretical•Jul 16, 2010•7 citations
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.
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 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.
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 explore the performance of the time-convolutionless (TCL) projection operator technique using the Fano-Anderson model as a test case. Comparing the exact TCL master equation with an expansion in powers of the strength of the system-environment coupling, we analyze the transient dynamics as well as the steady-state behavior. For a Lorentzian spectral density we demonstrate that the dimensionless expansion parameter corresponds to the ratio of the environmental correlation time to the relaxation time of the system, and we derive the convergence radius for the TCL expansion, which is seen to depend on the ratio of detuning and width of the spectral density. We further study the quantum non-Markovianity of the model based on the evolution of the Bures distance between quantum states and how it is represented by the second and fourth order of the expansion. Our results highlight both the strengths and the limitations of the TCL formalism in capturing key features of open quantum systems and, in particular, the challenges of accurately describing strongly coupled systems and non-Markovian dynamics.
We study the dynamics of open quantum systems driven by an external time-periodic force. Combining Floquet theory and the time-convolutionless projection operator technique we derive a time-local quantum master equation which exactly takes into account the periodic driving, while treating the system-environment interaction within second order in the coupling strength without performing the Markov approximation. The resulting equation of motion for the reduced density matrix is called Floquet time-convolutionless master equation. Employing the example of the driven spin-boson system, we demonstrate that this master equation is capable of describing strong non-Markovian effects, while yielding the Floquet-Lindblad master equation in the Markovian limit. A characteristic feature of memory effects in such driven dissipative systems is the emergence of sharp peaks of the trace-distance based non-Markovianity measure as a function of the driving amplitude, which can be traced back to quasienergy crossings leading to almost decoherence-protected subspaces through a quasienergy-induced dissipative decoupling mechanism.
Decoherence of the dispersive Jaynes–Cummings model which is placed in a lossy cavity is studied by means of the time-convolutionless non-Markovian quantum master equation. The effect of the interaction between a two-level atom and photon field on the damping operator of the quantum master equation is made clear by calculating the decay of coherence and entanglement of the relevant system. It is shown that although the interaction effect on the damping operator is small when the cavity loss has the Markovian property, it becomes significantly large when the cavity loss is subject to the non-Markovian process. Furthermore, the quantum channel of the two-level atom derived by the dispersive Jaynes–Cummings model is investigated. The decay of entanglement of the two two-level atoms is examined to show the importance of the interaction effect on the damping operator.