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arXiv:1207.5968 · Physical Review A

Quantification of memory effects in the spin-boson model

Govinda Clos, Heinz‐Peter Breuer

Published Jul 23, 2012• 88 citations
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Abstract

Employing a recently proposed measure for quantum non-Markovianity, we carry out a systematic study of the size of memory effects in the spin-boson model for a large region of temperature and frequency cutoff parameters. The dynamics of the open system is described utilizing a second-order time-convolutionless master equation without the Markov or rotating wave approximations. While the dynamics is found to be strongly non-Markovian for low temperatures and cutoffs, in general, we observe a special regime favoring Markovian behavior. This effect is explained as resulting from a resonance between the system's transition frequency and the frequencies of the dominant environmental modes. We further demonstrate that the corresponding Redfield equation is capable of reproducing the characteristic features of the non-Markovian quantum behavior of the model.

DOI
10.1103/physreva.86.012115

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We study quantum non-Markovian dynamics of the Caldeira-Leggett model, a prototypical model for quantum Brownian motion describing a harmonic oscillator linearly coupled to a reservoir of harmonic oscillators. Employing the exact analytical solution of this model one can determine the size of memory effects for arbitrary couplings, temperatures, and frequency cutoffs. Here, quantum non-Markovianity is defined in terms of the flow of information between the open system and its environment, which is quantified through the Bures metric as distance measure for quantum states. This approach allows us to discuss quantum memory effects in the whole range from weak to strong dissipation for arbitrary Gaussian initial states. A comparison of our results with the corresponding results for the spin-boson problem show a remarkable similarity in the structure of non-Markovian behavior of the two paradigmatic models.
We derive a time-convolutionless master equation for the spin-boson model in the weak coupling limit. The temporarily negative decay rates in the master equation indicate short time memory effects in the dynamics which is explicitly revealed when the dynamics is studied using the non-Markovian jump description. The approach gives new insight into the memory effects influencing the spin dynamics and demonstrates, how for the spin-boson model the the co-operative action of different channels complicates the detection of memory effects in the dynamics.
We study the effect of the rotating-wave approximation (RWA) and the secular approximation (SA) on the non-Markovian behavior in the spin-boson model at zero temperature. We find that both the RWA and SA lead to a dramatic reduction in the observed non-Markovianity. In general, non-Markovian dynamics is observed for the whole relaxation time of the system, whereas the RWA and SA lead to such dynamics only on the short time scale of the environmental correlation time. Thus, the RWA and SA are not necessarily justified in the studies of non-Markovianity, although they can estimate the state of the system precisely. Furthermore, we derive an accurate analytical expression for the non-Markovianity measure without the RWA or SA. This expression yields important insight into the physics of the problem.
We have studied the non-Markovianity of a dichotomously driven spin-boson model in the strong-coupling regime in both memory kernel and time convolutionless master-equation formulations. A strong correlation between the decay time of the environmental correlation function and the nonzero non-Markovianity is found in the absence of the external noise. Stochastic driving is shown to create strong non-Markovianity when the dynamics of the system without driving is Markovian. Also, exact analytical expressions for the trace distance distinguishability and the non-Markovianity were obtained for the certain range of the parameters that describe the system and its environment.
We investigate the effect of counter-rotating-wave terms on the non-Markovianity in quantum open systems by employing the hierarchical equations of motion in the framework of the non-Markovian quantum state diffusion approach. As illustrative examples, the non-Markovian memory effect of a qubit embedded in a bosonic or a fermionic environment with a detuned Lorentz spectrum at zero temperature is analyzed. It is found that the counter-rotating-wave terms are able to enhance the observed non-Markovianity whether the environment is composed of bosons or fermions. This result suggests that the rotating-wave approximation may reduce the non-Markovianity in quantum open systems. Moreover, we find that the modification of the non-Markovianity due to the different statistical properties of environmental modes becomes larger with the increase of the system-environment coupling strength.