Abstract
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 different ways of computing the time-evolution and combine their strengths. We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions. We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally "samples" it at nonzero characteristic frequencies. This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies. Second, we prove that the fixed-point equation sums up the time-domain gradient / Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects. Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel. For the transient generator this produces non-semigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.
Highlights
It is well known that the dynamics ρðt0Þ → ρðtÞ of the state of an open quantum system initially uncorrelated with its environment can be described equivalently by two exact but fundamentally different quantum master equations (QMEs)
This functional is closely related to the ordinary Laplace transform (4) of the memory kernel Kðt − sÞ, to which it reduces for constant c-number functions of time X 1⁄4 ωI in the limit t − t0 → ∞ for time-translational systems
The occupation numbers of reservoir modes are either 0 or 1 due to a total dynamical excitation cnounmstbraeirntd:†tdheþcRoudpωlibn†ωgb(ω3.4H) ceorenswerevesstutdhye the effects of energy-dependent coupling ΓðωÞ without initial reservoir statistics (T 1⁄4 0): We assume a Lorentzian profile of width γ whose maximum value Γ ≡ ΓðεÞ lies precisely at the atomic resonance: ΓðωÞ
Summary
It is well known that the dynamics ρðt0Þ → ρðtÞ of the state of an open quantum system initially uncorrelated with its environment can be described equivalently by two exact but fundamentally different quantum master equations (QMEs). The transformation connecting eigenvectors of Gð∞Þ and KðωÞ is found to be related to so-called initial-slip correction procedures [95,96,97,98,99] We show that both the stationary and the transient fixed-point equation are self-consistent expressions for the solution of the memory expansion discussed in Refs. IV we show that the fixed-point equation can be turned into two separate iterative numerical approaches for obtaining the transient and the stationary generator, respectively, from a given memory kernel This provides a new starting point for hybrid approaches in which the results of advanced time-nonlocal calculations [21,22,23,24] can be plugged into the time-local formalisms directly, bypassing the solution Πðt; t0Þ that ties Eqs.
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