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arXiv:2405.01722 · Quantum

Quantifying spectral signatures of non-Markovianity beyond the Born-Redfield master equation

A. Keefe, N. Agarwal, Archana Kamal

Published Sep 24, 2025• 7 citations
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Abstract

Memory or time-non-local effects in open quantum dynamics pose theoretical as well as practical challenges in the understanding and control of noisy quantum systems. While there has been a comprehensive and concerted effort towards developing diagnostics for non-Markovian dynamics, all existing measures rely on time-domain measurements which are typically slow and expensive as they require averaging several runs to resolve small transient features on a broad background, and scale unfavorably with system size and complexity. In this work, we propose a spectroscopic measure of non-Markovianity which can detect persistent non-Markovianity in the system steady state. In addition to being experimentally viable, the proposed measure has a direct information theoretic interpretation: a large value indicates the information loss per unit bandwidth of making the Markov approximation. In the same vein, we derive a frequency-domain quantum master equation (FD-QME) that goes beyond the standard Born-Redfield description and retains the full memory of the state of the reduced system. Using the FD-QME and the proposed measure, we are able to reliably diagnose and quantify non-Markovianity in several system-environment settings including those with environmental correlations and retardation effects.

DOI
10.22331/q-2025-09-24-1863

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Memory or time-non-local effects in open quantum dynamics pose theoretical as well as practical challenges in the understanding and control of noisy quantum systems. While there has been a comprehensive and concerted effort towards developing diagnostics for non-Markovian dynamics, all existing measures rely on time-domain measurements which are typically slow and expensive as they require averaging several runs to resolve small transient features on a broad background, and scale unfavorably with system size and complexity. In this work, we propose a spectroscopic measure of non-Markovianity which can detect persistent non-Markovianity in the system steady state. In addition to being experimentally viable, the proposed measure has a direct information theoretic interpretation: a large value indicates the information loss per unit bandwidth of making the Markov approximation. In the same vein, we derive a frequency-domain quantum master equation (FD-QME) that goes beyond the standard Born-Redfield description and retains the full memory of the state of the reduced system. Using the FD-QME and the proposed measure, we are able to reliably diagnose and quantify non-Markovianity in several system-environment settings including those with environmental correlations and retardation effects.
The basic features of the dynamics of open quantum systems, such as the dissipation of energy, the decay of coherences, the relaxation to an equilibrium or non-equilibrium stationary state, and the transport of excitations in complex structures are of central importance in many applications of quantum mechanics. The theoretical description, analysis and control of non-Markovian quantum processes play an important role in this context. While in a Markovian process an open system irretrievably loses information to its surroundings, non-Markovian processes feature a flow of information from the environment back to the open system, which implies the presence of memory effects and represents the key property of non-Markovian quantum behavior. Here, we review recent ideas developing a general mathematical definition for non-Markoviantiy in the quantum regime and a measure for the degree of memory effects in the dynamics of open systems which are based on the exchange of information between system and environment. We further study the dynamical effects induced by the presence of system-environment correlations in the total initial state and design suitable methods to detect such correlations through local measurements on the open system.
The precise characterization of dynamics in open quantum systems often presents significant challenges, leading to the introduction of various approximations to simplify a model. One commonly used strategy involves Markovian approximations, assuming a memoryless environment. In this study, such approximations are not used and an analytical dynamical depiction of an open quantum system is provided. The system under consideration is an oscillator that is surrounded by a bath of oscillators. The resulting dynamics are characterized by a second-order complex coefficient linear differential equation, which may be either homogeneous or inhomogeneous. Moreover, distinct dynamical regions emerge, depending on certain parameter values. Notably, the steady-state average excitation number (AEN) of the system shows rapid escalation with increasing non-Markovianity, reflecting the intricacies of real-world dynamics. In cases where there is detuning between the system frequency and the environmental central frequency within a non-Markovian regime, the AEN maintains its initial value for an extended period. Furthermore, the application of pulse control can effectively protect the quantum system from decoherence effects without using approximations. The pulse control can not only prolong the relaxation time of the oscillator, but can also be used to speed up the relaxation process, depending on the specifications of the pulse. By employing a kick pulse, the Mpemba effect can be observed in the non-Markovian regime in a surprisingly super-cooling-like effect.
Do phenomenological master equations with a memory kernel always describe a non-Markovian quantum dynamics characterized by reverse flow of information? Is the integration over the past states of the system an unmistakable signature of non-Markovianity? We show by a counterexample that this is not always the case. We consider two commonly used phenomenological integro-differential master equations describing the dynamics of a spin 1/21/21/2 in a thermal bath. By using a recently introduced measure to quantify non-Markovianity [Breuer et al., Phys. Rev. Lett. 103, 210401 (2009)] we demonstrate that as far as the equations retain their physical sense, the key feature of non-Markovian behavior does not appear in the considered memory kernel master equations. Namely, there is no reverse flow of information from the environment to the open system. Therefore, the assumption that the integration over a memory kernel always leads to a non-Markovian dynamics turns out to be vulnerable to phenomenological approximations. Instead, the considered phenomenological equations are able to describe time-dependent and unidirectional information flow from the system to the reservoir associated with time-dependent Markovian processes.
We investigate non-Markovianity measure using two-time correlation functions for open quantum systems. We define non-Markovianity measure as the difference between the exact two-time correlation function and the one obtained from quantum regression theorem in the Born-Markov approximation. Such non-Markovianity can easily be measured in experiments. We found that the non-Markovianity dynamics in different time scale crucially depends on the system-environment coupling strength and other physical parameters such as the initial temperature of the environment and the initial state of the system. In particular, we obtain the short-time and long-time behaviors of non-Markovianity for different spectral densities. We find that the thermal fluctuation always reduce the non-Markovian memory effect. Also, the non-Markovianity measure shows nontrivial initial state dependence in different time scales.
We investigate signatures of non-Markovianity in the dynamics of a periodically driven qubit coupled to a dissipative bosonic environment. We propagate the dynamics of the reduced density matrix of the qubit by integrating the numerically exact hierarchical equations of motion. Non-Markovian features are quantified by comparing on an equal footing the predictions from diverse and complementary approaches to quantum dissipation. In particular, we analyze the distinguishability of quantum states, the decay of the volume accessible to the qubit on the Hilbert space, the negativity of the canonical rates in the generalized Lindblad equation, and the relaxation of the memory kernels in the Nakajima-Zwanzig generalized quantum master equation. We study the effects of controlled driving on the coherent dynamics of the system. We show that a suitable external field can offset the ergodic relaxation of time-correlation functions, increase distinguishability over time, and strengthen non-Markovian effects witnessed by the canonical dissipation channels. We furthermore observe the phenomenon of eternal non-Markovianity for sufficiently small system-bath coupling and we discuss how this can be enhanced by modulating the frequency of the external drive. This work provides a broad theoretical analysis of quantum dissipation in the framework of open quantum dynamics and quantum information.
The dynamics of an open quantum system is non-Markovian when memory effect is non-negligible. The quantum memory effect is a very complex issue and presently there is no universally accepted method to quantify it. By examining the memory kernel of the Nakajima-Zwanzig master equation, we study the memory effect in a theoretical model consisting of a spin coupled to an environment of an Ising spin chain subject to a transverse field. We find that it can be dictated by either the time correlation in the environment, or the system-environment correlation, depending on the strength of the transverse field and the state of the environment. We analyze the information loss to the environment and backflow from it, and discover that the information flow is not always a reliable indicator for the nature of the dynamics.