Markov Approximation Research Articles

Controlling Matter Phases beyond Markov.

Controlling phase transitions in quantum systems via coupling to reservoirs has been mostly studied for idealized memory-less environments under the so-called Markov approximation. Yet, most quantum materials and experiments in the solid state, atomic, molecular and optical physics are coupled to reservoirs with finite memory times. Here, using the spectral theory of non-Markovian dissipative phase transitions developed in the companion paper [Debecker, Martin, and Damanet (to be published)], we show that memory effects can be leveraged to reshape matter phase boundaries, but also reveal the existence of dissipative phase transitions genuinely triggered by non-Markovian effects.

Unifying methods for optimal control in non-Markovian quantum systems via process tensors.

The large dimensionality of environments is the limiting factor in applying optimal control to open quantum systems beyond the Markovian approximation. Various methods exist to simulate non-Markovian systems, which effectively reduce the environment to a number of active degrees of freedom. Here, we show that several of these methods can be expressed in terms of a process tensor in the form of a matrix-product-operator, which serves as a unifying framework to show how they can be used in optimal control and to compare their performance. The matrix-product-operator form provides a general scheme for computing gradients using back propagation and allows the efficiency of the different methods to be compared via the bond dimensions of their respective process tensors.

Non-Markovian collective emission of giant emitters in the Zeno regime

We explore the collective Zeno dynamics of giant artificial atoms that are coupled, via multiple coupling points, to a common photonic or acoustic reservoir. In this regime, the establishment of atomic cooperativity and the revivification of exponential decay are highly intertwined, which is utterly beyond the non-Markovian regime with only retarded backaction. We reveal that giant atoms build up their collective emission smoothly from the decay rate of zero to that predicted by Markovian approximation and show great disparity between different waveguide QED setups. As a comparison, the steplike growth of instantaneous decay rates in the retardation-only picture has also been shown. All of these theoretical pictures predict the same collective behavior in the long time limit. From a phenomenological standpoint, we observe that the atomic superradiance exhibits significant directional property. In addition, the subradiant photons feature prolonged oscillation in the early stage of collective radiance, where the energy is exchanged remarkably between giant emitters and the field. Our results might be probed in state-of-art waveguide QED experiments and fundamentally broaden the fields of collective emission in systems with giant atoms. Published by the American Physical Society 2024

Efficient option pricing in the rough Heston model using weak simulation schemes

We provide an efficient and accurate simulation scheme for the rough Heston model in the standard (H>0) as well as the hyper-rough regime ( -1/2 $ ]]> H > − 1 / 2 ). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [Bayer and Breneis, arXiv:2309.07023], and provides a weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second-order weak convergence, while the computational cost increases linearly with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral's HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method's accuracy and efficiency.

Deep Learning-Based Vulnerability Detection and Mitigation in Virtualization Data Center

Virtualization is a critical technology that enables users to leverage the vast resources available within datacenters. Despite its numerous benefits, such as on-demand scalability, continuous availability, and cost efficiency, virtualization is susceptible to various security challenges, including intrusion, data compromise, and session hijacking. To address these threats, this study presents an innovative approach based on deep learning for detecting attacks and proactively isolating virtual machines (VMs) to mitigate their impact. The event sequences of VMs are transformed into event images using advanced techniques Integrated Gramian Markov Plot (IGMP). The proposed IGMP model comprises of the Gramian model with Markov estimate. The model uses the recurrence plot for the estimation of the IGMP in the virtualization process with the computation of data centers. Additionally, to improve the security IGMP model uses the aggregation signature generation model for the security features in the Virtual Machines. The proposed IGMP model uses the Deep learning models are then employed to extract meaningful features from these event images, which are subsequently classified into specific attack classes. Once an attack is predicted within the physical machine, the suspected VMs are immediately isolated to prevent further damage. Experimental results demonstrated that the high efficacy of the IGMP method, achieving an impressive attack prediction accuracy of 96%, surpassing existing approaches by at least 2%.

Heat currents in qubit systems

There is a current interest in quantum thermodynamics in the context of open quantum systems. An important issue is the consistency of quantum thermodynamics, in particular the second law of thermodynamics, i.e. the flow of heat from a hot reservoir to a cold reservoir. Here recent emphasis has been on composite system and in particular the issue regarding the application of local or global master equations. In order to contribute to this discussion we discuss two cases, namely as an example a single qubit and as a simple composite system two coupled qubits driven by two heat reservoirs at different temperatures, respectively. Applying a global Lindblad master equation approach we present explicit expressions for the heat currents in agreement with the second law of thermodynamics. The analysis is carried out in the Born–Markov approximation. We also discuss issues regarding the possible presence of coherences in the steady state.

Generalized kinetic theory of coarse-grained systems. I. Partial equilibrium and Markov approximations

The general kinetic theory of coarse-grained systems is presented in the abstract formalism of communication theory developed by Shannon and Weaver, Khinchin and Kolmogorov. The martingale theory shows that, under reasonable, general hypotheses, coarse-grained systems can be approximated by generalized Markov systems. For mixing systems, the Kolmogorov entropy production can be defined for nonstationary processes as Kolmogorov defined it for stationary processes.

Sudden change of genuine multipartite entanglement in non-Markovian dynamics

We investigate entanglement dynamics of bipartite as well multipartite systems beyond Markov approximation. We study two pairs of cavity–reservoir systems, modeled as four qubits and track the change of entanglement among cavity–cavity qubits, reservoir–reservoir qubits, and also for genuine entanglement of all four qubits. For cavity–cavity qubits, we find that non-Markovianity prolongs the life of entanglement besides collapse/revival phenomenon. For reservoir–reservoir qubits, entanglement sudden birth is delayed accordingly along with oscillations. For all four qubits, with a specific initial state, we find that genuine entanglement develops gradually, reaches to a maximum constant value and then freezes at this value for some time before decaying. This sudden change in dynamics of genuine entanglement occurs for a time window where neither cavities nor reservoirs are entangled. In contrast to Markov process, sudden change phenomenon may be recurrent in non-Markovian regime.

General framework for quantifying dissipation pathways in open quantum systems. II. Numerical validation and the role of non-Markovianity.

In the previous paper [C. W. Kim and I. Franco, J. Chem. Phys. 160, 214111-1-214111-13 (2024)], we developed a theory called MQME-D, which allows us to decompose the overall energy dissipation process in open quantum system dynamics into contributions by individual components of the bath when the subsystem dynamics is governed by a Markovian quantum master equation (MQME). Here, we contrast the predictions of MQME-D against the numerically exact results obtained by combining hierarchical equations of motion (HEOM) with a recently reported protocol for monitoring the statistics of the bath. Overall, MQME-D accurately captures the contributions of specific bath components to the overall dissipation while greatly reducing the computational cost compared to exact computations using HEOM. The computations show that MQME-D exhibits errors originating from its inherent Markov approximation. We demonstrate that its accuracy can be significantly increased by incorporating non-Markovianity by exploiting time scale separations (TSS) in different components of the bath. Our work demonstrates that MQME-D combined with TSS can be reliably used to understand how energy is dissipated in realistic open quantum system dynamics.

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

Effective mass approach to memory in non-Markovian systems.

Recent pioneering experiments on non-Markovian dynamics done, e.g., for active matter have demonstrated that our theoretical understanding of this challenging yet hot topic is rather incomplete and there is a wealth of phenomena still awaiting discovery. It is related to the fact that typically for simplification the Markovian approximation is employed and as a consequence the memory is neglected. Therefore, methods allowing to study memory effects are extremely valuable. We demonstrate that a non-Markovian system described by the Generalized Langevin Equation(GLE) for a Brownian particle of mass M can be approximated by the memoryless Langevin equationin which the memory effects are correctly reproduced solely via the effective mass M^{*} of the Brownian particle which is determined only by the form of the memory kernel. Our work lays the foundation for an impactful approach which allows one to readily study memory-related corrections to Markovian dynamics.

UAV-Assisted Task Offloading in Vehicular Edge Computing Networks

Vehicular edge computing (VEC) provides an effective task offloading paradigm by pushing cloud resources to the vehicular network edges, e.g., road side units (RSUs). However, overloaded RSUs are likely to occur especially in urban aggregation areas, possibly leading to greatly compromised offloading performance. Inspired by this, this paper explores this situation by introducing an unmanned aerial vehicle (UAV) to address the VEC overload problem. Specifically, we formulate a novel online UAV-assisted vehicular task offloading problem to minimize vehicular task delay under the long-term UAV energy constraint. To solve the formulated problem, we first decouple the long-term energy constraint based on the Lyapunov optimization technique. In this way, the problem can be solved in a real-time manner without requiring future information. Then, we construct a Markov chain based on Markov approximation optimization to find out the close-to-optimal UAV-assisted offloading strategies. Furthermore, we derive a mathematical analysis to rigorously demonstrate the offloading performance of the proposed algorithm. Additionally, the simulation results show that the proposed method outperforms the baselines by significantly reducing the vehicular task delay constrained by the long-term UAV energy budget under various system parameters, such as the energy budget and computation workloads.

Controllable spontaneous emission spectrum in an artificial giant atom: Dark lines and bound states

We study the spontaneous emission spectra of a few-level giant atom strongly coupled to a one-dimensional waveguide by employing variational and analytical approaches, which are beyond the rotating-wave approximation and the Markovian approximation. We show that the interference effects due to the multiple coupling points result in dark lines in spontaneous emission spectra of either two-level or three-level giant atoms regardless of whether there is a bound state or not. We illustrate that the generation of dark lines depends on the distance between the coupling points and their number. In the absence of a bound state, the dark lines can be used to suppress emission at a certain frequency in the spectrum. In the presence of a bound state, the total intensity of the emission spectrum can be significantly suppressed. The present results offer the possibility of control over the spontaneous emission with the dark lines due to the destructive interference in giant atoms. Published by the American Physical Society 2024

Optimal and instance-dependent guarantees for Markovian linear stochastic approximation

We study stochastic approximation procedures for approximately solving a d -dimen­sional linear fixed-point equation based on observing a trajectory of length n from an ergodic Markov chain. We first exhibit a non-asymptotic bound of the order t_{\mathrm{mix}}\frac{d}{n} on the squared error of the last iterate of a standard scheme, where t_{\mathrm{mix}} is a mixing time. We then prove a non-asymptotic instance-dependent bound on a suitably averaged sequence of iterates, with a leading term that matches the local asymptotic minimax limit, including sharp dependence on the parameters (d, t_{\mathrm{mix}}) in the higher-order terms. We complement these upper bounds with a non-asymptotic minimax lower bound that establishes the instance-optimality of the averaged SA estimator. We derive corollaries of these results for policy evaluation with Markov noise—covering the \mathrm{TD}(\lambda) family of algorithms for all \lambda \in [0, 1) —and linear autoregressive models. Our instance-dependent characterizations open the door to the design of fine-grained model selection procedures for hyperparameter tuning (e.g., choosing the value of \lambda when running the \mathrm{TD}(\lambda) algorithm).

Efficient Throughput Maximization in Dynamic Rechargeable Networks

In the Dynamic Rechargeable Networks (DRNs), to maximize the throughput by efficient energy allocation, the existing studies usually consider the spatio-temporal dynamic factors of the harvested energy, and seldom the network dynamic factors simultaneously, such as the time variable network resources and wireless interference. To take the network dynamic factors together, this paper studies the quite challenging problem, the network throughput maximization in the DRNs. We introduce the Time-Expanded Graph (TEG) to describe the above dynamic factors in an obvious way and design the Single Pair Throughput maximization (SPT) algorithm based on TEG. In the case of multiple pairs of source-targets, this paper introduces the Garg and Könemann's framework and then designs the Multiple Pairs Throughput (MPT) algorithm to maximize the overall throughput of all pairs. To reduce the time complexity, this paper proposes a Distributed and Parallel Throughput algorithm (DPT). In real applications, the network dynamic factors may not be known in advance. This paper proposes an Online Time-Span Algorithm (OTA) with Markov approximation and Lyapunov optimization, and conducts the extensive numerical evaluation based on the simulated data and the data collected by our real system. The numerical simulation results demonstrate the throughput improvement of our algorithms.

Adiabatic elimination in the presence of multiphoton transitions in atoms inside a cavity

Various approaches have been used in the literature for eliminating nonresonant levels in atomic systems and deriving effective Hamiltonians. Important among these are elimination techniques at the level of probability amplitudes, operator techniques to project the dynamics on to the subspace of resonant levels, Green’s function techniques, the James’ effective Hamiltonian approach, etc. None of the previous approaches is suitable for deriving effective Hamiltonians in intracavity situations. However, the James’ approach does work in the case of only two-photon transitions in a cavity. A generalization of the James’ approach works in the case of three-photon transitions in a cavity, but only under Raman-like resonant conditions. Another important approach for adiabatic elimination is based on an adaptation of the Markov approximation well-known in the theory of system–bath interactions. However, this approach has not been shown to work in intracavity situations. In this paper, we present a method of adiabatic elimination for atoms inside cavities in the presence of multiphoton transitions. We work in the Heisenberg picture, and our approach has the advantage that it allows one to derive effective Hamiltonians even when Raman-like resonance conditions do not hold.

Accurate estimates of dynamical statistics using memory.

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.

A mixed perturbative-nonperturbative treatment for strong light-matter interactions

Abstract The full information about the interaction between a quantum emitter and an arbitrary electromagnetic environment is encoded in the so-called spectral density. We present an approach for describing such interaction in any coupling regime, providing a Lindblad-like master equation for the emitter dynamics when coupled to a general nanophotonic structure. Our framework is based on the splitting of the spectral density into two terms. On the one hand, a spectral density responsible for the non-Markovian and strong-coupling-based dynamics of the quantum emitter. On the other hand, a residual spectral density including the remaining weak-coupling terms. The former is treated nonperturbatively with a collection of lossy interacting discrete modes whose parameters are determined by a fit to the original spectral density in a frequency region encompassing the quantum emitter transition frequencies. The latter is treated perturbatively under a Markovian approximation. We illustrate the power and validity of our approach through numerical simulations in three different setups, thus offering a variety of scenarios for a full test, including the ultra-strong coupling regime.

Autonomic IoT Application Placement in Edge/Fog Computing

Edge/Fog computing recently emerged as a novel distributed virtualized computing paradigm, where cloud services are extended to the edge of the network, thereby increasing network capacity and reducing latencies. In fog computing, applications are composed of microservices that are mapped to edge computing and communication devices (fog nodes). A crucial component in fog computing is placement algorithms that assign microservices to fog nodes since they determine the overall system performance in terms of energy consumption, communication costs, load balancing, and others. Placement strategies devised for cloud computing are generally centralized and, therefore, not well suited for decentralized fog systems. In this paper, we consider the joint optimization of two conflicting objectives, energy consumption at fog nodes and communication costs of applications, as a game between fog nodes and applications where each agent is modeled to control one objective. We follow a Markov approximation method for the design of a fully distributed autonomic service placement strategy without central coordination or global state information. Evaluation results show that the new approach provides a more optimal solution as compared to previous autonomic placement algorithms.

Pseudo rough Vol-of-Vol through Markovian Approximation

Pseudo rough Vol-of-Vol through Markovian Approximation