Including Memory Effects Research Articles

OQuPy: A Python package to efficiently simulate non-Markovian open quantum systems with process tensors.

Non-Markovian dynamics arising from the strong coupling of a system to a structured environment is essential in many applications of quantum mechanics and emerging technologies. Deriving an accurate description of general quantum dynamics including memory effects is, however, a demanding task, prohibitive to standard analytical or direct numerical approaches. We present a major release of our open source software package, OQuPy (Open Quantum System in Python), which provides several recently developed numerical methods that address this challenging task. It utilizes the process tensor approach to open quantum systems (OQS) in which a single map, the process tensor, captures all possible effects of an environment on the system. The representation of the process tensor in a tensor network form allows for an exact yet highly efficient description of non-Markovian OQS (NM-OQS). The OQuPy package provides methods to (1) compute the dynamics and multi-time correlations of quantum systems coupled to single and multiple environments, (2) optimize control protocols for NM-OQS, (3) simulate interacting chains of NM-OQS, and (4) compute the mean-field dynamics of an ensemble of NM-OQS coupled to a common central system. Our aim is to provide an easily accessible and extensible tool for researchers of OQS in fields such as quantum chemistry, quantum sensing, and quantum information.

Memristive switching of nanofluidic diodes by ionic concentration gradients

We demonstrate that nanofluidic diodes exhibit a memristive behavior that can be controlled not only by the amplitude and frequency of the driving oscillatory signal but also by changing the relative orientations of the externally imposed ionic concentration difference with respect to the pore charge density gradient. Because ionic concentration differences, memristive properties, and nanofluidic pores are of broad interest to electrochemical technology and membrane bioelectrochemistry, this study can be a useful contribution to the field. In our case, the memristor is a polymer membrane with current rectifying conical nanopores. The electrical interaction between the mobile ions in the aqueous solution and the pore surface charges leads to rich physico-chemical phenomena, including memory effects and the neuromorphic-like potentiation of the membrane conductance following voltage pulses (spikes). The multipore membrane can be used in the design of electrochemical circuits for signal conversion and information processing in iontronic hybrid devices.

A Fractional-Order Mathematical Model of Banana Xanthomonas Wilt Disease Using Caputo Derivatives

This article investigates a fractional-order mathematical model of Banana Xanthomonas Wilt disease while considering control measures using Caputo derivatives. The proposed model is numerically solved using the L1-based predictor-corrector method to explore the model’s dynamics in a particular time range. Stability and error analyses are performed to justify the efficiency of the scheme. The non-local nature of the Caputo fractional derivative, which includes memory effects in the system, is the main motivation for incorporating this derivative in the model. We obtain varieties in the model dynamics while checking various fractional order values.

Efficient Bayesian estimation of the generalized Langevin equation from data

Modeling non-Markovian time series is a recent topic of research in many fields such as climate modeling, biophysics, molecular dynamics, or finance. The generalized Langevin equation (GLE), given naturally by the Mori-Zwanzig projection formalism, is a frequently used model including memory effects. In applications, a specific form of the GLE is most often obtained on a data-driven basis. Here, Bayesian estimation has the advantage of providing both suitable model parameters and their credibility in a straightforward way. It can be implemented in the approximating case of white noise, which, far from thermodynamic equilibrium, is consistent with the fluctuation-dissipation theorem. However, the exploration of the posterior, which is done via Markov chain Monte Carlo sampling, is numerically expensive, which makes the analysis of large data sets unfeasible. In this work, we discuss an efficient implementation of Bayesian estimation of the GLE based on a piecewise constant approximation of the drift and diffusion functions of the model. In this case, the characteristics of the data are represented by only a few coefficients, so that the numerical cost of the procedure is significantly reduced and independent of the length of the data set. Further, we propose a modification of the memory term of the GLE, leading to an equivalent model with an emphasis on the impact of trends, which ensures that an estimate of the standard Langevin equation provides an effective initial guess for the GLE. We illustrate the capabilities of both the method and the model by an example from turbulence.

Numerical simulation of the time-fractional Fokker–Planck equation and applications to polymeric fluids

We introduce a new approach to the numerical approximation of the time-fractional Navier–Stokes–Fokker–Planck (TFNSFP) system, which involves the coupling of the incompressible Navier–Stokes equations to a time-fractional Fokker–Planck equation. The model arises in the context of dilute polymeric fluids, and it enhances the standard integer-derivative version of the model by including memory effects. The key challenge associated with the numerical solution of the TFNSFP system is that, in addition to it being nonlocal in time, it is, even in its simplest form, posed on a spatial domain that is the Cartesian product of two d-dimensional domains, for d∈{2,3}, the d-dimensional flow domain and the d-dimensional configuration space domain. We combine a kernel compression technique based on rational approximation of the integral kernel of the time-fractional derivative with a space-splitting method. By doing so, we transform the time-fractional partial differential equation into a fixed number of integer-order in time partial differential equations. The Fokker–Planck equation posed on the 2d-dimensional domain is decoupled into two d-dimensional problems, and a standard combination of the Hermite spectral method on the configuration space domain and a finite element method on the flow domain is then applied. Finally, we combine our numerical scheme for the time-fractional Fokker–Planck equation (TFFPE) with a standard Navier–Stokes solver. We propose an efficient implementation strategy based on the algebraic structure of the discretized time-fractional Fokker–Planck equation, which significantly reduces the computational cost.

FDTD Method for Electromagnetic Simulations in Media Described by Time-Fractional Constitutive Relations

In this paper, the finite-difference time-domain (FDTD) method is derived for electromagnetic simulations in media described by the time-fractional (TF) constitutive relations. TF Maxwell’s equations are derived based on these constitutive relations and the Grünwald–Letnikov definition of a fractional derivative. Then the FDTD algorithm, which includes memory effects and energy dissipation of the considered media, is introduced. Finally, one-dimensional signal propagation in such electromagnetic media is considered. The proposed FDTD method is derived based on a discrete approximation of the Grünwald–Letnikov definition of the fractional derivative and evaluated in a code. The stability condition is derived for the proposed FDTD method based on a numerical-dispersion relation. The obtained numerical results are compared with the outcomes of reference frequency-domain simulations, proving the accuracy of the proposed approach. However, high spatial resolution is required in order to obtain accurate results. The developed FDTD method is, unfortunately, computation and memory demanding when compared to the ordinary FDTD algorithm.

A case study of monkeypox disease in the United States using mathematical modeling with real data

In this article, we propose the mathematical modeling of monkeypox, a viral zoonotic disease, to study its near outbreaks in the United States. We use integer and fractional derivatives to simulate the transmission model dynamics effectively. The model parameter values are estimated using the Levenberg–Marquardt algorithm with the lsqcurvefit function in MATLAB. We derive the numerical solution of the integer-order model by using a neural network approach. For solving the fractional-order model, we use the finite-difference predictor–corrector method in the sense of the Caputo derivative. We perform several graphical simulations of future disease outbreaks. Implementing the Caputo fractional derivatives is motivated by its nonlocal nature, which includes memory effects in the model. The new data simulation and various solution approaches are the novelty of this research. This work aims to explore the transmission pattern and duration of a monkeypox outbreak in the United States.

Non‐linear modelling and distortion compensation in optical Fast‐orthogonal frequency division multiplexing systems

AbstractFast‐OFDM‐based intensity‐modulation and direct‐detection (IM/DD) has been proposed for deployment of cost‐efficient optical access networks, due to its implementation simplicity and high spectral efficiency. In this article, the accuracy of the generalised memory polynomial (GMP) for the non‐linear modelling of optical Fast‐OFDM links is studied, including memory effects and considering different model parameters. After model validation using measured data of a 10 km single mode fibre link, the GMP is used for performance investigations of a distortion compensation approach to optical Fast‐OFDM, for up to 16PAM modulation formats and different number of Fast‐OFDM subcarriers. This study firstly reports the performance results of optical 16PAM‐Fast‐OFDM systems using either 2PAM‐ or 4PAM‐based training signals for digital post‐distortion and FFT‐based channel estimation, and firstly investigates the influence of the zero padding (ZP) length on the performance of optical Fast‐OFDM. Excellent performance improvements are achieved using the proposed distortion compensation scheme, relative to conventional system implementation.

Ab initio real-time quantum dynamics of charge carriers in momentum space.

Application of the non-adiabatic molecular dynamics (NAMD) approach is limited to studying carrier dynamics in the momentum space, as a supercell is required to sample the phonon excitation and electron-phonon (e-ph) interaction at different momenta in a molecular dynamics simulation. Here we develop an ab initio approach for the real-time charge carrier quantum dynamics in the momentum space (NAMD_k) by directly introducing e-ph coupling into the Hamiltonian based on the harmonic approximation. The NAMD_k approach maintains the zero-point energy and includes memory effects of carrier dynamics. The application of NAMD_k to the hot carrier dynamics in graphene reveals the phonon-specific relaxation mechanism. An energy threshold of 0.2 eV-defined by two optical phonon modes-separates the hot electron relaxation into fast and slow regions with lifetimes of pico- and nanoseconds, respectively. The NAMD_k approach provides an effective tool to understand real-time carrier dynamics in the momentum space for different materials.

A Study on the Nonlinear Caputo-Type Snakebite Envenoming Model with Memory

In this article, we introduce a nonlinear Caputo-type snakebite envenoming model with memory. The well-known Caputo fractional derivative is used to generalize the previously presented integer-order model into a fractional-order sense. The numerical solution of the model is derived from a novel implementation of a finite-difference predictor-corrector (L1-PC) scheme with error estimation and stability analysis. The proof of the existence and positivity of the solution is given by using the fixed point theory. From the necessary simulations, we justify that the first-time implementation of the proposed method on an epidemic model shows that the scheme is fully suitable and time-efficient for solving epidemic models. This work aims to show the novel application of the given scheme as well as to check how the proposed snakebite envenoming model behaves in the presence of the Caputo fractional derivative, including memory effects.

Brief communication: An autonomous UAV for catchment-wide monitoring of a debris flow torrent

Abstract. Debris flows threaten communities in mountain regions worldwide. Combining modern photogrammetric processing with autonomous unoccupied aerial vehicle (UAV) flights at sub-weekly intervals allows mapping of sediment dynamics in a debris flow catchment. This provides important information for sediment disposition that pre-conditions the catchment for debris flow occurrence. At the Illgraben debris flow catchment in Switzerland, our autonomous UAV launched nearly 50 times in the snow-free periods in 2019–2021 with typical flight intervals of 2–4 d, producing 350–400 images every flight. The observed terrain changes resulting from debris flows exhibit preferred locations of erosion and deposition, including memory effects as previously deposited material is preferentially removed during subsequent debris flows. Such data are critical for the validation of geomorphological process models. Given the remote terrain, the mapped short-term erosion and deposition structures are difficult to obtain with conventional measurements. The proposed method thus fills an observational gap, which ground-based monitoring and satellite-based remote sensing cannot fill as a result of limited access, reaction time, spatial resolution, or involved costs.

Synchronization in a Multiplex Network of Nonidentical Fractional-Order Neurons

Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers are different. It is observed that the intralayer synchronization state occurs in weaker intralayer couplings when using nonidentical fractional-order derivatives rather than integer-order or identical fractional orders. Furthermore, the needed interlayer coupling strength for interlayer near synchronization decreases for lower fractional orders. The dynamics of the neurons in nonidentical layers are also considered. It is shown that in lower fractional orders, the neurons’ dynamics change to periodic when the near synchronization state occurs. Moreover, decreasing the derivative order leads to incrementing the frequency of the bursts in the synchronization manifold, which is in contrast to the behavior of the single neuron.

Excitation Energy Transfer Efficiency in Fluctuating Environments: Role of Quantum Coherence in the Presence of Memory Effects.

Several recent studies have interrogated the role of quantum coherence in affecting the transfer efficiency of an optical excitation to the designated "trap" state where the energy can be used for subsequent reactions, as in photosynthesis. However, these studies invoke a Markovian approximation for the time correlation function describing the environment-induced stochastic fluctuations. Here, we employ Kubo's quantum stochastic Liouville equation (QSLE) to include memory effects. We extend the existing QSLE scheme to introduce decay of a newly created excitation due to radiative and nonradiative channels and also by desired trapping toward the targeted chromophore. We show that the theoretical formalism based on the QSLE correctly reproduces the rate equation description in the Markovian limit, with the rate constants determined by an appropriate quantum limiting procedure. We find that under certain conditions, the efficiency of excitation transfer to the trap gains from the combined presence of quantum coherence and temporally correlated stochastic fluctuations. We work out different limiting situations in order to discover and quantify the optimum conditions for the energy transfer to the trapped state. We find that maximum energy transfer efficiency is achieved in the intermediate limit between coherent and incoherent transport.

Continuous-Variable Error Correction for General Gaussian Noises

Quantum error correction is essential for robust quantum-information processing with noisy devices. As bosonic quantum systems play a role in quantum sensing, communication, and computation, it is useful to design error-correction codes suitable for these systems against various different types of noises. While most efforts aim at protecting qubits encoded into the infinite-dimensional Hilbert space of a bosonic mode, Ref. [Phys. Rev. Lett. 125, 080503 (2020)] proposed an error-correction code to maintain the infinite-dimensional-Hilbert-space nature of bosonic systems by encoding a single bosonic mode into multiple bosonic modes. Enabled by Gottesman-Kitaev-Preskill states as ancilla, the code overcomes the no-go theorem of Gaussian error correction. In this work, we generalize the error-correction code to the scenario with general correlated and heterogeneous Gaussian noises, including memory effects. We introduce Gaussian preprocessing and postprocessing to convert the general noise model to an independent but heterogeneous collection of additive white Gaussian noise channels and then apply concatenated codes in an optimized manner. To evaluate the performance, we develop a theory framework to enable the efficient calculation of the noise SD after the error correction, despite the non-Gaussian nature of the codes. Our code provides the optimal scaling of the residue-noise SD with the number of modes and can be widely applied to distributed sensor networks, network communication, and composite quantum-memory systems.

Predicting entanglement and coherent times in FMO complex using the HEOM method

Fenna-Matthews-Olson (FMO) bacteriochlorophylls (BChls) are molecules responsible of the high efficiency energy transfer in the photosynthetic process of green sulfur bacteria, controversially associated to quantum phenomena of long lived coherence. This phenomenon is modelled using Quantum Open Systems (QOS) without include memory effects of the surrounding approximated as a phonon bath on thermal equilibrium. This work applies the Hierarchical Equations of Motion method (HEOM), a non-Markovian approach, in the modelling of the system evolution of FMO complex to perform predictions about the coherence times scales together with global and semi-local entanglement during the quantum excitation.

Solutions for a hyperbolic diffusion equation with linear reaction terms

General diffusion processes involve one or more diffusing species and are usually modelled by Fick’s law, which assumes infinite propagation velocity. In this article, searching for the effect of finite propagation speeds in a system with two reacting species, we investigate diffusing and reacting particles governed by a hyperbolic diffusion equation, that is, the Cattaneo equation, which describes a diffusion process with finite propagation velocity, in the presence of a constant external field and reaction terms. These reaction terms are linear and may be related to irreversible and reversible processes, including memory effects, depending on the choices of the reaction rates. We obtain exact solutions for the equilibrium concentrations and explore the rich variety of behaviours exhibited by the species involved in reaction processes. Our results may shine new light into systems with more than one kind of diffusing and reacting particles, as is the case in several industrial and biological process, when finite speeds and memory effects are involved.

Singularities of plane gravitational waves in Einstein’s general relativity

Similar to the Schwarzschild coordinates for spherical black holes, the Baldwin, Jeffery and Rosen (BJR) coordinates for plane gravitational waves are often singular, and extensions beyond such singularities are necessary, before studying asymptotic properties of such spacetimes at the null infinity of the plane, on which the gravitational waves propagate. The latter is closely related to the studies of memory effects and soft graviton theorems. In this paper, we point out that in the BJR coordinates all the spacetimes are singular physically at the focused point $u = u_s$, except for the two cases: (1) $\alpha =1/2, \; \forall \; \chi_n$; and (2) $\alpha =1, \; \chi_i = 0\; (i = 1, 2, 3)$, where $\chi_n$ are the coefficients in the expansion $\chi \equiv \left[{\mbox{det}}\left(g_{ab}\right) \right]^{1/4} = \left(u - u_s\right)^{\alpha}\sum_{n = 0}^{\infty}\chi_n \left(u - u_s\right)^n$ with $\chi_0 \not= 0$, the constant $\alpha \in (0, 1]$ characterizes the strength of the singularities, and $g_{ab}$ denotes the reduced metric on the two-dimensional plane orthogonal to the propagation direction of the wave. Therefore, the hypersurfaces $u= u_s$ already represent the boundaries of such spacetimes, and the null infinity does not belong to them. As a result, they cannot be used to study properties of plane gravitational waves at null infinities, including memory effects and soft graviton theorems.

From continuous-time random walks to controlled-diffusion reaction

Systems in nature that present anomalous diffusion phenomena due to irregularities of medium, traps or reactions process are reported daily. In this scenario, the diffusion with traps or localised–reactions emerge through various investigations that include numerical, analytical and experimental techniques. In this work, we construct a model which involves a coupling of two diffusion equations to approach the random walkers in a medium with localised reaction point (or controlled diffusion). We present the exact analytical solutions to the model. In the following article, we obtain the survival probability and mean square displacement. Moreover, we extend the model to include memory effects in reaction points. Thereby, we found a simple relation that connects the power-law memory kernels with anomalous diffusion phenomena, i.e. . The investigations presented in this work uses recent mathematical techniques to introduces a form to represent the coupled random walks in context of reaction–diffusion problem to localised reaction.

Mori-Zwanzig reduced models for uncertainty quantification

In many time-dependent problems of practical interest the parameters and/or initial conditions entering the equations describing the evolution of the various quantities exhibit uncertainty. One way to address the problem of how this uncertainty impacts the solution is to expand the solution using polynomial chaos expansions and obtain a system of differential equations for the evolution of the expansion coefficients. We present an application of the Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of such systems of differential equations. In particular, we construct reduced models for a subset of the polynomial chaos expansion coefficients that are needed for a full description of the uncertainty caused by uncertain parameters or initial conditions. Even though the MZ formalism is exact, its straightforward application to the problem of constructing reduced models for estimating uncertainty involves the computation of memory terms whose cost can become prohibitively expensive. For those cases, we present a Markovian reformulation of the MZ formalism which can lead to approximations that can alleviate some of the computational expense while retaining an accuracy advantage over reduced models that discard the memory altogether. Our results support the conclusion that successful reduced models need to include memory effects.

Inter-laboratory test for oxygen and hydrogen stable isotope analyses of geothermal fluids: Assessment of reservoir fluid compositions.

Knowledge of the accuracy and precision for oxygen (δ18 O values) and hydrogen (δ2 H values) stable isotope analyses of geothermal fluid samples is important to understand geothermal reservoir processes, such as partial boiling-condensation and encroachment of cold and reinjected waters. The challenging aspects of the analytical techniques for this specific matrix include memory effects and higher scatter of delta values with increasing total dissolved solids (TDS) concentrations, deterioration of Pt-catalysts by dissolved/gaseous H2 S for hydrogen isotope equilibration measurements and isotope salt effects that offset isotope ratios determined by gas equilibration techniques. An inter-laboratory comparison exercise for the determination of the δ18 O and δ2 H values of nine geothermal fluid samples was conducted among eleven laboratories from eight countries (CeMIEGeo2017). The delta values were measured by dual inlet isotope ratio mass spectrometry (DI-IRMS), continuous flow IRMS (CF-IRMS) and/or laser absorption spectroscopy (LAS). Moreover, five of these laboratories analyzed an additional sample set at least one month after the analysis period of the first set. Statistical evaluation of all the results was performed to obtain the expected isotope ratios of each sample, which were then subsequently used in deep reservoir fluid composition calculations. The overall analytical precisions of the measurements were ± 0.2‰ for δ18 O values and ± 2.0‰ for δ2 H values within the 95% confidence interval. The measured and calculated δ18 O and δ2 H values of water sampled at the weir box, separator and wellhead of geothermal wells suggest the existence of hydrogen and oxygen isotope-exchange equilibrium between the liquid and vapor phases at all sampling points in the well. Thus, both procedures for calculating the isotopic compositions of the deep geothermal reservoir fluid - using either the analytical data of the liquid phase at the weir box together with those of vapor at the separator or the analytical data of liquid and vapor phases at the separator -are equally valid.