Markovian Limit Research Articles

Spectral Map for Slow Collective Variables, Markovian Dynamics, and Transition State Ensembles.

Understanding the behavior of complex molecular systems is a fundamental problem in physical chemistry. To describe the long-time dynamics of such systems, which is responsible for their most informative characteristics, we can identify a few slow collective variables (CVs) while treating the remaining fast variables as thermal noise. This enables us to simplify the dynamics and treat it as diffusion in a free-energy landscape spanned by slow CVs, effectively rendering the dynamics Markovian. Our recent statistical learning technique, spectral map [Rydzewski, J. J. Phys. Chem. Lett. 2023, 14(22), 5216-5220], explores this strategy to learn slow CVs by maximizing a spectral gap of a transition matrix. In this work, we introduce several advancements into our framework, using a high-dimensional reversible folding process of a protein as an example. We implement an algorithm for coarse-graining Markov transition matrices to partition the reduced space of slow CVs kinetically and use it to define a transition state ensemble. We show that slow CVs learned by spectral map closely approach the Markovian limit for an overdamped diffusion. We demonstrate that coordinate-dependent diffusion coefficients only slightly affect the constructed free-energy landscapes. Finally, we present how spectral maps can be used to quantify the importance of features and compare slow CVs with structural descriptors commonly used in protein folding. Overall, we demonstrate that a single slow CV learned by spectral map can be used as a physical reaction coordinate to capture essential characteristics of protein folding.

Coarse-Graining Conformational Dynamics with Multidimensional Generalized Langevin Equation: How, When, and Why.

A data-driven ab initio generalized Langevin equation (AIGLE) approach is developed to learn and simulate high-dimensional, heterogeneous, coarse-grained (CG) conformational dynamics. Constrained by the fluctuation-dissipation theorem, the approach can build CG models in dynamical consistency (DC) with all-atom molecular dynamics. We also propose practical criteria for AIGLE to enforce long-term DC. Case studies of a toy polymer, with 20 CG sites, and the alanine dipeptide, with two dihedral angles, elucidate why one should adopt AIGLE or its Markovian limit for modeling CG conformational dynamics in practice.

Non-Hermitian pseudomodes for strongly coupled open quantum systems: Unravelings, correlations, and thermodynamics

The pseudomode framework provides an exact description of the dynamics of an open quantum system coupled to a non-Markovian environment. Using this framework, the influence of the environment on the system is studied in an equivalent model, where the open system is coupled to a finite number of unphysical pseudomodes that follow a time-local master equation. Building on the insight that this master equation does not need to conserve the hermiticity of the pseudomode state, we here ask for the most general conditions on the master equation that guarantee the correct reproduction of the system's original dynamics. We demonstrate that our generalized approach decreases the number of pseudomodes that are required to model, for example, underdamped environments at finite temperature. We also provide an unraveling of the master equation into quantum jump trajectories of non-Hermitian states, which further facilitates the utilization of the pseudomode technique for numerical calculations by enabling the use of easily parallelizable Monte Carlo simulations. Finally, we show that pseudomodes, despite their unphysical nature, provide a natural picture in which physical processes, such as the creation of system-bath correlations or the exchange of heat, can be studied. Hence, our results pave the way for future investigations of the system-environment interaction leading to a better understanding of open quantum systems far from the Markovian weak-coupling limit. Published by the American Physical Society 2024

Quantum Chaos and Coherence: Random Parametric Quantum Channels

The survival probability of an initial Coherent Gibbs State (CGS) is a natural extension of the Spectral Form Factor (SFF) to open quantum systems. To quantify the interplay between quantum chaos and decoherence away from the semi-classical limit, we investigate the relation of this generalized SFF with the corresponding l1-norm of coherence. As a working example, we introduce Parametric Quantum Channels (PQC), a discrete-time model of unitary evolution mixed with the effects of measurements or transient interactions with an environment. The Energy Dephasing (ED) dynamics arises as a specific case in the Markovian limit. We demonstrate our results in a series of random matrix models.

From stochastic Hamiltonian to quantum simulation: exploring memory effects in exciton dynamics

The unraveling of open quantum system dynamics in terms of stochastic quantum trajectories offers a picture of open system dynamics that consistently considers memory effects stemming from the finite correlation time of environment fluctuations. These fluctuations significantly influence the coherence and energy transport properties of excitonic systems. When their correlation time is comparable to the timescale of the Hamiltonian evolution, it leads to the departure of open system dynamics from the Markovian limit. In this work, we leverage the unraveling of exciton dynamics through stochastic Hamiltonian propagators to design quantum circuits that simulate exciton transport, capturing finite memory effects. In addition to enabling the synthesis of parametrizable quantum circuits, stochastic unitary propagators provide a transparent framework for investigating non-Markovian effects on exciton transport. Our analysis reveals a nuanced relationship between environment correlation time and transport efficiency, identifying a regime of ‘memory-assisted’ quantum transport where time-correlated fluctuations allow the system to reach higher efficiency. However, this property is not universal and can only be realized in conjunction with specific features of the system Hamiltonian.

Memory effects in the efficiency control of energy transfer under incoherent light excitation in noisy environments.

Fluctuations in energy gap and coupling constants between chromophores can play an important role in absorption and energy transfer across a collection of two-level systems. In photosynthesis, light-induced quantum coherence can affect the efficiency of energy transfer to the designated "trap" state. Theoretically, the interplay between fluctuations and coherence has been studied often, employing either a Markovian or a perturbative approximation. In this study, we depart from these approaches to incorporate memory effects by using Kubo's quantum stochastic Liouville equation. We introduce the effects of decay of the created excitation (to the ground state) on the desired propagation and trapping that provides a direction of flow of the excitation. In the presence of light-induced pumping, we establish a relation between the efficiency, the mean survival time, and the correlation decay time of the bath-induced fluctuations. A decrease in the steady-state coherence during the transition from the non-Markovian regime to the Markovian limit results in a decrease in efficiency. As in the well-known Haken-Strobl model, the ratio of the square of fluctuation strength to the rate plays a critical role in determining the mechanism of energy transfer and in shaping the characteristics of the efficiency profile. We recover a connection between the transfer flux and the imaginary part of coherences in both equilibrium and excited bath states, in both correlated and uncorrelated bath models. We uncover a non-monotonic dependence of efficiency on site energy heterogeneity for both correlated and uncorrelated bath models.

Theory of phonon sidebands in the absorption spectra of moiré exciton–polaritons

Excitons in twisted bilayers of transition metal dichalcogenides have strongly modified dispersion relations due to the formation of periodic moiré potentials. The strong coupling to a light field in an optical cavity leads to the appearance of moiré polaritons. In this paper, we derive a theoretical model for the linear absorption spectrum of the coupled moiré polariton–phonon system based on the time-convolutionless (TCL) approach. Results obtained by numerically solving the TCL equation are compared to those obtained in the Markovian limit and from a perturbative treatment of non-Markovian corrections. A key quantity for the interpretation of the findings is the generalized phonon spectral density. We discuss the phonon impact on the spectrum for realistic moiré exciton dispersions by varying twist angle and temperature. Key features introduced by the coupling to phonons are broadenings and energy shifts of the upper and lower polariton peak and the appearance of phonon sidebands between them. We analyze these features with respect to the role of Markovian and non-Markovian effects and find that they strongly depend on the twist angle. We can distinguish between the regimes of large, small, and intermediate twist angles. In the latter phonon effects are particularly pronounced due to dominating phonon transitions into regions which are characterized by van Hove singularities in the density of states.

Strongly coupled fermionic probe for nonequilibrium thermometry

We characterise the measurement sensitivity, quantified by the quantum Fisher information (QFI), of a single-fermionic thermometric probe strongly coupled to the sample of interest, a fermionic bath, at temperature T. For nonequilibrium protocols, in which the probe is measured before reaching equilibrium with the sample, we find new behaviour of the measurement sensitivity arising due to non-Markovian dynamics. First, we show that the QFI displays a highly non-monotonic behaviour in time, in contrast to the Markovian case where it grows monotonically until equilibrium, so that non-Markovian revivals can be exploited to reach a higher QFI. Second, the QFI rate is maximised at a finite interrogation time t∗ , which we characterize, in contrast to the solution t∗→0 known in the Markovian limit (Pavel Sekatski and Martí Perarnau-Llobet 2022 Quantum 6 869). Finally, we consider probes make up of few fermions and discuss different collective enhancements in the measurement precision.

Quantum master equation for the vacuum decay dynamics

The quantum master equation required to describe the dynamics of gravity-related vacuum decay is still challenging. We aim to study this issue. Our model consists of the spacetime and scalar field with self-interaction potential. The environment is chosen as spacetime while the system is formed by the vacua of the scalar field. We demonstrate that the quantum dynamics of the vacua can be described by the Redfield equation, which can depict the evolution of both coherence and the comoving volume fraction of the vacuum. Under the Markovian limit, coherence monotonically decreases with time, leading to the initial quantum state to decohere into a classical state. This helps the understanding of the decoherence of the universe. We also highlight that in certain circumstances, the evolution of the vacuum system may display non-Markovian dynamics. In specific scenarios, the classical limit of the quantum master equation is consistent with the classical master equation. In the steady state, the dominant vacuum corresponds to the smallest cosmological constant, and various dS vacua can reach equilibrium states.

Chiral and nonreciprocal single-photon scattering in a chiral-giant-molecule waveguide-QED system

We study chiral and nonreciprocal single-photon scattering in a chiral-giant-molecule waveguide-QED system. Here, the giant molecule consists of two coupled giant atoms, which interact with two linear waveguides, forming a four-port quantum device. We obtain the exact analytical expressions of the four scattering amplitudes using a real-space method. Under the Markovian limit, we find that the single-photon scattering behavior is determined by the coupling strength between the giant atoms and the waveguides, the coupling strength between the two giant atoms, and the nondipole effect caused by the phase accumulation of photons traveling between the coupling points. It is also found that chiral and nonreciprocal single-photon scattering can be realized by introducing the chiral coupling to break the symmetry in the coupling configuration between the giant molecule and the waveguides. In addition, an ideal chiral emitter-waveguide coupling enables a directional single-photon routing. In the non-Markovian regime, the scattering spectra are characterized by more abundant structures with multiple peaks and dips. In particular, we demonstrate that the non-Markovian retarded effect can induce the nonreciprocal single-photon scattering. Our results have potential applications in the design of optical quantum devices involving giant atoms, which can provide an efficient platform for studying chiral quantum optics.

Modeling spin relaxation in complex radical systems using MolSpin.

Spin relaxation is an important aspect of the spin dynamics of free radicals and can have a significant impact on the outcome of their spin-selective reactions. Examples range from the use of radicals as spin qubits in quantum information processing to the radical pair reactions in proteins that may allow migratory birds to sense the direction of the Earth's magnetic field. Accurate modeling of spin relaxation, however, is non-trivial. Bloch-Redfield-Wangsness theory derives a quantum mechanical master equation from system-bath interactions in the Markovian limit that provides a comprehensive framework for describing spin relaxation. Unfortunately, the construction of the master equation is system-specific and often resource-heavy. To address this challenge, we introduce a generalized and efficient implementation of BRW theory as a new feature of the spin dynamics toolkit MolSpin which offers an easy-to-use approach for studying systems of reacting radicals of varying complexity.

Tunable single-photon nonreciprocal scattering and targeted router in a giant atom-waveguide system with chiral couplings

We investigate the single-photon scattering properties of a driven three-level giant atom chirally coupled to two waveguides simultaneously in both the Markovian and the non-Markovian regimes. It is shown that under the Markovian limit, the chiral photon-atom interactions enable nonreciprocal scattering in a single waveguide and targeted photon routing with a probability of 100% in two waveguides, while the presence of the driving field and the giant atom structure introduce a more tunable parameter to manipulate the single-photon scattering behaviors. We also examine how the non-reciprocity and routing capability are influenced by the imperfect chirality and the atomic dissipation. In the non-Markovian regime, we show that the scattering behaviors are more complicated. The non-Markovicity induced non-reciprocity and photon routing are demonstrated in this paper. We believe that those results have potential applications in quantum network engineering.

Lagrangian Stochastic Model for the Motions of Magnetic Footpoints on the Solar Wind Source Surface and the Path Lengths of Boundary-driven Interplanetary Magnetic Field Lines

In this work, we extend Leighton’s diffusion model describing the turbulent mixing of magnetic footpoints on the solar wind source surface. The present Lagrangian stochastic model is based on the spherical Ornstein–Uhlenbeck process with drift that is controlled by the rotation frequency Ω of the Sun, the Lagrangian integral timescale τ L, and the root-mean-square footpoint velocity V rms. The Lagrangian velocity and the positions of magnetic footpoints on the solar wind source surface are obtained from the solutions of a set of stochastic differential equations, which are solved numerically. The spherical diffusion model of Leighton is recovered in the singular Markov limit when the Lagrangian integral timescale tends to zero while keeping the footpoint diffusivity finite. In contrast to the magnetic field lines driven by standard Brownian processes on the solar wind source surface, the interplanetary magnetic field lines are smooth differentiable functions with finite path lengths in our model. The path lengths of the boundary-driven interplanetary magnetic field lines and their probability distributions at 1 au are computed numerically, and their dependency with respect to the controlling parameters is investigated. The path-length distributions are shown to develop a significant skewness as the width of the distributions increases.

Effective master equations for two accelerated qubits

We revisit the problem involving two constantly accelerating Unruh-DeWitt detectors using Open Effective Field Theory methods. We study the time evolution of the joint detector state using a Markovian approximation which differs from the standard one taken in the literature. We show that this Markovian limit already implies the complete positivity of the dynamical evolution map without invoking the rotating wave approximation (RWA), in contrast to standard derivations of open system master equations. By calculating explicitly the domain of validity of this Markovian approximation, we argue that the lack of complete positivity in the usual microscopic derivation stems from the (subtle) fact that the Redfield equation is used outside its domain of validity. We give two well-known cases studied in the literature that violate the validity of the Markovian approximation: (i) the ``stacked trajectory'' limit (when detector trajectories are taken to be on top of one another), and (ii) large gap-to-acceleration ratio. Since Markovian dynamics with or without RWA can lead to different qualitative predictions for entanglement dynamics, our work emphasizes the need to properly track the regime of validity of all approximations.

Collective quantum beats from distant multilevel emitters

We analyze the dynamics of quantum beats in a system of two V-type three-level atoms coupled to a waveguide. We show that quantum beats can be collectively enhanced or suppressed, akin to Dicke super- and subradiance, depending on the interatomic separation and the initial correlations between the atoms. In particular, the interference properties of the collective beats are determined by the distance between the atoms modulo the beat wavelength. We study the collective atomic and field dynamics, illustrating a crossover from a Markovian to a non-Markovian regime as the atomic separation becomes sufficiently large to bring memory effects of the electromagnetic environment into consideration. In such a non-Markovian regime, collective quantum beats can be enhanced beyond the Markovian limit as a result of retardation effects. Our results demonstrate the rich interplay between multilevel and multiatom quantum interference effects arising in a system of distant quantum emitters.

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [Comment. Math. Helv. 95 (2020), pp. 593–633]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain–Kontorovich [Ann. of Math. (2) 180 (2014), pp. 137–196], Huang [An improvement to Zaremba’s conjecture, ProQuest LLC, Ann Arbor, MI, 2015] and Kan [Mat. Sb. 210 (2019), pp. 75–130]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [Amer. J. Math. 120 (1998), pp. 691-721]. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in this paper in a way that is straightforward to implement. These estimates apparently cannot be obtained by other known methods.

Non-Markovian Abraham-Lorentz-Dirac equation: Radiation reaction without pathology

Motion of a point charge emitting radiation in an electromagnetic field obeys the Abraham-Lorenz-Dirac (ALD) equation, with the effects of radiation reaction or self-force incorporated. This class of equations describing backreaction, including also the equations for gravitational self-force or Einstein's equation for cosmology driven by trace anomaly, contain third-order derivative terms. They are known to have pathologies like the possession of runaway solutions, causality violation in pre-acceleration and the need for an extra second-order derivative initial condition. In our current program we reexamine this old problem from the perspective of non-Markovian dynamics in open systems, applied earlier to backreaction problems in the early universe. Here we consider a harmonic atom coupled to a scalar field, which acts effectively like a supra-Ohmic environment, as in scalar electrodynamics. Our analysis shows that a) there is no need for specifying a second derivative for the initial condition; b) there is no pre-acceleration. These undesirable features in conventional treatments arise from an inconsistent Markovian assumption: these equations were regarded as Markovian ab initio, not as a limit of the backreaction-imbued non-Markovian equation of motion. If one starts with the full non-Markovian dynamical equation and takes the proper Markovian limit judiciously, no harms are done. Finally, c) There is no causal relation between the higher-derivative term in the equation of motion and the existence of runaway solutions. If the charge has an effective size greater than this critical value, its dynamics is stable. When this reasonable condition is met, radiation reaction understood and treated correctly in the non-Ohmic non-Markovian dynamics still obeys a third-order derivative equation, but it does not require a second derivative initial condition, and there is no pre-acceleration.

Benchmarking the cosmological master equations

Master equations are commonly employed in cosmology to model the effect of additional degrees of freedom, treated as an “environment”, onto a given “system”. However, they rely on assumptions that are not necessarily satisfied in cosmology, where the environment may be out of equilibrium and the background is dynamical. In this work, we apply the master-equation program to a model that is exactly solvable, and which consists of two linearly coupled scalar fields evolving on a cosmological background. The light field plays the role of the system and the heavy field is the environment. By comparing the exact solution to the output of the master equation, we can critically assess its performance. We find that the master equation exhibits a set of “spurious” terms that explicitly depend on the initial conditions, and which arise as a consequence of working on a dynamical background. Although they cancel out in the perturbative limit of the theory (i.e. at leading orders in the interaction strength), they spoil resummation. However, when those terms are removed, the master equation performs impressively well to reproduce the power spectra and the amount of the decoherence of the light field, even in the strongly decohered regime. We conclude that master equations are able to perform late-time resummation, even though the system is far from the Markovian limit, provided spurious contributions are suppressed.

Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model

We present simulations of nonequilibrium dynamics of quantum field theories on digital quantum computers. As a representative example, we consider the Schwinger model, a ($1+1$)-dimensional U(1) gauge theory, coupled through a Yukawa-type interaction to a thermal environment described by a scalar field theory. We use the Hamiltonian formulation of the Schwinger model discretized on a spatial lattice. With the thermal scalar fields traced out, the Schwinger model can be treated as an open quantum system and its real-time dynamics are governed by a Lindblad equation in the Markovian limit. The interaction with the environment ultimately drives the system to thermal equilibrium. In the quantum Brownian motion limit, the Lindblad equation is related to a field theoretical Caldeira-Leggett equation. By using the Stinespring dilation theorem with ancillary qubits, we perform studies of both the nonequilibrium dynamics and the preparation of a thermal state in the Schwinger model using IBM's simulator and quantum devices. The real-time dynamics of field theories as open quantum systems and the thermal state preparation studied here are relevant for a variety of applications in nuclear and particle physics, quantum information and cosmology.

The memory effects in the Langevin description of nuclear fission

We have calculated the fission width Γf within a one-dimensional Langevin approach with the potential energy given by simple two-parabolic (Kramers) potential and found out that for excitation energies slightly above the fission barrier the Γf(E) calculated in constant energy regime is substantially smaller than Γf(T) calculated in constant temperature regime and popular Bohr-Wheeler or Kramers approximations. At small excitations Γf(E) is by one order of magnitude smaller than Γf(T).In order to clarify the role of memory effects we have carried out the calculations of fission width within the Markovian and non-Markovian Langevin approaches. It is shown that the magnitude of memory effects depends very much on the value of the damping of collective motion parameter η. In the low damping region, the fission width Γfgrows as a function of η and decreases as a function of the relaxation time τ. In the high damping region, the tendency is opposite. The fission width Γfdecreases as a function of η and increases as a function of τ. Such dependence is common both for small and large excitations. For the excitation energies above 10 MeV the memory effects are negligibly small and the Markovian limit is quite accurate.