non-Markovian Limit Research Articles

Geometrical parameter dependency of non-Markovian and quantum speed limit measures in the spontaneous emission dynamics of a qubit near a plasmonic nanodisk

Geometrical parameter dependency of non-Markovian and quantum speed limit measures in the spontaneous emission dynamics of a qubit near a plasmonic nanodisk

Spontaneous emission of a quantum emitter near a graphene nanodisk under strong light-matter coupling

We investigate the dynamical evolution of the spontaneous emission of a two-level quantum emitter near a graphene nanodisk. We employ the macroscopic quantum electrodynamics methodology to study the reversible population dynamics of the excited state of the quantum emitter in the non-Markovian limit. Our results indicate that the quantum-emitter--nanodisk interaction enters the strong light-matter coupling regime, as manifested in the excited-state probability where Rabi oscillations and population trapping effects are observed. The level of non-Markovianity is estimated by computing three different well-established measures and relate them to the quantum speedup due to the strong-coupling interaction of the emitter with the graphene nanodisk. Importantly, high values of the non-Markovianity and quantum speedup measures are achieved when the emitter-nanodisk interaction is strong, whereas decreasing coupling strength leads to smaller values.

The Markov-quantile process attached to a family of marginals

Let µ = (µt)t∈R be any 1-parameter family of probability measures on R. Its quantile process (Gt)t∈R : ]0, 1[ → RR, given by Gt(α) = inf{x ∈ R : µt(]−∞, x]) > α}, is not Markov in general. We modify it to build the Markov process we call “Markov-quantile”. We first describe the discrete analogue: if (µn)n∈Z is a family of probability measures on R, a Markov process Y = (Yn)n∈Z such that Law(Yn) = µn is given by the data of its couplings from n to n + 1, i.e. Law((Yn, Yn+1)), and the process Y is the inhomogeneous Markov chain having those couplings as transitions. Therefore, there is a canonical Markov process with marginals µn and as similar as possible to the quantile process: the chain whose transitions are the quantile couplings. We show that an analogous process exists for a continuous parameter t: there is a unique Markov process X with the measures µt as marginals, and being a limit for the finite dimensional topology of quantile processes where the past is made independent of the future at finitely many times (many non-Markovian limits exist in general). The striking fact is that the construction requires no regularity for the family µ. We rely on order arguments, which seems to be completely new for the purpose. We also prove new results the Markov-quantile process yields in two contemporary frameworks: – In case µ is increasing for the stochastic order, X has increasing trajectories. This is an analogue of a result of Kellerer dealing with the convex order, peacocks and martingales. Modifiying Kellerer’s proof, we also prove simultaneously his result and ours in this case. – If µ is absolutely continuous in Wasserstein space P2(R) then X is solution of a Benamou–Brenier transport problem with marginals µt. It provides a Markov probabilistic representation of the continuity equation, unique in a certain sense.

Accurate Truncations of Chain Mapping Models for Open Quantum Systems.

The dynamics of open quantum systems are of great interest in many research fields, such as for the interaction of a quantum emitter with the electromagnetic modes of a nanophotonic structure. A powerful approach for treating such setups in the non-Markovian limit is given by the chain mapping where an arbitrary environment can be transformed to a chain of modes with only nearest-neighbor coupling. However, when long propagation times are desired, the required long chain lengths limit the utility of this approach. We study various approaches for truncating the chains at manageable lengths while still preserving an accurate description of the dynamics. We achieve this by introducing losses to the chain modes in such a way that the effective environment acting on the system remains unchanged, using a number of different strategies. Furthermore, we demonstrate that extending the chain mapping to allow next-nearest neighbor coupling permits the reproduction of an arbitrary environment, and adding longer-range interactions does not further increase the effective number of degrees of freedom in the environment.

Diffusion approximations in the online increasing subsequence problem

The online increasing subsequence problem is a stochastic optimisation task with the objective to maximise the expected length of subsequence chosen from a random series by means of a nonanticipating decision strategy. We study the structure of optimal and near-optimal subsequences in a standardised planar Poisson framework. Following a long-standing suggestion by Bruss and Delbaen (2004), we prove a joint functional limit theorem for the transversal fluctuations about the diagonal of the running maximum and the length processes. The limit is identified explicitly with a Gaussian time-inhomogeneous diffusion. In particular, the running maximum converges to a Brownian bridge, and the length process has another explicit non-Markovian limit.

Constructing many-body dissipative particle dynamics models of fluids from bottom-up coarse-graining

Since their emergence in the 1990s, mesoscopic models of fluids have been widely used to study complex organization and transport phenomena beyond the molecular scale. Even though these models are designed based on results from physics at the meso- and macroscale, such as fluid mechanics and statistical field theory, the underlying microscopic foundation of these models is not as well defined. This paper aims to build such a systematic connection using bottom-up coarse-graining methods. From the recently developed dynamic coarse-graining scheme, we introduce a statistical inference framework of explicit many-body conservative interaction that quantitatively recapitulates the mesoscopic structure of the underlying fluid. To further consider the dissipative and fluctuation forces, we design a novel algorithm that parameterizes these forces. By utilizing this algorithm, we derive pairwise decomposable friction kernels under both non-Markovian and Markovian limits where both short- and long-time features of the coarse-grained dynamics are reproduced. Finally, through these new developments, the many-body dissipative particle dynamics type of equations of motion are successfully derived. The methodologies developed in this work thus open a new avenue for the construction of direct bottom-up mesoscopic models that naturally bridge the meso- and macroscopic physics.

Quantum Coherence and Its Signatures in Extended Quantum Systems.

Quantum coherence controls almost every aspect of static and dynamic response of a strongly correlated quantum system, although details of this control have not been elucidated in many cases. We find that, in the presence of a significant static off-diagonal coupling, quantum coherence can survive much longer than the bath correlation time in a noisy environment. In the presence of time correlated noise (non-Markovian limit), coherence could propagate through the excited bath states, and this plays a nontrivial role in giving rise to a noncanonical temperature dependence of population distribution in an extended conjugated polymer-like system. The quantum coherence through the excited bath states vanishes in the high temperature limit, giving rise to the equilibrium Boltzmann distribution. Second, we discuss a role of quantum coherence in exciton localization that bears resemblance to Anderson localization. Calculations have been carried out not only with a chain of conjugated polymer but also with dimer and trimer subunits of the Fenna-Matthews-Olson (FMO) complex. We derive an analytical expression of the relation between steady state coherence and excitionic population distribution. We analytically showed that steady state coherence in equilibrium bath states is governed by interchromophoric coupling (J) whereas coherence in excited bath states is dictated by fluctuation strength (Vd) for the spatially correlated bath model. For the spatially uncorrelated bath model, we observe that at the low temperature limit coherence in the excited bath states dominates over coherence in equilibrium bath states and vice versa. In the study of the localization problem, we analytically show that, in the limit of a negligibly small fluctuation rate (bd), the diffusion coefficient is exactly proportional to the fluctuation rate, leading to complete breakdown of the Haken-Strobl-Silbey Markovian prediction.

Non-Markovian barrier crossing with two-time-scale memory is dominated by the faster memory component.

We investigate non-Markovian barrier-crossing kinetics of a massive particle in one dimension in the presence of a memory function that is the sum of two exponentials with different memory times, [Formula: see text] and [Formula: see text] . Our Langevin simulations for the special case where both exponentials contribute equally to the total friction show that the barrier-crossing time becomes independent of the longer memory time if at least one of the two memory times is larger than the intrinsic diffusion time. When we associate memory effects with coupled degrees of freedom that are orthogonal to a one-dimensional reaction coordinate, this counterintuitive result shows that the faster orthogonal degrees of freedom dominate barrier-crossing kinetics in the non-Markovian limit and that the slower orthogonal degrees become negligible, quite contrary to the standard time-scale separation assumption and with important consequences for the proper setup of coarse-graining procedures in the non-Markovian case. By asymptotic matching and symmetry arguments, we construct a crossover formula for the barrier crossing time that is valid for general multi-exponential memory kernels. This formula can be used to estimate barrier-crossing times for general memory functions for high friction, i.e. in the overdamped regime, as well as for low friction, i.e. in the inertial regime. Typical examples where our results are important include protein folding in the high-friction limit and chemical reactions such as proton-transfer reactions in the low-friction limit.

Delocalization and Quantum Entanglement in Physical Systems.

Quantum coherence and entanglement in an extended interacting system where energy levels are nondegenerate and coupled to a dissipative environment is a common occurrence in nature, like in photosynthetic reaction systems and conjugated polymers. The temperature dependence of quantum coherence in a trimer complex (first three subunits of the Fenna-Matthews-Olson complex) is studied using a temperature-dependent quantum stochastic Liouville equation. In the non-Markovian limit, the lowering of temperature induces long-lasting quantum coherence that, in turn, leads to delocalization, whose length grows. The entanglement and coherence length determine the nature of the dynamic localization.

Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37–54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded $$G{/}GI{/}n+GI$$ queue with stationary arrivals, nonexponential service times, n identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits.

Concept of quantum timing jitter and non-Markovian limits in single-photon detection

Single atoms form a model system for understanding the limits of single photon detection. Here, we develop a non-Markov theory of single-photon absorption by a two-level atom to place limits on the absorption (transduction) time. We show the existence of a finite rise time in the probability of excitation of the atom during the absorption event which is infinitely fast in previous Markov theories. This rise time is governed by the bandwidth of the atom-field interaction spectrum and leads to a fundamental jitter in time-stamping the absorption event. Our theoretical framework captures both the weak and strong atom-field coupling regimes and sheds light on the spectral matching between the interaction bandwidth and single photon Fock state pulse spectrum. Our work opens questions whether such jitter in the absorption event can be observed in a multi-mode realistic single photon detector. Finally, we also shed light on the fundamental differences between linear and nonlinear detector outputs for single photon Fock state vs. coherent state pulses.

Effects of dynamic disorder on exciton migration: Quantum diffusion, coherences, and energy transfer.

We study excitation transfer and migration in a one-dimensional lattice characterized by dynamic disorder. The diagonal and off-diagonal energy disorders arise from the coupling of system and bath. We consider both same bath (when baths are spatially correlated) and independent bath (when baths are completely uncorrelated) limits. In the latter case, all diagonal and off-diagonal bath coupling elements fluctuate independently of each other and the dynamics is complicated. We obtain time dependent population distribution by solving Kubo's quantum stochastic Liouville equation. In the Markovian limit, both energy transfer dynamics and mean square displacement of the exciton behave the similar way in same and independent bath cases. However, these two baths can give rise to a markedly different behavior in the non-Markovian limit. We note that previously only the same bath case has been studied in the non-Markovian limit. The other main results of our study include the following. (i) For an average, non-zero off-diagonal coupling value J, exciton migration remains coherent in same bath case even at long times while it becomes incoherent in independent bath case in the Markovian limit. (ii) Coherent transfer is manifested in an oscillatory behavior of the energy transfer dynamics accompanied by faster-than diffusive spread of the exciton from the original position. (iii) Agreement with available analytical expression of mean squared displacement is good in Markovian limit for independent bath (off-diagonal fluctuation) case but only qualitative in non-Markovian limit for which no complete analytical solution is available. (iv) We observe transition from coherent to incoherent transport in independent bath (diagonal fluctuation) case when the bath is made progressively more Markovian. We present an analytical study that shows coherence to propagate through excited bath states. (v) The correlation time of the bath plays a unique role in dictating the diffusive spread that is not anticipated in a Markovian treatment.

Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pumping

We report a theoretical study of a quantum optical model consisting of an array of strongly nonlinear cavities incoherently pumped by an ensemble of population-inverted two-level atoms. Projective methods are used to eliminate the atomic dynamics and write a generalized master equation for the photonic degrees of freedom only, where the frequency-dependence of gain introduces non-Markovian features. In the simplest single cavity configuration, this pumping scheme gives novel optical bistability effects and allows for the selective generation of Fock states with a well-defined photon number. For many cavities in a weakly non-Markovian limit, the non-equilibrium steady state recovers a Grand-Canonical statistical ensemble at a temperature determined by the effective atomic linewidth. For a two-cavity system in the strongly nonlinear regime, signatures of a Mott state with one photon per cavity are found. Nous présentons l'étude d'un modèle d'optique quantique consistant en un réseau de cavités fortement non linéaires en présence de pompage incohérent induit par un ensemble d'atomes à deux niveaux, avec inversion de population. Nous appliquons une méthode projective afin d'éliminer les degrés de liberté atomiques, et dérivons une équation maîtresse généralisée contenant uniquement les degrés de liberté photoniques, dans laquelle le pompage dépendant de la fréquence induit des effets non markoviens. Dans le cas simple d'une cavité, cette méthode de pompage induit de nouveaux effets de bistabilité et permet la création d'états de Fock avec un nombre défini de photons. Dans le cas de plusieurs cavités dans un régime faiblement non markovien, l'état stationaire hors équilibre prend la forme d'un ensemble grand-canonique, dont la température effective est définie par la largeur du spectre d'émission des atomes. Dans une configuration à deux cavités, en régime fortement non linéaire, nous observons la signature d'un état de Mott avec un photon par site.

Review of mixing length estimates and effects of toroidicity in a fluid model for turbulent transport in tokamaks

Basic aspects of turbulent transport in toroidal magnetized plasmas are discussed. In particular the fluid closure has strong effects on zonal flows which are needed to create an absorbing boundary for long wave lengths and also to obtain the Dimits nonlinear upshift. The fluid resonance in the energy equation is found to be instrumental for generating the L–H transition, the spin-up of poloidal rotation in internal transport barriers, as well as the nonlinear Dimits upshift. The difference between the linearly fastest growing mode number and the corresponding longer nonlinear correlation length is also addressed. It is found that the Kadomtsev mixing length result is consistent with the non-Markovian diagonal limit of the transport at the nonlinearly obtained correlation length.

Non-equilibrium spin-boson model: Counting statistics and the heat exchange fluctuation theorem

We focus on the non-equilibrium two-bath spin-boson model, a toy model for examining quantum thermal transport in many-body open systems. Describing the dynamics within the noninteracting-blip approximation equations, applicable, e.g., in the strong system-bath coupling limit and/or at high temperatures, we derive expressions for the cumulant generating function in both the Markovian and non-Markovian limits by energy-resolving the quantum master equation of the subsystem. For a Markovian bath, we readily demonstrate the validity of a steady-state heat exchange fluctuation theorem. In the non-Markovian limit a "weaker" symmetry relation generally holds, a general outcome of microreversibility. We discuss the reduction of this symmetry relation to the universal steady-state fluctuation theorem. Using the cumulant generating function, an analytic expression for the heat current is obtained. Our results establish the validity of the steady-state heat exchange fluctuation theorem in quantum systems with strong system-bath interactions. From the practical point of view, this study provides tools for exploring transport characteristics of the two-bath spin-boson model, a prototype for a nonlinear thermal conductor.

Relevance of quantum mechanics on some aspects of ion channel function

Mathematical modeling of ionic diffusion along K ion channels indicates that such diffusion is oscillatory, at the weak non-Markovian limit. This finding leads us to derive a Schrödinger–Langevin equation for this kind of system within the framework of stochastic quantization. The Planck's constant is shown to be relevant to the Lagrangian action at the level of a single ion channel. This sheds new light on the issue of applicability of quantum formalism to ion channel dynamics and to the physical constraints of the selectivity filter. To cite this article: S. Roy, R. Llinás, C. R. Biologies 332 (2009).

Non-Markovian noise mediated through anomalous diffusion within ion channels

It is evident from a wide range of experimental findings that ion channel gating is inherently stochastic. The issue of "memory effects" (diffusional retardation due to local changes in water viscosity) in ionic flow has been recently addressed using Brownian dynamics simulations. The results presented indicate such memory effects are negligible, unless the diffusional barrier is much higher than that of free solute. In this paper using differential stochastic methods we conclude that the Markovian property of exponential dwell times gives rise to a high barrier, resulting in diffusional memory effects that cannot be ignored in determining ionic flow through channels. We have addressed this question using a generalized Langevin equation that contains a combination of Markovian and non-Markovian processes with different time scales. This approach afforded the development of an algorithm that describes an oscillatory ionic diffusional sequence. The resulting oscillatory function behavior, with exponential decay, was obtained at the weak non-Markovian limit with two distinct time scales corresponding to the processes of ionic diffusion and drift. This will be analyzed further in future studies using molecular dynamics simulations. We propose that the rise of time scales and memory effects is related to differences of shear viscosity in the cytoplasm and extracellular matrix.

Population trapping due to cavity losses

In population trapping the occupation of a decaying quantum level keeps a constant nonzero value. We show that an atom-cavity system interacting with an environment characterized by a nonflat spectrum, in the non-Markovian limit, exhibits such a behavior, effectively realizing the preservation of nonclassical states against dissipation. Our results allow us to understand the role of cavity losses in hybrid solid state systems and pave the way to the proper description of leakage in the recently developed cavity quantum electrodynamic systems.

Exponential decay in a spin bath

We show that the coherence of an electron spin interacting with a bath of nuclear spins can exhibit a well-defined purely exponential decay for special (``narrowed'') bath initial conditions in the presence of a strong applied magnetic field. This is in contrast to the typical case, where spin-bath dynamics have been investigated in the non-Markovian limit, giving superexponential or power-law decay of correlation functions. We calculate the relevant decoherence time ${T}_{2}$ explicitly for free-induction decay and find a simple expression with dependence on bath polarization, magnetic field, shape of the electron wave function, dimensionality, total nuclear spin $I$, and isotopic concentration for experimentally relevant heteronuclear-spin systems.

Reexamination of spin decoherence in semiconductor quantum dots from the equation-of-motion approach

The longitudinal and transversal spin decoherence times, ${T}_{1}$ and ${T}_{2}$, in semiconductor quantum dots are investigated from the equation-of-motion approach for different magnetic fields, quantum dot sizes, and temperatures. Various mechanisms, such as the hyperfine interaction with the surrounding nuclei, the Dresselhaus spin-orbit coupling together with the electron--bulk-phonon interaction, the $g$-factor fluctuations, the direct spin-phonon coupling due to the phonon-induced strain, and the coaction of the electron--bulk- and/or surface-phonon interaction together with the hyperfine interaction are included. The relative contributions from these spin decoherence mechanisms are compared in detail. In our calculation, the spin-orbit coupling is included in each mechanism and is shown to have marked effect in most cases. The equation-of-motion approach is applied in studying both the spin relaxation time ${T}_{1}$ and the spin dephasing time ${T}_{2}$, either in Markovian or in non-Markovian limit. When many levels are involved at finite temperature, we demonstrate how to obtain the spin relaxation time from the Fermi golden rule in the limit of weak spin-orbit coupling. However, at high temperature and/or for large spin-orbit coupling, one has to use the equation-of-motion approach when many levels are involved. Moreover, spin dephasing can be much more efficient than spin relaxation at high temperature, though the two only differ by a factor of 2 at low temperature.