Restart Expedites Quantum Walk Hitting Times.

Classical first-passage times under restart are used in a wide variety of models, yet the quantum version of the problem still misses key concepts. We study the quantum hitting time with restart using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e., cases where the particle evades detection, while maintaining the ballistic propagation which is important for a fast search. We find profound effects of quantum oscillations on the restart problem, namely, a type of instability of the mean detection time, and optimal restart times that form staircases, with sudden drops as the rate of sampling is modified. In the absence of restart and in the Zeno limit, the detection of the walker is not possible, and we examine how restart overcomes this well-known problem, showing that the optimal restart time becomes insensitive to the sampling period.

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model.

We study a two state "jumping diffusivity" model for a Brownian process alternating between two different diffusion constants, D+>D-, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D-⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks.

It is recognized now that a variety of real-life phenomena ranging from diffusion of cold atoms to the motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.

Aging Wiener-Khinchin Theorem.

The Wiener-Khinchin theorem shows how the power spectrum of a stationary random signal I(t) is related to its correlation function ⟨I(t)I(t+τ)⟩. We consider nonstationary processes with the widely observed aging correlation function ⟨I(t)I(t+τ)⟩∼t(γ)ϕ(EA)(τ/t) and relate it to the sample spectrum. We formulate two aging Wiener-Khinchin theorems relating the power spectrum to the time- and ensemble-averaged correlation functions, discussing briefly the advantages of each. When the scaling function ϕ(EA)(x) exhibits a nonanalytical behavior in the vicinity of its small argument we obtain the aging 1/f-type of spectrum. We demonstrate our results with three examples: blinking quantum dots, single-file diffusion, and Brownian motion in a logarithmic potential, showing that our approach is valid for a wide range of physical mechanisms.

Aging and nonergodicity beyond the Khinchin theorem.

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.

Ergodic properties of Brownian motion under stochastic resetting.

We study the ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin- or fat-tailed distributions the normalized or non-normalized invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equation describing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this nonequilibrium process which should hold far beyond the case of Brownian motion analyzed here.

Infinite invariant density determines statistics of time averages for weak chaos.

Weakly chaotic nonlinear maps with marginal fixed points have an infinite invariant measure. Time averages of integrable and nonintegrable observables remain random even in the long time limit. Temporal averages of integrable observables are described by the Aaronson-Darling-Kac theorem. We find the distribution of time averages of nonintegrable observables, for example, the time average position of the particle, x[over ¯]. We show how this distribution is related to the infinite invariant density. We establish four identities between amplitude ratios controlling the statistics of the problem.

Measurement-induced quantum walks.

We investigate a tight-binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured "trajectory," and a combination of classical and quantum mechanical properties for the walk are observed. We explore the effects of the measurements on the spreading of the packet on a one-dimensional line, showing that except for the Zeno limit, the system converges to Gaussian statistics similarly to a classical random walk. A large deviation analysis and an Edgeworth expansion yield quantum corrections to this normal behavior. We then explore the first passage time to a target state using a generating function method, yielding properties like the quantization of the mean first return time. In particular, we study the effects of certain sampling rates that cause remarkable changes in the behavior in the system, such as divergence of the mean detection time in finite systems and decomposition of the phase space into mutually exclusive regions, an effect that mimics ergodicity breaking, whose origin here is the destructive interference in quantum mechanics. For a quantum walk on a line, we show that in our system the first detection probability decays classically like (time)^{-3/2}. This is dramatically different compared to local measurements, which yield a decay rate of (time)^{-3}, indicating that the exponents of the first passage time depend on the type of measurements used.

Time transformation for random walks in the quenched trap model.

We investigate subdiffusion in the quenched trap model by mapping the problem onto a new stochastic process: Brownian motion stopped at the operational time S(α) = ∑(x=-∞)(∞) (n(x))(α) where n(x) is the visitation number at site x and α is a measure of the disorder. In the limit of zero temperature we recover the renormalization group solution found by Monthus. Our approach is an alternative to the renormalization group and is capable of dealing with any disorder strength.

Residence time statistics for N renewal processes.

We present a study of residence time statistics for N renewal processes with a long tailed distribution of the waiting time. Such processes describe many nonequilibrium systems ranging from the intensity of N blinking quantum dots to the residence time of N Brownian particles. With numerical simulations and exact calculations, we show sharp transitions for a critical number of degrees of freedom N. In contrast to the expectation, the fluctuations in the limit of N→∞ are nontrivial. We briefly discuss how our approach can be used to detect nonergodic kinetics from the measurements of many blinking chromophores, without the need to reach the single molecule limit.

Fluctuations of time averages for Langevin dynamics in a binding force field.

We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean (eq) for overdamped brownian motion in a binding potential U(x). Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large nonergodic fluctuations and show how they are related to a superaging correlation function.

Hitchhiker model for Laplace diffusion processes.

Brownian motion is a Gaussian process describing normal diffusion with a variance increasing linearly with time. Recently, intracellular single-molecule tracking experiments have recorded exponentially decaying propagators, a phenomenon called Laplace diffusion. Inspired by these developments we study a many-body approach, called the Hitchhiker model, providing a microscopic description of the widely observed behavior. Our model explains how Laplace diffusion is controlled by size fluctuations of single molecules, independently of the diffusion law which they follow. By means of numerical simulations Laplace diffusion is recovered and we show how single-molecule tracking and data analysis, in a many-body system, is highly nontrivial as tracking of a single particle or many in parallel yields vastly different estimates for the diffusivity. We quantify the differences between these two commonly used approaches, showing how the single-molecule estimate of diffusivity is larger if compared to the full tagging method.

Infinite ergodic theory for heterogeneous diffusion processes.

We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as D(x)∼|x-x[over ̃]|^{2-2/α} in the vicinity of a point x[over ̃], where α can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation, Itô, Stratonovich, or Hänggi-Klimontovich, so the existence of an infinite density and the density's shape are both related to the considered interpretation and the structure of D(x).

Probing dynamics of single molecules: Nonlinear spectroscopy approach.

A two level model of a single molecule undergoing spectral diffusion dynamics and interacting with a sequence of two short laser pulses is investigated. Analytical solution for the probability of n=0,1,2 photon emission events for the telegraph and Gaussian processes is obtained. We examine under what circumstances the photon statistics emerging from such pump-probe setup provides new information on the stochastic process parameters and what are the measurement limitations of this technique. The impulsive and selective limits, the semiclassical approximation, and the fast modulation limit exhibit general behaviors of this new type of spectroscopy. We show that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful results, the information on the photon statistics is contained in the molecule's dipole correlation function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained within the selective limit depends on the both spectral shifts and rates and exhibits oscillations, which are not found in the corresponding line shape.

Fractional Langevin equation: overdamped, underdamped, and critical behaviors.

The dynamical phase diagram of the fractional Langevin equation is investigated for a harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) alpha_{c}=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase, (ii) alpha_{R}=0.441... marks a transition to a resonance phase when an external oscillating field drives the system, and (iii) alpha_{chi_{1}}=0.527... and (iv) alpha_{chi_{2}}=0.707... mark transitions to a double-peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over and underdamped regimes, with or without resonances, show behaviors different from normal.

Strong correlations between fluctuations and response in aging transport.

The correlations between the response of a random walker to an external driving field F switched on at time t_{a}, with the particle's fluctuations in the aging period (0,t_{a}) are investigated. Using the continuous time random walk and the quenched trap model, it is shown that these correlations remain finite even in the asymptotic limit t_{a}-->infinity. Linear response theory gives a relation between the correlations, the fractional diffusion coefficient, and the field F , thus generalizing the Einstein relation. In systems which exhibit aging, fluctuations in the aging period can be used to statistically predict the nonidentical response of particles to an external field.

Conditional 1/f

We demonstrate that the measurement of 1/f^{α} noise at the single molecule or nano-object limit is remarkably distinct from the macroscopic measurement over a large sample. The single-particle measurements yield a conditional time-dependent spectrum. However, the number of units fluctuating on the time scale of the experiment is increasing in such a way that the macroscopic measurements appear perfectly stationary. The single-particle power spectrum is a conditional spectrum, in the sense that we must make a distinction between idler and nonidler units on the time scale of the experiment. We demonstrate our results based on stochastic and deterministic models, in particular the well-known approach of superimposed Lorentzians, the blinking quantum dot model, and deterministic dynamics generated by a nonlinear mapping. Our results show that the 1/f^{α} spectrum is inherently nonstationary even if the macroscopic measurement completely obscures the underlying time dependence of the phenomena.

Quantum walks: The first detected passage time problem.

Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.

Erratum: Aging Wiener-Khinchin theorem and critical exponents of 1/f

This corrects the article DOI: 10.1103/PhysRevE.94.052130.

Photon counting statistics for blinking CdSe-ZnS quantum dots: a Lévy walk process.

We analyze photon statistics of blinking CdSe-ZnS nanocrystals interacting with a continuous wave laser field, showing that the process is described by a ballistic Lévy walk. In particular, we show that Mandel's Q parameter, describing the fluctuations of the photon counts, is increasing with time even in the limit of long time. This behavior is in agreement with the theory of Silbey and co-workers (Jung et al. Chem. Phys. 2002, 284, 181), and in contrast to all existing examples where Q approaches a constant, independent of time in the long time limit. We then analyze the distribution of the time averaged intensities, showing that they exhibit a nonergodic behavior, namely, the time averages remain random even in the limit of a long measurement time. In particular, the distribution of occupation times in the on-state compares favorably to a theory of weak ergodicity breaking of blinking nanocrystals. We show how our data analysis yields information on the amplitudes of power-law decaying on and off time distributions, information not available using standard data analysis of on and off time histograms. Photon statistics reveals fluctuations in the intensity of the bright state indicating that it is composed of several states. Photon statistics exhibits a Lévy walk behavior also when an ensemble of 100 dots is investigated, indicating that the strange kinetics can be observed already at the level of small ensembles.