Eli Barkai

Bar Ilan University | BIU · Department of Physics

ArXiv preprint arXiv:2407.08899

We investigate a quantum walk on a ring represented by a directed triangle graph with complex edge weights and monitored at a constant rate until the quantum walker is detected. To this end, the first hitting time statistics are recorded using unitary dynamics interspersed stroboscopically by measurements, which are implemented on IBM quantum compu...

We investigate a system of Brownian particles weakly bound by attractive parity-symmetric potentials that grow at large distances as $V(x) \sim |x|^\alpha$, with $0 < \alpha < 1$. The probability density function $P(x,t)$ at long times reaches the Boltzmann-Gibbs equilibrium state, with all moments finite. However, the system's relaxation is not ex...

Investigating the dynamics of chromatin and the factors that are affecting it, has provided valuable insights into the organization and functionality of the genome in the cell nucleus. We control the expression of Lamin-A, an important organizer protein of the chromatin and nucleus structure. By simultaneously tracking tens of chromosomal loci (tel...

Repeatedly monitored quantum walks with a rate 1/𝜏 yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small...

In biological, glassy, and active systems, various tracers exhibit Laplace-like, i.e., exponential, spreading of the diffusing packet of particles. The limitations of the central limit theorem in fully capturing the behaviors of such diffusive processes, especially in the tails, have been studied using the continuous time random walk model. For cas...

In biological, glassy, and active systems, various tracers exhibit Laplace-like, i.e., exponential, spreading of the diffusing packet of particles. The limitations of the central limit theorem in fully capturing the behaviors of such diffusive processes, especially in the tails, have been studied using the continuous time random walk model. For cas...

Stretched-exponential relaxation is a widely observed phenomenon found in ordered ferromagnets as well as glassy systems. One modeling approach connects this behavior to a droplet dynamics described by an effective Langevin equation for the droplet radius with an r2/3 potential. Here, we study a Brownian particle under the influence of a general co...

Recurrence time quantifies the duration required for a physical system to return to its initial state, playing a pivotal role in understanding the predictability of complex systems. In quantum systems with subspace measurements, recurrence times are governed by Anandan-Aharonov phases, yielding fractionally quantized recurrence times. However, the...

The non-Markovian continuous-time random walk model, featuring fat-tailed waiting times and narrow distributed displacements with a non-zero mean, is a well studied model for anomalous diffusion. Using an analytical approach, we recently demonstrated how a fractional space advection diffusion asymmetry equation, usually associated with Markovian L{...

We introduce a novel time-energy uncertainty relationship within the context of restarts in monitored quantum dynamics. Initially, we investigate the concept of "first hitting time" in quantum systems using an IBM quantum computer and a three-site ring graph as our starting point. Previous studies have established that the mean recurrence time, whi...

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 th...

Since the times of Holtsmark (1911), statistics of fields in random environments have been widely studied, for example in astrophysics, active matter, and line-shape broadening. The power-law decay of the two-body interaction of the form 1/|r|δ, and assuming spatial uniformity of the medium particles exerting the forces, imply that the fields are f...

First-passage time statistics in disordered systems exhibiting scale invariance are studied widely. In particular, long trapping times in energy or entropic traps are fat-tailed distributed, which slow the overall transport process. We study the statistical properties of the first-passage time of biased processes in different models, and we employ...

Diffusion occurs in numerous physical systems throughout nature, drawing its generality from the universality of the central limit theorem. Approximately a century ago it was realized that an extension to this type of dynamics can be obtained in the form of “anomalous” diffusion, where distributions are allowed to have heavy power-law tails. Owing...

Particles anomalously diffusing in contact with a thermal bath are initially released from an asymptotically flat potential well. For temperatures that are sufficiently low compared to the potential depth, the dynamical and thermodynamical observables of the system remain almost constant for long times. We show how these stagnated states are charac...

Since the times of Holtsmark (1911), statistics of fields in random environments have been widely studied, for example in astrophysics, active matter, and line-shape broadening. The power-law decay of the two-body interaction, of the form $1/|r|^\delta$, and assuming spatial uniformity of the medium particles exerting the forces, imply that the fie...

We study ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin/fat tailed distributions, the normalized/non-normalised invariant density of this process. The former case corresponds to known results in the resetting literature and the latter t...

Quantum walks underlie an important class of quantum computing algorithms, and represent promising approaches in various simulations and practical applications. Here we design stroboscopically monitored quantum walks and their subsequent graphs that can naturally boost target searches. We show how to construct walks with the property that all the e...

We study non-normalizable quasi-equilibrium states (NNQE) arising from anomalous diffusion. Initially, particles in contact with a thermal bath are released from an asymptotically flat potential well, with dynamics that is described by fractional calculus. For temperatures that are sufficiently low compared to the potential depth, the properties of...

We study the motion of an overdamped particle connected to a thermal heat bath in the presence of an external periodic potential in one dimension. When we coarse-grain, i.e., bin the particle positions using bin sizes that are larger than the periodicity of the potential, the packet of spreading particles, all starting from a common origin, converg...

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 maintain...

We study optimal restart times for the quantum first hitting time problem. Using a monitored one-dimensional lattice quantum walk with restarts, we find an instability absent in the corresponding classical problem. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that...

We study the motion of an overdamped particle connected to a thermal heat bath in the presence of an external periodic potential in one dimension. When we coarse-grain, i.e., bin the particle positions using bin sizes that are larger than the periodicity of the potential, the packet of spreading particles, all starting from a common origin, converg...

Exponential, and not Gaussian, decay of probability density functions was studied by Laplace in the context of his analysis of errors. Such Laplace propagators for the diffusive motion of single particles in disordered media were recently observed in numerous experimental systems. What will happen to this universality when an external driving force...

Extreme value (EV) statistics of correlated systems are widely investigated in many fields, spanning the spectrum from weather forecasting to earthquake prediction. Does the unavoidable discrete sampling of a continuous correlated stochastic process change its EV distribution? We explore this question for correlated random variables modeled via Lan...

Extreme value (EV) statistics of correlated systems are widely investigated in many fields, spanning the spectrum from weather forecasting to earthquake prediction. Does the unavoidable discrete sampling of a continuous correlated stochastic process change its EV distribution? We explore this question for correlated random variables modeled via Lan...

Transport in disordered media, such as those involving charge carriers in amorphous semiconductors, or contaminants in hydrogeological systems, are often described by time scale-free processes. We study the statistical properties of the first passage time of biased processes in different models, and employ the big jump principle that shows the domi...

We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a fi...

We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a fi...

We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models...

We consider the effect of periodic conditional no-click measurements on a quantum system. What is the final state of such a driven system? When the system has some symmetry built into it, the final state is a dark state provided that the initial state overlaps with this nondetectable fragment of the Hilbert space. We find two classes of such states...

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 first detected passage time under restart protocol using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e. cases where the particle is not d...

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, sho...

The virial theorem, and the equipartition theorem in the case of quadratic degrees of freedom, are handy constraints on the statistics of equilibrium systems. Their violation is instrumental in determining how far from equilibrium a driven system might be. We extend the virial theorem to nonequilibrium conditions for Langevin dynamics with nonlinea...

We design monitored quantum walks with the aim of optimizing state transfer and target search. We show how to construct walks with the property that all the eigenvalues of the non-Hermitian survival operator, describing the mixed effect of unitary dynamics and the back-action of measurement, coalesce to zero, corresponding to an exceptional point w...

We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models...

Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion...

It is well known that for sub-recoiled laser cooled atoms L\'evy statistics and deviations from usual ergodic behaviour come into play.Here we show how tools from infinite ergodic theory describe the cool gas.Specifically, we derive the scaling function and the infinite invariant density of a stochastic model for the momentum of the atoms using two...

The virial theorem, and the equipartition theorem in the case of quadratic degrees of freedom, are handy constraints on the statistics of equilibrium systems. Their violation is instrumental in determining how far from equilibrium a driven system might be. We extend the virial theorem to nonequilibrium conditions for Langevin dynamics with nonlinea...

Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion...

The velocity distribution of a classical gas of atoms in thermal equilibrium is the normal Maxwell distribution. It is well known that for sub-recoiled laser cooled atoms L\'evy statistics and deviations from usual ergodic behaviour come into play. Here we show how tools from infinite ergodic theory describe the cool gas. Specifically, we derive th...

The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and they are solely dominated by the largest summand called the big jump. Recently, the principle was introduced into...

With subrecoil-laser-cooled atoms, one may reach nanokelvin temperatures while the ergodic properties of these systems do not follow usual statistical laws. Instead, due to an ingenious trapping mechanism in momentum space, power-law-distributed sojourn times are found for the cooled particles. Here, we show how this gives rise to a statistical-mec...

Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The related discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements (MNM) until the first...

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, sho...

We consider the extreme value statistics of correlated random variables that arise from a Langevin equation. Recently, it was shown that the extreme values of the Ornstein-Uhlenbeck process follow a different distribution than those originating from its equilibrium measure, composed of independent and identically distributed Gaussian random variabl...

In this Colloquium we discuss the anomalous kinetics of atoms in dissipative optical lattices, focusing on the ``Sisyphus" laser cooling mechanism. The cooling scheme induces a friction force that decreases to zero for high atomic momentum, which in turn leads to unusual statistical features. We study, using a Fokker-Planck equation describing the...

We consider the extreme value statistics of N independent and identically distributed random variables, which is a classic problem in probability theory. When N → ∞, fluctuations around the maximum of the variables are described by the Fisher–Tippett–Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiti...

We investigate extreme value theory for physical systems with a global conservation law which describe renewal processes, mass transport models and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a non-analytical point in the middle of the support. We expose exact rel...

The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and they are solely dominated by the largest summand called the big jump. Recently, the principle was introduced into...

A classical random walker starting on a node of a finite graph will always reach any other node since the search is ergodic, namely it fully explores space, hence the arrival probability is unity. For quantum walks, destructive interference may induce effectively non-ergodic features in such search processes. Under repeated projective local measure...

We investigate the effect of conditional null measurements on a quantum system and find a rich variety of behaviors. Specifically, quantum dynamics with a time independent $H$ in a finite dimensional Hilbert space are considered with repeated strong null measurements of a specified state. We discuss four generic behaviors that emerge in these monit...

With subrecoil-laser-cooled atoms one may reach nano-Kelvin temperatures while the ergodic properties of these systems do not follow usual statistical laws. Instead, due to an ingenious trapping mechanism in momentum space, power-law-distributed sojourn times are found for the cooled particles. Here, we show how this gives rise to a statistical-mec...

A classical random walker starting on a node of a finite graph will always reach any other node since the search is ergodic, namely it is fully exploring space, hence the arrival probability is unity. For quantum walks, destructive interference may induce effectively non-ergodic features in such search processes. Under repeated projective local mea...

Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion...

We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the nth attempt. We calculate as well the mean and mean square, both of the...

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 equilib...

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann–Gibbs...

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep compared to temperature, physical observables like the mean square displacement are essentially time-independent over a long time interval, the stagnation epoch. However the standard Boltzmann-Gibbs (BG) distr...

Randomly repeated measurements during the evolution of a closed quantum system create a sequence of probabilities for the first detection of a certain quantum state. The resulting discrete monitored evolution for the return of the quantum system to its initial state is investigated. We found that the mean number of measurements until the first dete...

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 e...

Fractional kinetic equations employ noninteger calculus to model anomalous relaxation and diffusion in many systems. While this approach is well explored, it so far failed to describe an important class of transport in disordered systems. Motivated by work on contaminant spreading in geological formations, we propose and investigate a fractional ad...

We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n$th attempt. We calculate as well the mean and mean square both of th...

We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a "light"cone -v0t<x<v0t. In particular we study this density close to its maximum in the vicinity of the light cone. The spreading de...

We investigate extreme value theory for physical systems with a global conservation law which describes renewal processes, mass transport models, and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support. We expose exact re...

We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker is detected. This is a quantum version of the first-passage problem. We focus on the total probability Pdet that the particle is eventually detected in some target state, for example, on a node rd on a graph, after an arbitrary number of de...

We consider an overdamped Brownian particle subject to an asymptotically flat potential with a trap of depth U0 around the origin. When the temperature is small compared to the trap depth (ξ=kBT/U0≪1), there exists a range of timescales over which physical observables remain practically constant. This range can be very long, of the order of the Arr...

We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is...

We show that the mean time, which a quantum particle needs to escape from a system to the environment, is quantized and independent from most dynamical details of the system. In particular, we consider a quantum system with a general Hermitian Hamiltonian Ĥ and one decay channel, through which probability dissipates to the environment with rate Γ....

We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate 1/τ. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state |ψin〉 of the walker is orthogonal to the detected st...

We study the ballistic L\'evy walk and obtain the far-tail of the distribution for the walker's position. When the position is of the order of the observation time, its distribution is described by the well-known Lamperti-arcsine law. However this law blows up at the far-tail which is nonphysical, in the sense that any finite time observation will...

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 mod...

We investigate a generic discrete quantum system prepared in state |ψin〉 under repeated detection attempts, aimed to find the particle in state |d〉, for example, a quantum walker on a finite graph searching for a node. For the corresponding classical random walk, the total detection probability Pdet is unity. Due to destructive interference, one ma...

We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory. When $N\to\infty$, small fluctuations around the renormalized maximum of the variables are described by the Fisher-Tippett-Gnedenko theorem, which states that the distribution of maxima converge...

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator...