arXiv subjects

Eli Barkai

Publications and source records attributed to Eli Barkai.

At least 37 records · Page 2Linked to original sources

arXiv2025

Restart uncertainty relation for monitored quantum dynamics

We introduce a novel time-energy uncertainty relation 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, which represents the time taken to return to the initial state, is quantized as an integer multiple of the sampling time, displaying pointwise discontinuous transitions at resonances. Our findings demonstrate that, the natural utilization of the restart mechanism in laboratory experiments, driven by finite data collection time spans, leads to a broadening effect on the transitions of the mean recurrence time. Our newly proposed uncertainty relation captures the underlying essence of these phenomena, by connecting the broadening of the mean hitting time near resonances, to the intrinsic energies of the quantum system and to the fluctuations of recurrence time. This work not only contributes to our understanding of fundamental aspects related to quantum measurements and dynamics, but also offers practical insights for the design of efficient quantum algorithms with mid-circuit measurements.

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arXiv2024

A tale of three approaches: dynamical phase transitions for weakly bound Brownian particles

We investigate a system of Brownian particles weakly bound by attractive parity-symmetric potentials that grow at large distances as $V(x) \sim |x|^α$, with $0 < α< 1$. 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 exponential, as is usual for a confining system with a well-defined equilibrium, but instead follows a stretched exponential $e^{- \mathrm{const} \, t^ν}$ with exponent $ν=α/(2+α)$. This problem is studied from three perspectives. First, we propose a straightforward and general scaling rate-function solution for $P(x,t)$. This rate-function, which is an important tool from large deviation theory, also displays anomalous time scaling and a dynamical phase transition. Second, through the eigenfunctions of the Fokker-Planck operator, we obtain, using the WKB method, more complete solutions that reproduce the rate function approach. Finally, we show how the alternative path-integral formalism allows us to recover the same results, with the above rate-function being the solution of the classical Hamilton-Jacobi equation describing the most probable path. Properties such as parity, the role of initial conditions, and the dynamical phase transition are thoroughly studied in all three approaches.

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arXiv2024

Instability in the quantum restart problem

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 change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus $τ$, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time, is controlled not only by the restart time, as in classical problems, but also by the sampling time $τ$. We provide numerical evidence that this global minimum occurs for the $τ$ minimizing the mean hitting time, given restarts taking place after each measurement. Last but not least, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time $τ$.

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arXiv2024

First Hitting Times on a Quantum Computer: Tracking vs. Local Monitoring, Topological Effects, and Dark States

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 is recorded using unitary dynamics interspersed stroboscopically by measurements, which is implemented on IBM quantum computers with a midcircuit readout option. Unlike classical hitting times, the statistical aspect of the problem depends on the way we construct the measured path, an effect that we quantify experimentally. First, we experimentally verify the theoretical prediction that the mean return time to a target state is quantized, with abrupt discontinuities found for specific sampling times and other control parameters, which has a well-known topological interpretation. Second, depending on the initial state, system parameters, and measurement protocol, the detection probability can be less than one or even zero, which is related to dark-state physics. Both, return-time quantization and the appearance of the dark states are related to degeneracies in the eigenvalues of the unitary time evolution operator. We conclude that, for the IBM quantum computer under study, the first hitting times of monitored quantum walks are resilient to noise. Yet, a finite number of measurements leads to broadening effects, which modify the topological quantization and chiral effects of the asymptotic theory with an infinite number of measurements. Our results point the way for the development of novel quantum walk algorithms that exploit measurement-induced effects on quantum computers.

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arXiv2024

Laplace's first law of errors applied to diffusive motion

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 cases when the jump length distribution is super-exponential, e.g., a Gaussian, we use large deviations theory and relate it to the appearance of exponential tails. When the jump length distribution is sub-exponential the packet of spreading particles is described by the big jump principle. We demonstrate the applicability of our approach for finite time, indicating that rare events and the asymptotics of the large deviations rate function can be sampled for large length scales within a reasonably short measurement time.

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arXiv2024

Stretched-exponential relaxation in weakly-confined Brownian systems through large deviation theory

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 a $r^{2/3}$ potential. Here, we study a Brownian particle under the influence of a general confining, albeit weak, potential field that grows with distance as a sub-linear power law. We find that for this memoryless model, observables display stretched-exponential relaxation. The probability density function of the system is studied using a rate function ansatz. We obtain analytically the stretched-exponential exponent along with an anomalous power-law scaling of length with time. The rate function exhibits a point of nonanalyticity, indicating a dynamical phase transition. In particular, the rate function is double-valued both to the left and right of this point, leading to four different rate functions, depending on the choice of initial conditions and symmetry.

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arXiv2023

Controls that expedite first passage times in disordered systems

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 employ the big jump principle that shows the dominance of the maximum trapping time on the first passage time. Inspired by the restart paradigm, we demonstrate that the removal of this maximum significantly expedites transport. As the disorder increases, the system enters a phase where the removal shows a dramatic effect. Our results show how we may speed up transport in strongly disordered systems exploiting scale invariance. In contrast to the disordered systems studied here, the removal principle has essentially no effect in homogeneous systems; this indicates that improving the conductance of a poorly conducting system is, theoretically, relatively easy as compared to a homogeneous system.

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arXiv2023

Fractional Advection Diffusion Asymmetry Equation, derivation, solution and application

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 Lévy flights, describes the spreading of a packet of particles. Since we use Gaussian statistics for jump lengths though fat-tailed distribution of waiting times, the appearance of fractional space derivatives in the kinetic equation demands explanations provided in this manuscript. As applications we analyse the spreading of tracers in two dimensions, breakthrough curves investigated in the field of contamination spreading in hydrology and first passage time statistics. We present a subordination scheme valid for the case when the mean waiting time is finite and the variance diverges, which is related to Lévy statistics for the number of renewals in the process.

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arXiv2023

Statistics of Long-Range Force Fields in Random Environments: Beyond Holtsmark

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 fat-tailed distributed, and in general are described by stable Lévy distributions. With this widely used framework, the variance of the field diverges, which is non-physical, due to finite size cutoffs. We find a complementary statistical law to the Lévy-Holtsmark distribution describing the large fields in the problem, which is related to the finite size of the tracer particle. We discover bi-scaling, with a sharp statistical transition of the force moments taking place when the order of the moment is $d/δ$, where $d$ is the dimension. The high-order moments, including the variance, are described by the framework presented in this paper, which is expected to hold for many systems. The new scaling solution found here is non-normalized similar to infinite invariant densities found in dynamical systems.

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arXiv2023

Ergodic properties of Brownian motion under stochastic resetting

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 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 non-equilibrium process which should hold far beyond the case of Brownian motion analyzed here.

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arXiv2023

Non-normalizable quasi-equilibrium states under fractional dynamics

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 the system remain almost constant in time. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate the ensemble averages of observables. We obtain analytical estimates of the duration of NNQE, depending on the fractional order, from approximate theoretical solutions of the FTFPE. We study and compare two types of observables, the mean square displacement typically used to characterize diffusion, and the thermodynamic energy. We show that the typical time scales for stagnation depend exponentially on the activation energy in units of temperature multiplied by a function of the fractional exponent.

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arXiv2023

Brownian particles in periodic potentials: coarse-graining versus fine structure

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, converges to a normal distribution centered at the origin with a mean-squared displacement that grows as $2 D^* t$, with an effective diffusion constant that is smaller than that of a freely diffusing particle. We examine the interplay between this coarse-grained description and the fine structure of the density, which is given by the Boltzmann-Gibbs (BG) factor $e^{-V(x)/k_B T}$, the latter being non-normalizable. We explain this result and construct a theory of observables using the Fokker-Planck equation. These observables are classified as those that are related to the BG fine structure, like the energy or occupation times, while others, like the positional moments, for long times, converge to those of the large-scale description. Entropy falls into a special category as it has a coarse-grained and a fine structure description. The basic thermodynamic formula $F=TS - E$ is extended to this far-from-equilibrium system. The ergodic properties are also studied using tools from infinite ergodic theory.

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arXiv2023

Anomalous statistics of laser-cooled atoms in dissipative optical lattices

Diffusion occurs in numerous physical systems throughout nature, drawing its generality from the universality of the central limit theorem. Around 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. Due to a unique feature of its momentum-dependant dissipative friction force, laser-cooled atomic ensembles can be used as a test bed for such dynamics. The interplay between laser cooling and anomalous dynamics bears deep, predictive implications for fundamental concepts in both equilibrium and non-equilibrium statistical physics. The high degree of control available in cold-atom experiments allows for tuning of the parameters of the friction force, revealing transitions in the dynamical properties of the system. Rare events, in both the momentum and spatial distributions, are described by non-normalized states using tools adapted from infinite ergodic theory. This leads to new experimental and theoretical results, illuminating the various features of the system.

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arXiv2022

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 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.

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arXiv2022

Designing exceptional-point-based graphs yielding topologically guaranteed quantum search

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 eigenvalues of the non-Hermitian survival operator, describing the mixed effects of unitary dynamics and the back-action of measurement, coalesce to zero, corresponding to an exceptional point whose degree is the size of the system. Generally, the resulting search is guaranteed to succeed in a bounded time for any initial condition, which is faster than classical random walks or quantum walks on typical graphs. We then show how this efficient quantum search is related to a quantized topological winding number and further discuss the connection of the problem to an effective massless Dirac particle.

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arXiv2022

Exponential Tails and Asymmetry Relations for the Spread of Biased Random Walks

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 is applied? Using the ubiquitous continuous time random walk with bias, and the Crooks relation in conjunction with large deviations theory, we derive two properties of the positional probability density function $P_F(x,t)$ that hold for a wide spectrum of random walk models: (I) Universal asymmetric exponential decay of $P_F(X,t)$ for large $|X|$, and (II) Existence of a time transformation that for large $|X|$ allows to express $P_F(X,t)$ in terms of the propagator of the unbiased process (measured at a shorter time). These findings allow us to establish how the symmetric exponential-like tails, measured in many unbiased processes, will transform into asymmetric Laplace tails when an external force is applied.

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arXiv2022

Discrete Sampling of Extreme Events Modifies Their Statistics

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 Langevin dynamics for a particle in a potential field. For potentials growing at infinity faster than linearly and for long measurement times, we find that the EV distribution of the discretely sampled process diverges from that of the full continuous dataset and converges to that of independent and identically distributed random variables drawn from the process's equilibrium measure. However, for processes with sublinear potentials, the long-time limit is the EV statistics of the continuously sampled data. We treat processes whose equilibrium measures belong to the three EV attractors: Gumbel, Fréchet, and Weibull. Our work shows that the EV statistics can be extremely sensitive to the sampling rate of the data.

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arXiv2022

Local equilibrium properties of ultraslow diffusion in the Sinai model

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 finite system of growing size. This is due to time scale separation: Inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top $10^{17}$ steps and thereby also study finite-time crossover phenomena.

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