Search

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 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. However, 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.

Experimental data obtained in single-particle tracking experiments are challenging to interpret. The authors propose an approach for determining the dynamics of the stochastic motion of molecules based on the power spectrum, relevant to various non-stationary scale-free random walks.

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.

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: generic dark states that are found also for nonperiodic measurements, and Floquet dark states that are directly controlled by the periodicity of the measurements and which do not rely on the underlying symmetry of the Hamiltonian. A different behavior is found in the absence of dark manifolds, where for specific periodicities of the measurements we find nontrivial oscillatory dynamics, controlled by the measurement rate. Finally, when the control parameters are tuned, the eigenvalues of the survival operator coalesce to zero, and then one finds exceptional points with a large degeneracy. The physical meaning of this special type of degeneracy is that the null measurement process becomes impossible, implying that detecting the state is guaranteed. We analyze these effects with a nonperturbative method based on a classical charge picture.

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.

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 measurements, made on a target state, the final detection of the system is not guaranteed since the Hilbert space is split into a bright subspace and an orthogonal dark one. Using this we find an uncertainty relation for the deviations of the detection probability from its classical counterpart, in terms of the energy fluctuations.

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 (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker–Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker–Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.

We consider an overdamped Brownian particle subject to an asymptotically flat potential with a trap of depth U_{0} around the origin. When the temperature is small compared to the trap depth (ξ=k_{B}T/U_{0}≪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 Arrhenius factor e^{1/ξ}. For these quasiequilibrium states, the usual Boltzmann-Gibbs recipe does not work since the partition function is divergent due to the flatness of the potential at long distances. However, we show that the standard Boltzmann-Gibbs statistical framework and thermodynamic relations can still be applied through proper regularization. This can be a valuable tool for the analysis of metastability in the nonconfining potential fields that characterize a vast number of systems.

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 P_{det} that the particle is eventually detected in some target state, for example, on a node r_{d} on a graph, after an arbitrary number of detection attempts. Analyzing the dark and bright states for finite graphs and more generally for systems with a discrete spectrum, we provide an explicit formula for P_{det} in terms of the energy eigenstates which is generically τ independent. We find that disorder in the underlying Hamiltonian renders perfect detection, P_{det}=1, and then expose the role of symmetry with respect to suboptimal detection. Specifically, we give a simple upper bound for P_{det} that is controlled by the number of equivalent (with respect to the detection) states in the system. We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya's optimal detection for a classical random walk.

We investigate a generic discrete quantum system prepared in state |ψ_{in}〉 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 P_{det} is unity. Due to destructive interference, one may find initial states |ψ_{in}〉 with P_{det}

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. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

In a growing number of strongly disordered and dense systems, the dynamics of a particle pulled by an external force field exhibits superdiffusion. In the context of glass-forming systems, supercooled glasses, and contamination spreading in porous media, it was suggested that this behavior be modeled with a biased continuous-time random walk. Here we analyze the plume of particles lagging far behind the mean, with the single big jump principle. Revealing the mechanism of the anomaly, we show how a single trapping time, the largest one, is responsible for the rare fluctuations in the system. These nontypical fluctuations still control the behavior of the mean square displacement, which is the most basic quantifier of the dynamics in many experimental setups. We show how the initial conditions, describing either the stationary state or nonequilibrium case, persist forever in the sense that the rare fluctuations are sensitive to the initial preparation. To describe the fluctuations of the largest trapping time, we modify Fréchet's law from extreme value statistics, taking into consideration the fact that the large fluctuations are very different from those observed for independent and identically distributed random variables.

We discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long-time properties, this process exhibits aging: events counted initially in a time interval [0,t] statistically strongly differ from those observed at later times [t_{a},t_{a}+t]. The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous-time random walk. By this, we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking, along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.

The power spectrum of quantum dot (QD) fluorescence exhibits $1/{{f}^{\beta }}$ noise, related to the intermittency of these nanosystems. As in other systems exhibiting $1/f$ noise, this power spectrum is not integrable at low frequencies, which appears to imply infinite total power. We report measurements of individual QDs that address this long-standing paradox. We find that the level of $1/{{f}^{\beta }}$ noise decays with the observation time. The change of the spectrum with time places a bound on the total power. These observations are in stark contrast with most measurements of noise in macroscopic systems which do not exhibit any evidence for non-stationarity. We show that the traditional description of the power spectrum with a single exponent β is incomplete and three additional critical exponents characterize the dependence on experimental time.

The problem of the detection statistics of a quantum walker has received increasing interest. We investigate the effect of employing a moving detector, using a projective measurement approach with fixed sampling time , with the detector moving right before every detection attempt. For a tight-binding quantum walk on the line, the moving detector allows one to target a specific range of group velocities of the walker, qualitatively modifying the behavior of the quantum first-detection probabilities. We map the problem to that of a stationary detector with a modified unitary evolution operator and use established methods for the solution of that problem to study the first-detection statistics for a moving detector on a finite ring and on an infinite 1D lattice. On the line, the system exhibits a dynamical phase transition at a critical value of , from a state where the probability of detection decreases exponentially in time and the total detection probability is very small, to a state with power-law decay and a significantly higher total probability to detect the particle. The exponent describing the power-law decay of the detection probability at this critical is 10/3, as opposed to 3 for every larger . In addition, the moving detector strongly modifies the Zeno effect. [ABSTRACT FROM AUTHOR]

We consider super-diffusive Lévy walks in dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called ‘infinite density’, describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Lévy walk. We briefly discuss the consequences of non-universality for the d  >  1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian. [ABSTRACT FROM AUTHOR]

Local diffusion coefficients in disordered materials such as living cells are highly heterogeneous. We consider finite systems with quenched disorder in order to investigate the effects of sample disorder fluctuations and confinement on single-particle diffusivity. While the system is ergodic in a single disorder realization, the time-averaged mean square displacement depends crucially on the disorder; i.e., the system is ergodic but non-self-averaging. Moreover, we show that the disorder average of the time-averaged mean square displacement decreases with the system size. We find a universal distribution for diffusivity in the sense that the shape of the distribution does not depend on the dimension. Quantifying the degree of the non-self-averaging effect, we show that fluctuations of single-particle diffusivity far exceed the corresponding annealed theory and also find confinement effects. The relevance for experimental situations is also discussed. [ABSTRACT FROM AUTHOR]

We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) . Comparison of annealed (renewal) continuous time random walks (CTRWs) with logarithmic waiting time distribution and Sinai diffusion in quenched random landscapes reveals striking similarities, despite the great differences in their physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time-averaged and ensemble-averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal, with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble-averaged MSD and time-averaged MSD. [ABSTRACT FROM AUTHOR]