The first detection of a quantum particle on a graph has been shown to depend sensitively on the distance ξ between detector and initial location of the particle, and on the sampling time τ . Here we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest-neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the large ξ limit, the small τ limit and the power law decay with attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width, non-analytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value which is distance independent.Introduction: Recent experimental advances have made it possible to measure quantum walks at the single particle level [1][2][3][4]. A related advance is the quantum first detection problem which has drawn considerable theoretical attention [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], as it deals with the basic issue of when the particle will first be detected in a target state. Originally the topic emerged in the context of quantum search algorithms. Given a graph, and an Hamiltonian H, the presence or absence of a particle starting on node |x i is recorded at a node |x d sampled with period τ . H, τ and the measurement process [23], define the problem, which differs markedly from the corresponding wellstudied classical first-passage-time problem [24][25][26][27][28].Recently, a quantum renewal equation which relates the statistics of first detection times to the quantum evolution operator of the measurement-free system was derived [21,22]. This equation was used investigate the statistics of the first detected return, i.e. when |x i = |x d . The present Letter focuses on the first detected arrival, |x i = |x d . The questions to be tackled are: Given τ and the tight-binding Hamiltonian on an infinite line, what are the basic properties of the first detection probability? More specifically: Will the particle always be detected? If not, then what is the optimal sampling rate for which the total detection probability is maximized? The existence of an optimum is expected, since the Zeno effect [29,30] suppresses detection for too frequent measurements, while a too large τ aids the escape from the detector. What is the general behavior of the detection probability at attempt number n, which we denote F n ? What is its asymptotics for small and large n and how does it depend on the initial distance ξ = |x d − x i | of the particle from the detector? ...
The ergodicity breaking parameter is a measure for the heterogeneity among different trajectories of one ensemble. In this report this parameter is calculated for fractional Brownian motion with a random change of time scale, often called "subordination". We proceed to show that this quantity is the same as the known CTRW case.PACS numbers: 05.40.Fb,05.40.Fb "Weak ergodicity breaking" is a term which occurred in the spotlight a few years ago, see for instance [1][2][3][4][5][6] with respect to observables showing aging behavior. The term "aging" applies to the specific behavior pertinent to the values of an observable measured at two distinct instants of time, like the mean squared displacement (MSD) in the time interval between t 1 and t 2 , ∆X 2 (t 1 , t 2 ) as depending on the age t 1 of the system (assumed to be prepared at t = 0) at the beginning of observation. Aging here essentially refers to a non-stationarity of the observed process, in which ∆X 2 (t 1 , t 2 ) cannot be expressed via the time lag τ = t 2 − t 1 only and keeps a considerable dependence on t 1 as a stand-alone variable.Let us concentrate now on the ergodicity properties the squared displacement during the time interval τ in a measurement starting at t. In the case when the data acquisition in each single run of the process takes time from t = 0 to t = T , the time average of ∆X 2 (t, t + τ ) can be performed:The process is considered ergodic provided ∆X 2 (τ ) = ∆X 2 (t, t + τ ) . The discussion of the ergodicity implies that the corresponding limit does exist in some probabilistic sense (say, in probability) and is equal to the ensemble mean. Note that the time-average over the data acquisition interval removes the explicit t-dependence. For non-stationary processes the ensemble mean at any t depends on t, while the time-integration over the data acquisition interval removes this dependence. Therefore, even provided the integral above converges in a whatever sense, it cannot converge to all ∆X 2 (t, t + τ ) simultaneously, and our process is trivially non-ergodic.Whether the random process is stationary or not, one can ask how diverse are its different realizations, i.e. how different are two trajectories of the process with respect to some time-averaged observable O(τ ; t) (in our case the MSD, O(τ ; t) = ∆X 2 (t, t + τ )). To this purpose one considers a fixed-T approximation to eq.(1)At any finite T the value of O(τ ) T is a random variable. For both, stationary and non-stationary processes O(τ, t), one can consider a parameter describing the strength of fluctuations of O(τ ) T . As a measure of homogeneity or heterogeneity of different trajectories one can take the relative dispersion of the O(τ ) T in different realizations of the process:The parameter J (often called "ergodicity breaking parameter") shows how different are different trajectories of the process with respect to the observable O. For stationary processes, vanishing of J in the long time limit indeed implies ergodicity [7], and its non-vanishing witnesses against erg...
We consider scaled Brownian motion (sBm), a random process described by a diffusion equation with explicitly time-dependent diffusion coefficient D(t) = D0t α−1 (Batchelor's equation) which, for α < 1, is often used for fitting experimental data for subdiffusion of unclear genesis. We show that this process is a close relative of subdiffusive continuous-time random walks and describes the motion of the center of mass of a cloud of independent walkers. It shares with subdiffusive CTRW its non-stationary and non-ergodic properties. The non-ergodicity of sBm does not however go hand in hand with strong difference between its different realizations: its heterogeneity ("ergodicity breaking") parameter tends to zero for long trajectories.PACS numbers: 05.40. Fb,05.10.Gg Anomalous diffusion is a generic name for a class of transport processes which are close to diffusion in their origin (i.e. can be represented via generalized random walk schemes or Langevin equations) but do not lead to the mean squared displacement growing as the first power of time(with D being the diffusion coefficient and d the dimension of space), as predicted by the Fick's laws. Within the random walk schemes such deviations from the normal diffusion picture can arise either due to broad distributions of the waiting times between the steps (continuous time random walk models, CTRW), or due to slow decay of correlations between steps, or both, see [1] for a review, leading to the change of the power law in the time dependence of the mean squared displacement,The processes with α < 1 are called subdiffusion, the ones with α > 1 are termed superdiffusion. In the first case the formal diffusion coefficient D in Eq.(1) vanishes in the long time limit; in the second case it diverges. The single trajectory dynamics in normal diffusion is described by the Langevin equatioṅwith white, delta-correlated Gaussian noise ξ(t), ξ(t) = 0, ξ(t)ξ(t ) = δ(t−t ); the time-dependence of the probability density function (PDF) of the process or the one of its transition probabilities is given by the Fick's second law (diffusion equation)(both equations given here in one dimension). The description of anomalous diffusion of different origins often follows by modification of one of the equations above.In experiments, many processes of anomalous diffusion of unknown origin, i.e. when the observable of interest which cannot be fitted to the solutions of Eq. (1), are fitted to the results obtained for the so-called scaled Brownian motion (sBm) [2], a diffusion process with explicitly time-dependent diffusion coefficient D(t) = D 0 t α−1 . Numerical simulations of Ref. [3] show that, at least for the case of FRAP (fluorescence recovery after photobleaching), the fits may be astonishingly good, independent on the true nature of the simulated process (percolation, CTRW, etc.). This means that the form of FRAP recovery curves, if they hint onto anomalous diffusion, hardly depends on the origin of the corresponding anomaly. Using sBm model for calculating other properties ma...
How long does it take a quantum particle to return to its origin? As shown previously under repeated projective measurements aimed to detect the return, the closed cycle yields a geometrical phase which shows that the average first detected return time is quantized. For critical sampling times or when parameters of the Hamiltonian are tuned this winding number is modified. These discontinuous transitions exhibit gigantic fluctuations of the return time. While the general formalism of this problem was studied at length, the magnitude of the fluctuations, which is quantitatively essential, remains poorly characterized. Here, we derive explicit expressions for the variance of the return time, for quantum walks in finite Hilbert space. A classification scheme of the diverging variance is presented, for four different physical effects: the Zeno regime, when the overlap of an energy eigenstate and the detected state is small and when two or three phases of the problem merge. These scenarios present distinct physical effects which can be analyzed with the fluctuations of return times investigated here, leading to a topology-dependent time-energy uncertainty principle.
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.
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