Quantum walks: The first detected passage time problem

Abstract

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.

First Detected Arrival of a Quantum Walker on an Infinite Line

“…3. When ξ = 0, then re iβ = 1 and we recover the known result [5,21,22]. In the limit of large ξ or small τ we find a remarkably simple relation to the incidence time:…”

“…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.…”

Stochastic resetting and applications

“…For example, the spectral properties of quantum systems subjected to resetting have been studied in [166]. Another quantum setup where resetting can play an important role is the quantum random walk subjected to perturbations caused by repeated measurements [167][168][169][170].…”

Large fluctuations of the first detected quantum return time

“…A well investigated approach is to consider measurements on the original node, repeated stroboscopically until the first detection. The problem is to find the distribution of the number of attempts n till first detection [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. This sheds light on the backfire of measurement on unitary evolution, and on time processes in quantum mechanics.…”

“…The goal of this paper is to investigate the fluctuations of the number of measurements needed till the first return is recorded, namely the variance of n. It was shown previously that these can be very large close to critical sampling parameters even for small systems [1,13]. Here we provide formulas for the blow-ups of the fluctuations, using an elegant mapping of the quantum problem to 18 Measurements M are performed every τ units of time, between which the system evolves unitarily withÛ (τ ).…”