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 $\tau$, 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 demonstrate 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 $\tau$ 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 $(\mbox{time})^{(-3)}$ with superimposed oscillations, with exceptional behavior when the sampling period $\tau$ times the tunnelling rate $\gamma$ is a multiple of $\pi /2$. The amplitude of the power law decay is suppressed as $\tau \to 0$ due to the Zeno effect. Our work presented here, is an extended version of Friedman et al. arXiv:1603.02046 [cond-mat.stat-mech], and it predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like $(\mbox{time})^{-3/2}$, as elucidated by Schrödinger in $1915$.

arXiv commentAn extended version of our letter arXiv:1603.02046 [cond-mat.stat-mech]

References in corpus (25)

• Realization of quantum walks with negligible decoherence in waveguide lattices
• Realization of a quantum walk with one and two trapped ions
• Persistence and First-Passage Properties in Non-equilibrium Systems
• An example of the difference between quantum and classical random walks
• Strongly Correlated Quantum Walks in Optical Lattices
• Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks
• Time-of-arrival in quantum mechanics
• Quantum Control by Imaging : The Zeno effect in an ultracold lattice gas
• Measurement of Time-of-Arrival in Quantum Mechanics
• Hitting time for quantum walks on the hypercube
• Recurrence for discrete time unitary evolutions
• Recurrence and Pólya number of quantum walks
• Experimental quantum-walk revival with a time-dependent coin
• Quantum walks with infinite hitting times
• Detection of a quantum particle on a lattice under repeated projective measurements
• Quantum walks on quotient graphs
• Recurrence properties of unbiased coined quantum walks on infinite $d$-dimensional lattices
• Hitting time for the continuous quantum walk
• Survival of classical and quantum particles in the presence of traps
• Quantum Renewal Equation for the first detection time of a quantum walk
• Survival probability in a one-dimensional quantum walk on a trapped lattice
• Generalized Kac's Lemma for Recurrence Time in Iterated Open Quantum Systems
• Quantized recurrence time in iterated open quantum dynamics
• Unidirectional quantum walks: evolution and exit times
• Quantum Persistence: A Random Walk Scenario

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