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arXiv:1302.4669

Quantum First-Passage Time: Exact Solutions for a Class of Tight-Binding Hamiltonian Systems

V. Ranjith, Naresh Kumar

Published Feb 19, 2013• 0 citations
arXivHTMLPDF

Abstract

The Schrödinger integral-equation approach for calculating the classical first-passage time (C-fpt) probability density is extended to the case of quantum first-passage time (Q-fpt). Using this extension, we have calculated analytically the Q-fpt probability density for a class of few-site/state tight-binding Hamiltonian systems, e.g., a qubit, as well as for an infinite 1D lattice. The defining feature of such a quantum system is that the passage across the boundary between a subspace (omega) and its complement (omega-bar) is through a unique pair of "door-way" sites such that the first departure from (arrival at) omega corresponds to the first arrival at (departure from) omega-bar. The Q-fpt probability density so derived remains positive over the time interval in which it also normalizes to unity. These conditions of positivity and normalization define the physical time domain for the Q-fpt problem. This time domain is found to remain finite for the few-site/state Hamiltonian systems considered here, which is quite unlike the case for the diffusive C-fpt problems. The door-way sites and the associated Q-fpt probability density derived here should be relevant to inter-biomolecular/nanostructural electron transport phenomena.

DOI
10.48550/arxiv.1302.4669

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The Schrödinger integral-equation approach for calculating the classical first-passage time (C-fpt) probability density is extended to the case of quantum first-passage time (Q-fpt). Using this extension, we have calculated analytically the Q-fpt probability density for a class of few-site/state tight-binding Hamiltonian systems, e.g., a qubit, as well as for an infinite 1D lattice. The defining feature of such a quantum system is that the passage across the boundary between a subspace (omega) and its complement (omega-bar) is through a unique pair of "door-way" sites such that the first departure from (arrival at) omega corresponds to the first arrival at (departure from) omega-bar. The Q-fpt probability density so derived remains positive over the time interval in which it also normalizes to unity. These conditions of positivity and normalization define the physical time domain for the Q-fpt problem. This time domain is found to remain finite for the few-site/state Hamiltonian systems considered here, which is quite unlike the case for the diffusive C-fpt problems. The door-way sites and the associated Q-fpt probability density derived here should be relevant to inter-biomolecular/nanostructural electron transport phenomena.
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-passage phenomena play a fundamental role in classical stochastic processes. We here exactly solve a quantum first-passage time problem for quantum diffusion driven by measurement noise, a generalization of classical Brownian motion. Such continuous monitoring may trap the measured quantum system in a decoherence-free subspace, a fraction of the available state space that is isolated from the surroundings, and thus plays an important role in quantum information science. We analytically determine the first-passage time distribution, whose form neither depends on the system Hamiltonian nor on the measurement operator, and is therefore universal. These results provide a general framework to investigate the first-passage statistics of diffusive quantum trajectories.
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Abstract We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a process free of measurements, i.e. the solution of the Schrödinger equation, and the statistics of first detection events on a site. For stroboscopic measurements a quantum renewal equation yields basic properties of quantum walks. For example, for a tight binding model on a ring we discover critical sampling times, diverging quantities such as the mean time for first detection, and an optimal detection rate. For a quantum walk on an infinite line the probability of first detection decays like ( t i m e ) − 3 with a superimposed oscillation, critical behavior for a specific choice of sampling time, and vanishing amplitude when the sampling time approaches zero due to the quantum Zeno effect.