Quantum time of arrival distribution in a simple lattice model

TL;DR

This paper investigates the quantum time of arrival distribution for a particle in a lattice model, revealing power-law tails and proposing a perturbative method that aligns well with numerical results.

Contribution

It introduces a perturbative approach to compute quantum TOA distributions in a lattice model, capturing key features like power-law tails.

Findings

Power-law tails in the TOA distribution.

Good agreement between perturbative and exact numerical results.

Insights into survival probability in quantum detection scenarios.

Abstract

Imagine an experiment where a quantum particle inside a box is released at some time in some initial state. A detector is placed at a fixed location inside the box and its clicking signifies arrival of the particle at the detector. What is the \emph{time of arrival} (TOA) of the particle at the detector ? Within the paradigm of the measurement postulate of quantum mechanics, one can use the idea of projective measurements to define the TOA. We consider the setup where a detector keeps making instantaneous measurements at regular finite time intervals {\emph{till}} it detects the particle at some time $t$, which is defined as the TOA. This is a stochastic variable and, for a simple lattice model of a free particle in a one-dimensional box, we find interesting features such as power-law tails in its distribution and in the probability of survival (non-detection). We propose a perturbative calculational approach which yields results that compare very well with exact numerics.