Arrival time from the general theory of quantum time distributions

TL;DR

This paper advances the quantum theory of arrival time distributions by deriving a simple analytical expression for the arrival time of wave packets in one dimension, accounting for non-Hermitian effects due to detection.

Contribution

It develops a generalized framework for quantum arrival times, incorporating non-Hermitian operators from projection formalism, providing a practical analytical formula for arrival time distributions.

Findings

Derived a simple analytical expression for arrival time distribution.

Applied the formalism to wave packets in one dimension.

Revealed the role of non-Hermitian operators in quantum detection.

Abstract

We further develop the general theory of quantum time distributions introduced in arXiv:2010.07575 and apply it to find the distribution of arrival times at the detector. Even though the Hamiltonian in the absence of detector is hermitian, the time evolution of the system before detection involves dealing with a non-hermitian operator obtained from the projection of the hermitian Hamiltonian onto the region in front of the detector. Such a formalism eventually gives rise to a simple and physically sensible analytical expression for the arrival time distribution, for arbitrary wave packet moving in one spatial dimension with negligible distortion.