Exact Energy-Time Uncertainty Relation for Arrival Time by Absorption

TL;DR

This paper establishes an exact energy-time uncertainty relation for particle arrival times using an absorption model, providing new bounds and proposing an experimental realization with trapped ions.

Contribution

It proves a precise energy-time uncertainty relation for arrival times with absorption, including bounds involving absorption probability, and suggests a feasible experimental setup.

Findings

Derived uncertainty bounds: ΔT ΔE ≥ √p ħ/2 and <T> ΔE ≥ 1.37√p ħ.

Nearly minimal uncertainty achievable in a two-level system.

Proposed a trapped ion experiment to test the uncertainty relation.

Abstract

We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet. Under the sole assumption that the absorbing term vanishes on the initial wave function, we show that $\Delta T \Delta E \geq \sqrt p \hbar/2$ and $<T> \Delta E\geq 1.37\sqrt p\hbar$, where $<T>e$ denotes the mean arrival time, and $p$ is the probability for the particle to be eventually absorbed. Nearly minimal uncertainty can be achieved in a two-level system, and we propose a trapped ion experiment to realize this situation.