The semiclassical limit of a quantum Zeno dynamics

TL;DR

This paper investigates the semiclassical limit of quantum Zeno dynamics in cavity QED, revealing how symbols of truncated momentum operators converge to discontinuous functions in phase space as Planck's constant vanishes.

Contribution

It provides a detailed analysis of the asymptotic behavior of symbols related to quantum Zeno effects, introducing a smooth approximation involving the Airy function.

Findings

Limit of symbols is the discontinuous function pχ_D(x,p)

The symbol approximates pχ_D^{(N)}(x,p) with a smooth Airy-based function

Discussion of the dynamical implications of the asymptotic behavior

Abstract

Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant $\hbar\to0$ and large quantum number $N\to\infty$, with $\hbar N$ kept fixed. In a suitable topology, the limit is the discontinuous symbol $p\chi_D(x,p)$ where $\chi_D$ is the characteristic function of the classically permitted region $D$ in phase space. A refined analysis shows that the symbol is asymptotically close to the function $p\chi_D^{(N)}(x,p)$, where $\chi_D^{(N)}$ is a smooth version of $\chi_D$ related to the integrated Airy function. We also discuss the limit from a dynamical point of view.