The semiclassical limit of a quantum Zeno dynamics

Abstract

AbstractMotivated 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 $$\hbar \rightarrow 0$$ ħ → 0 and large quantum number $$N\rightarrow \infty $$ N → ∞ , with $$\hbar N$$ ħ N kept fixed. In a suitable topology, the limit is the discontinuous symbol $$p\chi _D(x,p)$$ p χ D ( x , p ) where $$\chi _D$$ χ 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 $$p\chi _D^{(N)}(x,p)$$ p χ D ( N ) ( x , p ) , where $$\chi _D^{(N)}$$ χ D ( N ) is a smooth version of $$\chi _D$$ χ D related to the integrated Airy function. We also discuss the limit from a dynamical point of view.