Temporal analysis of fractional revivals of molecular observables following impulsive alignment

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Laser-induced impulsive alignment of symmetric linear molecules leads to time-dependent observables which are analyzed in terms of their spectral components. Signals appear as sums over positive integers $\ensuremath{\ell}$ of periodic components ${A}_{2\ensuremath{\ell}}(t)$ with periods ${\ensuremath{\tau}}_{2\ensuremath{\ell}}={(2\ensuremath{\ell}c{B}_{0})}^{\ensuremath{-}1}$, where ${B}_{0}$ is the rotational constant in wave number units. Within each period ${\ensuremath{\tau}}_{2\ensuremath{\ell}}$, four fractional revivals at times $n{\ensuremath{\tau}}_{2\ensuremath{\ell}}/4$ $(n=0,1,2,3)$ exhibit constant successive phase shifts ${(\ensuremath{-}1)}^{\ensuremath{\ell}+p}\ensuremath{\pi}/2$ depending on the even-odd parity $(p=0,1)$ of the initial rotational states. This analysis gives a comprehensive account of the so-called high-order revivals usually discussed in terms of fractional revivals within the period ${\ensuremath{\tau}}_{2}$ of the average value $\ensuremath{\langle}{cos}_{\ensuremath{\vartheta}}^{2}\ensuremath{\rangle}(t)$ of ${cos}^{2}(\ensuremath{\vartheta})$, where $\ensuremath{\vartheta}$ is the angle between the molecular axis and the alignment direction. These considerations are illustrated by experiments and numerical calculations of laser-induced impulsive alignment and tunnel ionization of ${\text{CO}}_{2}$ molecules for which calculated ionization yields of fixed-in-space molecules are available.