Analysis of attosecond entanglement and coherence using feasible formulae

In recently published papers [M. J. J. Vrakking, Phys. Rev. Lett. 126, 113203 (2021)0031-900710.1103/PhysRevLett.126.113203; J. Phys. B 55, 134001 (2022)0953-407510.1088/1361-6455/ac6e17], Vrakking proposed an inventive scheme to control the entanglement or coherence of the vibrational states in a hydrogen molecular ion and a continuum electron, both of which were generated via the photoionization of a hydrogen molecule irradiated by a coherent pair of extreme ultraviolet (XUV) attosecond pulses and a few-femtosecond ultraviolet (UV) pulse. He clarified, for the first time to our knowledge, how the coherence of the XUV attosecond pulse pair is transferred to the molecular ion system accompanying a detached continuum electron by numerically solving a time-dependent Schrödinger equation (TDSE) governing the evolution of the ion and the electron in a rigorous manner. Nevertheless, it was not straightforwardly resolved how and why the specific characteristics of the resultant joint energy spectrogram emerged and how the entanglement or coherence was altered with the irradiation of the UV pulse. In this paper, we present an analytical solution of the TDSE using time-dependent perturbation theory, and we utilize the resultant solution to explain what causes the particular features in the entanglement or coherence between the electron and the ion spectra.