Photoabsorption Spectrum of Atom Hydrogen Driven by the Combination of a XUV Pulse and a Synthesized Optical Attosecond Pulse (SOAP)

We present a high-precision theoretical study of attosecond transient absorption spectroscopy (ATAS) of atomic hydrogen by numerically solving the time-dependent Schrödinger Equation (TDSE). A broadband extreme ultraviolet (XUV) attosecond pulse creates a wave packet of singly-excited bound states, which is subsequently probed by a time-delayed synthesized optical attosecond pulse (SOAP) with varying bandwidths and durations. When the SOAP has a narrow bandwidth (1.3–1.5 eV) and a long duration (~17 fs), the absorption spectrum exhibits conventional features, namely AC Stark shifts, half-cycle modulations (1.48 fs), and light-induced intermediate states, consistent with previous ATAS studies. In contrast, when the SOAP has a broad bandwidth (0.5–5.5 eV) and an attosecond duration (400 as), the dynamics are completely different. The spectrum reveals transverse wavelike modulations along the absorption lines and, remarkably, quantum beats with distinct frequencies, which are different from previous reports in hydrogen ATAS. To interpret these observations, we employ a dipole-control model. The model quantitatively reproduces the dominant modulation frequencies, identifying resonant couplings via two-photon processes (TPPs, 1.89 eV, period 2.18 fs) and three-photon processes (THPPs, 10.2 eV and 12.1 eV), as well as higher-order couplings. The validity of the δ-like pulse approximation is quantitatively assessed. The model remains accurate for pulse durations shorter than 700 as (bandwidth broader than 3.5 eV) but fails for longer pulses (exceeding 4 fs), where energy level splittings emerge. Our results demonstrate that the dipole-control model provides a reliable and intuitive framework for interpreting complex multiphoton interactions in ATAS, and highlight the unique capability of broadband SOAP probes to resolve attosecond-scale quantum beats inaccessible with conventional few-cycle infrared pulses.