Delay Zero Calibration in Attosecond Transient Absorption Spectrum(Invited)

The strong coupling of laser field with atoms and molecules can lead to the shift or even splitting of their energy levels. Observing the spectral changes through laser pump-probe experiments provides insights into the electron dynamics within atoms and molecules. Since the 1960s, ultrafast and intense laser technologies have continuously evolved. Especially in the early 21st century, scientists achieved the synthesis of ultrashort pulses reaching attosecond durations through high harmonic generation of noble-gas atoms under strong laser irradiation. The emergence of attosecond pulses allows for studying the ultrafast electronic dynamics of atoms and molecules within their natural time scales. Attosecond Transient Absorption Spectroscopy (ATAS) utilizes an isolated attosecond Extreme Ultraviolet (XUV) pulse as the probe light and another Infrared (IR) laser pulse with varying time delays as the pump light to obtain attosecond time-resolved electronic dynamics. In recent years, ATAS has found wide applications in various atomic, molecular, and solid-state systems. Among all these applications, how to accurately calibrate the delay-zero from experimental data is an important yet non-trivial task. Recently, HERRMANN J et al. have introduced a novel method to serve this purpose, taking full advantage of the multiphoton transitions in attosecond transient absorption spectroscopy. In their experiment, they observed quarter laser cycle (4 omega) oscillations in the transient absorption spectrum of helium, originating from four-photon coupling between high-order odd harmonics in the Attosecond Pulse Train (APT). By utilizing this highly nonlinear 4 omega signal to extract and calibrate the delay-zero, and comparing it with solutions of the time-dependent Schrodinger equation, the accuracy and effectiveness of this method were confirmed. In our work, we further extend this method to ATAS using isolated attosecond pulses as the probe light. Here, we have observed 2 omega(IR), 4 omega(IR), and 6 omega(IR) signals and, more importantly, we have found that the high-frequency signals are almost precisely located at the delay-zero, which provides a robust way for future experimental determination of the delay-zero. Specifically, we have employed a three-level model, which consists of the three lowest-energy states of the helium atom: the ground state 1s(2), the first excited state 1s2s, and the second excited state 1s2p, to simulate the ATAS of helium atoms. The dynamics of this system can be described by the discrete Schrodinger equation. We numerically solve the time-dependent Schrodinger equation using standard fourth or fifth-order Runge-Kutta algorithms. This approach allows us to capture the population dynamics of the states and subsequently calculate the time-dependent dipole moment of the system. By performing a Fourier transform of the dipole moment, we obtained the response function S(omega, tau) representing the strength of absorption across different spectral regions and its variation with the relative delay time between the pump and probe pulses. We plot the ATAS, focusing on the resonance peak corresponding to the 1s2p state and the positions of further emitting two (omega=Delta(ga)-2 omega(IR)) or four (omega=Delta(ga)-4 omega(IR)) IR photons. In addition, we have conducted wavelet analysis on the response function S(t) at these three specific positions to identify the main oscillation frequencies of the signal and their occurring time intervals. The results demonstrate that the absorption spectrum primarily oscillates at some certain periods corresponding to half, quarter, and one-sixth of the IR laser cycle, yielding oscillation frequencies of 2 omega(IR), 4 omega(IR), and 6 omega(IR), respectively. At omega=Delta(ga)-2 omega(IR), we observe that the center point of the 4 omega(IR) signal is close to the zero of the time delay. Therefore, in experiments, we can measure the transient absorption spectrum, extract the 4 omega(IR) signal at this position, determine its center point along the delay axis, and subsequently calibrate the delay-zero. Similarly, in case the IR laser is more intense and the signal at omega=Delta(ga)-4 omega(IR) is prominent, we can also utilize the 6 omega(IR) signal of the absorption spectrum to determine the position of the delay-zero. With extensive numerical simulations, either in resonance or with detuning, and for different carrier-envelope phases, we find that the 6 omega(IR) signal at omega=Delta(ga)-4 omega(IR) is the most robust, which makes it the best choice for calibrating the delay-zero. In summary, we have conducted a detailed wavelet analysis of the response function at some typical frequencies of the attosecond transient absorption spectrum of helium atoms. The results show that the center of the high-frequency oscillations provides a feasible approach for experimentally determining the delay-zero and is thus helpful to the correct interpretation of the experimental data, particularly in extracting time-related information such as response times of atoms and molecules to external stimulus, lifetimes of transient quantum states, to name only a few. We hope the theoretical predictions are observable in ATAS experiments and can be further extended to other types of pump-probe techniques.