# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
## 2 HATS for Atoms
### Attosecond-Resolved THz Generation from Argon
The experimentally determined \(\phi_{0}\) from the 18 th to 24 th harmonics are compared to semiclassical calculations to determine the absolute phase of the dual-color field. Using the harmonic chirp as a reference, the optimal phase for the maximum THz yield is estimated to be around \(0.8\pi\), which deviates significantly from zero as predicted by the four wave mixing model and from \(0.5\pi\) as predicted from the plasma current model. Based on the plasma current model, the electron undergoes a tunneling ionization carrying no internal velocity, after which the classical movement of the ionized electrons will form a residual current to emit the THz wave. Trying to reveal the effect of Coulomb field on electron dynamics, simulations on the THz modulation were carried out and compared by applying the current model with or without the Coulomb potential. As shown in Figure 3a,b, the optimal phase for THz generation is \(0.8\pi\) by analyzing the classical trajectories of the directly escaping and rescattered electrons, and this result is comparable to the  
Figure 2: (**a**) Typical experimental set-up for HATS generation and detection. (**b**) The experiment obtained optimal phase for even-harmonics (circles) are compared to the semiclassical calculation (dashed line). Optimal phase for THz yield is locked at \(0.8\pi\) (dotted dashed line). Reproduced with permission from [18]. Copyright IOP Publishing, 2013. 1D-TDSE calculations and the experimental observations. Meanwhile, by using the 3D-CTMC method, the current formed by the low energy electrons (<1.2 Up) as a function of the dual-color relative phase exhibits a maximum at 0.72\(\pi\), as illustrated in Figure 3c. Besides the classical trajectory methods, solutions on the 3D-TDSE of argon in strong laser fields also reveal a similar trend for THz yields peaking at around 0.75\(\pi\) when the Coulomb field is included. Based on the theoretical works on the 1D/3D classical trajectory and quantum analysis, we draw a conclusion that the escaping and rescattered electrons contribute to THz wave generation within dual-color laser field regime. These theoretical results are compared to the experimental observations to create a unified image of the rescattering events, which includes \"hard collision\" (recombine to the mother ions) of the rescattered electrons to emit high harmonics and the \"soft collision\" (forward rescattered electrons) to give birth to the THz emissions [17; 18]. The most common method in modeling strong field physics is to use the SFA, which is developed by [PERSON] in 1994 [32]. In SFA, the electron in the continuum can be treated as a free particle exposing in the external electric field, with the Coulomb potential being neglected. When the electrons are far from its parent ion during the propagation step (the second one of the three steps), the Coulomb interaction between the electrons and the ions are much weaker than the interaction between laser and electrons. That is why the Coulomb interaction is neglected in the SFA model, and, for the electrons with high kinetic energy (contribute to HHG), this model typically works. However, THz generation is mostly contributed by slow electrons, which could hardly be described under the SFA model (or other models developed for high energy electrons). The dynamics of the slow electrons is also one part of the whole strong field dynamics. That is why the studies on HATS could be helpful to give more insights on the rescattering process. In brief, the simultaneous measurement of THz wave and HHG enables the calibration of the absolute optimal phase for THz wave generation by taking advantage of the intrinsic chirp of high harmonics (the phase could also be inferred from the yield of high harmonics, as demonstrated by [PERSON] et al. [51]). HATS is useful in determining the phase of the dual-color laser field in an all-optical  
Figure 3: (**a**) THz yields by the escaping (black dotted line), the rescattering (red dashed line), and the total currents (green line) calculated from the plasma current model.