# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
## 2 HATS for Atoms
### THz Benchmark for the Long and Short Trajectories of HHG  
Typically, gas sources of high harmonics require high particle density by adiabatic expansion from a gas jet or gas cell to a vacuum chamber. The conversion efficiency of the high harmonics can be maximized via the phase-matching by adjusting the position (relative to the laser focus) and the backing pressure of the gas-supply instrument. In 2016, [PERSON] and his fellows reported the HATS detection of argon [59]. By controlling the position of the gas jet, dual-color relative phase dependent high harmonics and THz yields are studied by selecting only the short trajectory contributions, or both the short and long trajectory contributions. If the gas jet is placed at the down stream of the propagating direction, i.e., the phase-matching position, HHG from the short trajectory contribution is optimized and this results in a nearly linear relationship between the harmonic orders and the optimal phases. Under this condition, it can be found in Figure 4c that the optimal phase for the 26 th harmonic coincides the phase for maximum THz yields, similar to that previously reported [17]. Note that the laser intensity of [PERSON]'s experiment (\(4\times 10^{14}\) W/cm\({}^{2}\)) is about four times larger than the early one from [PERSON] [17]. Since the harmonic chirp is related to the cut-off energy, the optimal phases of the 26 th harmonic should be different under the two different experimental conditions. Hence, further investigations may be required to clarify whether and why THz and some specific order of HHG prefer the same dual-color phase. If the gas jet is placed closer to the focus, as shown in Figure 4e, the long trajectory contribution dominates for the low order harmonics (less than 30), while the short trajectory chirp appears significantly at larger harmonics. Meanwhile, in the harmonic yield, as shown in Figure 4e as a function of the relative two-color phase, a complex chirp can be seen due to the coexistence of the contributions from the long and the short trajectories. Harmonic phase of the short trajectory contribution could be balanced by the Gouy phase shift, the neutral gas dispersion, the dispersion of plasma (if the ionization is significant) and the phase of the atomic dipole moment [60], to give a clear linear relationship. It is interesting to investigate the spatially and spectrally resolved quantum paths of the long and short trajectories of electrons from experiment and theory [61; 62; 63; 64; 65]. The joint measurement of HATS is a supplementary to these results, which can use THz wave generation as a phase reference for the long and short trajectories of electrons. Especially for the dual-color laser generation case, the propagation of electron can be manipulated by the weak second harmonic, resulting in the simultaneous manipulation of the long and short trajectory with THz emission as a reference. If the driving laser has a cosine time-dependent shape, the last quarter period of electron dynamics could be traced by the long trajectory of the recombined electrons [66]. As demonstrated in Refs. [62; 65], simultaneous observation and manipulation on the long and short trajectory of electrons could serve as a supplement to the typical high harmonic detection with only the short trajectory contributions. We suggest a combined detection of both the short and long trajectories as well as the dual-color controlled HATS modulations to give deeper insights into on the dynamics of electrons in strong laser fields. When the intensity of the second harmonic is no more a perturbation, which occurs for example at about 0.1 of the intensity of the fundamental field, modulations exhibit an out-of-phase relationship between the yields of THz and different harmonics, as illustrated in Figure 4a,b. This could be understood within the schema of the classical three-step process. For strong second harmonic laser field condition, the modulations of even harmonics no longer originate from the perturbation action of the electron trajectories.