# High-Harmonic and Terahertz Spectroscopy (HATS): Methods and Applications
## 2 HATS for Atoms  
### Attosecond-Resolved THz Generation from Argon  
Typically, atoms driven by dual-color laser fields can produce few-cycle THz pulse with a pulse duration of several picoseconds. The femtosecond laser pulse determines the THz emission by the low frequency component within the laser-induced dipole of atoms or molecules. By delaying the relative phase between the two colors for about 300 attoseconds, THz output can be switched from on to off. It is surprised that the low frequency emissions could be manipulated by the sub-optical-cycle motion of electrons under the femtosecond strong laser field, which implies the existence of coherence in the electronic current. To understand the corresponding dynamics of THz generation, a clock with attosecond resolution is required. The phases of high harmonics carry the attosecond chirp and can be treated as a clock for ultrafast dynamics [46], because different orders of high harmonics are associated with their unique recombination and emission times of the corresponding classical trajectories [47]. Generally, for atoms that have no significant structure (minimum or maximum) on the photoionization cross section, the group delay between the neighboring harmonics, also called as the \"atto-chirp\", exhibits a linear dependence on the harmonic order [1, 48, 49]. Assuming the dual-color laser fields take the form of \(A(t)=A_{\omega}\cos\omega t+A_{2\omega}\cos(2\omega+\phi)\), where \(A_{\omega}\) and \(A_{2\omega}\) are the fundamental and the second  
Figure 1: (**a**) The up-conversion of the fundamental frequency (\(\omega\)) to generate the \(q\)th high harmonics \(\omega_{q}\); and (**b,c**) the general process of the down-conversion of laser frequency (\(\omega\)) to generate THz wave (\(\omega_{\rm THz}\)), with the help of the second harmonic pulse (\(\omega_{2}\)). harmonic electric fields with \(A_{2\omega}\ll A_{\omega}\), and \(\phi\) is the dual-color relative phase. The weak second harmonic introduces a perturbation on the propagation of the re-collision electrons [22], which can be written as  
\[\Delta_{2\omega}\approx\frac{A_{\omega}A_{2\omega}}{2\omega}(\sin \omega t_{r}\,-\,\sin\omega t_{i})\cos(2\omega t_{r}\,+\,\phi), \tag{1}\]  
with \(t_{i}\) and \(t_{r}\) being the ionization and recombination times of electrons, respectively. The ionization time \(t_{i}\) determines the subsequent dynamics of the electron, resulting in different accumulated energies, i.e., the different harmonic orders. The modulations of the even harmonics are due to the perturbation of the weak second harmonic field within the electron propagation, which is proportional to \(\sin^{2}\Delta_{2\omega}\)[18]. Thus, based on Equation (1), the output of one even-order harmonic reaches its maximum when the dual-color relative phase \(\phi_{max}\) equals to \(-2\omega t_{r}\). For the adjacent odd harmonics, emission time difference is approaching 45 attoseconds under the laser intensity \(3\times 10^{14}\) W/cm\({}^{2}\)[50], implying that, in the case of a simultaneous HATS detection, the chirp of high harmonics provides a clock to resolve the optimal phase of THz yields with several tens of attoseconds resolution. In 2012, [PERSON] and [PERSON] reported the first HATS detection from argon, solving the longstanding debate on THz optimal phase and finding out the effect of Coulomb potential on the electron trajectory [17; 18]. Dual-color fields are prepared by the Ti:Sapphire laser and its second harmonic introduced by a piece of thin \(\beta\) Barium Borate crystal (type I). The experimental set-up is illustrated in Figure 2a. The observed modulations of even harmonics and THz yields are fitted by \(I_{q}\propto a_{0}(q)+a_{1}(q)\phi+a_{2}(q)\cos^{2}[\phi-\phi_{0}(q)]\), with \(a_{0}\) the base line of modulation, \(a_{1}\) the variations from the circumstance and \(a_{2}\) the value of modulations from the dual-color relative phase.