High harmonic is a crucial technology for generating bright and coherent light sources in extreme ultraviolet (EUV) and X-rays for a wide range of applications, including material science, chemistry, and biology. High harmonic is also used in attosecond science, which studies the ultrafast dynamics of electrons in atoms and molecules on their natural timescale of attoseconds (10^-18 s). High harmonic has traditionally been studied in gases and solids, but recent research has shown that it can also be observed in liquids. High harmonic in liquids offers several advantages over traditional gas-phase and solid-phase high harmonic. Firstly, liquids have higher electron densities than gases. Second, liquids can withstand higher laser intensities and can repair damage automatically compare with solids. Thus, high harmonic in liquids is a promising candidate for a compact and brighter EUV source. Therefore, it is crucial to reveal the underlying mechanism of liquid-phase high harmonic. However, there are many fundamental questions in liquid-phase high harmonic. In theory, Zeng et al. conducted a study in 2020 to investigate liquid-phase high harmonic by using a disordered linear chain and proposed a formula that could quantify the cutoff energy. Subsequently, Xia et al. proposed a statistical two-level model and revealed the role of localized charge-resonance states in high harmonic from disordered liquids. In experiments, Luu et al. reported the observation and detailed characterization of high harmonic in the EUV region from liquid water. Here, we study the high harmonic from liquid water by solving the semiconductor Bloch equations (SBE) in length gauge and investigate the modulation of the spectral shift in harmonic spectra driven by two-color laser fields. Our findings indicate that manipulating the relative phase of two-color laser fields can control the frequency and yield of odd harmonics and allow for the fine tuning of the high harmonic spectrum. We believe that our primary findings will be helpful for future studies on strong-field and attosecond electron dynamics in liquids.

First, by solving the SBE in length gauge, the high harmonic from water driven by two-color laser fields consisting of a fundamental field and its second harmonic is studied. Then, time-frequency analysis of the calculated high-order harmonic current is carried out by wavelet transform to gain more information about the high harmonic process. After that, the contributions from positive and negative half-cycles in the time domain are artificially separated and respectively transformed into the frequency domain to see how the inter-half-cycle interference affects the frequency shift of different harmonic orders. Next, the frequency shift of H9 and H10 from positive and negative half-cycles in different phase differences is calculated. Furthermore, the time of re-encounter in the positive and negative half-cycles dictated by the motion of electrons and holes in real space is calculated.

A typical high harmonic spectrum (Fig. 2) of water shows that the harmonic spectrum contains both odd and even harmonics. The generation of even harmonics is the consequence of the asymmetric two-color field. A clear sign of a plateau is shown, followed by a cutoff at the 23rd harmonic. Another important feature of the spectrum is that the harmonics are all blue-shifted. It can be attributed to the nonadiabatic effect. The time-frequency analysis (Fig. 3) shows that the high-order harmonics are mainly generated in the rising edge of the fundamental pulse. In the rising edge of the laser field, the high harmonic possesses a positive chirp and thus leads to a blue-shifted spectrum. Furthermore, it is shown that by changing the relative phase of two-color laser fields, the even- and odd-harmonics are periodically modulated (Fig. 4). As the relative phase is tuned from 0 to π, the redshift of odd-harmonics increases, and the odd-harmonics yield increases first and then decreases. As the relative phase is tuned from 0 to 0.5π(0.6π-0.9π), the contribution of the positive half-cycle to odd-order high harmonic is greater (less) than that of the negative half-cycle (Fig. 5). As the relative phase is tuned from 0 to π, high harmonic produced in the negative half-cycle is blue-shifted first and then red-shifted, while high harmonic produced in the positive half-cycle is always red-shifted (Fig. 6). According to the different contributions from different half-cycles (Fig. 5) and the phase difference results from the time interval between two adjacent half-cycles (Table 1), the interference between the positive and negative half-cycles (Eq. 8) is analyzed. In conclusion, these phenomena can be attributed to inter-half-cycle interference between positive and negative half-cycles.

In this study, high harmonic in liquid water driven by two-color laser fields consisting of a fundamental field and its second harmonic is investigated. Our analysis focuses on the spectral blueshift, and time-frequency analysis reveals that the high harmonics are mainly generated during the rising edge of the fundamental pulse, resulting in a blue-shifted harmonic spectrum. Besides, by tuning the relative phase of two-color laser fields, both the amplitude and the center frequency of high harmonic can be modulated. These phenomena can be attributed to the interference of high harmonic emitted from positive and negative half-cycles. This study might shed new light on the attosecond electron dynamics in liquids and the tuning of liquid-phase high harmonic.