Impact of Statistical Fluctuations on High Harmonic Generation in Liquids

High harmonic generation (HHG) from gases and solids has been studied extensively. Whereas for liquids, it is far more challenging to understand the ultrafast dynamics with conventional methods. From a statistical perspective, we investigate the liquid-phase HHG theoretically by using a disordered linear chain. Our results reveal that (i) the harmonic spectra are characterized by a transition energy that separates the spectra into a low-energy region containing only odd harmonics and a high-energy region containing both even and odd harmonics with low yields; (ii) the transition energy depends on the fluctuation of structure and the field strength, but independent of the laser wavelength; (iii) the occurrence of dephasing is a natural result of the electron dynamics modulated by the long-range disorder. Furthermore, a simple formula is proposed to identify the transition energy, from which we correctly reproduce the experimental cutoff energies of HHG from liquid ethanol. Our results pave the way to better understand and control the HHG in liquids as another compact HHG source.

Figure 1

(a) Schematic diagram of a laser pulse interacting with a disordered chain. (b) High harmonic spectra calculated from ordered system $\sigma =0$ and disordered systems with different statistical fluctuations $\sigma =0.6$, 1.0, and 1.4, respectively. The green areas represent that the spectra contain both odd and even harmonics with comparable intensity. The black dashed lines indicate the transition energies $\mathrm{\Omega }$ predicted by Eq. (2). Parameters of the laser used are given in the text.

Figure 2

Transition energy as a function of the laser field strength and wavelength. (a) Comparison of harmonic spectra from a zero fluctuation (red) and a nonzero fluctuation system with $\sigma =1.4\mathrm{a}.\mathrm{u}.$ (blue) with four different field strengths $E_0=0.010$, 0.014, 0.018, and 0.022 a.u., of a $1.5\mu \mathrm{m}$ driving field. (b) Transition cutoff under different wavelengths $\lambda =1.0$, 1.3, 1.6, and $1.9\mu \mathrm{m}$ with the field strength $E_0=0.010\mathrm{a}.\mathrm{u}.$ and the fluctuation $\sigma =1.4\mathrm{a}.\mathrm{u}.$ The green areas and black dashed lines are the same as those in Fig. 1.

Figure 3

Cutoff energy dependence on peak laser field strength of liquid ethanol. Red dots represent the experimental data in Ref. [17], and the green line indicates the theoretical prediction by Eq. (2). The reference point we use here is identified by the black arrow. The laser wavelength is 1500 nm.

Figure 4

(a)–(d) Time-frequency analysis of HHG (logarithmic scale) for different fluctuations. (e),(f) Time-frequency analysis of HHG for different fluctuations, but artificially choosing a single state with eigenstate number $i=248$ as the initial state to evolve with time. $T=2\pi /\omega$ is the laser period.