Terahertz radiation induced by shift currents in liquids

Significance

Laser-induced ultrafast dynamics in liquids provides us a unique and efficient broadband laser-induced terahertz (THz) source. Yet, our understanding of the underlying physics in this disordered and complex system is incomplete. Here we developed a shift-current model, which could reproduce experimental measurements quantitatively. Nuclear quantum effect is found to play important roles in the THz radiation in H₂O and D₂O.

Abstract

Considerable progress has been made in the experimental studies on laser-induced terahertz (THz) radiation in liquids. Liquid THz demonstrates many unique features different from the gas and plasma THz. For example, the liquid THz can be efficiently produced by a monochromatic laser. Its yield is maximized with a longer driving-pulse duration. It is also linearly dependent on the excitation pulse energy. In two-color laser fields, an unexpected unmodulated THz field was measured, and its energy dependence of the driving laser is completely different from that of the modulated THz waves. However, the underlying microscopic mechanism is still unclear due to the difficulties in the description of ultrafast dynamics in complex disordered liquids. Here we propose a shift-current model. The experimental observations could be reproduced by our theory successfully. In addition, our theory could be further utilized to investigate the nuclear quantum effect in the THz radiation in H₂O and D₂O. This work provides fundamental insights into the origin of the THz radiation in bulk liquids.

Efficient and intense terahertz (THz) radiation source is the cornerstone of THz science and technology. It is widely used in communication, imaging, and explosive detection (1, 2). Laser-induced THz wave generation (TWG) from gases, solid crystals, and plasma has been extensively studied. Currently, there are still many limitations for the existing THz sources. For example, solid targets are fragile in intense lasers with high repetition rates due to the material damage. For the gas target, the photon energy conversion efficiency is not high enough because of the low density (3, 4). For the plasma, it often requires very high incident laser power (1, 2). Therefore, it is attractive to find an alternative type of THz source. In the past, liquids were not considered as a good THz source because they are strong absorptive media (5, 6). However, there are some exciting experimental developments recently in which TWG in a liquid line or film with thickness on the scale of $\mathrm{\mu }\mathrm{m}$ has been reported (7–15). In contrast to TWG in other media relying on two-color laser schemes, the liquid THz can be produced by a monochromatic laser efficiently. Its yield is linearly dependent on the driving pulse energy. A longer laser pulse duration can increase the THz yield obviously (8). The THz yield induced by a two-color laser field in nonliquid media depends on the relative phase (i.e., the phase delay) between the field components, and increases quadratically with the increase of the driving laser pulse energy. However, for liquid media, in addition to this phase-modulated contribution to the THz yield, an unmodulated THz signal which depends linearly on the laser pulse energy was measured (10). These studies open up a subarea of THz physics, and provides a potential direction for investigating THz devices.

Compared to the rapid progress in the experimental studies, theoretical development in liquid TWG is still in its infancy. Due to the complexity of electron dynamics in liquids, the physical mechanism behind liquid TWG is still not clear. There are some models to interpret TWG in other media, such as ponderomotive dynamic model (16–18), four-wave mixing model (19, 20), photocurrent model (21–23), full quantum model (24, 25), etc. Although these models can be used to explain TWG in gases, solids, and plasma, it is inappropriate for them to be directly generalized to disordered liquids. For example, the photocurrent model and four-wave mixing model predict that TWG cannot be efficiently produced in monochromatic lasers, contrary to the experimental results in liquids (7, 8). Essentially, the liquid molecule can not be treated as an isolated individual due to the molecule-to-molecule interaction in condensed phases. Moreover, the crystal energy-band theory does not work because the liquid is not periodic. These elements make the theoretical study of ultrafast dynamics in liquids extremely difficult. Currently, all the theoretical efforts are based on plasma dynamics plus interface effects (12) when the laser intensity is very high. However, the structure of liquids is completely neglected, thus the microscopic mechanism of TWG in the bulk liquid phase is still a mystery, which is crucial to understand the above features in liquid films.

Theoretical Model

In this article, a shift-current model is proposed to interpret the microscopic mechanism of TWG in liquids. As illustrated in Fig. 1, the eigenstate functions in liquids are localized in several molecular ranges due to the effect of disorder, which is known as Anderson localization (26). The laser-induced transition between different localized states is accompanied by the spatial displacement, which means that the current in liquids can be thought of as a shift current. From the dipole moment distribution in the lower right corner of Fig. 1, the energy differences between these states are in the THz range, which makes this unique current the origin of the TWG in liquids in monochromatic lasers. Even though we emphasize the contribution of the valence state, ionization process is necessary, otherwise there will be no net currents in a fully occupied valence band in liquids.

Fig. 1.

Sketch of laser-liquid interaction to generate THz waves. The eigenfunctions of the disordered liquid water are localized. The laser field induces the transition of these states, i.e., shift current. The energy difference between them is in the THz range as illustrated in the Lower Right, leading to THz emissions.

Numerical Method.

The current from the localized states can be numerically calculated by the acceleration of the bound states in the time-dependent Schrödinger equation (TDSE) (25, 27):

$$\begin{array}{c}\hfill J_A^{\mathit{intra}}\left(t\right)=-〈{\mathrm{\Phi }}^b(x,t)\left|\frac{\partial V}{\partial x}\right|{\mathrm{\Phi }}^b(x,t)〉,\end{array}$$ [1]

where b denotes the bound states, $\left|{\mathrm{\Phi }}^b(x,t)\right.〉=\sum _{i=1}^N\left|{\varphi }_i(x)\right.〉〈{\varphi }_i(x)|\psi (x,t)〉$, which is obtained by the projection method. In other words, the current can be regarded as the outcome of the accelerated motion of wave packets in the potential field. Moreover, the shift current is embedded in it, which can persist after the end of the laser because of the cumulative effect according to SI Appendix.

Analytic Method.

In order to deepen our understanding of the origin of THz radiation in disordered systems, we have developed an analytical approach for calculating the laser-induced current. It can be obtained by solving the Liouville equation of the density matrix (28). It is written as a superposition of a field-like current with laser frequency ω and a shift current with frequency ${\omega }_{\mathit{mn}}$:

$$\begin{array}{cc}\hfill J^{\mathrm{analytic}}\left(t\right)=& J_{\mathrm{field}-\mathrm{like}}\left(t\right)+J_{\mathrm{shift}}\left(t\right)\hfill \\ \hfill =& {\displaystyle \sum _n}{\displaystyle \sum _m}{\left|v_{\mathit{mn}}\right|}^2\frac{2\left(f_m-f_n\right)E_M}{{\omega }_{\mathit{mn}}({\omega }_{\mathit{mn}}+ħ\omega )}\left\{f_e\left(t\right)sin\left(\omega t\right)\right.\hfill \\ \hfill & +{\displaystyle \sum _{j=0}^N}{f}_e\left(jT+T/2\right)sin\left[{\omega }_{\mathit{mn}}\left(t-jT\right)\right]\hfill \\ \hfill & -{\displaystyle \sum _{j=0}^{N-1}}{f}_e\left(jT+T/2\right)\hfill \\ \hfill & \left.\times sin\left[{\omega }_{\mathit{mn}}\left(t-jT-T\right)\right]\theta \left(N-1\right)\right\},N=⌊t/T⌋,\hfill \end{array}$$ [2]

where ${\epsilon }_n({\epsilon }_m)$ is the $n(m)$th energy eigenvalue of the field-free Hamiltonian $H_0$, $v_{\mathit{mn}}$ is the velocity dipole matrix, ${\omega }_{\mathit{mn}}=\left({\epsilon }_n-{\epsilon }_m\right)$ is the energy difference, $f_n$ is the population of the nth eigenstate, $T=2\pi /\omega$ is the period of laser field, θ is the step function, ⌊⌋ is the rounding function, $E_M$ is the laser field strength, and $f_e(t)$ is the envelope of laser electric field. The weak magnetic components of the laser are neglected. The derivation process is from the simple case of DC field, then we extend it to a periodic field, finally to a pulsed laser field. Details may be referred to SI Appendix. In conjunction with Fig. 1, one can gain a clearer understanding of Eq. 2. The oscillation frequency of the shift current corresponds to the energy difference between two eigenstates, with their values in the THz frequency range, as evident from the dipole distribution in Fig. 1. From Eq. 2, one can also find that the shift current is proportional to the laser field strength, i.e., the THz intensity depends linearly on the intensity of the driving laser. The shift current also depends on the envelope, dipole, and the eigenenergy of the system, which can be used to study the nuclear quantum effects (NQEs). The polarization of the THz should be the same as the driving laser.

Results and Discussion

Comparison of Analytic and Numerical Methods in 1-D Liquid Model.

To study this shift-current model in detail, a solvable 1-D disordered atomic chain is used in the simulation. This chain embodies the essential characteristic of the short-range order and the long-range disorder of liquids, which has been successfully used to interpret high-order harmonic generation in liquids. The details of the numerical method can be found in ref. 27. After solving TDSE, the THz spectrum can be obtained by the Fourier transform of the total current $J_{\text{total}}\left(t\right)=〈\psi \left(x,t\right)|p|\psi \left(x,t\right)〉$, or the current in Eqs. 1 and 2 (25, 27). In our simulation, the laser pulse is a Gaussian-envelope field, the intensity is $3\times {10}^{12}$$\mathrm{W}/c{m}^2$, the wavelength $\lambda =800$$\mathrm{nm}$, the pulse duration [full width at half maximum (FWHM)] ${\tau }_{\omega }=58$ fs, the time step is 0.1 atomic unit (a.u.), and the total time is 500 optical cycles (o.c.). Fig. 2A illustrates the current obtained by Eqs. 1 and 2, respectively. One can observe that they agree with each other qualitatively. The difference in amplitude comes from the approximation of population $f_n$ in Eq. 2, indicating the validity of both methods. To distinguish the contributions of different components of the current, we decompose the current $J^{\mathrm{analytic}}(t)$ into a field-like current $J_{\mathrm{field}-\mathrm{like}}(t)$ and a shift current $J_{\mathrm{shift}}(t)$, as illustrated in Fig. 2B. In Fig. 2C, we also present the THz spectra by the total current in TDSE and the currents in Eqs. 1 and 2. Apparently, one can find a radiation in the range of $0\sim 5$ THz that is in qualitative agreement with the experimental results (8, 10). As we expect, the total THz radiation mainly originates from the intracurrent in the valence band. The analytical result by Eq. 2 with a small frequency shift roughly agrees with the numerical results. This deviation is attributed to the approximation that the energy difference remains a constant during the evolution process. However, they will be shifted in laser fields by the Stark effect. From Fig. 2C, the contribution of $J_{\mathrm{field}-\mathrm{like}}(t)$ is significantly smaller than that of $J_{\mathrm{shift}}(t)$. In other words, THz radiation is primarily generated by the shift current. Fig. 2D shows the THz spectrum under different laser intensities by Eq. 1. An important feature in Fig. 2D is that the peaks in the THz spectrum rise evenly with the same shape as the laser intensity increases.

Fig. 2.

(A) Laser-induced current by the TDSE method and analytic method. (B) Component decomposition of the laser-induced current by the analytic method. The laser intensity is $3\times {10}^{12}$$\mathrm{W}/c{m}^2$, ${\tau }_{\omega }=58$ fs. (C) THz spectra obtained by the currents in (A), (B), and the total current $J_{\text{total}}\left(t\right)$. (D) THz spectra and THz yield under different laser intensities. The normalized THz yield is obtained by the integration of spectrum in the range of 0 to 20 THz. The laser parameters can be found in the main text.

The inset in Fig. 2D is the THz yield as a function of the laser intensity. Obviously, the energy dependence is a linear function which is totally in agreement with the prediction in Eq. 2 and the striking experimental measurements (8, 9).

Liquid THz Radiation in Monochromatic Lasers.

In order to further test the validity of this theory, we have to rely on the method in Eq. 2 to compare with experimental results since the TDSE simulation for real liquids is impossible currently. The THz spectra of water by Eq. 2 and the experimental measurement (8) are shown in Fig. 3A. Apparently, the THz spectrum by our method reproduces quantitatively the experimental result. In the calculation, the dipoles are obtained via the path integral molecular dynamics (PIMD) (29, 30) and the Vienna Ab initio Simulation Package (VASP) (31). For more detailed computational information, please refer to SI Appendix. The NQEs of liquid water have received more attention. In refs. 32 and 33, the effect can be detected via the THz absorption spectrum. However, the isotope effect of water in the THz emission spectrum has not been reported, to our knowledge. This effect can be studied in Eq. 2 by using the ${\omega }_{\mathit{mn}}$, $v_{\mathit{mn}}$ obtained by PIMD plus VASP simulations of H₂O and D₂O. The total time duration for calculating the shift current is 6000 o.c. As Fig. 3A shows, the THz radiation of light water is stronger than that of heavy water in $0\sim 1$ THz. However, the THz spectrum of heavy water is slightly broader than that of light water. The reason is that the NQEs of light water are stronger than that of heavy water, leading to the range of ${\omega }_{\mathit{mn}}$ and the corresponding $v_{\mathit{mn}}$ in Eq. 2 different in H₂O and D₂O. This phenomenon is worth of further experimental confirmation which is important for us to learn the NQEs of liquid water. In conclusion, liquid TWG in monochromatic laser fields is induced by the shift current oscillating in the THz range, which vanishes in the laser-gas and laser-plasma interaction.

Fig. 3.

(A) Comparison of the THz spectra from liquid water by our theory and the experimental measurement. The laser pulse energy is 0.4 mJ. (B) THz yield as a function of the optical pulse duration. The experimental data are adopted from ref. 8.

To further verify the feature of TWG in liquids, we investigate the relationship between the THz energy and the pulse duration. The dependence in the liquid phase is different from that in the gas phase, which was demonstrated in related experiments (8, 14, 15). In order to better simulate the experimental results, an electric field whose envelope changes with the chirp is employed. The laser field is obtained by solving the nonlinear Schrödinger equation in media (34).

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E(t)=& \frac{E_0}{\sqrt{2{\phi }^{″}}}\frac{1}{\sqrt[4]{{\alpha }^2+{\beta }^2}}{e}^{-\frac{\alpha {\beta }^2}{{\alpha }^2+{\beta }^2}{t}^2}cos\left({\omega }_{0}t+\frac{{\alpha }^2\beta }{{\alpha }^2+{\beta }^2}{t}^2\right),\hfill \end{array}\end{array}$$ [3]

where $\alpha =\frac{1}{2}{\left(\frac{2\sqrt{2}\sqrt{ln2}}{{\tau }_p}\right)}^2$, $\beta =\frac{1}{2{\phi }^{″}}$, ${\tau }^{\prime }=\sqrt{1+\frac{a{\phi }^{″2}}{{\tau }_p^4}}{\tau }_p$, $a=64{\left(ln2\right)}^2$, ${\tau }_p$ is the duration of the chirp-free pulse, ${\tau }^{\prime }$ is the duration of the broadening pulse, and ${\phi }^{″}$ is the product of dispersion coefficient and propagation distance. The first-order dispersion coefficient ${\phi }^{\prime }$ and additional phase are ignored. In our calculation, ${\phi }^{″}$ can be derived according to the expanded pulse width. This formula describes the process of optical pulse broadening when the optical energy is fixed.

In our simulation, the laser intensity is $4.45\times {10}^{14}$$\mathrm{W}/\mathrm{c}{m}^2$, corresponding to the optical pulse energy $0.4$ mJ in the experiment (8). From the experimental curve in Fig. 3B, the THz yield first increases and subsequently decreases as the duration becomes longer. For the longer duration, the simulation by Eq. 2 agrees well with the experiment (8). In the case of short pulses, the peak laser intensity is high, and the deviation between theory and experiment may arise from the interplay of plasma effects and absorption by the media. Actually, the THz yield is controlled by the laser duration which further affects the amplitude of the post-pulse shift current as presented in Eq. 2. So, it can be considered as a cumulative effect of the shift current which is modulated by the optical pulse duration. In ref. 14, this phenomenon was explained as the result of the cascade ionization in liquids. Here, our model provides another perspective to understand the effect of optical pulse duration.

Liquid THz Radiation in Two-color Lasers.

TWG in the dichromatic field is practical for developing THz devices. A lot of experimental works demonstrated that the THz yield of gas media in the two-color field is much higher than that of the monochromatic field (20). Theoretically, the four-wave mixing model and the photocurrent model roughly interpret the mechanism of the THz yield enhancement in two-color fields (20, 21, 35). However, the modulated and unmodulated THz waves in liquid media in two-color fields are detected in the experiment (10). In Fig. 4A, the gray part corresponds to the unmodulated THz component, whereas the solid blue and dash red curves depict the modulated THz component by our theory and the experimental measurements, respectively. In addition, the results indicate that there were two significantly different THz energy dependencies in a water film (10). The photocurrent model (23) could be used to explain the modulation of the THz field. However, it is a mystery why the unmodulated THz field was also observed. To simulate the THz spectra in two-color fields, the laser field used in our work is $E(t)=E_{\omega }{e}^{-4ln2\frac{t^2}{{\tau }^2}}cos(\omega t)+E_{2\omega }{e}^{-4ln2\frac{{(t+\delta /\omega )}^2}{{\tau }^2}}cos(2\omega t+\delta )$, and δ is the relative phase. Considering that the energy of the doubling frequency field is about $10\%$ of the fundamental frequency field, the strength of the second field follows $E_{2\omega }=0.3E_{\omega }$. In order to explain the mechanism, we propose a more generalized model in which both photocurrent from quasi-free states and shift current from bound states are taken into account. The THz field in two-color fields can be written as (36–38)

$$\begin{array}{c}\hfill E_{\mathrm{THz}}(t)=E\left(t\right){\int }_{-\infty }^{t}w\left(t^{\prime }\right)d{t}^{\prime }+\frac{\partial J_{\mathrm{shift}}\left(t\right)}{\partial t},\end{array}$$ [4]

Fig. 4.

(A) THz yield as a function of the phase in the two-color field. The wavelength of the fundamental laser is 800 nm, the intensity is $4.45\times {10}^{14}$$\mathrm{W}/\mathrm{c}{m}^2$. The intensity of the double frequency laser is $4\times {10}^{13}$$\mathrm{W}/\mathrm{c}{m}^2$. The yield of photocurrent $Y_{\mathrm{photocurrent}}$ and shift current $Y_{\mathrm{shift}-\mathrm{current}}$ as a function of phase delay is presented in (B) and (C), respectively. (D) Dependence of the unmodulated and modulated THz yield on the laser intensity. The experimental data are adopted from ref. 10.

where $w(t)$ is the Keldysh ionization rate (39, 40). The first term is the contribution of the photocurrent, and the second term is the contribution of the shift current. The ionization potential and effective mass are calculated by a band structure of a 60-molecule supercell of liquid water using density functional theory (DFT) in the generalized gradient approximation (GGA) of Perdew, Burk, and Ernzerhof (PBE) (41, 42). $J_{\mathrm{shift}}\left(t\right)$ is the shift current in two-color fields which is given by taking into account the contribution of the $2\omega$ field in Eq. 2 as shown in SI Appendix. We present the THz yields by Eq. 4 as a function of the phase delay in Fig. 4A. One can find that the theoretical results agree well with the experimental observation in ref. 10.

In order to further analyze the sources of unmodulated contributions, we respectively computed the phase-dependent THz yield of the photocurrent and shift current. In Fig. 4B, the THz yield of photocurrent is fully modulated which aligns with our expectations. In Fig. 4C, the modulation of THz yield from the shift current is about 10%, indicating that the unmodulated THz radiation in liquids is attributed to the shift current. As illustrated in Fig. 4D, the energy dependence of the modulated THz field is a quadratic function, while the unmodulated one is a linear dependence. The former agrees with the expectation of the photocurrent model in ref. 22, while the latter agrees with the results by shift current in Eq. 2. This conclusion is in agreement with the experiment in ref. 10, which further verifies that the photocurrent plus shift-current models are the origin of TWG in two-color laser fields. This model can degenerate into the traditional photocurrent model in two-color fields for gases where the shift current vanishes.

Summary and Outlook

In summary, we addressed the fundamental question of the microscopic mechanism of THz wave generation in bulk liquids by including the structure information. In the case of monochromatic driving lasers, it demonstrates that the shift current is the origin of THz radiation. It answers why the THz field can be produced in liquids in monochromatic lasers. It can interpret quantitatively the THz energy dependence of driving laser intensity and duration. In addition, we find that the nuclear quantum effects play very important roles in THz emission from H₂O and D₂O, which has not been studied, to our knowledge. In the case of two-color driving lasers, we find that the THz signals originate from both photocurrent and shift current. The former is modulated as a function of phase delay, while the latter is almost unmodulated. These numerical and analytical results in this work are in good agreement with the experiments, which helps us understand the related phenomena of THz radiation from liquids. It provides a key to investigate electron dynamics in liquids.

Materials and Methods

The TDSE of the 1-D liquid chain (27) is solved numerically. The atomic spacing follows a truncated normal distribution. The number of atoms is 128, the average spacing is 10 a.u., and the SD σ is 1.4. The Crank-Nicholson method with periodic boundary conditions is used to evolve the wave function. The time step is 0.1 a.u., and the grid spacing is 0.1 a.u.

The PIMD (29, 30) is simulated in a supercell of 64 water molecules at 300 K with one standard atmospheric pressure. The time step is 0.25 fs, and the total duration is 4 ps. 32 beads are used. The DFT part is calculated by VASP with core electrons and nuclei described by the projector-augmented wave (PAW) method (43). Five configurations are superposed to obtain the converged THz spectrum.

Supplementary Material

Appendix 01 (PDF)

Dataset S01 (XLSX)

Data, Materials, and Software Availability

All other data are included in the manuscript and/or SI Appendix. Previously published data were used for this work (8, 10).