## Abstract

Under strong electromagnetic excitation, electron–hole (e-h) pairs may be generated in solids, which are subsequently driven to high energy and high momentum, producing high harmonics (HH) of the driving field. The HH efficiency depends on the degree of coherence between the driven electron and hole created by the laser field. Here, we disrupt this e-h coherence in an atomically thin semiconductor by photodoping via incoherent e-h pairs. We observe a strong, systematic harmonic order-dependent intensity reduction. This trend is explained by an exponential decay of the inter-band polarization, proportional to the sub-cycle excursion time of the e-h pair. Our study provides a platform to probe the importance of many-body effects, such as excitation density-dependent decoherence time for strongly driven electrons without the need of ultrashort laser pulses.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. INTRODUCTION

High-harmonic generation (HHG) is a coherent optical process [1–6] in which the incident photon energy is up-converted to the multiples of its initial energy. Under the influence of a strong laser field, electron-hole (e-h) pairs are generated in solids and subsequently driven to high energy and momentum within a fraction of the optical cycle. These dynamics encode the band structure [4,7–12], including Berry curvature, valleys, and non-trivial topological properties [13–20] of the source material, through both intraband current and interband polarization, into the high-harmonic spectrum. The underlying microscopic dynamics has been studied intensively during the past decade [11]. First, high harmonics (HHs) can be generated from the driven nonlinear intra-band current, associated with non-parabolic bands [4,5,7,21–23]. Second, HHs can arise from the dipole-response of the e-h pairs [21,22,24–28], which produce an interband polarization. The two sources are shown in Fig. 1(b) with the blue and red arrows, respectively. The unique feature of the interband channel is the particular sensitivity to coherence between the excited electrons and their associated hole, which is different from common interband-based incoherent processes, such as fluorescence emission.

A particular feature of solid-state systems that contrasts with the well-studied response in gases is that the above-mentioned coherence between the driven e-h pairs may be readily disrupted, either through intrinsic mechanisms, such as scattering with other coherently driven e-h pairs, or though extrinsic mechanisms, such as scattering with crystal defects, impurities, or photogenerated charge carriers. These processes can for example randomly change the phase of the interband polarization and thus generate incoherent charge carriers and impede interband HHG [29–31].

Numerical simulations using a phenomenological dephasing time ${T_2}$ for the e-h coherence suggest a clear high-harmonic (HH) order-dependent intensity reduction for decreasing e-h coherence [30–33]. which might allow to extract the e-h dephasing time from measured HH spectra [30].

While the constant depletion of the HH yield in ZnO under additional carrier injection has been demonstrated [34] and attributed to phase space-filling effects, experimental studies demonstrating and probing the e-h coherence time for strongly driven electrons have yet to be developed.

Here we intentionally disrupt the e-h coherence through photodoping and probe the underlying dynamics of the HHG process to probe the coherence lifetime in a controlled manner. As shown schematically in Fig. 1(a), we use two laser pulses: a strong mid-infrared (MIR) that drives the HHG process and a relatively weak visible pump pulse that perturbs the HHG process by resonantly injecting charge carriers from the valence to the conduction band.

## 2. EXPERIMENT

In our experiment, we use singly crystalline monolayer molybdenum disulfide (${\rm{Mo}}{{\rm{S}}_2}$, direct bandgap: 1.8 eV) as a model system to study the role of e-h dephasing in the HHG process. Monolayer ${\rm{Mo}}{{\rm{S}}_2}$ serves as an ideal material system as it has a reduced dielectric screening, strong many-body Coulomb interactions [35], and it has a strong dipole coupling with a band gap in the visible range, which results in ten distinct HH ranging from 1 eV to 4 eV when pumped with mid-infrared pulses having field strengths on the order of 1 V/nm [13,26]. A representative HH spectrum for the 5 µm drive field (cycle period of $T = 16.7 \;{\rm{fs}}$) with and without the 660 nm pump pulse is shown in Fig. 1(c). For a peak electric field strength of 0.6 V/nm (intensity $7 \times {10^{10}} \;{\rm{W/c}}{{\rm{m}}^2}$), we measure even- and odd-order harmonics, ranging from 5th to 16th harmonic order. The fluence $F$ of the resonant pump at 660 nm with a pulse duration of about 100 fs is selected below the damage threshold of ${\rm{Mo}}{{\rm{S}}_2}$. The delay in the MIR pulses with respect to the visible pulses is set to $\Delta t = 1 \;{\rm{ps}}$, to avoid their temporal overlap and keep the amount of photodoping constant. For larger delay times, we observe that the number of photo-injected carriers drops due to various relaxation processes (see Supplement 1). The observation of ten distinct harmonic orders enables us to obtain quantitative information about the response of different harmonics to various photodoping conditions.

When the resonant pump pulse is applied, we observe that all measured harmonics are significantly suppressed, with stronger suppression of the higher harmonic orders. The right panel shows the integrated spectral intensity as a function of the HH order, normalized to the condition without photodoping. Whereas the 5th order harmonic is reduced by 30%, higher orders are suppressed up to 85% for the 16th harmonic order. This characteristic trend is used below to extract the dephasing time ${T_2}$.

To systematically study the dependence of the HH, we focus on the variation of the initial photocarrier concentration $n$ and measure the corresponding HH spectrum from the delayed non-resonant strong-field drive. To obtain $n$, we first determine the absorbance as a function of pump fluence and assume that each absorbed photon generates an e-h pair. We then calculate $n$ at $\Delta t = 1\; {\rm{ps}}$, by taking rapid exciton–exciton annihilation into account (see [36] and Supplement 1). Figure 2(a) shows the obtained HH spectrum as a function of pump fluence. We observe that all measured HH orders decrease with increasing $n$. To compare different orders to one another, we plot the integrated and normalized spectral intensity in Fig. 2(b) for six different $n$. For $n = 0.96 \times {10^{12}}\;{\rm{c}}{{\rm{m}}^{- 2}}$ (blue circles) the HH intensity is reduced by 8% for the 5th harmonic and decreases almost monotonically to 15% for the highest observed harmonic. Increasing the photodoping to $n = 6.9 \times {10^{12}}\;{\rm{c}}{{\rm{m}}^{- 2}}$ (purple circles) results in a decrease of 30% for the 5th harmonic up to 65% for the 15th harmonic.

## 3. MICROSCOPIC PICTURE OF HHG UNDER PHOTODOPING

To investigate the role of photodoping in the HHG process, we begin by reexamining the details of the microscopic mechanism for solid-state HHG, illustrated in Fig. 1(b). Under the presence of the laser field, e-h pairs are mostly generated near the band gap at the $K$ and $K^\prime $ points (Step 1) where the dipole coupling between valence and conduction bands is maximized. Subsequently, the field transiently drives them to high energy and momentum states (Step 2), giving rise to coupled intraband motion (current, blue arrow), interband transitions (polarization, red arrows), and scattering events. The right panel in Fig. 1(b) shows schematically the interband polarization response during a quarter of an optical cycle. First, a polarization oscillating at low frequencies is generated, given by the band-gap energy, followed by high frequencies as the e-h pair is subsequently driven to higher energies. This polarization response is the dominant source for harmonics around and above the band gap (see Supplement 1), with an emission time defined by the recombination step in the semi-classical picture (Step 3) [8]. In this scheme, we note that harmonics originating from longer e-h trajectories are more sensitive to decoherence as the corresponding excursion time allows for more scattering events. By measuring the order-dependent variation in the harmonic intensity, we can infer the lifetime of the e-h coherence.

## 4. NUMERICAL MODEL SIMULATIONS

To extract a dephasing time from the order-dependent reduction in the HHG intensity, we simulate HHG using a tight-binding model Hamiltonian with a hexagonal graphene-like band structure with two different sub-lattices. Such a Hamiltonian exhibits broken inversion symmetry, leading to the production of both odd- and even-order harmonics [37]. The electron dynamics is determined by solving numerically the semiconductor Bloch equations (SBE) in the presence of the laser field, using the Houston representation [38]. Importantly, this formalism allows us to introduce phenomenologically an e-h dephasing time constant ${T_2}$ (see Supplement 1 and [31]). We assume that prior to the MIR field, all electrons were in the valence band. We note that the parameter ${T_2}$ accounts for both electron–electron and electron–phonon scattering as decoherence processes leading to the decay of the e-h polarization.

The results of the simulations are first analyzed by plotting time–frequency spectrograms, which provide an intuitive visualization of the dynamics of the e-h trajectories. Figures 3(b)–3(d) show the results for the dephasing times of ${T_2} = 10,3, {\rm{and}}\;2 \;{\rm{fs}}$, respectively. For our experimental parameters, and particularly for harmonics above the band gap, the interband polarization dominates over the intraband current, we thus limit our discussion in Figs. 3(b)–3(d) to the inter-band polarization. For a dephasing time longer than a quarter of the optical cycle, i.e., ${T_2} = 10 \;{\rm{fs}}$ [Fig. 3(b)], the e-h pair maintains its coherence up to the point of maximal e-h energy separation, i.e., 8 eV (blue star). Here, the e-h pair is generated when the electric field amplitude $|E(t)|$ has a peak (red brown) and the vector potential $|A(t)|$ has a minimum, and reaches its maximal energy-spacing after a quarter of the optical cycle, i.e., when $|A(t)|$ has its peak (blue star). Reducing the dephasing time to 3 fs in Fig. 3(c) and 2 fs in Fig. 3(c) suppresses the coherence for the longest short trajectories and thus, the maximal energy achieved is exponentially reduced. We observe for ${T_2} = 3 \;{\rm{fs}}$ a maximal photon energy of about 7 eV, which drops down to 6 eV for ${T_2} = 2 \;{\rm{fs}}$.

Figure 3(e) shows the total calculated HH spectrum for different ${T_2}$ values. Within our model a shorter dephasing time clearly suppresses higher orders more strongly.

To evaluate the trends for suppression across the different harmonic orders, we again integrate the spectral intensity and normalize it to ${T_2}({n_0}) = T/4$, as indicted in Fig. 3(f). Different colors show various dephasing times from ${T_2} = 4.25 \;{\rm{fs}}$ (red) to ${T_2} = 0.75 \;{\rm{fs}}$ (gray), with respect to ${T_2}({n_0}) = 4.5 \;{\rm{fs}}$. The intrinsic dephasing time ${T_2}({n_0})$ was set to about one-quarter of an optical cycle, as this value best reflects our experimental observations (see Supplement 1 for different normalizations). We note that the shape of the drop is also captured by a simple analytic description, which relays on the exponential decay of the interband polarization caused by e-h dephasing. The intensity of each harmonic order can be described as

We assume that ${\tau _{{\rm{HH}}}}$ scales linearly as a function of harmonic order. Equation (2) is used to qualitatively explain the harmonic order dependence in Figs. 2(b) and 3(f) (dashed lines). We note that the analytic model is jut used for a qualitative analysis. A full quantitative analysis is done by the full SBE model simulation.

Now, we compare the numerical simulation with the experimental results, shown in Fig. 2, and relate $n$ and ${T_2}$. As each curve in Fig. 3(f) is described by the single ${T_2}$ parameter, we can fit the numerical results directly to the experimental data. Thus, we take all measured harmonics for a fixed photodoping into account [Fig. 2(a)] to determine one characteristic ${T_2}$. Importantly, the numerical model captures the experimental data quite well with just one free parameter. This allows us to determine the relationship between $n$ and ${T_2}$ in Fig. 3(g). By increasing $n$, we observe that ${T_2}$ decreases by 40% for the maximal photocarrier doping of $6.9 \times {10^{12}} \;{\rm{c}}{{\rm{m}}^{- 2}}$.

Based on the simulations, we estimate an upper boundary of e-h coherence as a quarter of the optical cycle (i.e., 4.5 fs) for no or weak photodoping, which decreases to 2.5 fs for $n = 6.9 \times {10^{12}} \;{\rm{c}}{{\rm{m}}^{- 2}}$. For ${T_2}$ longer than a quarter of an optical cycle, the one-to-one relation between excursion times of the e-h pairs and the emitted photon energies will be lost, resulting in relatively flat or non-monotonic reduction of the HH yield with increasing order, as shown in Supplement 1.

## 5. CONCLUSION

In the present work, the effect of photodoping on the HHG process is explained as a consequence of an enhancement in the rate of e-h dephasing. Dephasing mainly affects interband-dominated harmonics around and above the band gap, as supported by our numerical simulations. In contrast, intra-band current may dominate HHG for harmonics at energies well below the band gap [30]. Assuming that scattering among the electrons does not decrease the intraband current significantly, these harmonics should actually be enhanced under photodoping because of the higher carrier densities, as also suggested in [34,39]. For still higher applied pump intensities, i.e., when $n \gtrsim {10^{13}}\, {\rm{c}}{{\rm{m}}^{- 2}}$, band-gap renormalization in monolayers [40], as well as state filling for peak pump fluence $F \gt {F_{{\rm{sat}}}} = 0.13 \;{\rm{mJ/c}}{{\rm{m}}^2}$ might become important for the overall HH intensity. The monotonic depletion of the HH intensity as a function of harmonic order has also been observed in our auxiliary measurements of bulk ${\rm{Mo}}{{\rm{S}}_2}$, indicating that strong excitonic effects and band-gap renormalization are not primarily responsible for the observed monotonic reduction of the HH intensity. Within this work we have treated dephasing phenomenologically within a single-electron picture, as used in SBE. Our work shows that many-body effects such as dephasing effects and scattering are crucial in solid-state HHG. We expect that future model simulations based on multi-electron systems might allow further insights to correlated electron dynamics in various systems.

In summary, we employed an all-optical approach based on HHG to measure the coherence of strongly driven e-h pairs in solid materials on sub-cycle timescales. In the experiment, we control the decay rate of the e-h coherence by photodoping monolayer ${\rm{Mo}}{{\rm{S}}_2}$ and measure the corresponding HHG response. This doping enables us to reveal the charge carrier dynamics and their correlations in the HHG process. We find that an increased charge-carrier concentration reduces the overall efficiency of HHG, with more prominent effects observed as the HH order increases. Within the framework of the semiconductor Bloch equations, we attribute this observation to enhanced dephasing, which produces an exponential decay of the inter-band polarization that scales with the excursion time of the e-h pair, rather than the phase-space filling effect. Our results highlight the importance of many-body effects, such as density-dependent decoherence in HHG, and advance understanding of coherence in solid-state HHG, with ramifications for compact all-optical solid-state spectroscopy, as well as application such as short-wavelength light sources and attosecond pulse generation. It also opens up the possibility of determining dephasing times in solids without the requirements of ultrashort laser pulses.

## Funding

U.S. Department of Energy; W. M. Keck Foundation.

## Acknowledgment

We thank Giulio Vampa, Ignacio Franco, Azar Oliaei Motlagh, and Hamed Koochaki Kelardeh for fruitful discussions. F.L. acknowledges support from a Terman Fellowship and startup funds from the Department of Chemistry at Stanford University. Y.K. acknowledges support from the Urbanek-Chorodow Fellowship from Stanford University. C.H. acknowledges support from the Humboldt Fellowship and the W. M. Keck Foundation. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division through the AMOS program.

## Disclosures

The authors declare no conflict of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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