We report on experimental investigation of nonperturbative high harmonic generation (HHG) in monolayer MoS2 in the ultraviolet spectral region driven by mid-infrared light. We study how the HHG is influenced by pre-excitation of the monolayer using resonant and near-resonant pulses in a pump–probe-like scheme. The resonant light creates high density exciton population. Due to ultrafast dephasing caused by electron–electron scattering, the HHG is suppressed in the presence of pre-excited carriers. In the case of near-resonant excitation with photon energy below the exciton transition, the dynamics of the observed suppression of the HHG yield contains a fast component, which is a consequence of momentum scattering at carriers, which are excited by two-photon transition when the two pulses temporally overlap in the sample. This interpretation is supported by comparing the experimental data with theoretical calculations of the two-photon absorption spectrum of the MoS2 monolayer. This work demonstrates a possibility to control HHG in low-dimensional materials on ultrashort timescales by combining the driving strong-field pulse with a weak near-resonant light.

High harmonic generation (HHG) in solids1–6 is a nonperturbative nonlinear optical process, during which a coherent light wave with high amplitude of electric field and low photon energy interacts with a solid-state material, typically a crystal. As a result, electron–hole wave packets are promoted to conduction and valence bands via quantum tunneling. This process occurs during a short time window close to the maxima of the oscillating field. The duration of the tunneling excitation window is typically few hundreds of attoseconds to single femtoseconds, depending on the frequency of the driving light. Subsequently, the carriers are coherently accelerated in the crystal by the oscillating field and propagate far from equilibrium positions. The electron and hole can eventually recombine, leading to generation of high energy photons via the interband mechanism of HHG.3–5,7 Due to the nonparabolicity of electron and hole dispersion in the material, the harmonically driven carriers undergo anharmonic motion, leading to the so-called intraband mechanism of HHG.8–10 HHG in solids has been used to investigate the band structure,11,12 Berry curvature,13 topological surface states,14 the dynamics of a photoinduced phase transition,15 or coherent phonon dynamics.16 

A crucial requirement for observation of a macroscopic wave containing harmonic frequencies is the coherence of the individual microscopic emission sources that contribute to the radiated field.17,18 However, when the charge carriers in a solid are accelerated to high energies, the momentum scattering causes fast decoherence. The most important scattering processes are electron–phonon scattering, electron–electron scattering, and scattering at ionized impurities. While the electron–phonon scattering times for electrons with low kinetic energy are typically long (hundreds of femtoseconds to picoseconds) compared to the time period of the mid-infrared light (several femtoseconds), the electron–phonon and electron–electron scattering times can become much shorter for electrons that are accelerated to high kinetic energies and/or at high densities of excited carriers. The latter process can be controlled independently of the HHG process using a second resonant pulse exciting the carriers prior to the impact of the strong infrared pulse, which drives the HHG. The suppression of the HHG yield by excited carriers was recently observed in ZnO,19 carbon nanotubes,20 or MoS2 monolayer,21–23 where the authors used infrared pulses to drive HHG. The carriers were excited by a resonant pump pulse19,21,22 or by a pulse with photon energy higher than the bandgap.23 HHG in ZnO was also modulated by pulses with photon energy lower than the bandgap,24 and the ultrafast response was interpreted as a consequence of nonlinear frequency mixing present in the sample for zero time delay between the two pulses.

In this paper, we show an experimental study of the effect of resonant and near-resonant pre-excitation on the HHG process in monolayer MoS2.25 The monolayer has an advantage of direct bandgap, strong optical nonlinear response, and the absence of propagation effects, which means that problems with phase matching and reabsorption are avoided. Therefore, this material is ideal for studying the microscopic mechanism of HHG and its control by optical pre-excitation. The harmonics are generated in a different regime than in previous studies.5,21–23,26–28 We use higher photon energy of the driving pulse of 0.62 eV (wavelength of 2000 nm) and observe harmonic spectra in the ultraviolet spectral region 2.7–6.2 eV (wavelength region of 200–450 nm). These photon energies correspond to interband transitions between higher lying bands in the MoS2 band structure,29 thus going beyond the two-band approximation. We observe that the HHG yield is suppressed by high density carriers excited in the sample prior or during the illumination by the strong infrared pulse, which drives the HHG. The dynamics of the change in the HHG yield as a function of the time delay between the pre-excitation and driving pulses differs significantly depending on the photon energy of the pre-excitation pulse. For non-resonant pre-excitation, the observed ultrafast suppression of the HHG yield is explained by momentum scattering of coherently driven electron–hole wavepackets at charge carriers produced by two-photon absorption induced simultaneously by the pre-excitation and the strong-field pulses. The spectrum of two-photon absorption in MoS2 is theoretically calculated and compared to the measured results.

In our experiments, we illuminate a MoS2 monolayer on the SiO2 substrate with mid-infrared pulses and measure the spectra of high harmonic frequencies reflected from the sample [see Fig. 1(a)]. The HHG spectra are measured as a function of the time delay between a femtosecond resonant or near-resonant pump pulse [tunable photon energy 1.25–2 eV and a full width at half maximum (FWHM) pulse duration of about 20 fs measured with spectral phase interferometry for direct electric field reconstruction (SPIDER)]30 and the mid-infrared high intensity pulse [central photon energy 0.6 eV, a FWHM pulse duration of 38 fs measured using third harmonic frequency-resolved optical gating (THFROG)31 and a peak intensity of up to 5 TW/cm2], which drives the HHG. Both pulses are generated from the output of a femtosecond ytterbium-based laser (Pharos SP-6W, Light Conversion, 170 fs, 1030 nm). The resonant and near-resonant pulses are generated in an in-house developed noncollinear optical parametric amplifier (NOPA), which uses femtosecond supercontinuum generated in a sapphire crystal as a seed and which is pumped by the second harmonics of the fundamental laser output (515 nm) generated in a BBO crystal. The mid-infrared pulses used to drive the HHG are obtained by the NOPA+DFG (difference frequency generation) setup, which is described in detail in Ref. 32. The time delay between the two pulses is controlled by a motorized translation stage (Thorlabs MTS25) with a minimal step size of 50 nm, which corresponds to the step in time of 0.33 fs. The spectra are measured using a grating spectrograph (Shamrock 163) and a cooled CCD camera (Andor iDUS 420). During the experiments, the samples are imaged in situ using an optical microscope setup to ensure the spatial overlap of the pre-excitation and the driving pulses and their position at the sample. We use linear polarizations of both pulses, which are independently controlled by broadband half wave plates (Thorlabs AQWP05M-980 and AQWP05M-1600). Our HHG detection setup has different sensitivities for two orthogonal polarization components due to the prisms used to filter out the third harmonic signal and due to the polarization dependence of the diffraction efficiency of the spectrometer grating. For this reason, in the measurement showing the dependence of the HHG yield on the direction of the polarization of the driving field, we detect the harmonic spectra separately for the two orthogonal polarization components, which are selected by a UV Glan-laser polarizer. The signals are corrected for different detection sensitivity, and the total HHG yield is obtained as their sum. The experiments are carried out at room temperature with the laser repetition rate of 25 kHz. The studied samples are prepared by gel-film assisted mechanical exfoliation from the bulk MoS2 crystal. The monolayers are then transferred to a SiO2 substrate, which is used due to its wide bandgap, leading to a negligible contribution to the generated harmonic spectra.

FIG. 1.

(a) Geometry of the experiment for investigation of the role of resonant and near-resonant excitation on the yield of high harmonic generation (HHG) in the MoS2 monolayer. (b) HHG spectrum generated in the MoS2 monolayer with the linear polarization of the driving field along the mirror symmetry axis of the crystal (armchair direction). (c) Dependence of the polarization unresolved HHG yield of each harmonic frequency on the orientation of the linear polarization of the driving field with respect to the mirror symmetry axis of the crystal.

FIG. 1.

(a) Geometry of the experiment for investigation of the role of resonant and near-resonant excitation on the yield of high harmonic generation (HHG) in the MoS2 monolayer. (b) HHG spectrum generated in the MoS2 monolayer with the linear polarization of the driving field along the mirror symmetry axis of the crystal (armchair direction). (c) Dependence of the polarization unresolved HHG yield of each harmonic frequency on the orientation of the linear polarization of the driving field with respect to the mirror symmetry axis of the crystal.

Close modal

The measured HHG spectrum for linear polarization of the driving pulse along the direction of the mirror symmetry axis of the MoS2 monolayer is shown in Fig. 1(b). We observe harmonic frequencies up to the 10th order corresponding to the photon energy of 6 eV. This is considerably higher photon energy than observed in previous studies of HHG in TMD materials.5,21–23,26 The reason is probably the shorter wavelength of the driving field compared to previous experiments in combination with a high field strength of about 3 GV/m used in our experiments, which can be applied to the monolayer due to the short pulse duration. The yield of HHG in the MoS2 monolayer and its polarization state is found to depend on the direction of linear polarization with respect to the mirror symmetry plane.26 In Fig. 1(c), we plot the HHG yield as a function of the angle θ between the direction of linear polarization of the driving light and the mirror symmetry plane of the monolayer. The direction of linear polarization is controlled by a rotation of a half wave plate. The two perpendicular polarization components (horizontal and vertical) of HHG radiation are detected separately. The resulting spectrum is obtained by summing the two polarization components with particular weights corresponding to the measured polarization dependence of the detection setup (prisms and spectrometer grating). The results thus represent the total HHG yield independent of the polarization state of the generated ultraviolet light. We observe harmonic orders of 5–9 at photon energies of 2.8–5.5 eV (higher photon energies are not transmitted through the polarizer used in the detection setup for these measurements). For photon energies below 4 eV, the maxima of odd orders are observed for θ = 30°, while the even orders have their maxima for θ = 0°. However, at higher photon energies, the maximum yield of even and odd orders switches. This behavior has been previously observed in the MoS2 monolayer, and it was attributed to the role of higher energy bands in the HHG process.29 

The main goal of this paper is to study the influence of the HHG yield by a resonant or near-resonant pump pulse applied to the sample prior to the illumination by the strong infrared pulse, which drives the HHG process. We measured the high harmonic spectra as a function of the time delay between the pump and the strong infrared pulses with the time step of 6.7 fs. The amplitudes of individual harmonic orders obtained by integrating the spectral window with the width of 0.3 eV around the maxima shown in Fig. 1(b) are plotted in Fig. 2(a). When the real carrier population is generated in the sample by a resonant pump pulse, the HHG yield decreases for all the harmonics. An interesting observation is the fact that the relative decrease is not monotonically changing with the harmonic order as observed in previous studies for harmonic photon energies up to 3.5 eV.21–23 The decrease in the HHG yield is caused by electron–electron scattering induced by the high density of excited carriers, which leads to extremely fast dephasing of the coherent electron–hole wave packets.21 The generated high harmonic radiation results from the macroscopic nonlinear current of coherently oscillating electrons. Once the quantum phase between the electron and the field of the driving infrared wave is lost, the electron does not further contribute to the generated coherent wave. As the density of the excited carriers decreases on picosecond time scales,21,22 the harmonic signal recovers to the value without the resonant pump after several tens of picoseconds.

FIG. 2.

(a) Relative change of the high harmonic yield for different harmonic orders as a function of the time delay between the resonant pre-excitation at photon energy of 1.91 eV and the infrared pulse. (b), (c), and (d) The same as in (a) with the pre-excitation pulse detuned from the exciton resonance to photon energies of (b) 1.65 eV, (c) 1.55 eV, and (d) 1.46 eV. The spectra of the excitation pulses are shown in the insets, where the energy of the lowest exciton resonance (1sA exciton) is indicated. The pump fluences are (a) 1.8 mJ/cm2, (b) 12.9 mJ/cm2, (c) 7.5 mJ/cm2 and (d) 3.8 mJ/cm2.

FIG. 2.

(a) Relative change of the high harmonic yield for different harmonic orders as a function of the time delay between the resonant pre-excitation at photon energy of 1.91 eV and the infrared pulse. (b), (c), and (d) The same as in (a) with the pre-excitation pulse detuned from the exciton resonance to photon energies of (b) 1.65 eV, (c) 1.55 eV, and (d) 1.46 eV. The spectra of the excitation pulses are shown in the insets, where the energy of the lowest exciton resonance (1sA exciton) is indicated. The pump fluences are (a) 1.8 mJ/cm2, (b) 12.9 mJ/cm2, (c) 7.5 mJ/cm2 and (d) 3.8 mJ/cm2.

Close modal

Unlike the other experiments in MoS2, we also focus on the experiments with the pump beam at photon energy below the lowest exciton resonance of MoS2 [data shown in Figs. 2(b)2(d)]. When the near-resonant pulse overlaps in time with the strong infrared pulse, the harmonic yield is significantly decreased. In this case, the strong suppression shows ultrafast dynamics, and the harmonic yield recovers after 80 fs to a value that is close to the yield without the pump pulse. The probability of carrier excitation depends on the photon energy and fluence. The results in Fig. 2 are measured for different pump fluences: (a) 1.8 mJ/cm2, (b) 12.9 mJ/cm2, (c) 7.5 mJ/cm2, and (d) 3.8 mJ/cm2. When normalized to the same fluence of the pump pulse, the suppression of the HHG yield is the highest for the resonant pump photon energy.

Figure 3(a) shows the maximum relative change of the harmonic emission yield as a function of the intensity of the off-resonant pulse for overlapping pump and infrared pulses (time delay 0 fs). The data are fitted by linear functions yn = anx. The harmonic intensity suppression is different for different harmonic orders; it is smallest for the 8th harmonic and for 5th, 6th, 7th, 9th, and 10th, it is monotonically increasing. After the short signal suppression, there is also a picosecond component of the change of HHG yield in the case of near-resonant excitation [see Fig. 3(b)]. This signal cannot be caused by the excitons generated by single-photon absorption process due to the fact that the photon energy is lower than the energy of the exciton resonance. The picosecond component is thus caused by carriers excited by two-photon absorption of the near-resonant pulse, which is confirmed by the quadratic dependence of its amplitude shown in the inset of Fig. 3(b).

FIG. 3.

(a) Relative decrease in the HHG yield for different harmonic orders as a function of the intensity of the off-resonant pump pulse at energy 1.46 eV. Measurements are performed for the two pulses overlapped on the sample (time delay 0 fs). The experimental data (points) are shown in log–log scale. The lines correspond to fits of the data with the linear functions yn = anx. (b) 5th harmonic yield as a function of the time delay between the pre-excitation and the infrared pulses for three different pump pulse intensities. The inset shows a relative decrease in the HHG yield 500 fs after pre-excitation for different harmonic orders as a function of the intensity of the off-resonant pump pulse.

FIG. 3.

(a) Relative decrease in the HHG yield for different harmonic orders as a function of the intensity of the off-resonant pump pulse at energy 1.46 eV. Measurements are performed for the two pulses overlapped on the sample (time delay 0 fs). The experimental data (points) are shown in log–log scale. The lines correspond to fits of the data with the linear functions yn = anx. (b) 5th harmonic yield as a function of the time delay between the pre-excitation and the infrared pulses for three different pump pulse intensities. The inset shows a relative decrease in the HHG yield 500 fs after pre-excitation for different harmonic orders as a function of the intensity of the off-resonant pump pulse.

Close modal

The ultrafast suppression of the HHG yield might be attributed to two effects. The first possible source is the optical Stark effect (OSE)33–37 induced by the near-resonant field. OSE was previously observed in 2D materials38–44 and leads to a transient shift of the excitonic resonance. The energy shift induced by OSE may increase the energy barrier that the electrons need to overcome during tunnel excitation, which is an important ingredient of HHG. As a consequence, the tunneling probability would be reduced, leading to a suppression of the harmonic yield. In two-level approximation representing a simplified model of the excitonic system in 2D transition metal dichalcogenides, the OSE induced energy shift scales with the field amplitude F, energy of the resonance E0, and the applied photon energy ℏω as ΔE|F|2E0ω. A theory of OSE based on the perturbative solution of semiconductor Bloch equations gives only a small correction to this dependence.43 The second possible source of the ultrafast HHG suppression is the enhanced generation of carriers due to two-photon excitation, which becomes strongly enhanced when the two pulses are present on the sample in the same time. The higher excited carrier density then leads to stronger suppression of the HHG via decoherence due to electron–electron scattering. To resolve the underlying physical mechanism, we study the dynamics of the HHG yield for different detuning of the near-resonant excitation from the exciton energy [see Figs. 2(b)2(d)]. Figure 4(a) shows the relative suppression of the harmonic yield as a function of pump peak intensity for three different near-resonant photon energies of the excitation pulse. The suppression is stronger as the detuning becomes larger. Therefore, the ultrafast HHG suppression cannot be explained by the OSE, for which the energy shift is indirectly proportional to the detuning energy. Another argument is based on the fact that the Stark energy shift in transition metal dichalcogenides is typically only several tens of meV.43 The estimated decrease in the tunneling probability induced by such a small increase in the exciton energy is only few percent, which is much less than the observed relative change of the HHG yield.

FIG. 4.

(a) The magnitude of the relative decrease in the HHG yield as a function of the detuning between the photon energy of the near-resonant pulse and the energy of the 1sA exciton resonance. The data are normalized by dividing the measured change of the HHG yield by the pump intensity (both in relative units). (b) The calculated absorption spectrum of MoS2 (black curve) and a sketch of two-photon generation of excitons for three different photon energies of the near-resonant pulse of 1.65 eV (dark red), 1.55 eV (red), and 1.46 eV (orange). The infrared photons with a photon energy of 0.62 eV are represented by the black arrows, while the near-resonant pump photons are represented by color arrows (the graph contains a break in the y-axis for clarity).

FIG. 4.

(a) The magnitude of the relative decrease in the HHG yield as a function of the detuning between the photon energy of the near-resonant pulse and the energy of the 1sA exciton resonance. The data are normalized by dividing the measured change of the HHG yield by the pump intensity (both in relative units). (b) The calculated absorption spectrum of MoS2 (black curve) and a sketch of two-photon generation of excitons for three different photon energies of the near-resonant pulse of 1.65 eV (dark red), 1.55 eV (red), and 1.46 eV (orange). The infrared photons with a photon energy of 0.62 eV are represented by the black arrows, while the near-resonant pump photons are represented by color arrows (the graph contains a break in the y-axis for clarity).

Close modal

This analysis suggests that the observed ultrafast suppression of the HHG yield induced by the near-resonant pump pulses is caused by the two-photon generation of excitons and the subsequent increase in the electron dephasing rate. The excitation process is illustrated in Fig. 4(b) together with the two-photon absorption coefficient in the MoS2 monolayer calculated by the theory presented in the following chapter. The excitonic levels45–47 form clear peaks in the two-photon absorption spectra. For the near-resonant pulse at 1.46 eV, the sum of photon energies for two-photon transition (infrared HHG driving pulse at 0.6 eV) gives 2.06 eV, which is partially overlapped with 1sB exciton resonance at EB = 2.03 eV. For the higher photon energies of 1.55 and 1.65 eV, the sum of the photon energies is shifting away from 1sB exciton resonance to the region, where the two-photon absorption coefficient is decreasing. This explains why the measured HHG suppression is stronger for lower pump photon energies corresponding to larger detuning from the exciton resonance.

The pump and the strong infrared pulses time durations are 20 fs and 38 fs (FWHM), respectively. However, the dynamics of the observed HHG suppression with near-resonant pump pulses of about 80 fs is approximately twice longer than the expected duration of the convolution of Gaussian envelopes, which is only about 43 fs. This can be explained by the fact that our pulses do not exactly resemble the Gaussian profile in time, but there are pronounced side wings in both pulses, leading to prolongation of the convolution function. The temporal response function of the setup is shown in Fig. 5, where we plot the measured high-order frequency mixing signal (dashed curve) in the phase-matched direction for the third order interaction ωpump + 2ωIR in comparison with the dynamics of HHG suppression (full curves).

FIG. 5.

Comparison of the transient change of the HHG yield with the near-resonant pump with a photon energy of 1.65 eV (solid curves) and the high-order frequency mixing signal (dashed curve).

FIG. 5.

Comparison of the transient change of the HHG yield with the near-resonant pump with a photon energy of 1.65 eV (solid curves) and the high-order frequency mixing signal (dashed curve).

Close modal

Recently, similar ultrafast suppression of the HHG yield has been measured in bulk ZnO.24 The phenomenon was interpreted by a decrease in the intensity of the infrared pulse, which is partially depleted in the crystal due to frequency mixing processes. The lower intensity of the infrared pulse should then lead to a suppression of the HHG. In our experiments, the HHG yield scales approximately with the third power of the intensity of the strong infrared pulse. To observe the suppression of the yield by 50%, the intensity of the infrared pulse would have to drop by ∼20%. However, the efficiency of nonlinear frequency mixing processes in the monolayer is only about 10−4 or less. Based on this argumentation, we conclude that the origin of HHG suppression in the time overlap region is the creation of electron–hole pairs by two-photon absorption of the weak near-resonant and strong off-resonant infrared pulses. This effect is not present in Ref. 24, where the sum of photon energies of the two pulses is not sufficient to create electron–hole pairs in ZnO via two-photon absorption and only multiphoton processes are available.

We consider the seven-band model of the TMD monolayer, which goes beyond the simplest two-band case (see the supplementary material for details). The rate of optical transitions, induced simultaneously by two photons, can be written as
(1)
Here, ⟨f|U|in⟩ is the matrix element of the two-photon process. It couples the initial state |in⟩ = |0⟩ (with occupied valence bands and empty conduction bands) and final exciton state |f⟩ = |q, n, τ, c, v⟩ (with the total momentum and discrete quantum number n) consisting of an electron in the conduction (c) band and a hole in the valence (v) band of the τ = ±1 valley. We skip the bands’ spin indices here because optical transitions conserve spin, i.e., couples only the bands with the same spin. Here, the operator U is responsible for the two-photon process. It corresponds to the second order processes in perturbation theory (see Refs. 48 and 49). The perturbation operator is taken in the form Hint(t) = −P · E, where P is the monolayer’s polarization operator and E is the in-plane electric field of incoming light pulses. Taking into account that optical pulses in the experiment are linearly polarized with polarization vectors e1, e2, field amplitudes |E1|, |E2|, and phases ϕ1, ϕ2, we write
(2)
For this particular case, the interaction term takes the form
(3)
Using the results of Refs. 48 and 49, we obtain the following expression for the transition rate Γq′,m to the excitonic state with energy Em(q′) due to two-photon absorption:
(4)
with the two-photon absorption amplitude
(5)
Here, the index ν denotes all the information about the excitonic states, i.e., ν = q, n, τ, c + j, vl. Here, Eν = En(q′) is the energy of the exciton in the τ valley with momentum q and quantum number n, consisting of an electron in c + j (j = 0, 1, …) conduction and a hole in vl (l = 0, 1, …) valence bands, respectively.
Using this approach, we derive the expression for the two-photon absorption coefficient (details can be found in the supplementary material),
(6)

Here, α = e2/ℏc is the fine structure constant. The parameter β(ω1, ω2) can be associated with the absorption coefficient for the monolayer TMD. To evaluate the numerical value of this coefficient, one needs to know the k · p coupling parameters γ2, γ4, γ5, γ6; the energies Ev−3, Ev, Ec, Ec+2 (see Ref. 50); numerical values of A, B, and δ parameters, which can be evaluated numerically using k ⋅ p parameters50 and the two-body approximation (see the supplementary data in Ref. 51); and the spectrum of excitons Em(0) in monolayer TMD.52 

To estimate the absorption coefficient, we can use the semi-analytical formula for the spectrum of the excitons,52 
(7)
where Eg is the single-particle bandgap in the system and Ry* = μe4/(22ɛ2) is the effective Rydberg energy for the exciton with reduced mass μ and dielectric constant ɛ of the medium surrounding the monolayer. Dimensionless parameters γ and δ depend on the ratio of the effective Bohr radius a0*=/m0e2 and effective in-plane screening length of the monolayer r0*=r0/ε. Taking into account the parameters for the considered system ɛ = 1.6, μ = 0.26m0, and r0 = 41.5 Å and the energy positions of the 1 s states for A and B excitons, E1sA=1.886 meV and E1sB=2.032 meV, respectively,43 we obtain Ry* ≈ 1.38 eV, γ ≈ 0.943, and δ ≈ 0.746. Using these parameters, we calculate the spectrum of the excitons. Then, using the Elliott type formula (6), we estimate the absorption coefficient for the first 5 s excitonic states (as an example) and the broadened delta function with Γ = 0.026 eV and present it in Fig. 4(b).

In conclusion, we experimentally investigate the ultrafast modulation of high harmonic generation in 2D transition metal dichalcogenide MoS2 using resonant and near-resonant light. We show that the pre-excitation of high density carriers leads to ultrafast dephasing and suppression of coherent interband polarization, which is responsible for macroscopic HHG. With the resonant pre-excitation, the dynamics of HHG suppression follows the recombination of excitons in the material. With the near-resonant pump pulse with photon energy lower than the exciton transition energy, the excitons are generated mainly via two-photon absorption driven by a combination of one photon from the pump pulse and one photon from the strong infrared driving pulse. This process occurs only when the pulses are overlapped in time on the sample. Combined with the short duration of pulses used in this study, this leads to ultrafast modulation of HHG at sub-100 fs timescales, thus enabling a new class of fast nonlinear optical devices working in the strong-field regime.

See the supplementary material for more details on the theory of two-photon absorption in the MoS2 monolayer.

The authors would like to acknowledge the support by Charles University (Nos. UNCE/SCI/010, SVV2020-260590, PRIMUS/19/SCI/05, GA UK 1190120) and the Czech Science Foundation (No. 23-06369S), co-funded by the European Union (ERC, eWaveShaper, No. 101039339). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

The authors have no conflicts to disclose.

Pavel Peterka: Formal analysis (equal); Investigation (equal); Writing – original draft (equal). Artur O. Slobodeniuk: Methodology (equal); Software (equal). Tomáš Novotný: Methodology (equal). Pawan Suthar: Methodology (equal). Miroslav Bartoš: Resources (lead). František Trojánek: Software (equal). Petr Malý: Writing – review & editing (supporting). Martin Kozák: Conceptualization (lead); Investigation (equal); Methodology (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
S.
Ghimire
,
A. D.
DiChiara
,
E.
Sistrunk
,
P.
Agostini
,
L. F.
DiMauro
, and
D. A.
Reis
, “
Observation of high-order harmonic generation in a bulk crystal
,”
Nat. Phys.
7
,
138
141
(
2011
).
2.
T. T.
Luu
,
M.
Garg
,
S. Y.
Kruchinin
,
A.
Moulet
,
M. T.
Hassan
, and
E.
Goulielmakis
, “
Extreme ultraviolet high-harmonic spectroscopy of solids
,”
Nature
521
,
498
502
(
2015
).
3.
G.
Vampa
,
T.
Hammond
,
N.
Thiré
,
B.
Schmidt
,
F.
Légaré
,
C.
McDonald
,
T.
Brabec
, and
P.
Corkum
, “
Linking high harmonics from gases and solids
,”
Nature
522
,
462
464
(
2015
).
4.
S.
Ghimire
and
D. A.
Reis
, “
High-harmonic generation from solids
,”
Nat. Phys.
15
,
10
16
(
2019
).
5.
N.
Yoshikawa
,
K.
Nagai
,
K.
Uchida
,
Y.
Takaguchi
,
S.
Sasaki
,
Y.
Miyata
, and
K.
Tanaka
, “
Interband resonant high-harmonic generation by valley polarized electron–hole pairs
,”
Nat. Commun.
10
,
3709
(
2019
).
6.
N.
Yoshikawa
,
T.
Tamaya
, and
K.
Tanaka
, “
High-harmonic generation in graphene enhanced by elliptically polarized light excitation
,”
Science
356
,
736
738
(
2017
).
7.
G.
Vampa
,
C.
McDonald
,
G.
Orlando
,
D.
Klug
,
P.
Corkum
, and
T.
Brabec
, “
Theoretical analysis of high-harmonic generation in solids
,”
Phys. Rev. Lett.
113
,
073901
(
2014
).
8.
D.
Golde
,
T.
Meier
, and
S. W.
Koch
, “
High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter-and intraband excitations
,”
Phys. Rev. B
77
,
075330
(
2008
).
9.
M.
Wu
,
S.
Ghimire
,
D. A.
Reis
,
K. J.
Schafer
, and
M. B.
Gaarde
, “
High-harmonic generation from Bloch electrons in solids
,”
Phys. Rev. A
91
,
043839
(
2015
).
10.
O.
Schubert
,
M.
Hohenleutner
,
F.
Langer
,
B.
Urbanek
,
C.
Lange
,
U.
Huttner
,
D.
Golde
,
T.
Meier
,
M.
Kira
,
S. W.
Koch
et al, “
Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations
,”
Nat. Photonics
8
,
119
123
(
2014
).
11.
G.
Vampa
,
T.
Hammond
,
N.
Thiré
,
B.
Schmidt
,
F.
Légaré
,
C.
McDonald
,
T.
Brabec
,
D.
Klug
, and
P.
Corkum
, “
All-optical reconstruction of crystal band structure
,”
Phys. Rev. Lett.
115
,
193603
(
2015
).
12.
P.
Suthar
,
F.
Trojánek
,
P.
Malý
,
T. J.-Y.
Derrien
, and
M.
Kozák
, “
Role of van Hove singularities and effective mass anisotropy in polarization-resolved high harmonic spectroscopy of silicon
,”
Commun. Phys.
5
,
288
(
2022
).
13.
T. T. H. J.
Wörner
and
H. J.
Wörner
, “
Measurement of the berry curvature of solids using high-harmonic spectroscopy
,”
Nat. Commun.
9
,
916
(
2018
).
14.
Y.
Bai
,
F.
Fei
,
S.
Wang
,
N.
Li
,
X.
Li
,
F.
Song
,
R.
Li
,
Z.
Xu
, and
P.
Liu
, “
High-harmonic generation from topological surface states
,”
Nat. Phys.
17
,
311
315
(
2021
).
15.
M. R.
Bionta
,
E.
Haddad
,
A.
Leblanc
,
V.
Gruson
,
P.
Lassonde
,
H.
Ibrahim
,
J.
Chaillou
,
N.
Émond
,
M. R.
Otto
,
Á.
Jiménez-Galán
et al, “
Tracking ultrafast solid-state dynamics using high harmonic spectroscopy
,”
Phys. Rev. Res.
3
,
023250
(
2021
).
16.
O.
Neufeld
,
J.
Zhang
,
U. D.
De Giovannini
,
H.
Hübener
, and
A.
Rubio
, “
Probing phonon dynamics with multidimensional high harmonic carrier-envelope-phase spectroscopy
,”
Proc. Natl. Acad. Sci. U. S. A.
119
,
e2204219119
(
2022
).
17.
G.
Wang
and
T.-Y.
Du
, “
Quantum decoherence in high-order harmonic generation from solids
,”
Phys. Rev. A
103
,
063109
(
2021
).
18.
I.
Floss
,
C.
Lemell
,
K.
Yabana
, and
J.
Burgdörfer
, “
Incorporating decoherence into solid-state time-dependent density functional theory
,”
Phys. Rev. B
99
,
224301
(
2019
).
19.
Z.
Wang
,
H.
Park
,
Y. H.
Lai
,
J.
Xu
,
C. I.
Blaga
,
F.
Yang
,
P.
Agostini
, and
L. F.
DiMauro
, “
The roles of photo-carrier doping and driving wavelength in high harmonic generation from a semiconductor
,”
Nat. Commun.
8
,
1686
(
2017
).
20.
H.
Nishidome
,
K.
Nagai
,
K.
Uchida
,
Y.
Ichinose
,
Y.
Yomogida
,
Y.
Miyata
,
K.
Tanaka
, and
K.
Yanagi
, “
Control of high-harmonic generation by tuning the electronic structure and carrier injection
,”
Nano Lett.
20
,
6215
6221
(
2020
).
21.
C.
Heide
,
Y.
Kobayashi
,
A. C.
Johnson
,
F.
Liu
,
T. F.
Heinz
,
D. A.
Reis
, and
S.
Ghimire
, “
Probing electron-hole coherence in strongly driven 2D materials using high-harmonic generation
,”
Optica
9
,
512
516
(
2022
).
22.
K.
Nagai
,
K.
Uchida
,
S.
Kusaba
,
T.
Endo
,
Y.
Miyata
, and
K.
Tanaka
, “
Effect of incoherent electron-hole pairs on high harmonic generation in atomically thin semiconductors
,” arXiv:2112.12951 (
2021
).
23.
Y.
Wang
,
F.
Iyikanat
,
X.
Bai
,
X.
Hu
,
S.
Das
,
Y.
Dai
,
Y.
Zhang
,
L.
Du
,
S.
Li
,
H.
Lipsanen
et al, “
Optical control of high-harmonic generation at the atomic thickness
,”
Nano Lett.
22
,
8455
8462
(
2022
).
24.
S.
Xu
,
H.
Zhang
,
J.
Yu
,
Y.
Han
,
Z.
Wang
, and
J.
Hu
, “
Ultrafast modulation of a high harmonic generation in a bulk ZnO single crystal
,”
Opt. Express
30
,
41350
41358
(
2022
).
25.
K. F.
Mak
,
C.
Lee
,
J.
Hone
,
J.
Shan
, and
T. F.
Heinz
, “
Atomically thin MoS2: A new direct-gap semiconductor
,”
Phys. Rev. Lett.
105
,
136805
(
2010
).
26.
H.
Liu
,
Y.
Li
,
Y. S.
You
,
S.
Ghimire
,
T. F.
Heinz
, and
D. A.
Reis
, “
High-harmonic generation from an atomically thin semiconductor
,”
Nat. Phys.
13
,
262
265
(
2017
).
27.
J.
Cao
,
F.
Li
,
Y.
Bai
,
P.
Liu
, and
R.
Li
, “
Inter-half-cycle spectral interference in high-order harmonic generation from monolayer MoS2
,”
Opt. Express
29
,
4830
4841
(
2021
).
28.
Y.
Kobayashi
,
C.
Heide
,
H. K.
Kelardeh
,
A.
Johnson
,
F.
Liu
,
T. F.
Heinz
,
D. A.
Reis
, and
S.
Ghimire
, “
Polarization flipping of even-order harmonics in monolayer transition-metal dichalcogenides
,”
Ultrafast Sci.
2021
,
9820716
.
29.
L.
Yue
,
R.
Hollinger
,
C. B.
Uzundal
,
B.
Nebgen
,
Z.
Gan
,
E.
Najafidehaghani
,
A.
George
,
C.
Spielmann
,
D.
Kartashov
,
A.
Turchanin
et al, “
Signatures of multiband effects in high-harmonic generation in monolayer MoS2
,”
Phys. Rev. Lett.
129
,
147401
(
2022
).
30.
E. M.
Kosik
,
A. S.
Radunsky
,
I. A.
Walmsley
, and
C.
Dorrer
, “
Interferometric technique for measuring broadband ultrashort pulses at the sampling limit
,”
Opt. Lett.
30
,
326
328
(
2005
).
31.
T.
Tsang
,
M. A.
Krumbügel
,
K. W.
DeLong
,
D. N.
Fittinghoff
, and
R.
Trebino
, “
Frequency-resolved optical-gating measurements of ultrashort pulses using surface third-harmonic generation
,”
Opt. Lett.
21
,
1381
1383
(
1996
).
32.
M.
Kozák
,
P.
Peterka
,
J.
Dostál
,
F.
Trojánek
, and
P.
Malý
, “
Generation of few-cycle laser pulses at 2 μm with passively stabilized carrier-envelope phase characterized by f-3f interferometry
,”
Opt. Laser Technol.
144
,
107394
(
2021
).
33.
S. H. C. H.
Townes
and
C. H.
Townes
, “
Stark effect in rapidly varying fields
,”
Phys. Rev.
100
,
703
(
1955
).
34.
V.
Ritus
, “
Shift and splitting of atomic energy levels by the field of an electromagnetic wave
,”
Sov. Phys. JETP
24
,
1041
1044
(
1967
).
35.
C.
Ell
,
J.
Müller
,
K.
Sayed
, and
H.
Haug
, “
Influence of many-body interactions on the excitonic optical Stark effect
,”
Phys. Rev. Lett.
62
,
304
(
1989
).
36.
D.
Chemla
,
W.
Knox
,
D.
Miller
,
S.
Schmitt-Rink
,
J.
Stark
, and
R.
Zimmermann
, “
The excitonic optical Stark effect in semiconductor quantum wells probed with femtosecond optical pulses
,”
J. Lumin.
44
,
233
246
(
1989
).
37.
N. B.
Delone
and
V. P.
Krainov
, “
AC Stark shift of atomic energy levels
,”
Phys.-Usp.
42
,
669
(
1999
).
38.
J.
Kim
,
X.
Hong
,
C.
Jin
,
S.-F.
Shi
,
C.-Y. S.
Chang
,
M.-H.
Chiu
,
L.-J.
Li
, and
F.
Wang
, “
Ultrafast generation of pseudo-magnetic field for valley excitons in WS2 monolayers
,”
Science
346
,
1205
1208
(
2014
).
39.
E. J.
Sie
,
J. W.
McIver
,
Y.-H.
Lee
,
L.
Fu
,
J.
Kong
, and
N.
Gedik
, “
Valley-selective optical Stark effect in monolayer WS2
,”
Nat. Mater.
14
,
290
294
(
2015
).
40.
E. J.
Sie
,
C. H.
Lui
,
Y.-H.
Lee
,
L.
Fu
,
J.
Kong
, and
N.
Gedik
, “
Large, valley-exclusive Bloch-Siegert shift in monolayer WS2
,”
Science
355
,
1066
1069
(
2017
).
41.
T.
LaMountain
,
H.
Bergeron
,
I.
Balla
,
T. K.
Stanev
,
M. C.
Hersam
, and
N. P.
Stern
, “
Valley-selective optical Stark effect probed by Kerr rotation
,”
Phys. Rev. B
97
,
045307
(
2018
).
42.
P. D.
Cunningham
,
A. T.
Hanbicki
,
T. L.
Reinecke
,
K. M.
McCreary
, and
B. T.
Jonker
, “
Resonant optical Stark effect in monolayer WS2
,”
Nat. Commun.
10
,
5539
(
2019
).
43.
A.
Slobodeniuk
,
P.
Koutenský
,
M.
Bartoš
,
F.
Trojánek
,
P.
Malý
,
T.
Novotný
, and
M.
Kozák
, “
Semiconductor Bloch equation analysis of optical Stark and Bloch-Siegert shifts in monolayer WSe2 and MoS2
,”
Phys. Rev. B
106
,
235304
(
2022
).
44.
A. O.
Slobodeniuk
,
P.
Koutenský
,
M.
Bartoš
,
F.
Trojánek
,
P.
Malý
,
T.
Novotný
, and
M.
Kozák
, “
Ultrafast valley-selective coherent optical manipulation with excitons in WSe2 and MoS2 monolayers
,”
Npj 2D Mater. Appl.
7
,
17
(
2023
).
45.
A.
Splendiani
,
L.
Sun
,
Y.
Zhang
,
T.
Li
,
J.
Kim
,
C.-Y.
Chim
,
G.
Galli
, and
F.
Wang
, “
Emerging photoluminescence in monolayer MoS2
,”
Nano Lett.
10
,
1271
1275
(
2010
).
46.
Y.
Niu
,
S.
Gonzalez-Abad
,
R.
Frisenda
,
P.
Marauhn
,
M.
Drüppel
,
P.
Gant
,
R.
Schmidt
,
N. S.
Taghavi
,
D.
Barcons
,
A. J.
Molina-Mendoza
et al, “
Thickness-dependent differential reflectance spectra of monolayer and few-layer MoS2, MoSe2, WS2 and WSe2
,”
Nanomaterials
8
,
725
(
2018
).
47.
R.
Frisenda
,
Y.
Niu
,
P.
Gant
,
A. J.
Molina-Mendoza
,
R.
Schmidt
,
R.
Bratschitsch
,
J.
Liu
,
L.
Fu
,
D.
Dumcenco
,
A.
Kis
et al, “
Micro-reflectance and transmittance spectroscopy: A versatile and powerful tool to characterize 2D materials
,”
J. Phys. D: Appl. Phys.
50
,
074002
(
2017
).
48.
L. D.
Landau
and
Lifshitz
,
Quantum Mechanics: Non-Relativistic Theory
,
vol. 3
(
Butterworth-Heinemann
,
1981
).
49.
J. J.
Sakurai
and
E. D.
Commins
, Modern quantum mechanics,
2020
.
50.
D. V.
Rybkovskiy
,
I. C.
Gerber
, and
M. V.
Durnev
, “
Atomically inspired k·p approach and valley Zeeman effect in transition metal dichalcogenide monolayers
,”
Phys. Rev. B
95
,
155406
(
2017
).
51.
Ł.
Kipczak
,
A. O.
Slobodeniuk
,
T.
Woźniak
,
M.
Bhatnagar
,
N.
Zawadzka
,
K.
Olkowska-Pucko
,
M.
Grzeszczyk
,
K.
Watanabe
,
T.
Taniguchi
,
A.
Babiński
et al, “
Analogy and dissimilarity of excitons in monolayer and bilayer of MoSe2
,”
2D Mater.
10
,
025014
(
2023
).
52.
M.
Molas
,
A.
Slobodeniuk
,
K.
Nogajewski
,
M.
Bartos
,
A.
Bala
,
K.
Babiński
,
T.
Watanabe
,
C.
Taniguchi
,
M.
Faugeras
et al, “
Energy spectrum of two-dimensional excitons in a nonuniform dielectric medium
,”
Phys. Rev. Lett.
123
,
136801
(
2019
).

Supplementary Material