Excitonic resonances control the temporal dynamics of nonlinear optical wave mixing in monolayer semiconductors

Monolayer semiconductors are an emerging platform for strong nonlinear light–matter interactions that are enhanced by the giant oscillator strength of tightly bound excitons. Little attention has been paid to the impact of excitonic resonances on the temporal dynamics of such nonlinearities, since harmonic generation and optical wave mixing are generally considered instantaneous processes. We find that a significant time difference, ranging from −40 to +120 fs, is necessary between two light pulses for optimal sum-frequency generation (SFG) and four-wave mixing (FWM) to occur from monolayer WSe2 when one of the pulses is in resonance with an excitonic transition. These resonances involve both band-edge A excitons and high-lying excitons that comprise electrons from conduction bands far above the bandgap. Numerical simulations in the density-matrix formalism rationalize the distinct dynamics of SFG and FWM. The interpulse delays for maximal SFG and FWM are governed primarily by the lifetime of the one-photon and two-photon resonant states, respectively. The method therefore offers an unconventional probe of the dynamics of excitonic states accessible with either one-photon or two-photon transitions. Remarkably, the longest delay times occur at the lowest excitation powers, indicating a strong nonlinearity that offers exploration potential for excitonic quantum nonlinear optics. Researchers show that resonant coupling of light pulses with excitonic transitions affects the optimal time difference between pulses for sum-frequency generation and four-wave mixing in monolayer WSe2.

Article https://doi.org/10.1038/s41566-022-01080-1 flopping of the excited-state population occurs even under comparatively weak pulsed-laser excitation 20 , and is associated with light-driven quantum interference between the excitation pathways of the bound band-edge A exciton (AX) and a two-photon resonant high-lying exciton (HX) involving upper conduction bands at the same ±K-points 19 . As shown via experiment and GW plus Bethe-Salpeter equation (GW-BSE) theory, the energy of the HX species happens to be close to twice that of the AX, forming an almost evenly spaced three-level system 19 . Such a ladder-type excitonic structure offers a new all-solid-state platform for coherent nonlinear optical processes such as electromagnetically induced transparency 21 . This three-level system is intrinsic to the unusual band structure of the monolayer 19 and differs fundamentally from conventional biexcitonic systems 22 , where a high density of excitons is needed for two excitons to meet in space to form a bound state that contributes to the nonlinear interaction 23,24 . Besides SHG and SFG, single-pass optical parametric amplification has also been demonstrated in TMDC monolayers 25 . Although excitonic enhancement of the nonlinear light-matter interaction appears to be well established, it is not at all clear whether and how excitonic transitions affect the temporal dynamics of nonlinear interactions. Figure 1a illustrates the setup for a two-colour excitation experiment using ultrafast laser pulses of 130-140 fs length and 80 MHz repetition rate at 713 nm (with an energy ħω pump = 1.739 eV) and 840 nm (ħω probe = 1.476 eV), impinging on an exfoliated WSe 2 monolayer with a time difference Δt = t probe − t pump . For notational convenience, we refer to the pulse resonant with the excitonic transition as the 'pump' pulse and the pulse detuned far from any excitonic resonance as the quantum picture of light-matter interactions, every interaction process inevitably changes the quantum state for both photons and electrons. In the density-matrix representation of the electronic quantum states, the expectation value for the observable polarization P (t) = ⟨μρ (t)⟩ (where μ is the dipole-moment matrix and ρ is the density matrix) therefore generally contains an integration over all possible quantum pathways, involving all transitions with non-zero dipole-moment matrix elements μ ij for all possible times and frequencies at which electric fields are present 1,5 . This approach has proved powerful for multidimensional coherent spectroscopy 6 , where a nonlinear response is excited by multiple laser pulses. An important consequence of this extended picture of wave mixing is that parametric interactions progress via diagonal elements ρ ii of the density matrix, which means that they can be sensitive not only to dephasing of transition dipoles but also to excited-state lifetimes as well as to state depletion and transition saturation.
Transition metal dichalcogenide (TMDC) monolayers have emerged as an intriguing two-dimensional medium to accommodate strong light-matter interactions [7][8][9][10][11] . As direct-gap semiconductors with weak dielectric screening, TMDC monolayers host strongly bound excitons near the K-points of the Brillouin zone [12][13][14] . Owing to broken inversion symmetry, the materials exhibit nonlinear responses of both even and odd order, suggesting a wide range of application potential in nonlinear wave mixing 7,11,15,16 . Among other effects, the giant excitonic enhancement in nonlinearity 7,15,17 near resonance exhibited by these materials has enabled SHG and SFG under continuous-wave irradiation 18 'probe' pulse, although we do not limit Δt to positive values nor do we restrict the probe beam to lower intensities than the pump beam. The sample is cooled to a temperature of 5 K using a microscope cryostat and excited through a long-working-distance microscope objective with a numerical aperture (NA) of 0.6. Details of the setup are shown in Supplementary Fig. 1. Figure 1b plots a representative spectrum of the upconverted radiation detected in back-reflected geometry through the same objective, for a time delay of Δt = 0 and using a 680 nm (1.82 eV) short-pass filter. Five characteristic spectral contributions are identified by their frequencies: SHG of the pump beam at 2ω pump , SFG of the pump and probe beams at ω pump + ω probe , SHG of the probe beam at 2ω probe , FWM at 2ω pump − ω probe and upconverted photoluminescence (UPL) of the so-called 'B exciton' close to the FWM peak 19 . As monolayer TMDCs have giant second-and third-order optical susceptibilities 7 , both second-order nonlinear processes, such as SFG, and third-order nonlinear processes, such as FWM, are very efficient 15,25,26 . Monolayer TMDCs therefore constitute broadband optical frequency mixers 27 . Excitation and detection through the high-NA objective bypasses most of the constraints on beam direction due to the conservation of momentum that is usually necessary for efficient nonlinear wave mixing 2 . Figure 1c shows a two-dimensional map of the upconverted radiation as a function of the pump-probe delay time Δt. As expected, neither the SHG of either the pump or probe beam nor the UPL show any dependence on the coincidence of pump and probe impulses. On the other hand, both SFG and FWM are sensitive to Δt. However, the maximum efficiency of these two processes does not occur at zero delay, which is surprising given that parametric, instantaneous nonlinear optical wave mixing should be most effective for an optimal temporal pulse overlap. As the time delay between the two laser impulses is sensitive to group delays accumulated in the optical setup, the zero delay is referenced to the SFG response of a 100-µm-thick beta-barium borate (BBO) crystal cut at 29.2°, which is placed next to the monolayer WSe 2 sample as detailed in Supplementary Fig. 1. Figure 1d shows the SFG intensity from the BBO crystal (grey spheres) as well as the spectrally integrated SFG (blue spheres) and FWM (orange spheres) intensities of the WSe 2 sample as a function of the delay time, along with Gaussian fits (lines). The dashed lines mark the centre of each Gaussian, with the FWM peaking at the delay time Δt FWM = 80 fs (orange) and the SFG peaking at Δt SFG = 120 fs (blue).
To examine the role of excitons in the retardation of these nonlinear processes, we perform the measurements as a function of the photon energy. Figure 2a shows the dependence of the time delay for maximal SFG (blue spheres) on the energy of the pump laser for a fixed probe energy ħω probe = 1.476 eV. Comparison with the photoluminescence (PL) spectrum (black curve) reveals that the largest Δt SFG of 120 fs is reached precisely when the pump laser is in resonance with the AX. Under off-resonance detuning, Δt SFG decreases to zero. Since both

Fig. 2 | Excitonic effects on the delay time for maximal SFG and FWM. a,
Comparison of the PL spectrum of monolayer WSe 2 (black curve) with the delay time for maximal SFG, Δt SFG , as a function of the pump photon energy (blue spheres). The power of the probe beam is set to 400 µW (176 µJ cm −2 ) at 1.476 eV and the pump beam is set to 3 µW (1.32 µJ cm −2 ). The blue line is a guide for the eye. b, Dependence of Δt SFG on the pump beam power for monolayer WSe 2 (blue spheres) and monolayer WS 2 (purple spheres) using ħω pump = 1.739 eV in resonance with the AX, ħω probe = 1.476 eV and a probe-beam power of 400 µW. c, PL spectrum of monolayer WSe 2 (black curve) compared with the delay time for maximal FWM, Δt FWM , as a function of the pump photon energy (orange spheres).
The measurement conditions are the same as in a. The orange line is a guide for the eye. The dashed line marks the two-photon resonance with an HX, which is congruent with the position of the dip in the SHG spectrum (see Supplementary  Fig. 4). d, Dependence of Δt FWM on the pump power for monolayer WSe 2 (orange spheres) and monolayer WS 2 (purple spheres) using ħω pump = 1.739 eV, ħω probe = 1.476 eV and a probe-beam power of 400 µW. The red and dark-red wave packets indicate the temporal relation of the pump and probe pulses. The grey dashed line marks the zero time delay. Throughout, the error bars are the standard deviation of the results of three independent measurements on the same sample position.
Article https://doi.org/10.1038/s41566-022-01080-1 the pump and probe pulses are short and therefore spectrally broad, the peak in the energy dependence of Δt SFG appears to be broader than the PL peak of the AX.
If the delay is caused by an excitonic resonance, it should depend on the excitation density. Figure 2b shows Δt SFG for monolayer WSe 2 (blue spheres) as a function of the pump power for ħω pump = 1.739 eV, ħω probe = 1.476 eV and a probe-beam power of 400 µW (0.18 mJ cm −2 ). On increasing the pump power by three orders of magnitude (below the Mott transition, see Supplementary Note 2), Δt SFG decreases from >120 fs to close to zero. Experimentally, this can lead to the counterintuitive phenomenon of an increasing SFG intensity with a decreasing pump power, as shown in Supplementary Fig. 2. It may be tempting to assign such a decrease in Δt SFG to the decrease of the AX lifetime with increasing excitation density, as observed in transient intra-and interband spectroscopy 28 and optical two-dimensional Fourier-transform spectroscopy 29 . However, as we demonstrate below, this decrease can also be rationalized in terms of the dynamics of Rabi flopping 30 . In contrast to the effect of varying the pump power, Supplementary Fig. 3a shows that comparable changes in the probe power do not affect the Δt SFG . To eliminate the possibility of optical artefacts and establish a further reference for the measurements, the experiments were repeated for monolayer WS 2 in place of the WSe 2 sample. Since the bandgap of monolayer WS 2 is approximately 0.3 eV larger than that of monolayer WSe 2 , the lasers will not be in resonance with excitonic transitions. Indeed, Δt SFG from monolayer WS 2 (purple spheres in Fig. 2b) is close to zero over the entire range of pump power.
The amplitudes of the FWM signals detected simultaneously are shown in Fig. 2c,d (orange spheres). In contrast to Δt SFG , the delay time for maximum FWM, Δt FWM , does not peak at the pump photon energy of the AX resonance (Fig. 2c), but rather at an energy that also gives rise to quantum interference in the SHG as shown in Supplementary Fig. 4 (refs. 19,20 ). This quantum interference arises due to the high-lying excitonic state HX, which lies at 3.44 eV and contributes to the delay Δt FWM through a two-photon resonance with the pump laser, as illustrated in the inset of Fig. 2c. As found for Δt SFG , Δt FWM decreases with increasing pump power (orange circles in Fig. 2d) and is independent of the probe power ( Supplementary Fig. 3b) and the probe photon energy ( Supplementary Fig. 5c). Unexpectedly, however, Δt FWM drops below zero in the high-pump-power regime: above 1 mW (0.44 mJ cm −2 ), the centre of the probe pulse must reach the sample 40 fs before the centre of the pump pulse for maximum efficiency of FWM. As reported in Supplementary Fig. 5a, the maximum values of |Δt SFG | and |Δt FWM | become even larger for WSe 2 monolayer samples with hexagonal boron nitride (hBN) encapsulation. This change may conceivably arise from a suppression of local sample inhomogeneities 31,32 and a modification of the radiative decay rate due to a cavity effect 33 . The fact that the maximal values of Δt SFG and Δt FWM appear at the lowest powers of both the pump and probe beams ( Supplementary Fig. 5b) demonstrates that strong nonlinear interactions extend into the regime of the low excitation densities, opening up avenues for further exploration of quantum nonlinear optics 34,35 .
While the change in the interpulse delay for SFG can, in principle, be rationalized through a pump-power dependence of a sufficiently long-lived excitonic resonance 29 , the negative Δt FWM is not straightforward to explain. To account for the excitonic resonances, instead of following the conventional analysis of the parametric light-matter interaction up to fixed nonlinear order 1,2 , we use a full time-dependent solution of the density-matrix dynamics in the presence of the laser field (see Methods for details). This approach uses a simplified state description borrowed from atomic optics and has been successfully employed before to describe the effect of Rabi flopping in the unusual SHG spectra from monolayer WSe 2 (ref. 20 ). For realistic excitation conditions, the laser fields are modelled as Gaussian pulses with parameters matching the experimental ones as illustrated in Fig. 3a. We describe the allowed eigenstates using a simple ladder-type three-level system shown in Fig. 3a, which consists of a ground state |1〉, the AX state |2〉 and the HX state |3〉. The detailed parameters used for the simulation, listed in the Supplementary Fig. 6 and Supplementary Table 1, are based on experimental conditions. Figure 3b shows a calculated optical wave-mixing spectrum for the same spectral range as probed experimentally (Fig. 1b), at zero time delay between the pump and probe pulses (ħω pump = 1.739 eV and ħω probe = 1.476 eV). Five prominent peaks are identified, including FWM (2ω pump − ω probe ), SHG (2ω pump and 2ω probe ), and SFG (ω pump + ω probe ). A six-wave-mixing feature (3ω pump − 2ω probe ) appears at a higher pump fluence (see Supplementary Fig. 8). Figure 3c shows the result of calculating the spectrally integrated SFG and FWM intensities as a function of the delay time between the pump and probe pulses. Here, the pump laser is set to be resonant with state |2〉, with a low power of 1 µW (0.44 µJ cm −2 ). The simulated maximal SFG intensity appears for a positive delay time of approximately 105 fs, which is close to the experimental value of 120 fs in Fig. 1d. The maximal FWM intensity appears at a delay time of 50 fs, which is somewhat smaller than the experimental result although it confirms the experimental trend that Δt FWM tends to be smaller than Δt SFG . Next, we investigate the influence of the pump power and photon energy on Δt SFG and Δt FWM . Figure 4a,c shows the dependence on the pump photon energy of Δt SFG and Δt FWM at a low pump power of 1 µW. The maximum SFG arises for a pump beam resonant with the |1〉 → |2〉 transition (the AX), whereas the maximum FWM is found when the pump laser is two-photon resonant with the |1〉 → |3〉 transition, that is, the HX. These resonant enhancements coincide with the experimental observations (Fig. 2a,c). Figure 4b,d plots the simulated pump-power dependence of Δt SFG and Δt FWM . Both Δt SFG and Δt FWM drop with increasing power, qualitatively matching with experiments. The simulation, however, shows that both Δt SFG and Δt FWM become negative at high pump powers, although only Δt FWM is found to do so in experiment. This divergence is further discussed below.
The simulations can be rationalized by studying the corresponding dynamics of the density-matrix elements, as shown in Fig. 4e,f for both low (1 µW, 0.44 µJ cm −2 ) and high (1 mW, 0.44 mJ cm −2 ) pump powers. We propose that the largest SFG and FWM intensities are obtained when the centre of the probe pulse, that is, where the field strength is highest, coincides with the largest transient population of the excitonic state. Since both the AX state |2〉 and the HX state |3〉 have short but finite lifetimes, the state populations accumulate over the short timescale of the pump-probe pulse sequence. In the low-pump-power regime, maximum state occupancy then appears at positive delay times as seen in Fig. 4e. The delay times for maximal population of states |2〉 and |3〉 match well with Δt SFG and Δt FWM , respectively. As the pump power increases, however, the laser field becomes sufficiently strong, so that Rabi flopping in the state populations becomes relevant on the timescale of the interpulse delay. A detailed time evolution of the state population as a function of the pump power is shown in Supplementary Fig. 9. The signatures of the onset of this Rabi flopping have been studied in the context of SHG, when the two-photon resonance of the |1〉 → |3〉 HX transition is driven 20 . Rabi flopping is directly observed experimentally in the frequency domain, by examining the fingerprints of quantum interference in the SHG spectra as a function of the pulse length and amplitude-the number of transparency dips in the spectrum relates directly to the number of Rabi cycles undergone by the driven system in a single laser pulse 20 . Since the early depopulation of the excited states associated with Rabi flopping shifts maximum occupancy to earlier times (Fig. 4f), Δt SFG and Δt FWM decrease with increasing pump power (Fig. 4b,d).
Unlike in the experiments, however, the simulated Δt SFG does turn negative. As we discussed in ref. 20 , these density-matrix simulations do not include effects of many-body interactions such as enhanced Auger recombination at high pump fluences and large exciton densities. We speculate that the efficient Auger-like exciton-exciton annihilation of band-edge excitons 28,36 in monolayer WSe 2 substantially raises the decay rate of state |2〉 at high exciton densities, and therefore prevents Δt SFG from becoming negative. As shown in Fig. 5a,b, the simulated delay time Δt SFG for maximum SFG intensity is very sensitive to the lifetime of state |2〉 and even approaches zero as this lifetime tends to zero. In stark contrast to this observation, Fig. 5c,d shows that Δt FWM is mainly sensitive to the lifetime of state |3〉 but is quite insensitive to that of state |2〉.
The different dependencies of Δt SFG and Δt FWM on the lifetimes of the two states imply that SFG and FWM can be used to probe the dynamics of one-photon allowed and two-photon allowed excitonic states simultaneously but independently of each other. The largest experimentally observed delay time of Δt SFG = 120 fs for bare WSe2 corresponds to an AX lifetime of approximately 200 fs (Fig. 5a), which matches well with the population relaxation and radiative lifetime measured by transient intra-and interband spectroscopy 28 as well as optical two-dimensional Fourier-transform spectroscopy 29 . Although excitonic dynamics of one-photon allowed transitions are straightforward to probe using a variety of conventional optical means, the dynamics of two-photon allowed dark excitonic states have remained much more challenging to access, and our approach provides a simple solution to this problem.
In summary, we explored the influence of the delay between two laser pulses on second-and third-order optical wave-mixing processes in monolayer WSe 2 . Substantial delays between the two pulses are found to be necessary to maximize the SFG and FWM signal intensities. Understanding these time delays poses challenges to the description Article https://doi.org/10.1038/s41566-022-01080-1 of spectroscopic observables in terms of the conventional picture of low-order perturbative nonlinear interactions 2 . The observations can, however, be rationalized in the framework of coherent population dynamics of the excitons under optical pumping as described within the density-matrix formalism, which computes the interaction of light and matter to arbitrary perturbative order. Further distinction between conventional parametric SFG and FWM and processes involving excitons will necessitate disentangling of the multitude of Liouville-space pathways embodied in the density-matrix time dynamics. The qualitative agreement between simulation and experiment suggests that the simplified ladder-type excitonic three-level system indeed captures the essential physical contributions to the delay effects observed in the optical wave mixing. Our work therefore demonstrates that nonlinear optical wave-mixing spectroscopy offers new opportunities for measuring the population dynamics of optically pumped excitonic states that are either one-or two-photon allowed, where the dynamics of the latter have proved challenging to access so far. The strong nonlinear interactions that enable the time delays to persist into the low-excitation regime promise to open up an as-yet unexplored regime of exciton-based quantum nonlinear optics in TMDC monolayers.

Online content
Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41566-022-01080-1.

Sample fabrication
The WSe 2 and WS 2 monolayers and thin layers of hBN were mechanically exfoliated from bulk single crystals (WSe 2 and WS 2 , HQ Graphene; hBN, National Institute for Materials Science) using Nitto tape and transferred to a silicon substrate with a 90 nm silicon dioxide layer through stamping 37 . The hBN encapsulated monolayer WSe 2 sample was fabricated using a pick-up method 38 . The sample was cooled to 5 K using a microscope cryostat.

Nonlinear spectroscopy
A schematic of the experimental setup is depicted in Supplementary  Fig. 1. A Ti:sapphire laser beam (Chameleon Ultra II, 80 MHz repetition rate, Coherent) was split using a 20:80 plate beamsplitter. The 20% reflected beam was used as the probe beam at the frequency ω probe . The 80% transmitted beam was used to pump an optical parametric oscillator with intra-cavity frequency doubling (OPO-X fs, APE), which generates the pump beam at the frequency ω pump . Both beams pass through custom-made prism-based pulse compressors to compensate their chirp, and a power-control unit was used to regulate the excitation power. The probe beam was directed through a delay stage and then combined with the pump beam through a 50:50 beamsplitter. The reflected beams were focused onto the sample surface inside a cryostat using a microscope objective (LUCPLFLN40X, 0.6NA, Olympus). The diameter of the laser spot was estimated to be 1.9 µm. The transmitted beams were focused into a BBO crystal that provides monitoring of the SFG signal measured using a photodiode, as detailed in Supplementary Note 1. The signal reflected from the sample passed through a 680 nm short-pass filter, was dispersed by a spectrometer (Acton SP2300, Princeton Instruments) and detected using a cooled CCD camera (PIXIS 100, Princeton Instruments). A 50:50 beamsplitter (BS3) was used to separate the excitation and emission beams.

Characterization of excitonic states
To characterize the AX of monolayer WSe 2 , a 488 nm continuous-wave laser was focused onto the sample to excite the PL. To characterize the dark HX of monolayer WSe 2 , an 80 fs pulsed Ti:sapphire laser (Mai Tai XF, 80 MHz repetition rate, Spectra-Physics) was focused onto the sample to drive the SHG. The PL signal after a 488 nm long-pass filter and the SHG signal after a 680 nm short-pass filter were measured using the same detection unit described for the nonlinear spectroscopy.

Density-matrix formalism simulation
The polarization in P (t) ∝ ∑ ij μ ij ρ ij (t) is understood to be the source term for scattered radiation. Specific information about the SFG and FWM spectra was obtained by analysing P(t) in the Fourier domain. We solved the time dynamics of the density matrix ρ in the framework of a Lindblad-type equation {Γ ,ρ } +Λ . Here, the density matrix is defined in the conventional way as ρ = ∑ i p i |i⟩⟨i| , with p i the probability for the system to be in state |i❭. The HamiltonianĤ is given by matrix elements H ii = E i and H ij = −ε(t)μ ij (i ≠ j), where E i are the eigenstate energies, ε(t) is the electric-field component of the incident light field and μ ij are the transition dipole moments linked to the ρ ij coherence. For simplicity, all vectorial properties of the field and the transition dipoles are neglected. The decay matrix Γ is diagonal with non-zero diagonal elements Γ ii = Γ i (i > 1). The repopulation term Λ describes the repopulation of the ground state with a single non-zero matrix element Λ 11 = ∑ i>1 Γ i ρ ii . ħ is the reduced Planck constant.

Data availability
Any additional data are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability
The Mathematica code used for the numerical simulations discussed in this paper is available from the corresponding authors upon reasonable request.