Resonant optical Stark effect in monolayer WS2

Breaking the valley degeneracy in monolayer transition metal dichalcogenides through the valley-selective optical Stark effect (OSE) can be exploited for classical and quantum valleytronic operations such as coherent manipulation of valley superposition states. The strong light-matter interactions responsible for the OSE have historically been described by a two-level dressed-atom model, which assumes noninteracting particles. Here we experimentally show that this model, which works well in semiconductors far from resonance, does not apply for excitation near the exciton resonance in monolayer WS2. Instead, we show that an excitonic model of the OSE, which includes many-body Coulomb interactions, is required. We confirm the prediction from this theory that many-body effects between virtual excitons produce a dominant blue-shift for photoexcitation detuned from resonance by less than the exciton binding energy. As such, we suggest that our findings are general to low-dimensional semiconductors that support bound excitons and other many-body Coulomb interactions.


Supplementary Note 1. Intervalley biexciton photoinduced absorption
Supplementary Figure 1 shows that for cross-circular polarizations ( ! ! ), a new photoinduced absorption band appears below the A-exciton resonance, at 631 nm. We assign the new absorption feature to the coherent creation of intervalley biexcitons. The σ-pump photon creates an exciton in the K'-valley. The subsequent σ+ probe photon can then access the | !! → | !! ! transition, which we observe as a photoinduced absorption band. Based on the separation between the A-exciton peak and this feature, we estimate the biexciton binding energy as 48 ± 5 meV. spectrum for t = 0 fs (dots) and best fit (green) to the sum of two Gaussians, the first located at the exciton absorption of 616 nm (blue) and the second indicates the intervalley biexciton photoinduced absorption feature at 631 nm (red). The ground state absorption spectrum is included (black) for reference.

Supplementary Note 2. Intervalley charged biexciton and trion binding energies
The valley selective optical Stark effect in monolayer WS 2 can also be observed at low temperature. Here, the narrower absorption linewidths may make it easier to quantify small optical Stark shifts. Supplementary Figure 2 shows the valley-resolved TA spectra in monolayer WS 2 at 78K for a photoexcitation wavelength of 596 nm, which is resonant with the A-exciton. Unfortunately, illumination of WS 2 in vacuum leads to unavoidable creation of a ground state-to-trion absorption feature at 608 nm, shown in Supplementary   Figure 3a, which clutters the spectra and complicates analysis. This trion absorption arises due to laser-induced removal of physisorbed oxygen from sulfur vacancies in WS 2 , as previously reported by Currie et al. 1 Based on the separation between the A-exciton and trion features, we estimate the trion binding energy as 41 ± 5 meV. For cross-circular polarizations ( ! ! ), a new photoinduced absorption band associated with a multiple particle complex, shown in Supplementary Figure 3b, is just evident at 613 nm. This is analogous to the intervalley biexciton feature observed at room temperature. We estimate the binding energy of this multiple particle complex of 59 ± 5 meV, which is larger than the biexciton binding energy observed at room temperature, and indicates that the 78K induced absorption may originate from a charged biexciton, i.e. an exciton-trion complex. 2 This is consistent with the more trionic character of the absorption spectrum.

Supplementary Note 5. Single wavelength OSE measurements
Here, we demonstrate that single wavelength, i.e. two-color pump probe, measurements of the OSE can be misleading, yielding artificial results. Supplementary Figure 6 summarizes photoinduced changes in transmission for 610 nm excitation and probed at a wavelength of 635 nm, which is just below the exciton resonance. The large decrease in transmission, i.e. increase in absorption, for cross-circularly polarized ( ! ! ) pulses may cause the observer to conclude that a very large red-shift is present. On the other hand, the small increase in transmission, i.e. decrease in absorption, for co-circularly polarized ( ! ! ) pulses may cause the observer to conclude that a small blue-shift occurs. The time dependence of this effect yields a large polarization centered at time zero, consistent with a coherent process driven by light-matter coupling. The surprising sub-linear power dependence seems to suggest that the observed effect goes beyond the typical OSE described by the two-level dressed exciton model. These observations are qualitatively similar, though opposite in direction, to recent reports of anomalous optical Stark shifts attributed to coupling between excitons and intervalley biexcitons. 4 However, the full transient absorption spectra we have shown, e.g. in the preceding section, show conclusively that a large red-shift is not present for cross-circularly polarized ( ! ! ) pulses. Instead, the increased absorption at 635 nm is caused by coherent creation of intervalley biexcitons, as detailed above and in the main text. Similarly, we have shown that a large blue-shift occurs for co-circularly polarized ( ! ! ) pulses, and that its fluence dependence is approximately linear with power. We therefore conclude that single wavelength measurements of the OSE can be misleading.

Supplementary Note 6. OSE observed with linearly polarized light
The valley-selective OSE remains present for linearly polarized pump and probe.
Supplementary Figure 7 shows the time-dependent photoinduced changes in transmission, and corresponding time-dependent absorption spectra, for resonant excitation of monolayer WS 2 with collinear polarization. Because linearly polarized light is a superposition of ! and ! polarizations, we expect it to experience changes to both the K-and K'-valleys. We have shown above that a blue-shift occurs for co-circular polarization and no shift occurs for cross-circular polarization. Therefore, both the K-and K'-valleys should blue-shift in response to a linearly polarized pump pulse and those shifts can be measured with a linearly polarized probe. This is indeed what we observe, as shown in Supplementary Figure 7.