Effects of Floquet Engineering on the Coherent Exciton Dynamics in Monolayer WS₂

Figure 2(a) shows the signal amplitude as a function of $t_{1}$ and $\hbar\omega_{3}$ (the energy of the emitted photons), in the absence of any pump pulse, with the sample temperature at 4 K. Two peaks at $\hbar\omega_{3}=2.07$ eV and $\hbar\omega_{3}=2.04$ eV are evident and attributed to the A-exciton and trion, respectively [38, 39, 40, 34]. The exciton peak decays as a function of $t_{1}$ due to decoherence of the ${\ket{g}\bra{X_{K^{\prime}}}}$ coherence, with the decoherence time, $T_{2}$, determined to be $360\pm 10$ fs. The 2D spectrum, obtained by Fourier transforming with respect to $t_{1}$, is shown in Appendix B and is similar to previously reported 2D spectra [31, 30, 32, 33, 34, 41, 37, 42].

The changes to the amplitude and phase of the signal induced by a $\sigma^{+}$ polarized pump pulse, with fluence of 751.7 µJ/cm² and detuned by 200 meV, can be seen in Fig. 2(b) and (c), respectively. For $t_{1}$ delays $<$ 250 fs, the pump arrives before $\textbf{{k}}_{1}$ and has no influence on the signal amplitude or phase. For $t_{1}$ delays $>$ 400 fs, the pump arrives after $\textbf{{k}}_{1}$, and before $\textbf{{k}}_{2}$ and $\textbf{{k}}_{3}$, such that the system is in a ${\ket{g}\bra{X_{K^{\prime}}}}$ coherence when the pump arrives (Fig. 1(c)). The pump induces both a drop in signal amplitude (evident for $t_{1}>250$ fs) and a phase shift that increase from 0 to $\pi$ between $t_{1}=250$ fs and $t_{1}=400$ fs.

It is evident in Fig. 2(c) that the phase shift arising from the AC-Stark shift varies as a function of $\hbar\omega_{3}$. Figure 2(e) plots the average phase shift between 400 and 500 fs for several pump fluences and clearly shows the significant decrease in phase shift as $\hbar\omega_{3}$ increases. The distribution of $\hbar\omega_{3}$ values comes from the inhomogeneous broadening of the exciton energy across the sample, as shown in Fig. 2(d), and leads to an increase in the magnitude of the pump detuning as $\hbar\omega_{3}$ increases. The $\hbar/omega_{3}$ dependence of the phase shift is thus consistent with the expectation that the magnitude of the AC-Stark shift scales inversely with pump detuning [13, 14].

The evolution of the measured phase difference as a function of $t_{1}$, obtained by taking a slice at the exciton energy ($\hbar\omega_{3}=2.072\pm 0.005$ eV), is shown in Fig. 3(a) for different pump fluences and polarizations (see Supplementary Material available at **** for the spectrally-resolved data). The induced phase shift increases as the $\textbf{{k}}_{1}$ pulse is scanned through the pump pulse, up to a total accumulated phase shift of $\pi$ for the highest fluence. The magnitude of this phase shift varies linearly with intensity, as expected for the AC-Stark shift and shown in Fig. 3(c). With the $\sigma^{-}$ polarized pump the AC-Stark shift occurs on the $X_{K}$ state, which should not affect the measured signal because the system is in a ${\ket{g}\bra{X_{K^{\prime}}}}$ coherence when the pump arrives and is thus insensitive to instantaneous changes in the other valley. The Bloch-Siegert effect will, however, cause a blue-shift of the $X_{K^{\prime}}$ state [16, 21, 22], and is the source of the phase shift observed in this case. This phase shift is much smaller than in the case of the $\sigma^{+}$ polarized pump, but still linear with fluence, as shown in Fig. 3(c).

The magnitude of the AC-Stark shift ($\hbar\omega_{ACS}$) and Bloch-Siegert shift ($\hbar\omega_{BS}$) are given by

where µ is the dipole matrix element for the $K$ and $K^{\prime}$ excitons and $E$ is the electric field amplitude [16]. With a red-detuned pump energy of $\hbar\omega_{p}=$ 1.87 eV and exciton energy $\hbar\omega_{X}=$ 2.07 eV, the expected ratio between the AC-Stark shift and Bloch-Siegert shift is $(\omega_{X}+\omega_{p})/(\omega_{X}-\omega_{p})=$ 19.7. The measured ratio of slopes in Fig. 3(c) is $8\pm 3$. This suggest the measured Bloch-Siegert shift is approximately a factor of two larger than would be expected based on the measured AC-Stark shift. This discrepancy cannot be explained by the polarization purity of the pump or MDCS pulses (as discussed in detail in the Supplementary Material), but may arise from Coulomb interactions and the presence of higher energy bands in WS₂, which are excluded from the two-level framework used to derive Eq. 1 and 2, and are expected to affect the AC-Stark and Bloch-Siegert shifts differently [21, 22, 23].

To observe the effect of the pump on the inter-valley coherence, the pump was shifted to arrive 150 fs after $\textbf{{k}}_{1}$ and $\textbf{{k}}_{2}$, which arrived simultaneously. The phase of the signal at the exciton energy is plotted in Fig. 3(b) as a function of $t_{2}$. Here, for $t_{2}<100$ fs the pump arrives after $\textbf{{k}}_{3}$ and actually acts on the $\ket{X_{K}}\bra{g}$ coherence. For $t_{2}>200$ fs the pump is acting on the inter-valley coherence. Figure 3(b) shows that phase shifts with approximately equal magnitude, but opposite sign, were observed for the two pump polarizations. The evolution of the inter-valley coherence $\ket{X_{K}}\bra{X_{K^{\prime}}}$ is given by $e^{i(\omega_{X_{K}}-\omega_{X_{K^{\prime}}})t_{2}}$ and shifts to either the $X_{K}$ or $X_{K^{\prime}}$ energy will lead to a phase shift. With the highest pump fluence of 875 µJ/cm² a $\pi$ phase shift of the inter-valley exciton coherence was realized.

The total phase shift, obtained by averaging the phase shift between $t_{2}=220$ fs and $t_{2}=300$ fs, is plotted as a function of pump fluence in Fig. 3(c) (green line). A linear trend is again observed, however, the slope is slightly smaller than for the case of the AC-Stark shift on the $\ket{g}\bra{X_{K^{\prime}}}$ coherence. In fact, the slope of the total phase shift when the pump is in $t_{2}$ and acting on the $\ket{X_{K}}\bra{X_{K^{\prime}}}$ coherence is equivalent to the difference between the isolated slopes of the AC-Stark effect and Bloch-Siegert effects (dashed line in Fig. 3(c)), acquired when the pump was in $t_{1}$. This arises because $X_{K^{\prime}}$, and $X_{K}$ are both blue-shifted (by the AC-Stark shift and Bloch-Siegert shift, respectivley), and thus the energy difference between them (and hence accumulated phase shift) is given by the difference between the two effects. That is, the two effects act to counter one another when rotating the phase of the $\ket{X_{K}}\bra{X_{K^{\prime}}}$ coherence.

The shaded regions in Fig. 3(a) and (b) indicate the delay range where the pump overlaps with the pulse being scanned ($\textbf{{k}}_{1}$ and $\textbf{{k}}_{3}$ respectively). In these regions the dynamics of the phase accumulation can be observed. The rate of change of the phase is determined by the induced energy shifts, i.e. the coherence evolves as $e^{i\Delta t_{n}}$. The time dependent energy shift can thus be determined by differentiating the measured phase shift: $\Delta(t_{n})=d\phi/dt_{n}$. This is shown in Fig. 3(d) for a $\sigma^{+}$ polarized pump in $t_{1}$. A Gaussian fit to the instantaneous AC-Stark shift (red lines) closely resembles the temporal profile of the measured cross-correlation between the pump and the MDCS excitation beams (black line). This indicates that within the temporal resolution afforded by the 31 fs $\textbf{{k}}_{1}$ pulse, the AC-Stark shift follows the intensity of the pump pulse envelope. The peak AC-Stark shift measured is $>$16 meV for the highest pump fluence of 751.7 µJ/cm^{2 11}1The measured evolution of the Stark shift arises from a convolution of the $\textbf{{k}}_{1}$ pulse and the pump-induced shift, hence the actual peak shift will be slightly higher than the one obtained directly from the measurement. Similarly, with the $\sigma^{+}$ polarized pump in $t_{2}$ the measured time dependent AC-Stark shift follows the cross-correlation between the pump and the MDCS beam (Fig. 3(e)). With the $\sigma^{-}$ polarized pump, the peak is broader and shifted to shorter $t_{2}$ times because in this case, for $t_{2}<100$ fs, the system is in a $\ket{X_{K}}\bra{g}$ coherence when the pump arrives, and $X_{K}$ is affected by the AC-Stark shift.

The dynamics of the AC-Stark and Bloch-Siegert shifts are modelled by incorporating these shifts as phenomenological perturbations to the state energies in simulations of the MDCS experiments (see Appendix D for details). There were no free parameters in the model (all were determined directly from experiment) and it was assumed that the AC-Stark and Bloch-Siegert shifts followed exactly the instantaneous intensity of the 100 fs pump pulse. The results are shown as dashed lines in Fig. 3(a) and (b). The close agreement between experiment and simulation further suggests that all details of the shifts are captured in the model and that the phase evolves smoothly as the energy shifts follow the instantaneous intensity of the pump pulse.

Despite the smooth evolution of the phase, the decrease in the signal amplitude seen in Fig. 2(b) suggests the pump causes a partial loss of coherence, and thus that the interaction of the red-detuned pump is not entirely adiabatic. Figure 4(a) and (b) show the normalised signal amplitude, integrated around the exciton energy, as a function of $t_{1}$ and $t_{2}$, respectively. In both cases, a decrease in the signal amplitude, is evident upon arrival of the pump pulse, which indicates additional decoherence of the $\ket{g}\bra{X_{K^{\prime}}}$ and $\ket{X_{K}}\bra{X_{K^{\prime}}}$ coherences, regardless of the pump polarization. After the pump pulse, the decay rate once again matches the unpumped data, as indicated by the slope of the data and fits in Fig. 4(a) and (b). The return of the decay rate after the pump pulse is even more clearly demonstrated in Fig. 4(c) and (d), which show the ratio of the pumped and unpumped amplitudes. It is clear that for $t_{2}>200$ fs, the curves are flat, indicating that the decay rates with and without the pump are the same ²²2For the pump in $t_{1}$, the curve for $t_{1}>400$ fs is flat for the simple reason that coherence time before the pump is increasing. This rules out the possibility that two-photon absorption of the pump-pulse is creating additional carriers that scatter and cause additional decoherence, as they would persist and affect the decoherence time well beyond the pulse overlap range. Thus, we conclude that the enhanced decoherence rate is consistent with a field driven effect persisting only over the finite pump duration. In the measurements of Ye et al. [4] they observe a decrease of the degree of linear polarization, which could reflect a loss of coherence. However, in that case they attribute those losses to the fact that the response is integrated in time, and includes emission before, during, and after the phase is rotated by the pump. They also have some contribution from the Gaussian distribution of pump intensity, which will cause different phase shifts. In our measurements, both of these effects are eliminated: the measured signal is proportional only to the coherence at the time the 30 fs $\textbf{{k}}_{3}$ pulse interacts with the sample, hence there is minimal temporal averaging. The spatial averaging over a distribution of pump intensities was minimised by ensuring that the pump spot size was larger than the MDCS spots and because the MDCS signal scales with the product of the electric field of each pulse, the effective area for the signal generation has a diameter a factor of $\sqrt{2}$ smaller than the individual beams. It is therefore clear that in our measurements there is some additional non-trivial decoherence when the pump pulse is present.

The magnitude of the extra coherence loss is linearly proportional to the pump fluence (see Supplementary Material) and is quantitatively consistent with power broadening. For the case of a constant wave or very long pump pulse, the power broadened linewidth is given by [45]:

where $\Omega=\absolutevalue{\mu}E$ is the resonant Rabi splitting [46] ³³3as opposed to the generalized Rabi splitting, which takes into account the detuning, and $\Gamma_{0}$ is the equilibrium homogeneous linewidth. The ratio $\frac{\Omega}{\Gamma_{0}}$ indicates that power broadening is effectively dependent on the number of Rabi oscillations that occur within the decoherence time. In the case of short pulses, where the pulse duration is less than the decoherence time, quantifying the power broadening can be more challenging [48]. The amount of broadening is still expected to depend on the Rabi frequency, and hence vary with the intensity across the pulse envelope [48]. However, the decoherence time is no longer the shortest relevant timescale. To place an upper limit on the peak power broadening, we calculated the Rabi frequency from Eq. 1, using the peak AC-Stark shift measured (16 meV), and in place of $\Gamma_{0}$ we use the inverse of the pump pulse duration: $=\frac{1}{100fs}=41$ meV. This gives a value of 6.9 meV for the power broadened linewidth, compared to the equilibrium value of 1.8 meV.

An experimental estimate of the average power broadening throughout the pump pulse is obtained from the data in Fig. 4(a) and (b). A linear fit to the data (on the semi-log axes) over the duration of the pulse yields an average decay rate that is 1.6-1.7 times faster than the unpumped decay. This suggests an average power broadened linewidth of $\sim$3 meV.

The power broadening was added to the simulations as a phenomenological contribution to the homogeneous linewidth, which follows the time dependent intensity of the pump pulse. This is an oversimplification, as the process for power broadening is not expected to be instantaneous. Nonetheless, using a peak broadening of 3.5 meV (i.e. at the peak of the pump pulse the power broadened linewidth is 5.3 eV) the simulated signal amplitude matches the dynamics of the measured amplitude for the $\sigma^{-}$ case exceptionally well, as shown in Fig. 4(c). This value for the peak broadening is consistent with the upper limit calculated above and the average linewidth determined by fitting a single decay to the data in the pump overlap region. For the other measurements, the dynamics and/or magnitude of the decrease in amplitude differs somewhat from the simulations using the same parameters. These differences highlight the fact that the dynamics of power broadening for short pulses, and with the valley selectivity of monolayer TMDCs, are more complicated than the simple picture considered. However, with our phenomenological model, and quantitative comparisons, we have demonstrated that power broadening is present and the associated loss of coherence presents a limitation when using the AC-Stark and Bloch-Siegert effects to control bandstructures and the phase of quantum coherences.