Coherent dynamics of Floquet-Bloch states in monolayer \nWS2\n reveals fast adiabatic switching

Abstract

Floquet engineering offers a path to optically-controlled materials, but experimental implementations have frequently relied on femtosecond pulses to achieve the high peak fields required to maximise interactions and obtain a measurable response. The Floquet formalism, however, is based on a continuous, periodic drive. Here we use femtosecond laser pulses to drive the optical Stark effect, a simple realization of Floquet engineering, in monolayer WS2. By monitoring the coherent evolution of the free-induction decay, we show that the system evolves adiabatically, and that the finite duration of the pulses does not introduce any effects beyond Floquet theory. Furthermore, we demonstrate that the induced energy shift follows a linear dependence on instantaneous intensity, even for ultrafast driving fields of fewer than 15 optical cycles. Non-equilibrium phases of matter can provide access to microscopic physics, and macroscopic properties, unavailable in extant materials1–5. Explorations of materials driven out of equilibrium has led to the discovery of a range of phenomena such as a light-driven anomalous Hall effect6,7, light-induced superconductivity8,9 and light-induced phase transitions10,11. Floquet engineering12, the process of using a periodic perturbation to reversibly manipulate bandstructure, is one avenue to control material properties. Floquet theory removes the time-dependence of a Hamiltonian, resulting in an infinite, periodic set of replica bands spaced by the frequency of the perturbation, known as Floquet sidebands13. The electromagnetic field oscillations of light can act as the periodic perturbation while careful selection of the energy and other properties can be used to manipulate interactions between the equilibrium bands and their Floquet replicas to control the resulting non-equilibrium properties. Floquet-Bloch bands generated in this manner have been observed, for example, as replicas of the surface states of a topological insulator in timeand angle-resolved photoemission spectroscopy (tr-ARPES) measurements13,14. The application of Floquet theory to both the Haldane model15 and HgTe/CdTe quantum wells16 predicted a reversible change in the topology of the bandstructure in both cases. Cold atom experiments have been used to simulate these predictions, showing changes to the topological phase via the Chern number17,18, and the emergence of chiral edge states19. Additionally, experiments with graphene and a circularly polarized mid-IR driving field appear to show the realization of a topologically non-trivial band structure and the opening of a gap, as evidenced by an anomalous Hall effect6,7,20. A topological phase transition is also predicted to arise in semiconducting monolayer transition metal dichalcogenides (TMDs) when driven with an optical field with frequency larger than the band gap21,22, but has yet to be observed experimentally. In order to elicit a response at a detectable level, ultrafast laser pulses with high peak intensities are commonly used in these experiments because the interactions between states typically scales with the driving field amplitude. Floquet theory, however, is predicated on a continuous, monochromatic drive12,16. The discrete nature of these pulses then raises new questions: How short can pulses be before Floquet theory ceases to be applicable? When does the bandstructure (i.e., the eigenstates of the system) cease to evolve adiabatically? The optical Stark effect (equivalently, the AC Stark effect) is a well-understood phenomenon23 that emerges naturally from Floquet theory24. It provides an ideal model to test the limits of the Floquet formalism for short pulses and the adiabaticity of turning on the periodic driving field on femtosecond timescales. In TMD monolayers21,22,25–28, a valley-selective optical Stark shift has been observed, with the magnitude of the shift obeying the expected linear dependence on pump intensity, and inverse dependence on the detuning of the pump from the transition energy29. Coherence between valleys was shown to be maintained to some extent, though not quantified, and the phase rotated by the shift25. However, little consideration has been given to the duration of the driving field, with the majority of these measurements using pulses longer than 100fs. Experiments investigating the optical Stark effect in GaAs quantum wells/bulk GaAs found that the adiabatic approximation holds with a >100 fs sub-bandgap pump pulse30–32. A Green’s function analysis, in agreement, found that the width of the Gaussian envelope was the best predictor of convergence to the case of a continuous drive33, and Floquet analysis applied to the turning on of a pulse showed a similar result34. Here, we drive the optical Stark effect in monolayer WS2 with red-detuned pump pulses as short as 34 fs (~15 optical cycles) and probe the response with 25 fs pulses, resonant with the exciton transition. By varying the pump pulse duration while maintaining constant pulse energy, we show that the observed shift scales linearly with peak intensity, as predicted by Floquet theory, and, more significantly, follows (within our time resolution) the instantaneous intensity of the pulse envelope, showing no deviation from Floquet theory and the adiabatic approximation. Furthermore, we observe coherent dynamics of the free induction decay, revealing a dephasing time of 38.8 fs at room temperature and an adiabatic shift of the transition energy induced by the pump pulse, as identified by the smooth and continuous evolution of the macroscopic coherence through the dynamic shift. Monolayer WS2 has a direct gap at the K/K’ symmetry points in the Brillouin zone, and a large exciton binding energy, ~320 meV35. Due to its two constituent elements and hexagonal lattice the crystal structure lacks inversion symmetry, which introduces significant Berry curvature around the K/K’ valleys. Spin-orbit coupling splits the valence and conduction bands, which leads to a spin-valley locking of opposite sign. Consequently, circularly-polarized light selectively excites one of the two energetically-degenerate valleys, enabling so-called valleytronic applications36. This also leads to an optical Stark effect that is valley selective, when driven by a circularly polarized laser field. To investigate the effect of ultrashort pulses in driving Floquet Bloch bands, we characterized the Stark shift in a monolayer of WS2 on a SiO2/Si substrate using a 25 fs probe pulse, resonant with the A-exciton energy (2.025 eV). The shift was induced by the sub-bandgap (1.850 eV) pump pulse whose duration was varied from 34 to 65 fs by altering the spectral bandwidth. The dynamics associated with the observed shift were determined by varying the delay between the pump and probe pulses. Details are provided in the Supplementary Information (SI). Figure 1: (a) Measured transient reflectance from 34 fs, 50 μJ/cm2 pump for co-circularly polarized probe; (b) Maximum signal at pulse overlap (relative delay t = 0) for 20 μJ/cm2 (red) and 50 μJ/cm2 (blue) pump fluence. Probe fluence was 3.5 μJ/cm2. The equilibrium optical properties of the exfoliated WS2 monolayer were characterized by spatially-resolved reflectance contrast and photoluminescence measurements performed at room temperature, as reported previously37, and are shown in supplementary Fig. S1. To account for the influence of the intervening 300 nm SiO2 layer and Si substrate, a 4-Lorentzian model was used to extract the equilibrium permittivity of the WS2 layer using the transfer matrix method. The resulting equilibrium reflectance and absorption spectra, and the complex permittivity of the monolayer, are presented in Fig. S2. This enabled the conversion of the measured transient reflectance data (dR/R) to the transient absorption of the WS2 monolayer (See SI for details). The transient reflectance measured with co-circularly polarized pump and probe pulses is shown in Error! Reference source not found.(a) for a 34 fs pump pulse duration. The spectral response when pump and probe overlap in time (t = 0) is highlighted in Fig. 1(b), and shows a positive peak (increased reflectance due to decreased absorption) on the low-energy side of the exciton and a negative peak (decreased reflectance due to increased absorption) on the high-energy side. This dispersive profile arises from the transition being blueshifted, as a result of the pump-induced optical Stark effect, and is consistent with previous observations21,22. The magnitude of the observed shift at t = 0 is 0.68 meV, as determined by the spectral weight transfer (SWT) calculated from the transient absorption spectrum22,32 (See SI for details). For relative delays t > 0 (i.e. for the pump pulse arriving first) there is no signal beyond the pulse overlap regime, confirming the expectation that the pump pulse is not generating any significant exciton population. Conversely, for relative delays t < 0 a response with fringes is observed [see Fig. 1(a)], persisting beyond -100 fs, well beyond the pulse overlap regime. The fringes represent a spectral interferogram arising from the interference between two pulses separated in time: the residual probe pulse, and a coherent response generated when the driving field (pump beam) is present. In this case, the probe beam arrives first and excites a coherent superposition between ground and A-exciton states. This leads to a macroscopic polarization in the monolayer that remains in phase with the initial laser pulse for a time limited by decoherence (due to the finite homogeneous linewidth) and dephasing (due to the finite inhomogeneous linewidth). This macroscopic coherence can re-radiate as coherent photoluminescence, in the same direction as the probe, with a p phase shift. This process, also known as free-induction decay, occurs whenever there is coherent resonant excitation. In the absence of the pump this coherent emission adds to the probe, giving the pump-