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1 DOI: 1.138/NMAT4156 Valley-selective optical Stark effect in monolayer WS Edbert J. Sie, 1 James W. McIver, 1, Yi-Hsien Lee, 3 Liang Fu, 1 Jing Kong, 4 and Nuh Gedik 1, 1 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 139, USA Department of Physics, Harvard University, Cambridge, MA 138, USA 3 Material Sciences and Engineering, National Tsing-Hua University, Hsinchu 313, Taiwan 4 Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 139, USA (Dated: October 4, 14) 1. Semi-classical and quantum description of the optical Stark effect. Time-resolved, valley-specific α spectra 3. Pump polarization-dependent, valley-specific α spectra 4. Finding the energy shift E from the measured α spectra 5. Estimating the energy shift E from semi-classical theory 1. Semi-classical and quantum description of the optical Stark effect For the purpose of understanding the optical Stark effect in atoms, it is often sufficient to write down the semiclassical form of the Hamiltonian, in which light is represented by the classical fields as external perturbation. By diagonalizing the Hamiltonian, the atomic energy levels can be obtained and the optical Stark effect is apparent in the light-induced change of the energy spectrum. The creation of the Floquet states can also be seen from the energy denominator of the shift as will be described below. In the quantum description, on the other hand, light is represented by quantized oscillators, from which the Floquet states are seen in the quantized photon energy spacing in the energy spectrum. In the following, we will begin by discussing the semi-classical picture of the interaction, and later compare with the quantum picture. Similar to the static Stark effect, the optical Stark effect also results from induced energy repulsion between two states. To understand this, we consider a two-level atomic system of states a and b with respective energies of E a and E b (Fig S1a), and we put this atom in the presence of static electric field E. Since the atomic orbitals have a definite parity, either even or odd under spatial inversion symmetry, the application of first-order perturbation with E keeps the energy levels unchanged. Hence, the Stark effect in atoms emerges from the secondorder perturbation with E, inducing a hybridization between states a and b that results in shifted energy levels by as much as b ˆpE a E b E a = M ab E E b E a E b = (1) b ˆpE a E a = = M ab E E a E b E b E a where ˆp is the electric dipole moment operator of the atom, E is the static electric field strength, and M ab is the polarization matrix element between a and b. Such energy shifts result in a wider separation of the energy levels, also known as the state repulsion, with magnitude E = M ab E E b E a () where the magnitude is quadratic in E and is inversely proportional to the energy separation E b E a before the application of the field (Fig S1a). In the optical Stark effect, the perturbation is written as Ĥ (t) =ˆpE(t), where E(t) =E cos πνt is the oscillating electric field with amplitude E and frequency ν. Here, we can use the standard time-dependent perturbation theory [1] to find the shift of the energy levels ( ) E b (t) =H ab(t) i(eb E a )t 1 exp i t H ab(t ) exp ( i(e b E a )t ) (3) dt where E a (t) has a similar expression after switching the index. It is insightful to write the perturbation Hamiltonian as Ĥ ab (t) =ˆpE ( e iπνt + e iπνt) /. In this way, we can identify the perturbation as coherent absorption and emission of light from the atomic states. By evaluating equation (3), we can obtain the time-averaged shift of the energy levels [ ] M ab E E Ēb = 1 Ēa = 1 M ab E b E a hν + E b E a + hν [ M ab E E M ab E b E a hν + E b E a + hν ] (4) where E = E / is the time-averaged value of the electric field squared. These expressions are consistent with the case of the static electric field in equation (1) in which NATURE MATERIALS Macmillan Publishers Limited. All rights reserved.

2 DOI: 1.138/NMAT4156 FIG. S1. (a) Energy level diagram of the static Stark effect in two-level atoms. (b) Semi-classical and (c) quantum picture of the optical Stark effect. (d) Schematic of the three Hamiltonian terms in describing the coherent light-matter interaction. hν and E E. Furthermore, by comparing the energy denominator with the static case, we can see that the two terms in Ēb correspond to the state repulsion of b by two photon-dressed states. These two states originate from the coherent absorption and emission of light from state a, as shown in Fig S1b. Similarly, the two terms in Ēa are the state repulsion of a by the two photon-dressed states of b. These photon-dressed states are the Floquet states that we identified in the main text. In our experiment, hν is detuned only slightly below E b E a such that the contribution from the first term to the energy shift dominates, and that of the second term can be neglected. As a result, the energy level separation increases by Ē = M ab E (5) E b E a hν This is the semi-classical result of the optical Stark shift that we obtained for equation () in the main text. As can be seen from this expression, the energy shift is linearly proportional to the light intensity and is inversely proportional to the light energy detuning from the E b E a transition. On the other hand, the fully quantized description of NATURE MATERIALS 14 Macmillan Publishers Limited. All rights reserved.

3 DOI: 1.138/NMAT4156 SUPPLEMENTARY INFORMATION3 the optical Stark effect can be described by using the standard Jaynes-Cummings model [] where light is represented by quantized oscillators with energy of hν. The Hamiltonian is given by Ĥ = 1 (E b E a )ˆσ z +hνâ + â+ 1 ω R (ˆσ+â +ˆσ â +) (6) where the three terms correspond to the two-level atom, the photon reservoir, and the atom-photon interactions, respectively (Fig S1d), ω R is the atom-photon coupling strength (Rabi frequency), and the Pauli matrices and the ladder operators are given by ˆσ z = b b a a ˆσ = a b ˆσ + = b a â n = n n 1 â + n = n +1 n +1 (7) where n is the number of photons. Due to the interaction term, this Hamiltonian only couples states a, n +1 and b, n, which we can use as the basis for the Hamiltonian matrix ( H = hν n + 1 )( ) ( ) (8) Eb E a hν ω R n +1 ω R n +1 (Eb E a hν) This Hamiltonian matrix can be diagonalized to give the energies ( E n,± = hν n + 1 ) ± 1 (9) (E b E a hν) +( ω R ) (n + 1) As can be seen from this expression, the Floquet states emerge naturally through the discrete energy term in units of hν (Fig S1c). Additionally, even in the absence of external perturbation of light (n = ), states a and b can still exhibit hybridization through the vacuum field, for which the coupling strength is ω R [3]. The optical stark shift of a b optical transition can also be evaluated, after taking a weak field approximation in which ω R E b E a hν, giving E n 1 n( ω R ) E b E a hν (1) where ( ω R ) is proportional to E. Thus, both the semi-classical and the quantum description of coherent light-matter interaction give consistent picture about the Floquet states and the energy shift due to the optical Stark effect.. Time-resolved valley-specific α spectra FIG. S. (a) Measured absorbance of monolayer WS (black) and a rigidly shifted one (dashed) to simulate the optical Stark effect. (b) The simulated change of absorption induced by the pump pulse. (c) Valley-specific α(ω, t) spectra measured using pump-probe pulses of the same (left panel) and opposite (right panel) helicities. It shows a peak shift only at the K valley ( t = fs) and common background signals from the photoexcited excitons ( t > fs). The equilibrium absorption spectrum of our sample (Fig Sa, black), as measured using differential reflectance microscopy, shows an absorption peak at E =. ev, which is also consistent with other measurements [4]. Optical Stark effect gives rise to an energy shift of this peak (simulated in Fig Sa, dashed), which can be measured as induced absorption α at slightly higher energy (simulated in Fig Sb, red shaded). This α spectrum is the signature of the optical Stark effect in our experiment. In Fig Sc, we present a pair of experimentally measured α spectra as a function of pump-probe time delay. Here, we used left circularly polarized (σ ) pump pulses with photon energy of 1.8 ev (below the absorption peak), fluence of 6 J/cm, and duration of 16 fs at FWHM. The probe helicity is tuned to be the same (σ, left panel) and opposite (σ +, right panel) to the pump helicity. On the left panel, the spectra show a distinct feature centered within the pump pulse duration ( t = fs), which consists of positive α above the original absorption peak and negative below it. On the right panel, however, the spectra show only a faint NATURE MATERIALS Macmillan Publishers Limited. All rights reserved.

4 DOI: 1.138/NMAT and premature signal during the pump pulse duration, in contrast to that of the left panel. At t > fs, both panels share common spectra, where α is negative at the absorption peak and positive below it. The α spectra at t = fs shown on the left panel indicate that the absorption peak is shifted to higher energy. The absence of this feature on the right panel indicates that the shift is well isolated within only the lightdriven valley. The non-zero α values at t > fs in both panels, when the pump pulse no longer persist, are induced by photoexcited excitons, due to excitonic bleaching and biexcitonic absorption [5, 6], and they are present in both valleys. These excitons, however, do not contribute to the optical Stark effect and just add common background signals in both valleys. The different signatures of α spectra (Fig Sc, left panel) that are induced by the pump pulse ( t = fs) and by the photoexcited excitons ( t > fs) prove that the observed valley-selective energy shift is driven coherently by light, and not by photoexcited excitons. In order to eliminate slight contributions from the photoexcited excitons at t =fs,wetake.5 α(ω, t = 4 fs) from the right panel of Fig Sc as the background (BG) signal. This BG signal will be used in section S3. 3. Pump polarization-dependent valley-specific α spectra In the following discussion, we only consider results taken at t = fs where the optical Stark effect takes place. We present the valley specific α spectra, before (Fig S3a) and after (Fig S3b) BG subtraction, as a function of pump polarization. We used σ and σ + probe pulses to monitor the K valley (left panels) and K valley (right panels), respectively. The latter figures show a clear transition of α value: largest when the pump pulses have the same helicities with the probe pulses, halved when linearly polarized, and nearly zero for opposite helicites, as is also summarized in Fig S3d. This results from the shifting of energy levels as a function of pump polarization, as is schematically depicted in Fig S4. This behavior arises from the selection rule in the matrix element M v, which is responsible for the previously reported valley-selective photoluminescence (see the main text). 4. Finding the energy shift E from the measured α spectra We can represent the absorption peak by a Gaussian lineshape. The resulting absorption spectrum due to an energy shift of E can be expressed as α(ω, E) =A exp ( (ω E) c ) (11) where A is the absorption peak, c is the FWHM/ ln and = 1. Fig S5a shows the absorption spectra before FIG. S3. (a) Valley-specific α(ω, t = ) spectra measured as a function of pump polarization, before and (b) after the background (BG) signal subtraction. These spectra show that the energy shifts are well isolated in either valley. (c,d) Spectral polarization cuts from the corresponding panels in (a,b) for a clearer comparison. 4 NATURE MATERIALS 14 Macmillan Publishers Limited. All rights reserved.

5 DOI: 1.138/NMAT4156 SUPPLEMENTARY INFORMATION5 FIG. S4. Band diagrams showing how the energies of the conduction band (CB) and the valence band (VB) are shifted as a function of pump polarization. measure the full spectrum of α, which can be performed in transient absorption spectroscopy [5], and verify if the measured α spectrum profile results from a peak shift (Fig S5b). To determine the shift in the peak position, we can integrate the area of this α(ω, E) curve in the range of ω, which is referred in the main text as the spectral weight transfer (SWT), and by using equations (S11) and (S1) it can be evaluated as ( ω ) α(ω, E)dω = c α(ω) E dω = A E = A E dx exp( x) (13) That is, by measuring the SWT and the absorption peak A, we can extract the energy shift E due to the optical Stark effect. The measured SWT is plotted in Fig 3b,c in the main text with a new energy scale as determined by equation (S13). 5. Estimating the energy shift E from semi-classical theory In theory, the optical Stark shift is completely described by the transition dipole moment M ab, pump field strength E and pump detuning, which can be calculated and compared with the data in Fig 3c. The energy shift has a simple expression (equations (S5) and (S1)) given by E = ( ω R) (14) FIG. S5. (a) Absorption spectra before (black) and after (red) the absorption peak is shifted by E. (b) The change of the absorption spectrum due to shifted energy. (grey shaded) and after (red curve) the peak is shifted by E. We can then obtain the change of the absorption spectrum (Fig S5b) before and after the shift α(ω, E) =α(ω, E) α(ω) ( ) α(ω) α(ω E) = E E = α ω E = ω c α(ω) E (1) where we only keep terms in the first order of E. This result has been routinely used to estimate the energy shift E via optical Stark effect through measuring the change in the absorption α at a particular energy ω [7]. This method, however, is insensitive to distinguishing a peak shift from a peak broadening. Thus, it is crucial to where ω R (= M ab E ) is the Rabi frequency. We can estimate for M ab from the study of quantum well semiconductors [8] given as ( ) eh (M ab ) E g = (15) E m c where E (= ev) is the transition energy of the exciton, E g (=.3 ev) is the quasiparticle band gap [9], m c (=.3 m ) is the effective mass of the conduction electron [1]. This gives M ab = 56 Debye for monolayer WS (1 Debye = Cm), which is about twice the value obtained for GaAs quantum wells [11]. For the maximum energy shift of 18 mev, as measured at = 18 mev, we used pump fluence of 1 J/cm with pulse width of 16 fs (obtained from Fig 1e). This gives the peak irradiance of I = fluence/width = ɛ ce / from which we can determine the field strength of E = 75 MV/m and the Rabi frequency of ω R = 87 mev. This calculation yields the energy shift of E = 1 mev which is consistent with the measured value of 18 mev from our experiments. NATURE MATERIALS Macmillan Publishers Limited. All rights reserved.

6 DOI: 1.138/NMAT In comparison to the optical Stark effect, the energy shift resulted if we were applying a static field instead of the optical field is given by E = (M abe ) E (16) As can be seen, in order to obtain the same energy shift, we need E = 15 MV/m, which is very difficult to achieve in dc. Note also that the application of a static electric field in monolayer WS will not lift the intervalley degeneracy. gedik@mit.edu [1] Bransden, B. H. & Joachain, C. J. Physics of Atoms and Molecules. nd ed. (Prentice Hall, New Jersey, 3). [] Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theories with application to beam maser. Proc. IEEE 51, (1963). [3] Brune, M. et al. Quantum Rabi oscillation: A direct test of field quantization in a cavity. Phys. Rev. Lett. 76, (1996). [4] Zhao, W. et al. Evolution of electronic structure in atomically thin sheets of WS and WSe. ACS Nano 7, (13). [5] Sie, E. J., Lee, Y.-H., Frenzel, A. J., Kong, J. & Gedik, N. Biexciton formation in monolayer MoS observed by transient absorption spectroscopy. Preprint at (13). [6] Mai, C. et al. Many-body effects in valleytronics: Direct measurement of valley lifetimes in single-layer MoS. Nano Lett. 14, -6 (14). [7] Joffre, M., Hulin, D., Migus, A. & Antonetti, A. J. Mod. Opt. 35, (1988). [8] Asada, M., Kameyama, A. & Suematsu, Y. Gain and intervalence band absorption in quantum-well lasers. IEEE J. Quantum Electron., (1984). [9] Chernikov, A. et al. Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS. Phys. Rev. Lett. 113, 768 (14). [1] Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R. Theory of neutral and charged excitons in monolayer transition metal dichalcogenides. Phys. Rev. B 88, (13). [11] Schmitt-Rink, S. & Chemla, D. S. Collective excitations and the dynamical Stark effect in a coherently driven exciton system. Phys. Rev. Lett. 57, (1986). 6 NATURE MATERIALS 14 Macmillan Publishers Limited. All rights reserved.