Optical manipulation of valley pseudospin

Transcription

1 Optical manipulation of valley pseudospin Ziliang Ye, Dezheng Sun, and Tony F. Heinz Departments of Applied Physics and Photon Science, Stanford University, 348 Via Pueblo Mall, Stanford, CA 9435, USA SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, CA 9425, USA I. Quantum beat model We analyze our experiment on valley pseudospin rotation within a model based on the phenomenon of quantum beats. This process describes the interference between two spontaneous emission pathways - - in our case, excitons in the two valleys - - that share a common ground state 1. Because the emission from each valley corresponds to opposite circularly polarized states, a coherent superposition state with equal weightings in the two valleys will produce linearly polarized emission, with a polarization orientation that depends on the phase difference between the two valley components. Our control pulse changes this relative phase and, hence, the valley pseudospin and the orientation of the linearly polarized emission. Our ability to control the valley excitation follows from the optical selection rules for the creation and emission of excitons in either valley 2 : g ε! R µ!" K 1 = g ε! L µ!" K 2 = D g ε! L µ!" K 1 = g ε! R µ!" K 2 =. (SI1) Here g denotes the ground state of the system; K 1 and K 2 are states with excitons excited in the indicated valley, corresponding to K and K states in the main text; ˆε L/R = ( x ± iy) / 2 is the polarization vector for left/right circular polarized light;! µ is the electric- dipole operator; and D is the magnitude of the transition dipole moment, which is the same for both valleys. NATURE PHYSICS 1

2 At t =, we resonantly excite the sample with a light pulse with a linearly polarized electric field. As a result, based on the selection rules given above, we prepare the system in a superposition state of excitons in the K1 and K2 valleys: ψ (t = ) = α 1 K 1 + α 2 K 2. (SI2) For the specific case of excitation by an x - polarized electric field,! E x = E x = E ( ˆε L + ˆε R ) / 2, we have coefficients of α 1 = α 2 = 1/ 2. We treat the excitation pulse creating this state as having negligible duration. (Although not of importance for our discussion, this simplification introduces some inaccuracy in modeling the rising edge of the response induced by the control pulse.) For the present, we also neglect the effect of population decay and decoherence, the influence of which are introduced below. We can then describe the subsequent evolution of the system in terms of the transition energies E1 and E2 for excitons in the K1 and K2 valleys by ψ (t) = α 1 e iφ 1(t) K 1 + α 2 e iφ 2 (t) K 2, (SI3) where the phase is given by φ 1/2 (t) =! 1 E 1/2 (t ')dt '. t (SI4) In our experiment, the exciton transition energies are shifted by the control laser pulse through the optical Stark effect and are therefore time dependent. The indicated adiabatic description of the response of the system is justified by the fact that the shifts in the exciton energy occur on a time scale that is slow compared to the oscillation frequency. We now consider the resulting fluorescence emission. The time- dependent fluorescence intensity for emission with linear polarization along a direction at angle θ direction from the x - axis is given by I θ (t) g ˆε θ! µ ψ (t) 2, (SI5) where ε! θ = (e iθ ε! L + e iθ ε! R ) / 2 is the polarization vector for the measured emission direction. From the selection rules given above, we then find I θ (t) (α 1 De i(φ 1(t) θ ) + α 2 De i(φ 2 (t)+θ ) ) (α 1 + α 2 + α 1 α * 2 e i(φ 1(t) φ 2 (t) 2θ ) + c.c) (SI6) We can generalize this result to describe an ensemble of excitons through use of the density matrix formalism with respect to states corresponding to excitons in the two valleys: 2 NATURE PHYSICS

3 I θ (t) (ρ 11 + ρ ρ 12 cos(φ 1 (t) φ 2 (t) 2θ)), (SI7) 2 2 where ρ 11 = α 1, ρ 22 = α 2, and ρ 12 = α 1 α * 2. (In writing the latter relation, we assumed for simplicity that α 1 and α 1 are real, as is the case for our x - polarized excitation.) We see that when the valley pseudospin is rotated by an angle Δφ = φ 1 φ 2, then the experimental signature of the measured angular distribution of PL intensity will be rotated by an angle of θ = Δφ / 2. In our study, the phase difference is induced by an intense circularly polarized laser pulse through the optical Stark effect: Δφ(t) = φ 1 (t) φ 2 (t) =! 1 t ΔE 12 (t ' τ )dt ', (SI8) where τ is the delay between the excitation and control pulses and ΔE 12 (t) represents the temporal profile of the induced energy shift for the control pulse at zero time delay. Based on the measured duration of the control pulse, we model ΔE 12 (t) as a Gaussian function with a full- width at half maximum of 1 fs. In the following, we introduce two phenomenological decay times: T1 to describe the exciton population lifetime (which is dominated by non- radiative decay channels) and T2 to characterize the intervalley decoherence time. We then can write I θ (t) e t/t 1 (ρ 11 + ρ 22 ) + 2e t/t 2 ρ 12cos(φ 1 (t) φ 2 (t) 2θ), (SI9) where the superscript for the density matrix elements denotes their initial values immediately after the excitation pulse. As we do not time resolve the PL in our measurements, we use the time integral of the above expression to predict the measured Stokes parameters. Note that the first term contributes a background independent of the analyzer angle, while the second term generates the polarization anisotropy that yields a finite value for the Stokes parameter S2. We can establish relations of parameters within our model using a reference experiment carried out without the control pulse (for which Δφ = ). In this case, we observe a peak value of the normalized Stokes parameter S2 =.2 as we scan the analyzer angle. Within our model, this imposes the following constraint on the ratio between initial density matrix elements: T 2 / T 1 =.2(ρ 11 + ρ 22 ) / 2ρ 12. This relation accounts for the incomplete degree of linear polarization observed experimentally in the reference experiment. If a fully coherent state is established initially, then the decoherence time T2 would need to be about 5 times shorter than the population decay time. (If only partial coherence is established initially because of the off- resonant character of the excitation, then this ratio would be smaller.) Using this constraint, we obtain for the normalized Stokes parameter ( θ = 45! ): NATURE PHYSICS 3

4 S 2 =.2T 2 1 e t/t 2 sin(δφ(t))dt. The best fit of S 2 as a function of the delay τ between the excitation and control pulses (Fig. 4 of main text) yieldst 2 = (35 ± 5) fs and ΔE 12 () = (13 ± 2) mev. This Stark shift agrees with the value inferred from the frequency- domain measurements (Sects. III and IV below). We can also calculate the average orientation of the linear polarized emission θθ! and the average degree of the linear polarization νν using well- known relations for the Stokes parameters 3 : θ = arg(s 1 + is 2 ) / 2 and v = S S 2, where we have S 1 =.2T 2 1 e t/t 2 cos(δφ(t))dt. We plot these quantities in Fig. S1 as a function of the time delay τ of the control pulse with respect to the excitation pulse. The maximum rotation angle is found to be 22 (Fig. S1a), in agreement with the experimental result of Fig. 3 of main text. a Rotation angle (degree) b Degree of linear polarization Excitation-control delay τ (fs) Excitation-control delay τ (fs) 8 Fig. S1. Calculated effect of the control pulse on the linear polarized photoluminescence based on the quantum- beat model with the experimentally determined value of the decoherence time of T 2 = 35 fs. a, the angle of rotation of the polarization and b, the degree of linear polarization.as a function of the excitation- control delay time ττ. II. Reduction in the degree of linear polarization by the control pulse In both Fig. 3 in the main text and Fig. S1b, we observe a decrease in the degree of linear polarization of the photoluminescence by the application of the control pulse. This phenomenon is analogous to the Hanle effect in which polarization of PL is reduced when the transition dipole moment precesses around a real magnetic field 4. In our experiment, the coherently excited excitons emit photons with different polarization as the valley pseudospin rotates around the pseudomagnetic field. Just as in the conventional Hanle effect, the PL signal in our measurement reflects a summation over emission times, which leads to a reduction in the degree of linear polarization compared to the situation without pseudospin rotation. In addition to this factor, there is also an averaging effect arising from the finite spatial extent of the control beams. The finite diameter of the control beam implies that the phase difference between two valleys also exhibits a spatial dependence: 4 NATURE PHYSICS

5 Δφ(t,r) =! 1 t ΔE 12 (t ' τ,r)dt ', where ΔE 12 (r) describes the spatial profile of the control beam. The PL signal thus samples a range of phase shifts over the 1.7- μm diameter of the excitation beam. Although the control beam (2.1 μm diameter) is larger than the excitation beam, this spatial variation still leads to a modest decrease in the degree of linear polarization. In our modeling for Fig. 4, both the temporal and spatial averaging effects have been accounted for. The predicted reduction in the degree of depolarization calculated in Fig. S1b agrees with the experimental result in Fig. 3. In order to show that this depolarization effect is intrinsic to the rotation, we carried out a control experiment in which we compared two conditions for valley pseudospin rotation, one with a linearly polarized control pulse and the other without the control pulse (i.e., at negative time delay). The corresponding measurements of the polarization (Fig. S2) agree well with one another. This result demonstrates that the depolarization is induced primarily by the averaging effects associated with the rotation, rather than by any inherent pump- generated decoherence process. S Positive Delay Negative Delay Analyzer angle (degree) = 5 fs = -2 fs Fig. S2. Variation of S 2 with the angle θ of the analyzer for linearly polarized control pulses at a positive 5- fs delay (blue curve) and a circularly polarized control light at a negative 2- fs delay (grey curve). The good agreement of the response indicates that a linearly polarized control pulse does not introduce additional depolarization. As discussed in the text, depolarization only occurs under valley pseudospin rotation. III. Frequency- domain measurement of the optical Stark effect In the frequency domain, we can determine the optical Stark shift by comparing the reflection contrast spectra with and without the pump pulse. The reflection contrast spectrum in the absence of pump excitation is measured by a white light source with a spectrometer. The reflection contrast with the control pulse (pump on) is obtained Reflection Contrast (a.u.) Photon Energy (ev) Experiment: with pump Fit Experiment: without pump Fit 1.78 Fig. S3. The reflectivity contrast spectra for monolayer WSe 2 with (green circles) and without (blue solid curve) pump excitation. The blue and red dashed curves are fits based on the thin- film model described in the text and yield an optical Stark shift of mev. (The error bar corresponds to the uncertainty in the fitting procedure.) 5 NATURE PHYSICS 5

6 by adding up the pump- induced change to the unpumped reflection spectrum. This allows us to measure the effect of the pump pulse using the lock- in technique described in Methods section of the main text. The reflection contrast spectra with and without pump excitation are plotted below. In order to explain the shape of the observed shape of the reflection contrast spectra, we must take into account the fact that our WSe2 sample was supported on a silicon wafer covered by a 27- nm thick oxide film. This gives rise to optical interference effects that significantly alter the spectral characteristics of the WSe2 monolayer. In order to determine the magnitude of the optical Stark shift, we have therefore applied a thin- film model to retrieve the optical response of the WSe2 monolayer from the experimental data. Based on the standard thin- film treatment 5, we can write the reflectivity from the monolayer on the oxidized silicon substrate at normal incidence as where 2 r R WSe2 = r WSe2 = 1 e i ( β 1+β ) 2 + r 2 e i ( β 1 β ) 2 + r 3 e i ( β 1+β ) 2 + r 1 r 2 r 3 e i ( β 1 β 2 ) e i ( β 1+β ) 2 + r 1 r 2 e i ( β 1 β ) 2 + r 1 r 3 e i ( β 1+β ) 2 + r 2 r 3 e i ( β 1 β 2 ) 2 ; (SI1) r 1 = n air n WSe2 n air + n WSe2, r 2 = n WSe 2 n SiO 2 n WSe2 + n SiO2, r 3 = n SiO 2 n Si n SiO2 + n Si, β WSe2 = 2πn WSe2 d WSe2, λ 1, β SiO2 = 2πn SiO2 d SiO2 λ 1. (SI11) Here n denotes the complex (in- plane) refractive index of the materials, λ is the vacuum wavelength of light, and the thicknesses of the layers are = 27 nm and d WSe2 =.65 nm. The required wavelength- dependent refractive indices for SiO2 and Si are taken from the literature 6,7. For the WSe2 monolayer, we use a model of the dielectric function that includes a Lorentzian resonance (for the exciton), together with an off- resonant contribution from the higher lying electronic transitions 8 that is unchanged with and without pump excitation. We apply the same formalism with n WSe2 = 1 to obtain the background reflectivity R bg for the bare substrate. We then calculate reflection contrast spectrum, δ R = (R WSe2 R bg ) / R bg, to compare with the experimental results. We determine the parameters for the Lorentzian excitonic feature and the background contribution to the dielectric function of the WSe2 layer from a fit of experimental reflection contrast spectrum in the absence of pump excitation. To describe the influence of the optical Stark effect, we blue shift the Lorentzian peak in the WSe2 dielectric function and slightly broaden the peak, while conserving the oscillator strength of the transition. The best fit to the experimental data (Fig. S3) implies a shift of the excitonic resonance of 2 ±.5 mev. The corresponding d SiO2 6 NATURE PHYSICS

7 imaginary parts of the inferred dielectric functions are plotted in the Fig 1c of the main text. IV. Coherence effects in the frequency- domain measurement of the optical Stark shift The coherence time for the excitonic transitions in the WSe2 monolayer is comparable to the duration of the pump pulse used to induce the optical Stark shift in our frequency- domain study. In analyzing the transient absorption spectrum, even when characterized by a very short probe pulse, we must therefore consider the influence of coherence, since the response of the system defining the absorption spectrum continues after the probe pulse has subsided. This leads to the observation of a reduced spectral shift compared to optical Stark shift for a longer pump pulse of the same peak intensity. In keeping with the usual measurement conditions and terminology, we describe this latter shift as the intrinsic optical Stark effect. Here relate what we measure by pump- probe spectroscopy to the intrinsic optical Stark shift. We note that the intrinsic optical Stark shift is the parameter relevant for our analysis of the pseudospin rotation: it defines the instantaneous frequency shifts of the excitonic transition at a given pump intensity and is the parameter that enters into the quantum- beat model of the valley pseudospin rotation described above. To capture the underlying physics, we apply a simple model to describe the absorption spectra in the presence of the optical Stark shift induced by the control pulse 9. We probe the system response by a white- light probe. We treat the probe as an ideal delta- function pulse, E(t) = E δ ( t), since the finite duration of the probe pulse is relatively unimportant in our case. The response of the material after the excitation pulse is described by a single resonance with a coherence time T. In the rotating wave approximation at the unperturbed transition frequency, we can then write the polarization produced in the medium by the delta- function probe pulse at time t = as P(t) = P e t/t e i/! t ΔE(t')dt'. (SI12) Here ΔE(t) is the change in the exciton transition energy induced by the pump pulse through the optical Stark effect. It is treated as scaling linearly with the pump intensity, with the coefficient of proportionality defining the intrinsic optical Stark effect. In the transient regime, this shift produces a dynamic phase change in the medium when the pump pulse is present. Since the probe polarization decays over the coherence time T, if the pump pulse is shorter than or comparable to T, then the influence of the optical Stark effect will be reduced and a decreased optical Stark shift will be seen in the pump- probe spectra compared to the case of a long pump pulse. NATURE PHYSICS 7

8 For a thin sample with weak absorption, the absorption spectrum can be determined based on the energy loss in the driving electric field. This leads to an absorption spectrum proportional to ω Re(E! * (ω ) P! (ω )), where E! (ω ) and P! (ω ) are the Fourier transforms of E(t) and P(t), the probe electric field and induced polarization (both in the rotating frame). In Fig. S4, we show by numerical simulations how the duration (and diameter) of the pump beam can affect the Stark shift observed in a pump- probe measurement. In the left figure, we compare a series of simulated spectral shifts for different pump pulse durations, all with the same peak intensity. In the simulation, the optical Stark shift (defining the instantaneous frequency of the excitonic transition) is taken as proportional to the instantaneous intensity of the pump beam. The exciton coherence time is 78 fs, as inferred from the PL linewidth (17 mev), which is dominated by inhomogeneous broadening. The probe pulse is treated as an ideal white light pulse of negligible duration, which arrives at time zero, i.e., at the peak of the pump pulse. When the pump pulse is much longer than the coherence time of the exciton, the observed Stark shift approaches its intrinsic value of 1 mev. However, as the duration of the pump pulse is decreased to the coherence time of the exciton, the system is influenced by the entire temporal profile of the pump pulse, leading to a reduced observed Stark shift and a distorted line shape. Further, in the spatial dimension, the probe beam also samples a significant part of the Gaussian profile of the pump beam. This results in a further reduction of the observed Stark shift compared with that expected from the peak intensity at the center of the beam, as illustrated by the simulation in Fig. S4b for the experimental pump (d = 2.1 μμμμ) and probe (1.7 μμμμ) beam diameters. a Absorption (a.u.) unshifted Δt = 5 fs Δt = 1 fs Δt = 2 fs Δt = 3 fs Δt = 5 fs Δt = 1 fs b Absorption (a.u.) unshifted Δt = 1 fs, d = 1 µm Δt = 1 fs, d = 2.1 µm Frequency (mev) Frequency (mev) 3 4 Fig. S4. Simulations of the effect of the exciton coherence on the optical Stark shift observed in spectrally resolved pump- probe measurement of the optical absorption. The probe pulse is taken as having negligible duration. a, The optical Stark shifts induced by a series of pump beams of the same peak intensity, but with pulse durations ΔΔtt varying from 5 fs to 1 fs. b, The optical Stark shifts for Gaussian pump beams with the same peak intensity at the center, but different diameters d. Results are shown for our experimental conditions (d = 2.1 μμμμ) and in the limit of a large beam (d = 1 μμμμ) for a probe beam of 1.7 μμμμ diameter. 8 NATURE PHYSICS

9 V. Photoluminescence spectra collected for various analyzer directions In Fig. S5, we present raw emission spectra as a function of analyzer angle for three different polarizations of the control beam. From these data we wish to determine the angular shift of the exciton emission feature (as in Fig. 3 of the main text). We see immediately from the figure that the exciton emission (around 715 nm) has a significant linearly polarize component and that the orientation of the linearly polarized emission shifts with the helicity of the control radiation, as indicated by the changing analyzer angle dependence in the three panels. We further note that the spectral characteristics of the exciton emission do not change with this shift, nor does the background response exhibit any dependence on the analyzer angle. For these reasons, the precise fashion in which the exciton emission spectra are treated in the spectral domain is not critical to our analysis. Fig. S5: False color representation of the PL spectra from the WSe 2 monolayer with linearly polarized excitation as a function of analyzer angle θ for different control pulse polarizations. The strong feature around 715- nm corresponds to emission from the A exciton. From left to right, we show, respectively, data for control pulses with right circular, linear, and left circular polarization. VI. Two- photon photoluminescence spectrum As discussed in the main text, for high intensities of the control pulse, we begin to see emission induced by two- photon absorption. This two- photon PL (TPL) interferes with our measurement of the PL from our excitation pulse, which we are using to track manipulation of the valley pseudospin. In Fig. S6 we show the TPL for the highest control fluence (1 mj/cm 2 ). Compared with the one- photon PL from the excitation beam under our experimental conditions, the TPL accounts for less than 1% of the total PL signal in our measurements. Due to the quadratic power dependence of TPL, the two- photon contribution increases quickly for higher control pulse fluence. The TPL then competes with the one- photon PL and would complicate the interpretation of the PL in terms of the valley pseudospin on which our experiment is based. Consequently, we limit the control beam fluence to less than 1 mj/cm 2 in all of our measurements. NATURE PHYSICS 9

10 Intensity (Counts) One-photon PL Two-photon PL Fig. S6. Comparison of PL from the WSe 2 monolayer induced by two- photon PL from the control pulse (red symbols) and by the desired one- photon process (black symbols) from the excitation pulse. These data, collected at the highest control fluence (1 mj/cm 2 ) used in our experiment, demonstrate that the TPL remains slight and does not influence our analysis Energy (ev) 1 NATURE PHYSICS

11 References: 1. Svanberg, S. Atomic and Molecular Spectroscopy, Springer - Verlag Berlin Heidelberg (24). 2. Xiao, D., Liu, G.- B., Feng, W., Xu, X. & Yao, W. Coupled Spin and Valley Physics in Monolayers of MoS 2 and Other Group- VI Dichalcogenides. Phys. Rev. Lett. 18, (212). 3. Collett, E. Field Guide to Polarization. (SPIE Press, 25). 4. Meier, F. & Zakharchenya, B. P. Optical Orientation. (Elsevier, 212). 5. Ni, Z. H. et al. Graphene Thickness Determination Using Reflection and Contrast Spectroscopy. Nano Letters 9, 2758 (27). 6. Edwards, D. F. & Ochoa, E. Infrared refractive index of silicon. Appl. Opt. 19, (198). 7. Maliston, I. H. Interspecimen Comparison of the Refractive Index of Fused Silica. J. Opt. Soc. Am. 55, (1965). 8. Li, Y. et al. Measurement of the optical dielectric function of monolayer transition- metal dichalcogenides: MoS 2, MoSe 2, WS 2, WSe 2. Phys. Rev. B 9, (214). 9. Misewich, J., Glownia, J. H., Rothenberg, J. E. & Sorokin, P. P. Subpicosecond uv kinetic spectroscopy: Photolysis of thallium halide vapors. Chemical Physics Letters 15, (1988). NATURE PHYSICS 11