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## Creator

T. Uto, B. Evrard, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), M. Kroner, A. İmamoğlu

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© 2024 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Interaction-Induced ac Stark Shift of Exciton-Polaron Resonances](https://mdr.nims.go.jp/datasets/f701ee42-bb97-493d-b996-1d24de8e8649)

## Fulltext

Supplementary material:Interaction induced ac-Stark shift of exciton-polaron resonancesI. DEVICE AND ELECTRON DENSITYESTIMATIONOur device structure is sketch in Fig. 1 of the maintext. It consists of a monolayer MoSe2 encapsulated inhBN flakes with a thicknesses estimated from their op-tical contrast to be dtop ≈ 21 ± 3 nm (top layer) anddbot ≈ 42 ± 5 nm (bottom layer). Underneath the lowerhBN, a few-layer graphite flake is used as a gate to elec-trostatically dope the sample. All flakes were mechani-cally exfoliated from bulk crystals, and stacked using astandard dry-transfer technique [1] with a poly(bisphenolA carbonate) film on a polydimethylsiloxane (PDMS)stamp. The heterostructure was then deposited on aSi/SiO2 (285 nm) substrate, and the graphite and MoSe2flakes were then contacted using optical lithography andelectron beam metal deposition.To introduce electrons in the TMD, we ground it whileapplying a voltage V to the graphite gate. Neglect-ing the quantum capacitance of the doped TMD mono-layer, we obtain the charge density from the relationnee = C(V −V0), where e is the electron charge, V0 is thegate voltage at the onset of doping and C = ϵ0ϵ⊥hBN/dbotis the geometric capacitance. For the hBN out-of-planedielectric constant we use the value ϵ⊥hBN = 3.5 [2, 3].The assumption of a negligible quantum capacitance hasbeen verified in previous studies of Van der Waals het-erostructures, where the charge density could be inferede.g. from the filling of a Moiré superlattice [4–6] or ofLandau levels [7]. It is further supported in our sam-ple by the observation of a linear dependence of the APoscillator strength fAP ∝ (V − V0) at low density (SeeSec. IV).II. EXPERIMENTAL SETUPThe sample is loaded in a dry cryostat (attodry800)with free space optical access. Our main light source isa pulsed Ti:sapphire laser (Tsunami, Spectra-physics),with a repetition rate of 76MHz and a pulse durationof ≈ 140 fs (FWHM, before pulse-shaping). The pulsesare split into the pump and probe paths, before beingrecombined, as shown in Fig. 1 of the main text. Forthe pump beam, we reduce the bandwidth using a pulse-shaper, typically to ≈ 5meV, resulting in a duration of≈ 270 fs. We show a typical time profile obtained usingan autocorrelator in Fig. S1, from which we can extractthe pulse duration using a Gaussian fit.For the probe pulse, we use a non-linear crystal fiber(femtowhite 800, NKT photonics) to generate a broadcontinuum spanning the exciton/polarons resonances.Both pulses are focused near the diffraction limit onto1.0 0.5 0.0 0.5 1.0Time [ps]0.00.20.4Intensity [a.u.]FIG. S1. Time profile of the pump pulse measured using anautorcorrelator. From a Gaussian fit, we obtain a duration of≈ 270 fs (FWHM).0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Exciton density [cm 2] 1e12012345Exciton shift [meV]FIG. S2. Exciton energy shift as a function of the densityof excitons generated by the pump beam. The y error barscorrespond to the statistical error estimated by repeating themeasurement 3 times. The x error bars correspond to the sys-tematic error arising from the uncertainty on the various pa-rameters (in particular the exciton (non)radiative decay rate)which are used to estimate the exciton density. The blueline is a linear fit (with the confidence interval shown as ablue cone), from which we extract the interaction strengthUex ≈ 0.09± 0.03µeVµm2.the sample using a microscope objective with NA≈ 0.8(LT-APO/VISIR/0.82). We typically use average powerof ∼ 102µW for the pump and below microwatt for theprobe. For the pump, this results in peak intensity oforder ∼ 1 GW/cm2, and fluences of order ∼ 102uJ/cm2.III. ESTIMATION OF THE EXCITON DENSITYAND INTERACTION STRENGTHIn order to estimate the exciton-exciton interactionstrength, we need to evaluate the density of excitons nex2generated by the pump beam. From the optical Blochequation, we can relate nex to the flux of photon Ii im-pinging on the TMD [8]nex =2IiR0γr(1 + δ2ex/γ̃2), (1)where γr (γnr) is the (non-)radiative decay rate, γ̃ =(γnr+γr)/2 is the dephasing rate and R0 = γ2r /(γr+γnr)2the TMD reflectance on resonance. From a transfermatrix simulation [8] we obtain γr = 2.4 ± 0.4meV,γnr = 0.6± 0.3meV, hence R0 = 0.65± 0.13 and we candeduce the field on the TMD (and hence Ii) as a func-tion of the incoming field. The error bar comes from theuncertainty on the hBN thicknesses and refractive index(see Sec. I) which are inputs of the TMM simulation.We show in Fig. S2 the measured exciton light shift∆ as a function of the exciton density. In this mea-surement the light intensity is kept to a fixed valueIpeak ≈ 1.7 ± 0.3GW/cm2 (≈ 170 uW average power)and we scan the detuning δ in a range 30− 110meV. Weobserve a linear dependence ∆ = Uexnex and extract theinteraction strength from a fit, Uex ≈ 0.09±0.03µeVµm2.IV. FITTING OF THE REFLECTION DATADue to the interference of the light reflected at thedifferent dielectric interfaces of our stack, our reflectionspectrum are fitted by a sum of a Lorentzian and a dis-persive LorentzianS(E) = A[cos θσ22(E − E0)2 +σ24+ sin θσ(E − E0)(E − E0)2 +σ24].(2)Here E is the reflected photon energy, and A, σ, E0 arefitting parameters which depend on ne. The last fittingparameter θ is independent on ne; in practice, we let itfree and verify that we obtain the same values for all fits.We always fit independently the AP and RP resonance.We show in Fig. S3 the results of the fit as a function ofthe gate voltage.In addition we also fit the data with the reflection spec-trum predicted by transfer matrix simulation [8], with theAP/RP energy and (non)radiative decay rates taken asfitting parameter. From this analysis, we obtain the ratioof the oscillator strength fAP,RP/fex = γr,AP/RP/γr,ex,shown in Fig. S3d, which we then use to infer the in-teraction strength ratio UAP/Uex from a measurement ofthe light shift as described in the main text.V. COUPLING TO THE BIEXCITON STATEIn this section, we report on our observation of the ex-citon light shift (at charge neutrality), for cross-circularlypolarized pump and probe laser. In this situation, which0 5V [V]0102030Amplitude×10aRPAP0 5V [V]051015Line shift [meV]b0 5V [V]51015Linewidth [meV]c0 5V [V]0.000.250.500.751.00r,P/r,XdFIG. S3. Results of the fitting of the AP and RP reflectionspectrum. We use the fitting function (2). In a, b, c, weshow the fitted amplitude A(V ) (multiplied by ten for theAP), resonance shift E0(V ) − E0(0) and linewidth σ(V ) asa function of the gate voltage V for the AP (red line) andexciton/RP resonance (blue line). The onset of doping occursat V0 ≈ 0.35V. The last panel d show γr,AP/γr,ex (red) andγr,RP/γr,ex (blue), the radiative decay rate of the AP and RP,normalized to that of the exciton. The black line is the sum(γr,AP + γr,RP)/γr,ex, showing a small decay at low densityhas been studied in [9, 10], a new contribution to the lightshift arise from the coupling to a biexciton sate, a boundstate of two excitons in opposit valleys. The energy levelstructure to consider is shown in Fig. S4a and in thedressed state basis in b. When the pump laser is de-tuned from the exciton transition by the biexciton bind-ing energy δex = Ebinding, we expect an avoided cross-ing between the dressed states |X,n⟩ and |XX,n− 1⟩,where X and XX respectively designate the exciton andbiexciton states, and the integers n, n − 1 correspondsto the number of pump photons [11]. Indeed, we ob-serve a change of sign of the light shift as we sweepthe pump-laser detuning across the biexciton resonanceδex = Ebinding, as shown in Fig. S4 c ,d. The amplitudeof the shift always remain much smaller than the excitonlinewidth and decrease near resonance, where we mostlyobserve a broadening of the transition and did not resolvethe expected Autler-Towns splitting. From a heuristic fit∆ex,⊥ = a arctan[(δex−Ebinding)/b]+ c , of the light shiftat zero time delay (panel c), we extract the biexcitonbinding energy Ebinding = 29 ± 1.5meV. As mention inthe main text, this value is slightly larger than the es-timates of [9, 10], which could be due to the presenceof residual charges in devices without electrical gates,screening the Coulomb interaction and thereby reducing3|GS|XK |XK′|XX+EexEbinding|GS, n|XK, n 1|XK ′, n|XX, n 1exexE binding1 0 1Delay  [ps]0.10.00.10.2ex,() [meV]ex [meV]c da b25 30ex [meV]0.10.00.1ex, [meV]25 30FIG. S4. Exciton light shift in cross-polarization. The rele-vant energy levels are shown in a. The σ+-polarized pumplaser (red arrow) drives the transition between the biexci-ton state |XX⟩ and the K′-valley exciton |XK′⟩ producedby the σ−-polarized probe laser (blue). Near the resonanceδex ≈ Ebinding, the dominant contribution to the light shift ofthe K′-valley exciton comes from level repulsion between thedressed state |XX,n− 1⟩ and |XK′ , n⟩, as shown in b. Thisresults in a sign change of the light shift ∆ex,⊥, plotted as afunction of the pump-probe delay τ in c. In d, we show thelight shift at zero time-delay as a function of the pump detun-ing, which is fitted (solid blue line) to extract the biexcitonbinding energy Ebinding.the biexciton binding energy.VI. FIT OF THE LIGHT SHIFT WAVELENGTHDEPENDENCEIn this section we provide more details on the fitting ofthe light shift ∆AP as a function of the pump detuning.We focus on the analysis of the AP, but the data forthe exciton in co-polarization is similar to that of theAP, while the case of cross-polarization is described inSec.V.Quite generally, from second order perturbation the-ory, the light shift can be expanded as [12]∆AP =AδAP+Bδ2AP+Cδ2RP+DδAP − Ebinding, (3)where the first term corresponds to the usual ac-Starkshift arising from light-matter dressing, the second andthird terms respectively capture the AP-AP and AP-RPinteraction, and the last term can arise from a couplingto a charged biexciton.For the small detunings (compared to the exciton Ry-dberg) that we investigate, we expect the interactionterms to dominate over the light-matter dressing. Fur-thermore, we also expect the AP-AP interaction to playa dominant role compared to the AP-RP interaction. In-deed, our pump laser being red-detuned from the APresonance, we always have δAP < δRP , and we typicallyhave nAP ≪ nRP. This strong inequality is howevernot always satisfied, in particular at low density whenfAP ≪ fRP. On the other hand, this regime also corre-sponds to the situation where the exciton content is verylarge for the RP and conversely very small for the AP,resulting in UAP−AP ≫ UAP−RP. We therefore alwaysexpect C/δ2AP = UAP−APnAP ≫ D/δ2RP = UAP−RPnRP.Finally, same-valley excitons are not expected to form astable bound state due to Pauli blocking, and thereforewe do not expect the last contribution in (3) to arisefor co-circular pump and probe polarization. For cross-circular polarization, at charge neutrality a biexciton ex-ists and we report on his contribution to the AC-Starkshift in Sec.V. In the presence of electrons, charged biex-citon have been observed in WS2 [13]. However, Pauliblocking is inhibiting the formation of a charged biexcionfor Molybdenum compounds [9], where the resident elec-tron and that of the exciton occupy the same lower con-duction band (contrary to Tungsten compounds, wherebright excitons have their electron in the upper conduc-tion band).From all these considerations, we expect the AP lightshift to follow a law ∆AP = B/δ2AP in the regimes ex-plored in our work. As shown in Fig. S5, we can recoverthis result from a fit to our data, without prior knowl-edge.First, in panel a, we show the case of co-circularly po-larized pump-probe. A one-parameter fitB/δ2AP capturesvery well our measurements; the largest deviation fromthe fit, observed at small detuning, could be due to ab-sorption. In contrast our data cannot be reproduced bya fit in A/δAP or C/δ2RP. Finally, in a three parameterfit including these three terms, the AP-AP interactionterms contributes to at least 93% of the light shift (atthe largest detuning).We observe a similar behavior for cross-polarization(up to a sign change of B). In that case, we also attempta two-parameters fit D/(δAP − Ebinding), which yields apoor agreement with the data, thereby excluding a possi-ble shift induced by the transition to a charged biexcitonstate.VII. SIMULATION OF THE LINE DISTORTIONIt can be seen in figure 1 c of the main text that theeffect of the pump on the exciton resonance is more com-plex than a bare line shift. We observe (i) a broad-ening of the resonance and (ii) at negative time delay,the appearance of side peaks, both on the blue and redside of the exciton resonance. In an adiabatic approxi-4101 2 × 101 3 × 1014 × 101 6 × 101AP [meV]10 1100AP [meV]aA/ APB/ 2APC/ 2RPA/ AP + B/ 2AP + C/ 2RP101 2 × 101 3 × 1014 × 101 6 × 101AP [meV]10010 1AP [meV]bB/ 2APD/( AP Ebinding)FIG. S5. Fits of the wavelength dependence of the AP lightshift, for co- (a) and cross- (b) circularly pump-probe polar-ization. In both panel, we combine two data sets, one ob-tain with low pump intensities (at small detuning, blue filleddata points) and one obtained at larger pump intensities (atlarger detuning, white filled data points). The latter are thenrescaled by the intensity ratio. In this way, we are able tokeep a decent signal-to-noise ratio as we increase the detun-ing, while remaining in the linear-in-intensity regime closer toresonance.mation, We have an instantaneous and local light shift∆(x, t) ∝ Ipump(x, t). The duration of the pump pulseis about twice that of the probe pulse, and both beamsare focused to the same spot size. These two effects re-sults in an inhomogeneous (in both time and space) lightshift, which readily explains (i). As mentioned in themain text, the fact that the resonance almost perfectlyrecovers excludes non-coherent effects, such as heatingor photo doping, as significant broadening mechanisms.The appearance of side peaks (ii) is a more complex butwell known artifact of pump-probe experiments, whichis studied in details in e.g. [14, 15]. Briefly, the sidepeaks can be understood as the free induction decay ofa real population of exciton injected by the probe andperturbed by the pump - which explains why this effectsshows up when the probe comes before the pump, i.e. forτ ≲ 0. A quantitative treatment requires the resolutionof the semiconductor optical Bloch equation beyond theadiabatic approximation. Here, we propose a very sim-ple approach, which can capture the appearance of bothFIG. S6. Simulation of the pump-probe experiments takinginto account the finite duration of the laser pulses, which re-sults in a broadening of the resonance and side bands at neg-ative time delay.blue and red detuned peaks, but which is not meant tobe quantitatively accurate. We consider that the polar-ization generated by the probe pulse readP (t) = cos[(ω +∆0gpump(t)2)t]×P0[gprobe(t− τ) + P1Θ(t− τ)e−γ(t−τ)](4)where gpump (gprobe) is the envelope of the pump (probe)electric field, which we take to be the hyperbolic secantfunction with width σpump (σprobe). We considered thatthe pump (probe) pulse arrives at t = 0 (t = τ). Thefree induction decay responsible for the side peaks corre-sponds to the second term in (4), where Θ is a step func-tion (we take the hyperbolic tangent with width σprobe)and γ corresponds to the exciton decay rate. We Fouriertransform P (t) and shows its modulus square in Fig. S6,for ω = 2400 rad.ps−1; ∆0/ω = 2×10−3; σpump = 0.4 ps;σprobe = 0.1 ps; P1/P0 = 0.1; γ = 1.4 ps−1. These pa-rameters are not determine after a precise calibration,there chosen are in a realistic range to observe clear sidepeaks, as can be seen in Fig. 4. In particular we have cho-sen σprobe/σpump slightly smaller than in the actual ex-periments, for the purpose of having brighter side peaks.To conclude we point out that these artifact are morepronounced for the exciton resonance, compared of theAP. This could be due to a smaller ratio P1/P0 for theAP.VIII. REPULSIVE POLARON LIGHT SHIFTIn the main text we focused our analysis of the AC-Stark shift of the attractive polaron. Here we breiflydiscuss the case of the repulsive polaron (RP).Similarly to the exciton or AP, we observe a blue shiftof the RP for co-circularly polarized pump and probe51.6401.6451.6501.6551.6601.665E [eV]Co-pol.a0.10.20.40.6RP [meV]c1/ 2RP1/ 2AP2 1 0 1 2 [ps]1.6401.6451.6501.6551.6601.665E [eV]Cross-pol.b50 60 70 80 90 100RP [meV]-0.1-1RP [meV]d1/ 2RP1/ 2AP01R0.00 0.25 0.50 0.75 1.00ne [×1012cm 2]2.01.51.00.50.00.51.01.52.0RP [meV]eCo-pol.Cross-pol.FIG. S7. Repulsive polaron light shift. Evolution of the reflection spectrum as a function of the pump-probe delay, in co-(a) and cross-circular polarization (b). At these relatively large electronic density ne ≈ 1012cm−2, the detuning dependance ofthe light shift (c and d) is better fitted by a 1/δ2AP instead of 1/δ2RP . This observation indicates that AP-RP interactions arethe leading contribution to the light shift in the detuning range explored here. This effect, together with an overall increase ofthe interaction due to the polaronic dressing, contributes to an increase of the light shift with the electronic density (e). Thesedata were obtained in the same experimental condition as for the AP light shift shown in Fig. 3 and 4 of the main text, i.e.with Ipk ≈ 1.7GWcm−2 and δAP ≈ 25meV, δRP ≈ 50meV, (note that these are the detuning at ne → 0, they slightly changewith increasing ne, see Fig. S3b).pulse, and a red shift for cross-circular polarization, seeFig. S7a,b. The former is driven by repulsive interac-tion, as described for the exciton or AP. The latter canhave two origins, the coupling to the biexciton (discussedin Sec.V), in particular at low densities when the exci-ton content of the RP is large and attractive interaction(as observed for the AP) at larger densities when thepolaronic dressing is more prominent. In that second sit-uation, due to the transfer of oscillator strength from theRP to the AP for increasing ne (see Fig. S3), both RP-RPand AP-RP interactions should be taken into account:∆RP = URP−RPnRP + UAP−RPnAP . (5)The role of AP-RP interaction is further enhanced fora red-detuned pump, such that δAP < δRP. The rela-tive contribution of the RP-RP and AP-RP interactioncan be deduced from the dependence of the light shifton the pump detuning. We observe in Fig. S7 cd, thatfor both co- and cross-polarization, the RP light shiftis indeed better fitted by a 1/δ2AP ∝ nAP law insteadof 1/δ2RP ∝ nRP, showing that at such electronic den-sity ne ≈ 1012cm−2, the AP-RP interactions are theleading contribution to the light shift. For increasingelectronic density, the enhancement of the interactioncounter-acts the reduction of the oscillator strength andwe observe an increase of the light shift in both polariza-tion (Fig. S7 e). Note that we restricted our analysis todensities ne ⪅ 1012 cm−2 as the RP resonance becomestoo broad for larger densities and the fit unreliable.[1] P. Zomer, M. Guimarães, J. Brant, N. Tombros, andB. Van Wees, Applied Physics Letters 105, 013101(2014).[2] A. Laturia, M. L. Van de Put, and W. G. Vandenberghe,npj 2D Materials and Applications 2, 6 (2018).[3] K. K. Kim, A. Hsu, X. Jia, S. M. Kim, Y. Shi, M. Dres-selhaus, T. Palacios, and J. Kong, ACS nano 6, 8583(2012).6[4] E. C. Regan, D. Wang, C. Jin, M. I. B. Utama, B. Gao,X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta,M. Blei, J. D. Carlström, K. Watanabe, T. Taniguchi,S. Tongay, M. Crommie, A. Zettl, and F. Wang, Nature579, 359 (2020).[5] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak,K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan,and K. F. Mak, Nature 579, 353 (2020).[6] Y. Shimazaki, I. Schwartz, K. Watanabe, T. Taniguchi,M. Kroner, and A. Imamoğlu, Nature 580, 472 (2020).[7] T. Smoleński, O. Cotlet, A. Popert, P. Back, Y. Shi-mazaki, P. Knüppel, N. Dietler, T. Taniguchi, K. Watan-abe, M. Kroner, and A. Imamoglu, Phys. Rev. Lett. 123,097403 (2019).[8] G. Scuri, Y. Zhou, A. A. High, D. S. Wild, C. Shu,K. De Greve, L. A. Jauregui, T. Taniguchi, K. Watan-abe, P. Kim, et al., Physical review letters 120, 037402(2018).[9] K. Hao, J. F. Specht, P. Nagler, L. Xu, K. Tran, A. Singh,C. K. Dass, C. Schüller, T. Korn, M. Richter, et al.,Nature communications 8, 15552 (2017).[10] C.-K. Yong, J. Horng, Y. Shen, H. Cai, A. Wang, C.-S.Yang, C.-K. Lin, S. Zhao, K. Watanabe, T. Taniguchi,et al., Nature Physics 14, 1092 (2018).[11] C. Cohen-Tannoudji and S. Reynaud, Journal of PhysicsB: Atomic and Molecular Physics 10, 345 (1977).[12] M. Combescot, Physics reports 221, 167 (1992).[13] J. B. Muir, J. Levinsen, S. K. Earl, M. A. Conway, J. H.Cole, M. Wurdack, R. Mishra, D. J. Ing, E. Estrecho,Y. Lu, et al., Nature Communications 13, 6164 (2022).[14] S. Koch, N. Peyghambarian, and M. Lindberg, Journalof Physics C: Solid State Physics 21, 5229 (1988).[15] H. Haug and S. W. Koch, Quantum theory of the op-tical and electronic properties of semiconductors (WorldScientific Publishing Company, 2009).https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1038/s41586-020-2092-4https://doi.org/ 10.1038/s41586-020-2085-3https://doi.org/ 10.1038/s41586-020-2191-2https://doi.org/ 10.1103/PhysRevLett.123.097403https://doi.org/ 10.1103/PhysRevLett.123.097403 Supplementary material:Interaction induced ac-Stark shift of exciton-polaron resonances  Device and electron density estimation Experimental setup Estimation of the exciton density and interaction strength Fitting of the reflection data Coupling to the biexciton state Fit of the light shift wavelength dependence Simulation of the line distortion Repulsive polaron light shift References