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

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Interaction induced AC-Stark shift of exciton-polaron resonancesT. Uto,1, 2, ∗ B. Evrard,1, ∗ K. Watanabe,3 T. Taniguchi,3 M. Kroner,1 and A. İmamoğlu11Institute for Quantum Electronics, ETH Zürich, CH-8093 Zürich, Switzerland2Institute of Industrial Science, The University of Tokyo,4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan3Research Center for Electronic and Optical Materials, NIMS, 1-1 Namiki, Tsukuba 305-0044, JapanLaser induced shift of atomic states due to the AC-Stark effect has played a central role incold-atom physics and facilitated their emergence as analog quantum simulators. Here, we explorethis phenomena in an atomically thin layer of semiconductor MoSe2, which we embedded in aheterostructure enabling charge tunability. Shining an intense pump laser with a small detuningfrom the material resonances, we generate a large population of virtual collective excitations, andachieve a regime where interactions with this background population is the leading contribution tothe AC-Stark shift. Using this technique we study how itinerant charges modify – and dramaticallyenhance – the interactions between optical excitations. In particular, our experiments show that theinteraction between attractive polarons could be more than an order of magnitude stronger thanthose between bare excitons.Introduction — Atomically thin transition metaldichalcogenides (TMDs) and their van der Waals het-erostructures constitute a versatile platform for explo-ration of phenomena at the frontier of many-body physicsand quantum optics [1–3]. Arguably, the most signifi-cant feature of this new platform is the weak dielectricscreening and relatively heavy band-mass of electronsand holes, leading to strong Coulomb interactions andappearance of tightly bound excitons as elementary opti-cal excitations. On the one hand, the small Bohr radius(aex ∼ 1nm) of excitons implies strong coupling to light,which ensures that a pristine monolayer TMD realizes anatomically-thin mirror in the absence of a cavity [4, 5] andexhibits large normal-mode splitting between exciton-polariton modes when embedded inside a cavity [6, 7].On the other hand, electron- or hole-exchange based in-teraction between two tightly bound excitons is drasti-cally reduced, leading to a predominantly linear opticalresponse [5, 8, 9]. Therefore, strong exciton-photon cou-pling is fundamentally linked to weak exciton-exciton in-teractions in TMDs, which in turn constitutes a majorchallenge to the prospect of engineering nonlinear opticaldevices [10–13].Different approaches that could meet this challengeby enhancing exciton-exciton interactions without sacri-ficing strong light-matter coupling have been explored.While promising results are reported [9, 14–16], thelarge uncertainty in the determination of the underly-ing exciton-exciton interaction strength has been a majorhindrance in assessing and comparing these approaches.Measurement of the interaction-induced blue-shift underdirect resonant excitation leads to generation of a sizeabledark exciton population, rendering the extracted interac-tion strength unreliable. A partial remedy is provided bystudying nonlinear response of exciton-polaritons; how-ever, recent theoretical work showed that interactions be-tween exciton-polaritons can be drastically different fromthose between bare excitons [17].In this Letter, we introduce a novel method to reli-ably measure exciton-exciton interactions, based on thelight shift of the excitonic resonances in response to anintense red-detuned femtosecond laser pulse. Previously,the AC-Stark shift of excitons in TMDs was studied forlarge pump detuning [18, 19], in a regime well capturedby the simple picture of a dressed two-level system, sim-ilar to that of a single atom in an off-resonant light field.For very large detunings, comparable to the band gap,the Bloch-Siegert shift becomes significant and has beenobserved in [20]. Here, we are interested in the oppo-site limit, where the pump detuning from the excitonicresonances is much smaller than the exciton binding en-ergy: in this limit, the pump pulse generates a largepopulation of virtual excitations that exist only duringthe pump-pulse duration. The interactions between thisbackground of pump-generated virtual excitations and atest excitation produced by the probe pulse provide thedominant contribution to the light shift [21–26].A possible avenue to enhance exciton-exciton interac-tions is to embed them in a two dimensional degener-ate electron system (2DES). Following seminal studieson III-V quantum wells [27–29], recent work establishedthat dynamical screening of excitons in a doped TMDsby the 2DES modifies the nature of elementary opticalexcitations, leading to the formation of attractive- andrepulsive-exciton-polarons (AP and RP) [30–32]. Ar-guably, the principal result of our work is the use of theAC-Stark effect to measure the bare AP interactions asa function of ne, where we demonstrate a dramatic en-hancement of the polaron-polaron interaction, up to afactor ∼ 35 as compared to interactions between bareexcitons, for ne ≤ 2 × 1011 cm−2. This behavior wastheoretically predicted in [9], but was not experimentallyobserved.Our experiments are performed on a monolayer MoSe2encapsulated in hBN, at cryogenic temperature (T ≲10K). Our setup is sketched in Fig. 1a. A mode-locked20 1 2Density ne [×1012 cm 2]1.611.631.651.67Energy [eV]×10RPAPExc.2 0 2Delay [ps]1.641.651.66Energy [eV]ex [meV]ex[meV]0.1155020 1000 20 40R/Racd ebNonline a r  Fib e rPu ls e  Sha p e rSp e c t ro - m e te rhBNhBNMoSe 2GrVs ub s t ra teTi-Sa p  La s e rΔtDe la y lineHWP LPQWP4  K|0⟩ |𝑋⟩ |𝑋𝑋⟩ |𝑁:𝑋⟩ |𝑁+1:𝑋⟩ Ω2𝛿 𝑈ex  2𝑈ex  N(N-1)2𝑈ex  N(N+1)FIG. 1. a Sketch of the pump-probe setup and the van derWalls heterostructure. b Reflection spectrum of the measureddevice as a function of the electron density ne. To enhancevisibility, the signal is multiplied by 10 for E < 1.64 eV. cSchematic of the energy levels showing the usual AC-Starkshift for the first two-levels and the interaction induced shiftwhich increases with the exciton density. d Reflection spec-trum at charge neutrality, as a function of the delay betweenthe co-circularly-polarized pump and probe pulses. From a fit(dashed line) we extract the amplitude of the light shift. Thelatter is shown in e as a function of the pump detuning fromthe exciton resonance (δex), and well fitted by ∆ex = B/δ2ex(solid line). Throughout the manuscript, error bars show thestatistical error corresponding to two standard deviations ob-tained from a set of a few repetitions of the experiment.laser delivers∼ 100 fs pulses. With a pulse shaper we nar-row the bandwidth of the pump (increasing the durationby a factor ≲ 2), while a non-linear crystal fiber gener-ates a white-light continuum to probe the exciton and APtransition. Both pulses are focused near the diffractionlimit onto the sample (more details about the sample andexperimental setup in the Supplementary Material (SM)[33]).The reflection spectrum as a function of the density neis shown in Fig. 1b, the exciton line smoothly evolves intothe RP and a red-shifted AP resonance emerges [30–32].Exciton light shift — To benchmark our method, wefirst focus on the excitonic light shift at charge neutral-ity, for red-detuned pump laser, co-circularly polarizedwith the probe. For zero delay between the two pulses(τ = 0) we observe a blue shift and a broadening of theexciton line (Fig. 1 c). The latter stems from the aver-aging over a spatial and time dependent light shift, sincethe pump and probe lasers have comparable spot sizeand duration. For τ ≲ 0 we also observe the emergenceof weak sidebands, a common artifact of pump-probe ex-periments, which can be understood as the free induc-tion decay of probe-generated excitons perturbed by thepump (for more details see e.g. [26, 34] or the SM [33])For detunings δex = Eex − Epump large compared tothe Rabi frequency of the pump laser (Ωmax), the lightshift can be expanded as [23]∆ex ≈ Aδex+Bδ2ex. (1)Here, the first term corresponds to the usual AC-Starkshift of a two-level system [35]. The second term arisesfrom many-body effects, namely Coulomb interactionand Pauli blocking, due to the pump-laser-generated pop-ulation of virtual excitons. These interaction effects areusually described within a Hatree-Fock approximation[24–26], in which case we can write B/δ2ex = Uexnex wherenex ∝ δ−2ex is the exciton density and Uex is an effectiveexciton-exciton interaction strength. In a regime of inter-mediate detunings, ℏΓex,Ωmax ≪ |δex| ≪ Ex, where Γexis the exciton radiative decay and Ex the exciton bindingenergy, the interaction-induced light shift is expected todominate over the single-particle response. This is theregime explored throughout this Letter. Figure 1d showsthat the exciton light shift is indeed well captured by a1/δ2ex dependence. From this measurement we extractUex ≈ 0.09 ± 0.03µeVµm2 (for details on the calibra-tion of nex, see the SM [33]). Our estimate is consis-tent with previous measurements [5, 8, 9], albeit an or-der of magnitude lower than the theoretical expectation,∼ 3Exa2ex ∼ 1µeVµm2 [36, 37].For cross-circularly polarized pump and probe lasers,producing excitons in opposite valleys, the exchange in-teraction is suppressed, leaving a negligible direct interac-tion shift [36, 37]. On the other hand, two opposit-valleyexcitons can bind into a biexciton. The pump drives thetransition from a “probe exciton” to the biexciton, result-ing in an additional contribution to the light shift, pre-viously investigated in [38, 39]. We report similar resultsin the SM [33], although we point out that we obtain abiexciton binding energy Ebinding = 29±1.5meV slightlylarger than the values reported in [38, 40] while being ingood agreement with another recent measurement [41](the discrepancy could stems from residual charges inungated devices).31.62 1.63E [eV]101Probe delay [ps] a1.62 1.63E [eV]b1.62 1.63E [eV]c0 1 2Ipk [GW.cm 2]0.02.55.07.510.0AP [meV]dAP [meV]AP [meV]0.11e10 20 30 50FIG. 2. AC-Stark shift of the attractive polaron (AP) reso-nance for co-circularly polarized pump and probe lasers. Ina-c, we show the AP resonance as a function of the pump-probe delay τ for increasing pump laser intensity (Ipk ≈0.7; 1.3; 2GW/cm2). The shift at τ = 0 is plotted in d asa function of the intensity, showing deviation from a lineardependence in Ipk (dashed line). Here the pump laser de-tuning from AP is δAP ≈ 13meV and the electron density isne ≈ 1.7 × 1012 cm−2. In e, we show the δAP dependence ofthe shift, which is well fitted by B/δ2AP, shown as a blue line.Here, ne ≈ 0.17× 1012 cm−2 and Ipk ≈ 0.4GWcm−2 so thatthe Ipk dependence is within the linear regime.1 2ne [×1012cm 2]1.61.82.02.22.4AP [meV]a1 2ne [×1012cm 2]102030UAP/UexbFIG. 3. a Electron density dependence of the attractive po-laron (AP) light shift for co-circularly polarized pump andprobe beam. From this data and a measurement of theAP oscillator strength [33], the AP-AP interaction strengthUAP is extracted and compared to the exciton-exciton in-teraction strength Uex in b. Here, δAP ≈ 25meV andIpk ≈ 1.7GWcm−2.Attractive polaron light shift — Having established ourapproach to measure interactions between excitons, wenow turn to the main results of our paper, where weinvestigate the interactions between APs in an electrondoped TMD. Figure 2a-c shows the AP light shift forco-circularly-polarized pump and probe lasers: despiteits relatively low oscillator strength (fAP), the large blueshift of the AP well exceeds its linewidth. We remarkthat the AP resonance is symmetric around τ = 0, indi-cating that the pump laser does not generate incoherentAP population or quench the 2DES. This observationshould be contrasted with resonant pump-probe experi-ments carried out using AP-polaritons [9]. However, forincreasing pump intensity Ipk we observe in Fig. 2a-c anoverall reduction of fAP together with a small red shift(≈ 0.8meV at most), both independent on τ . This lat-ter observation suggests that part of the pump-pulse en-ergy is absorbed by the TMD, which then relaxes on atime scale much longer than the pulse repetition rate.Thereby, depending on the pump intensity, we effectivelychange the steady-state of the TMD, presumably its tem-perature and/or charge density - both of which poten-tially leading to a reduction of fAP. As a consequence,we observe a sub-linear increase of the light shift withincreasing Ipk (Fig. 2d). We point out that the regimeof linear dependence of ∆AP with Ipk increases with in-creasing detuning, and we are able to observe the onsetof a saturation in (Fig. 2d) only because the system isdriven close to resonance (δAP ≈ 13meV). In the fol-lowing we focus on a range of pump-laser detunings andintensities where the reduction of fAP is negligible and∆AP ∝ Ipk.Figure 2d shows the dependence of the AP light shifton the detuning from the AP resonance (δAP = EAP −Epump) for ne ≈ 0.17× 1012 cm−2. It is well reproducedby a ∆AP ∝ 1/δ2AP law, which demonstrates that it origi-nates predominantly from AP-AP interactions. We pointout that considering the small fAP as compared to thatof the RP oscillator strength, particularly for low ne, onecould expect that the pump generates more RP than AP(despite a smaller detuning to the latter). However, alight shift dominated by the RP population would scaleas ∆AP ∝ nRP ∝ 1/δ2RP. Even at the lowest electrondensities (ne ≈ 0.17 × 1012cm−2) we measured, we donot observe such a detuning dependence and the devi-ation from ∆AP ∝ 1/δ2AP law remains negligible [33].This observation suggests that the AP-RP interactionsare much weaker than the AP-AP interaction, especiallyfor low ne, which is consistent with the fact that the RPhas a dominant exciton content in that regime.Since fAP increases linearly with ne, we would nor-mally expect ∆AP to also increase linearly with ne – in-deed, ∆AP = UAPnAP and nAP ∝ fAP ∝ ne. At a firstglance, this is indeed what we observe in fig.3a. How-ever, we also find that unlike fAP, a linear ne fit to ∆APyields a finite value for ne = 0: this striking observa-410 30 70AP [meV]-1-0.5-0.1AP, [meV]a2.5 0.0 2.5Probe delay [ps]0.40.20.0AP, [meV]1.01.72.6b1 2Density [×10 12cm 2]0.40.30.20.1AP, [meV]c1 2Density [×10 12cm 2]1.11.00.90.8UAP,/EexdFIG. 4. Light shift of the attractive polaron (AP) for cross-circularly polarized pump and probe beam. a Detuning δAPdependence of the light shift at an electron density of ne ≈ 1.0 × 1012 cm−2. For large δAP (open symbols) we use the fullbandwidth of the pump and Ipk ≈ 5.3GW/cm−2. To approach the AP resonance, we reduce the bandwidth and consequentlyIpk by a factor of ≈ 0.4; we then re-scale the data (full symbols), ensuring that the two measurements match at intermediatedetunings (δAP ≈ 25; 30meV). The data is well fitted by a B/δ2AP law, shown as a blue line. b Time dependence of the lineshift for various ne at δAP ≈ 25meV. The red line is a fit, from which we extract the shift at τ = 0. The latter is shownin c as a function of ne. d The ratio of the interaction between opposite valley APs and that of same valley excitons forIpk ≈ 1.7GW/cm−2.tion can be explained as an increase of the interactionstrength UAP with decreasing ne that becomes prominentfor ne ≤ 2×1011 cm−2, where it counteracts the effect ofdecreasing nAP or fAP. To highlight this feature, we com-pare the renormalized AP light shift ∆AP/fAP with thatof the exciton ∆ex/fex, for the same pump intensity andat a wavelength such that δex = δAP. In this way, we ob-tain the interaction ratio UAP/Uex = ∆AP/fAP×fex/∆exwhich we plot in Figure 3b. We observe a dramatic en-hancement of the AP-AP interactions – up to a factor 35– as ne is lowered.In a simple but far-reaching ansatz, the AP wave-function is described as a superposition of a zero-momentum bare exciton plus an unperturbed 2DES, andan exciton scattered into a finite momentum state whilegenerating a single particle-hole excitation in the 2DESof the conduction band of the opposite valley [27–30, 42].The latter contribution could also be considered as a su-perposition of trion-hole pairs. For low ne, the proba-bility of finding a bare exciton (quasi-particle weight) inan AP excitation is small. Consequently, an AP exci-tation predominantly generates a collective excitation oftightly bound trions with radius aT ∼ 2nm, each sur-rounded by a Fermi sea hole of extent on the order ofthe inverse Fermi wavevector k−1F . The depletion of the2DES around the trion leads to an effective repulsive in-teraction between two APs, through a partial suppressionof hybridization of the bare exciton and collective trion-hole excitations. The expansion of the depleted region∝ k−1F ∝ n−1/2e as ne decreases can thus partially com-pensate for the reduction of fAP and consequently nAP,insuring the persistence of a significant AP light shift forlow ne.The mechanism outlined above only takes place forsame-valleys APs [9, 43]. In cross-polarized configura-tion, we also observed an AP light shift ∆AP,⊥, albeitmuch smaller in magnitude and of the opposite sign. Fig-ure 4a shows the detuning dependence of ∆AP,⊥ whichis consistent with a ∆AP,⊥ ∝ −1/δ2AP law, pointing outagain to an interaction between the probe and pumplaser induced APs. More importantly, we emphasizethat our light shift data cannot be fitted with ∆AP,⊥ ∝1/(δAP − E0); which could have emerged from couplingto a putative charged biexciton resonance[33, 40, 43].Figure 4b shows the time dependence of the AP lightshift at various densities. For τ > 0 (pump beforeprobe), we observe a continuous red shift of the APline, which increases together with the density ne, possi-bly due to residual pump-induced high momentum APs.We emphasize that this shift also exists in co-circularly-polarized pump-probe measurements but remains neg-ligible as compared to the AC-Stark shift at τ = 0.To fit the data and extract the coherent response, weuse a sum of a Gaussian and a piecewise linear func-tion. Contrary to the co-circularly-polarized case, theamplitude of the Gaussian term (coherent response) in-creases approximately linearly with the electron densityne as shown in Fig. 4 c; here, we discarded low densityne < 8× 1012 cm−2 data for which the fit was unreliable.After proper normalization by fAP, we extract the in-teraction strength between opposite-valley APs, UAP,⊥,which we compare to the same-valley exciton interac-tion Uex in Fig. 4d: we observe almost no dependenceof UAP,⊥, which remains comparable (in absolute value)to Uex for all ne.To explain this observation, we consider a σ−-polarizedpump laser producing APs in the K ′-valley. Polaronformation promotesK-valley electron to high momentum5states with k ∼ 1/aT , thereby reducing the phase-spacefilling (at low k) for a probe-generated K-valley AP. Forlow nE where kFaT ≪ 1 this mechanism could explainthe attractive interactions between opposite-valleys APs.However, further investigations are needed to confirmthis hypothesis.Conclusion and Outlook — Our work establishes theAC-Stark effect as a novel approach to measure the inter-action between optical excitations in TMDs monolayers.An extension of this technique to assess the modificationof interactions due to the formation of moiré heterostruc-tures in (twisted) heterobilayers presents no difficulties.By using a detuned pump laser, we generate a large vir-tual exciton or AP population and thereby almost fullysuppress dark-exciton generation, which plagued previ-ous studies [9]. An exciting application of the techniquewe developed would rely on a Laguerre-Gauss pumpbeam to shift away the AP resonance, except in a smallregion around the beam’s vortex. 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