# Fileset [2024A00865G_Trilayer_Manuscript.pdf](https://mdr.nims.go.jp/filesets/dc6a350e-a16b-4b61-80e6-6f8479186ffb/download) ## Creator Liuxin Gu, Lifu Zhang, Ruihao Ni, Ming Xie, Dominik S. Wild, Suji Park, Houk Jang, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Mohammad Hafezi, You Zhou ## Rights This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1038/s41566-024-01434-x[In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Giant optical nonlinearity of Fermi polarons in atomically thin semiconductors](https://mdr.nims.go.jp/datasets/57f71c49-c4c9-4aef-b703-5e5cce84d584) ## Fulltext 1 Giant optical nonlinearity of Fermi polarons in atomically thin semiconductors Liuxin Gu1,*, Lifu Zhang1,*, Ruihao Ni1, Ming Xie2, Dominik S. Wild3, Suji Park4, Houk Jang4, Takashi Taniguchi5, Kenji Watanabe6, Mohammad Hafezi7, You Zhou1,8,† 1Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA 2Condensed Matter Theory Center, University of Maryland, College Park, MD 20742, USA 3Max Plank Institute of Quantum Optics, Garching, 85748, Germany 4Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA 5Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 6Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan 7Joint Quantum Institute (JQI), University of Maryland, College Park, MD 20742, USA 8Maryland Quantum Materials Center, College Park, Maryland 20742, USA *These authors contributed equally to this work. †To whom correspondence should be addressed: youzhou@umd.edu mailto:youzhou@umd.edu2 Realizing strong nonlinear optical responses is a long-standing goal of both fundamental and technological importance. Recently significant efforts have focused on exploring excitons in solids to achieve nonlinearities even down to few-photon levels. However, a crucial tradeoff arises as strong light-matter interactions require large oscillator strength and short radiative lifetime of excitons, which limits their nonlinearity. Here we experimentally demonstrate strong nonlinear optical responses with large oscillator strength by exploiting the coupling between excitons and carriers in an atomically thin semiconductor. By controlling the electric field and electrostatic doping of trilayer WSe2, we observe the hybridization between intralayer and interlayer excitons and the formation of Fermi polarons. Substantial optical nonlinearity is observed under continuous wave and pulsed laser excitation, where the Fermi polaron resonance blueshifts by as much as ~10 meV. Intriguingly, we observe a remarkable asymmetry in the optical nonlinearity between electron and hole doping, which is tunable by the applied electric field. We attribute these features to the optically induced valley polarization due to the interactions between excitons and free charges. Our results establish atomically thin heterostructures as a highly versatile platform for engineering nonlinear optical response with applications to classical and quantum optoelectronics. 3 Nonlinear optical phenomena lie at the heart of classical and quantum optics, with applications ranging from data communications to quantum control4,5. Developing physical systems with stronger optical nonlinearity while reducing their power requirement holds the promise for more efficient optoelectronics and may unlock new technologies such as single-photon switches and transistors6–8. In recent years, significant efforts have been devoted to investigating excitons in semiconductors as a solid-state medium for realizing strong optical nonlinearity1–3. Van der Waals heterostructures based on atomically thin transition metal dichalcogenides (TMDs) have emerged as a new platform for fundamental studies of excitons and for engineering optical responses9,10. Excitons in such two-dimensional materials are highly tunable with rich spin-valley physics and possess characteristics promising for optical nonlinearity, such as strong light-matter interactions and weak screening of Coulomb potential10. However, a major challenge in achieving high nonlinearity under low excitation power arises from the balance between the strengths of exciton-photon and exciton-exciton interactions11–13. For instance, intralayer excitons in these materials exhibit large oscillator strength but experience weak interactions, dominated by exchange interactions and limited by their short lifetime14. On the other hand, interlayer excitons in TMD heterostructures have longer lifetimes and experience interactions due to their finite electric dipole moment15,16. Unfortunately, spatial separation of the electron-hole wavefunctions leads to weaker absorption of incident photons. Several recent studies demonstrated enhanced the interlayer exciton absorption via its hybridization with intralayer excitons in MoS21,17–19, although the oscillator strength of such hybrid excitons is still an order of magnitude weaker than the intralayer ones. So far, strong nonlinearity in the intralayer excitons has not yet been realized. In this study, we report giant nonlinear optical responses of intralayer charged excitons, i.e., Fermi polarons, on the order of several millielectronvolts under photon flux of 1013 photons per second 4 per square micrometer. The strong nonlinearity is highly tunable by doping and electric field, based on which we attribute the nonlinearity to an optically induced valley polarization resulting from exciton-carrier scattering. Results Electrical control of excitons In our experiments, we encapsulate exfoliated WSe2 homotrilayersinside two layers of hBN in a dual-gate geometry to independently control the overall doping levels and the displacement field (Fig. 1a & b). Figure 1c shows the photoluminescence (PL) map of the sample by varying the electric field while keeping the samples undoped. We observe strong emission from excitons XI, whose energy linearly shifts with electric field from 1.58 eV to 1.50 eV, and PL peaks at 1.71, 1.55 and 1.52 eV that remains constant with varying electric field. We attribute the peak at 1.71 eV to intralayer momentum-direct exciton XA at the K-K transition, based on their strong absorption (Fig. 1d) and zero Stark shift. The lower energy of XI indicates that they are momentum-indirect excitons at the band edge, corresponding to transition across the indirect gap, located at the valence band Γ and conduction band Q valleys, according to the band structure calculations20–22. From the linear Stark shift, we estimate the electric dipole of XI to be around 0.78 𝑛𝑚 ∙ 𝑒 (see Methods). The corresponding vertical displacement of the electron-hole pair in XI is approximately half of the distance between the top and bottom tungsten layers, indicating the electrons or holes are partially layer delocalized. Next, we measure the reflectance of the trilayer under an electric field (Fig. 1d). In addition to the intralayer XA, we observe an additional reflectance contrast at 1.78 eV (IXD), which exhibits a substantial Stark effect of almost 100 meV (additional devices shown in Fig. S1). The finite 5 reflection contrast and linear Stark effect of IXD suggest that it corresponds to interlayer exciton at the direct K-K transition with larger oscillator strength than those momentum-indirect excitons, XI, observed in PL. From the slope of the Stark effect, we estimate the electron-hole displacement to be 1.35 𝑛𝑚. Interestingly, as the energy of IXD approaches that of the intralayer exciton XA under a higher electric field, we observe an apparent anti-crossing behavior of XA and IXD near the electric field of 0.05 V/nm. We note that the levels are not fully avoided and there is always finite reflection from XA at 1.71 eV for all electric fields, which will be discussed in more depth later. To quantitatively understand the avoided crossing, we extract the exciton energies by fitting reflectance spectra and then model the anti-crossing using a simple coupled oscillator model, from which we estimate a coupling strength of W = 10 ± 2 meV between XA and IXD (Fig. S2). We further characterize how electrostatic gating modifies intralayer excitons XA and their hybridization with interlayer excitons. Figure 1d shows the doping-dependent reflectance spectra of the sample under zero electric field. Upon doping, the reflectance from neutral XA diminishes as they lose their oscillator strength, and the charged intralayer excitons emerge and shift to lower energies, with similar behaviors observed in PL (Fig. S3). The redshift of charged excitons with increasing doping levels can be explained in terms of attractive Fermi polaron23, or an increase in the exciton-trion energy splitting with increasing Fermi energy24. The similar doping dependence of reflectance and PL is due to the absence of Pauli blocking in both cases when the free carriers do not reside in the K valley25. As we focus on the highly doped regime where excitons interact with a large number of carriers, we refer to these charged excitons as Fermi polarons in later discussions23. 6 Giant optical nonlinearity Next, we study the excitons’ nonlinear optical response by measuring the sample’s reflectance spectra under different laser pumping. Figures 2a-c show the reflectance spectra of the sample, probed with a halogen lamp, while we excite the system with a 635nm continuous wave (CW) laser of different power. When the trilayers are electron-doped or intrinsic, optical pumping does not alter the reflectance spectra significantly. Intriguingly, however, in the hole-doped regime, optical pumping leads to a dramatic blueshift of the Fermi polaron XA+, on the order of a few millielectronvolts, and a slight linewidth increase under tens of microwatts excitation (Fig. 2b). The oscillator strength of XA+, extracted from the reflection spectra, remains almost constant at low excitation power and decreases at higher power (Fig. S4). We note that PL signals are more than four orders of magnitude weaker than the reflected light and therefore negligible. To visualize the doping dependence of nonlinearity, we measure the relative change in the sample’s reflectance induced by the optical pumping, ∆𝑅/𝑅 = 𝑅𝑝/𝑅𝑛𝑝 − 1, where Rp and Rnp are the sample’s reflectance spectra with and without the laser pumping, respectively. Figure 2d shows the reflectance spectra ∆𝑅/𝑅 under symmetric gating with zero electric field, where we observe a striking asymmetry between the electron and hole sides. We briefly note that our observed nonlinearity of intralayer excitons is orders of magnitude stronger than that of dipolar interlayer excitons in bilayer TMDs19,26 (see Table S1 for detailed comparison with other systems). Another critical distinction is that the nonlinearity in bilayer TMDs occurs for interlayer excitons in the intrinsic regime1,19. In addition to non-resonant excitation, we also probe the optical nonlinearity by resonantly exciting XA with a pulsed laser (718 to 730 nm wavelength, ~100 ps pulse duration) and observe qualitatively similar nonlinear behaviors with a ~10 meV blueshift of XA+ and an increase in its 7 linewidth, only on hole-doped side (Figs. 2e & S5). To investigate how excitation photon energies impact the nonlinearity, we tune the pulsed laser energy across the XA+ resonance. Resonant excitation results in a stronger blueshift than higher energy excitation, while there is no noticeable shift in XA+ when the photon energy falls below XA+ (Fig. S6). Additionally, we observe no significant wavelength dependence when the photon energy exceeds XA+ (Fig. S6). Notably, the pulsed excitation generates an order of magnitude smaller blueshift than the CW laser at lower power with its peak power comparable to the CW laser’s average power (Fig. S8). Optically induced valley polarization The observed Stark effect and anti-crossing in WSe2 trilayers can be understood by examining their crystal and band structure. In trilayers, each monolayer is rotated 180 degrees, resulting in alternating K and K' points between layers27 (Figs. 3a, b). Here we mainly consider the hole tunnelling process, which is predicted to be much stronger than electron tunnelling by density functional calculations28. The sizeable spin-orbit coupling in the valence band dictates that the direct tunneling between the neighboring layers would be much weaker than that between the top and bottom layers across the middle layer (Fig. 3a, b). Such tunneling leads to finite oscillator strength of interlayer K-K excitons IXD and their avoided crossing with intralayer XA29. Indeed, the experimentally extract coupling strength between XA and IXD is consistent with the calculated interlayer coupling strength of holes at the K valley22. This picture is further corroborated by our measured dipole moment of IXD being close to the distance between the top and bottom layers, and explains our observation that the level crossing at XA is not fully avoided, since IXD does not couple to intralayer excitons in the middle layer. 8 The observed optical nonlinearity and their doping dependence cannot be simply explained by heating or carrier injection from the laser since both effects result in redshift of the intralayer excitons (Figs. 1e & S9). Therefore, we examine how the interactions among the elementary excitations in the systems, i.e., intralayer excitons XA, momentum-indirect excitons XI, and free carriers, may give rise to the observed nonlinearity. First, we extract a large interaction strength g ~ 2 meVµm2 from the linear fit of energy blueshift vs. XA+ density at low power30 (Fig. S8). It is apparent that excitonic interactions (i.e. between XA-XA and XA-XI) alone cannot produce the observed nonlinearity. On the one hand, the XA-XA interactions among intralayer excitons are repulsive but weak, which scales linearly with density with a coefficient of gex∼ αEBR2∼1.9µeVµm2, where α is a constant (~6), EB ≈ 100 meV is the exciton binding energy and R (∼1.78 nm) denotes the exciton Bohr radius in trilayers31. Given the short lifetime of XA (∼picoseconds)32,33 and thus the small density (∼109 cm-2, see Methods), the amount of blueshift is expected to be orders of magnitude smaller than the experimental values (and comparable to monolayers and bilayers20,31). On the other hand, although XA can acquire a dipole moment via its hybridization with IXD and experience XA-XI dipolar interaction with XI, such interactions should have persisted in the intrinsic and electron-doped regime. Furthermore, we estimate an upper limit of dipolar interaction strength gd ~ 1.8µeVµm2, orders of magnitude smaller than the experimental interaction strength g (see Methods). This conclusion is supported by the experimental observation of much weaker nonlinearity of XI than XA+, even when XI is fully polarized by an external electric field (see Methods and Fig. S10 for additional discussion). Therefore, we attribute the observed nonlinearity to a valley polarization created from the interactions between XA and free carriers. In particular, excitons created by optical pumping may 9 induce a nonequilibrium valley population imbalance of resident carriers between K and Γ valleys, via mechanisms such as exciton-carriers scattering34,35. Importantly, under zero electric field, the energy difference between K and Γ valleys is rather small in trilayers22, on the order of tens of millielectronvolts, based on first-principle calculations and transport studies (Fig. 3c). As a result, electrostatically doped holes at the Γ point could be efficiently scattered into the K valley by excitons such as XA and XI, via Coulombic and exchange interactions34,35 (Fig. 3d). This net accumulation of valley population at K (and K') induces phase space filling and, consequently, the observed blueshift of XA+. Such a scattering process would also introduce additional dephasing, which explains the increase in the XA+ linewidth. Meanwhile, the oscillator strength remains unchanged at low excitation power, since it depends on the total doping level, but decreases at higher power due to saturation. In addition to the exciton-carrier scattering, the net valley polarization may also be created by the electric field induced by optical pumping. For instance, it has been suggested in bilayer TMDs that the generation of dipolar XI excitons may create a non-zero displacement field due to spontaneous symmetry breaking1. Such a displacement field can introduce a relative energy shift between K and Γ valleys22, thus creating a valley polarization. To further investigate such possible valley polarization, we probe the population of the resident carriers in the K and K’ valleys under circularly polarized resonant excitation. As shown in Fig. S11, we observe a stronger blueshift of XA+ in the K than K’ valley when exciting excitons in the K valley, consistent with our optically induced valley polarization picture. The proposed valley polarization mechanism also explains the strong electron-hole asymmetry of optical nonlinearity. Intrinsic trilayer WSe2 exhibit weak nonlinearity, since no valley polarization can be created without resident carriers. In the electron-doped case, the valley polarization of 10 resident carriers is prevented by the much larger energy splitting, on the order of hundreds of millielectronvolts, between Q (band minimum) and the K valleys (where XA resides) in the conduction band22. We also briefly note that in TMD monolayers, optical pumping with circularly polarized light can polarize carriers in K vs. K’ valleys32–38. The generation of valley imbalance in monolayers has been attributed to mechanisms, such as different inter-valley vs. intra-valley carrier relaxation rates, and different indirect excitons and spin-forbidden dark excitons relaxation rates39,40. Unlike in monolayer, where a spin flip of carriers is required for scattering between K and K’ valleys, the Γ point is spin-degenerate so that the intervalley scattering between K and Γ in trilayers can happen via a spin-conserving process, such as direct Coulomb and exchange interactions between holes and excitons. We note that XI likely plays a crucial role in exciton-electron scattering owing to their larger population than XA (see Fig. S10 for estimated XI density). Since relaxation processes of XA to XI and XI to the ground state both involve finite momentum transfer, they can facilitate the valley polarization process via exciton-carrier scattering. This suggests that the dynamics of valley polarization could occur at a timescale similar to the XI lifetime, on the order of nanoseconds, which may explain our observation of weaker nonlinearity created by the pulsed laser than CW with the same peak power (Fig. S8). To further corroborate our hypothesis, we control the relative energies of Γ and K valleys by applying an external electric field20,22, and study the resulting changes in both the energies and optical nonlinearity of Fermi polarons. First, we measure the reflectance of the XA+ and XA− as we change the electric field under fixed doping levels (Fig. 4a & b). With an increasing field, the energy of the XA+ blueshift at a smaller electric field and then switches to redshift above an electric field of ~0.05 V/nm (Fig. 4a). The electric field shifts the valence band maximum from Γ to K due 11 to the distinct orbit characters of the eigenstates22. While the eigenstate at K becomes layer-polarized with higher energies with the field, the wavefunction at Γ point features large interlayer coupling with small energy change under the field22. This shift from Γ to K leads to more phase space-filling in K and a blueshift of XA+ , consistent with our proposed mechanism for optical nonlinearity (Fig. S13). Above an electric field of ~0.05 V/nm, XA+ begins to redshift. This electric field is comparable to the critical field under which band edge shifts from Γ to K, as determined from transport studies22 (note the literature reported field differs from our definition by a factor of the relative dielectric constant). In stark contrast, the energy shift of the XA− is much smaller in electron-doped region (Fig. 4b), where the band edge remains at the Q point. The small energy variation of XA− can be explained by the avoided crossing similar to the undoped case (Fig. S1). Intriguingly, the electric field also dramatically changes the nonlinear responses of XA+ . As shown in Fig. 4c, the reflectance change induced by optical pumping, ∆R/R shows a clear flip in the color contract at ~0.05 V/nm, which coincides with the crossover field between the blueshift and redshift of XA+ in Fig. 4a. The magnitude of blueshift and redshift is similar under the same excitation condition, around 10 meV (Fig. 4d). Such electric-field control of optical nonlinearity provides additional supporting evidence for our proposed mechanism involving valley polarization (Fig. 4e, f). Figure 4e shows the modified band structure above the critical field22. Under optical excitation, the same exciton-hole scattering process can induce a population transfer of holes from K to Γ, and reduce the XA+ energies (Fig. 4f). This also explains the similar critical electric fields where both nonlinearity and electric-field susceptibility turn from blue- to redshift. Conclusion 12 Our results demonstrating highly nonlinear excitons with large oscillator strengths open avenues for engineering exciton-carrier interactions in atomically thin heterostructures to explore strongly interacting many-body physics and develop novel optoelectronics. By designing the atomic and electronic structures of the heterostructures, one may engineer the strong interactions between bright, dark excitons, and free charges26. The excitonic and free charge populations are highly tunable, and elucidating the complex interactions between intralayer, interlayer excitons, and charges will be of significant future interest for studying many-body physics in a hybrid Fermi-Bose system41–43, from both theoretical and experiment perspectives. These strongly interacting optical excitations can be used to realize active nonlinear metasurfaces based on spatial confinement of excitons by moiré superlattice and local electrostatic gate44–46. Combining strong nonlinearity with spatial confinement could also further boost nonlinearity and allow exploring quantum optical effects, including nonclassical light sources and few-photon nonlinearity11,47. While current experiments require 103-104 photons to shift the resonance by a linewidth, improving materials quality and engineering photonic environment could lower required photon count significantly by reducing the Fermi polaron linewidth. Finally, the demonstrated optical control of exciton resonances could enable novel nonlinear optoelectronic devices such as all-optical switching, nonlinear optomechanical resonators48,49, and optical limiting devices4,50. Acknowledgements: This research is primarily supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences Early Career Research Program under Award No. DE-SC-0022885. The fabrication of samples is supported by the National Science Foundation CAREER Award under Award No. DMR-2145712. This research used Quantum Material Press (QPress) of the Center for 13 Functional Nanomaterials (CFN), which is a U.S. Department of Energy Office of Science User Facility, at Brookhaven National Laboratory under Contract No. DE-SC0012704. K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan for hBN synthesis. Author contributions Y.Z. and L.G. conceived the project. L.G. fabricated the samples and performed the experiments. L.Z., R.N., S.P., and H.J. assisted with sample fabrication. L.Z. and R.N. helped with optical measurements. M.X., D.S.W. M.H. and Y.Z. contributed to the data analysis and theoretical understanding. T.T. and K.W. provided hexagonal boron nitride samples. L.G. and Y.Z. wrote the manuscript with extensive input from the other authors. Competing financial interest The authors declare no competing financial interests. Additional Information Supplementary information is available in the online version of the paper. Reprints and permission information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to youzhou@umd.edu. mailto:youzhou@umd.edu14 References: 1. Datta, B. et al. Highly nonlinear dipolar exciton-polaritons in bilayer MoS2. Nat. Commun. 13, 1–7 (2022). 2. Delteil, A. et al. 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Dual-gated WSe2 homotrilayer van der Waals heterostructure and their optical characteristics under gating at T = 4 K. a, a schematic of the trilayer vdWs heterostructure. The homotrilayer WSe2 is encapsulated with two hBN of 15~20 nm thick. b, an optical image of the homotrilayer WSe2 device (Scale bar: 5 𝜇𝑚). The trilayer and neighboring bilayer regions are enclosed by the yellow and green dashed line, respectively. c, Photoluminescence spectra of the WSe2 trilayer under an electric field. The bright emission exhibiting a Stark shift under an electric field at 1.5 to 1.58 eV corresponds to the indirect exciton XI. The upper weaker emission at 1.7 eV 20 corresponds to the momentum direct K-K intralayer exciton XA. d, Reflectance spectra of the WSe2 trilayer under an electric field. The high energy momentum direct K-K interlayer exciton shows a Stark shift of ~100 meV and begins to hybridize with XA when their energies become degenerate around the electric field of 0.05 V/nm. e, Doping-dependent reflectance spectra of the WSe2 trilayer. With increasing doping concentration, the intralayer trion or Fermi polaron (XA-/XA+) shifts toward lower energy. 21 Figure 2. Nonlinearity in hole-doped homotrilayer WSe2 at T = 4 K . a, b, c, Reflectance contrast R/R0 of the trilayer under 0 𝜇W, 10 𝜇W, 30 𝜇W CW (635nm) laser excitation with different doping levels, where R0 is the reflectance of a reference region near the trilayer region with bare graphite/hBN/graphite on SiO2 on the sample. Intriguingly the exciton blueshifts when the sample is hole-doped. d, Relative change in the reflectance induced by 30 𝜇W of CW laser pumping under different doping. The color map is obtained by normalizing the reflectance change induced by the CW excitation with respect to the reflectance without optical pumping, ∆𝑅/𝑅 = 𝑅(30𝜇𝑊)𝑅(𝑁𝑜 𝑃𝑢𝑚𝑝)− 1 . Therefore, a positive value (red color) at high energy and negative 22 value (blue) at low energy indicates a blueshift. The apparent discontinuity near 0V is due to the high excitation power and finite voltage step used, and such transition becomes smoother at lower excitation power (see Fig. S7 for additional data). e, Blueshift of the XA+ as a function of the pulsed laser excitation (718-720nm, resonant with XA+) power, with a hole doping level of 7 × 1012𝑐𝑚−2. The seeming redshift in the peak energy below 1 μW is due to fluctuations. In fact, XA+ consistently shifts toward higher energy with increasing power (see Fig. S8 for additional data and analysis). 23 Figure 3. Electronic band structure of trilayer WSe2. a, The crystal structure of natural trilayer WSe2 dictates alternating K and K' valleys among neighboring layers. The strong spin-orbit coupling of holes leads to weak tunneling among neighboring layers but strong tunnelling between the top and bottom layer. b, The tunnelling of holes between top and bottom layer results in the hybridization of intralayer K-K excitons XA and interlayer K-K excitons IXD. c, d, Band structures and carrier populations of hole-doped trilayer WSe2 in the absence of electric field(E = 0). c, Optical excitation generates both momentum direct intralayer XA+ and momentum indirect XI of higher population. Intralayer Fermi polaron XA+ can interact with XI and the free holes in the system. d, Under strong optical excitation, the interaction between intralayer 24 excitons and free charges can induce a population transfer of carriers from the Γ to the K valley. The energy difference between Γ and K is small. The additional free carriers in K valley leads to phase space filling and optically induced blueshift of XA+. We note that in this nonequilibrium state, the dashed lines do not represent the Fermi level but are indications for the carrier population. 25 Figure 4. Electric-field dependent exciton energy and nonlinearity in homotrilayer WSe2 at T = 4 K. a, b, Electric field dependence of the intralayer Fermi polaron reflectance contrast R/R0 in trilayer with a (a) hole and (b) electron doping density of 4.9× 1012𝑐𝑚−2. c, Reflectance change induced by a pulsed laser excitation of 12 𝜇W power. The color map is obtained the same way as Fig. 2d. Under a small electric field, XA+ shows a blueshift but it begins to redshift under 26 excitation at higher electric field. d, Line plots of the XA+ power-dependent peak shift under different electric fields. The XA+ shows a blue shift of ~10 meV under zero applied electric field and a redshift of similar magnitude under large electric field. The corresponding electric fields for these linecuts are indicated by the dashed lines in (c). e, f, An electric field induces a shift of the valence band edge from Γ to K valley. Under strong optical pumping, a net valley polarization is induced by exciton-carrier scattering with increased carriers at the Γ point, which leads to the optically induced redshift of XA+ at higher electric field. In the nonequilibrium states, the dashed lines are used as indications for the carrier populations, and do not represent the Fermi level. 27 Methods: Device fabrication Graphite, hBN flakes are mechanically exfoliated from the bulk crystals onto the silicon chip with SiO2 layer. Some of the exfoliated homotrilayer WSe2 flakes were provided by the Quantum material press (QPress) facility in Center for functional nanomaterials (CFN) at Brookhaven national laboratory (BNL). The thickness of the hBN flakes and WSe2 layer numbers are estimated based on the colour contrast under the optical microscopy. The heterostructure is assembled in a transfer station built by Everbeing Int’l Corp., which use PDMS(Polydimethylsiloxane) and PC (Polycarbonate) as stamp and transfer all the flakes in a dry transfer method onto a silicon chip with 285nm SiO2 layer. Then the electrical contacts are patterned by electron-beam lithography and a liftoff process where we deposited 5nm of Cr and 80nm of Au by thermal evaporation. Optical spectroscopy The optical measurements were performed in our home-built confocal microscope with Attodry 4K cryostat. The apochromatic objective equipped in the chamber has numerical aperture NA=0.82. The PL measurement is performed with a 635nm diode laser excitation. The reflectance measurement is performed using either a halogen lamp (from Thorlabs) or a supercontinuum white laser (from YSL Photonics Inc.) as the excitation source. The diode laser has a diffraction-limited spot size while the beam diameter of the white laser is slightly larger, around 1𝜇𝑚. The white laser has a pulse duration of ~60 ps with variable repetition rate up to 40 MHz. Our power-dependent reflectance is measured under both continuous wave and pulsed excitation. In CW measurements, we illuminate the sample with the halogen lamp as the probe and use the CW (635nm) diode laser as excitation with a power ranging from 0.02 to 30 microwatts. In the pulsed resonant excitation 28 case, we excite the system with a supercontinuum white laser filtered to 718-730nm wavelength range. We vary the incident power and directly measure the reflected white laser signal from the sample. In both cases, our reflectance spectral is normalized by dividing the reflected light intensity from the sample trilayer area by the reflected light intensity from the nearby bare hBN on SiO2 area. The spectra is measured by a Horiba iHR320 spectrometer using a 300mm/line grating and a Synapse-Plus back-illuminated deep depletion CCD camera. Doping Density and Electric field The doping density and electric field are determined by considering the heterostructure as a parallel capacitor30. The applied electric field is calculated as 𝐸 =𝐷𝜀0𝜀𝑊𝑆𝑒2, while the D displacement field is determined by 𝐷 =12(𝐶𝑇𝑜𝑝 ∙ 𝛥𝑉𝑇𝐺 − 𝐶𝐵𝑜𝑡𝑡𝑜𝑚 ∙ 𝛥𝑉𝐵𝐺). The top and bottom capacitance are given by 𝐶𝑇𝑜𝑝(𝐵𝑜𝑡𝑡𝑜𝑚) = 𝜀0𝜀ℎ𝐵𝑁𝑡ℎ𝐵𝑁 , where t is the thickness of top and bottom hBN. 𝛥𝑉𝑇 and 𝛥𝑉𝐵 is the applied top and bottom gate voltage relative to the offset voltage to the band edge, respectively. The total doping density in the system can be determined as 𝑛 = 1𝑒∙ (𝐶𝑇𝑜𝑝 ∙ 𝛥𝑉𝑇𝐺 + 𝐶𝐵𝑜𝑡𝑡𝑜𝑚 ∙𝛥𝑉𝐵𝐺) . We use 𝜀𝑊𝑆𝑒2 = 7, 33,41 𝜀ℎ𝐵𝑁 = 3, 41 in our case. The thicknesses of hBN layers are extracted by atomic force microscope measurements. Extract Dipole Distance d The energy shift as a function of the electric field of the IXD and XI obeys the law of the stark shift, which can be described as: ∆𝐸 = |𝑒 ∙ 𝑑 ∙ 𝐸| = |𝑒 ∙ 𝑑 ∙𝜀ℎ𝐵𝑁𝜀𝑊𝑆𝑒2∙𝑉𝐵𝐺𝑡𝐵𝐺| (1) Thus the dipole moment of excitons can be calculated as 29 𝑑 = |∆𝐸𝑒∙𝜀𝑊𝑆𝑒2𝜀ℎ𝐵𝑁∙𝑡𝐵𝐺𝑉𝐵𝐺| (2) Estimate of exciton density In the steady state, the exciton densities of both intra- and inter-layer excitons can be estimated by 𝑛𝑋 = 𝐼𝛼𝜏/𝐴ℏ𝜔, where I is the pump power, A is the pump beam size, ℏ𝜔 is the photon energy, 𝛼 is the sample’s absorbance at the pump wavelength, and 𝜏 is the lifetime of respective excitons. The lifetime of interlayer excitons XI is estimated from time-dependent PL measurements, while we take the lifetime of XA from literature. The density of XI can be estimated from an independent method by measuring their PL and considering dipolar interactions using a simple capacitor model in which the exciton density is proportional to the blue shift induced by the repulsion between excitons20: 𝑛𝑥 =𝜀0𝜀𝑡𝑚𝑑∙𝛥𝐸𝑑𝑒2 , where 𝛥𝐸 is the blueshift of the emission energy and d is the dipole moment of XI, estimated to be ~0.78nm from the Stark effect. Estimate of exchange and dipolar interaction strength The exchange interaction strength can be estimated from gex∼ αEBR2, where α is a constant, EB is the exciton binding energy and R denotes the exciton Bohr radius in trilayers. For our calculation, we adopt α = 6, EB of ~100 meV, R of 1.78 nm, which are values from literature31, to estimate the interaction strength. Meanwhile, XA could acquire finite electrical dipoles via its hybridization with IXD. We note that to the first order, there should be zero net dipoles and weak dipolar interactions under symmetric gating. However, the emergence of local net dipoles is plausible due to spontaneous symmetry 30 breaking. In light of this, we calculate an upper limit for the dipolar interactions, assuming all dipoles are aligned. Using a parallel plate model, the dipolar interaction strength is given by 𝑔𝑑 =𝑒2 𝑑𝜀0𝜀𝑇𝑀𝐷 , where ed is the dipole moment of XA. This dipole moment, ed, acquired from the hybridization with IXD, can be estimated from the composition percentage of each exciton species in the coupled oscillator model (Fig. S2). The value peaks when XA become degenerate with IXD reaching ~0.7 𝑛𝑚 ∙ 𝑒, and leading to an estimated dipolar interaction strength of gd ~1.7 μeVμm2. Estimate of photon numbers required to shift the resonance by a linewidth The photon numbers are calculated from 𝑛 =Pτ𝐸 where P is the power of the laser, τ is the Fermi polaron lifetime, and 𝐸 =h𝑐𝜆 is the photon energy. For pulsed laser excitation, we convert the average pumping power (𝑃𝐴𝑣𝑔) into peak power 𝑃 =𝑃𝐴𝑣𝑔𝑓𝑅∙𝑡𝑝, with 𝑓𝑅 is repetition rate and 𝑡𝑝 is the pulse duration. Data availability Source data are provided with this paper. All other data are available from the corresponding authors upon reasonable request.