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Felix Schäfer, Henry Mittenzwey, Markus Stein, Oliver Voigt, Lara Greten, Daniel Anders, Isabel Müller, Florian Dobener, Marzia Cuccu, Christian Fuchs, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Alexey Chernikov, Kerstin Volz, Andreas Knorr, Sangam Chatterjee

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[Distinct Rabi splitting in confined systems of MoSe2 monolayers and (Ga,In)As quantum wells](https://mdr.nims.go.jp/datasets/36eaeec4-bb11-41b3-8553-aea62965de88)

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Distinct Rabi splitting in confined systems of MoSe2 monolayers and (Ga,In)As quantum wellsArticle https://doi.org/10.1038/s41467-025-63098-7Distinct Rabi splitting in confined systems ofMoSe2 monolayers and (Ga,In)Asquantum wellsFelix Schäfer 1,7, Henry Mittenzwey2,7, Markus Stein 1 , Oliver Voigt2,Lara Greten 2, Daniel Anders1, Isabel Müller1, Florian Dobener 1,Marzia Cuccu3, Christian Fuchs4, Kenji Watanabe 5, Takashi Taniguchi 6,Alexey Chernikov 3, Kerstin Volz4, Andreas Knorr 2 & Sangam Chatterjee 1Rabi splitting is a defining signature of strong light-matter interaction, emer-ging when a two-level system is resonantly driven by an optical field, resultingin a spectral doublet separated by the Rabi energy. In solid-state systems, Rabisplitting occurs at exciton resonances, where it is shaped by many-bodyinteractions intrinsic to the material. Here, we investigate the Rabi splittingdynamics in two paradigmatic two-dimensional semiconductors: a hBN-encapsulated MoSe2 monolayer and a (Ga,In)As multiple quantum well struc-ture. In MoSe2, strong Coulomb interactions dominate over light-mattercoupling, while in the quantum wells, both interactions are of comparablestrength. While both systems exhibit clear Rabi splitting under resonantexcitation, their behavior diverges under increased excitation strength. MoSe2displays sublinear Rabi splitting due to excitonic correlations, whereas (Ga,In)As quantum wells reveal additional spectral resonances and coherent opticalgain, indicating a transition beyond the simple two-level regime. These con-trasting behaviors are quantitatively captured by a unified microscopic many-body theory based on Heisenberg equations of motion and an excitonexpansion. Our findings elucidate the impact of many-body interactions oncoherent exciton dynamics and establish a framework for tailoring strong-fieldoptical responses in two-dimensional materials.The coherent nonlinear interaction of light and matter at the quan-tum level unveils rich dynamics with fascinating effects across var-ious fields of research1–4: from atomic5 and molecular physics6 tocondensed matter systems7 and quantum optics8,9. The observationof level-splitting phenomena associated with electronic transitionsdressed by photons is among the most fundamental. For example,the interaction of photons in resonance with the transition energy ofa quantum two-level system (2-LS) results in the splitting of itsabsorption into distinct sidebands. Such characteristic spectralmanifestations are termed Autler-Townes splitting, dynamical StarkReceived: 31 January 2025Accepted: 5 August 2025Check for updates1Institute of Experimental Physics I and Center for Materials Research (LaMa), Justus-Liebig-University Giessen, Heinrich-Buff-Ring 16, D-35392Giessen, Germany. 2Nichtlineare Optik und Quantenelektronik, Institut für Physik und Astronomie (IFPA), Technische Universität Berlin, D-10623Berlin, Germany. 3Dresden Integrated Center for Applied Physics and Photonic Materials (IAPP) and Würzburg-Dresden Cluster of Excellence ct.qmat,Technische Universität Dresden, D-01062 Dresden, Germany. 4Structure & Technology Research Laboratory (WZMW), Philipps-University Marburg, Hans-Meerwein-Straße 6, D-35032 Marburg, Germany. 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.7These authors contributed equally: Felix Schäfer, Henry Mittenzwey. e-mail: markus.stein@exp1.physik.uni-giessen.deNature Communications |         (2025) 16:8109 11234567890():,;1234567890():,;http://orcid.org/0000-0002-0458-9225http://orcid.org/0000-0002-0458-9225http://orcid.org/0000-0002-0458-9225http://orcid.org/0000-0002-0458-9225http://orcid.org/0000-0002-0458-9225http://orcid.org/0000-0002-0616-9881http://orcid.org/0000-0002-0616-9881http://orcid.org/0000-0002-0616-9881http://orcid.org/0000-0002-0616-9881http://orcid.org/0000-0002-0616-9881http://orcid.org/0000-0001-9796-3257http://orcid.org/0000-0001-9796-3257http://orcid.org/0000-0001-9796-3257http://orcid.org/0000-0001-9796-3257http://orcid.org/0000-0001-9796-3257http://orcid.org/0000-0003-1987-6224http://orcid.org/0000-0003-1987-6224http://orcid.org/0000-0003-1987-6224http://orcid.org/0000-0003-1987-6224http://orcid.org/0000-0003-1987-6224http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0009-0001-1712-4590http://orcid.org/0009-0001-1712-4590http://orcid.org/0009-0001-1712-4590http://orcid.org/0009-0001-1712-4590http://orcid.org/0009-0001-1712-4590http://orcid.org/0000-0002-0237-5880http://orcid.org/0000-0002-0237-5880http://orcid.org/0000-0002-0237-5880http://orcid.org/0000-0002-0237-5880http://orcid.org/0000-0002-0237-5880http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-63098-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-63098-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-63098-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-63098-7&domain=pdfmailto:markus.stein@exp1.physik.uni-giessen.dewww.nature.com/naturecommunicationssplitting or Rabi splitting. Originally reported for optically drivenmolecules in the gas phase10, these spectral splittings have beenattributed to the atomic states being ”dressed” by the light field. Theemerging states are then separated by the Rabi energy, which scaleswith the transition dipole moment and the driving field amplitude.The emergence of sufficiently sharp optical transitions due toimproved material quality enables similar research on condensedmatter systems11–13. Notably, these are inherently Coulomb-interacting many-body systems. They natively exhibit more intri-cate dynamics due to electron-hole collisions and excitation-induceddephasing14–19. By tailoring these interactions—e.g., introducing freecharges, applying strain gradients, or modifying the environment—such effects can be harnessed to enhance device performance.Unfortunately, such many-body effects also render the observationof splitting phenomena challenging. Consequently, observations ofRabi oscillations in condensed matter systems are relatively rare20–23.However, theoretical studies predict that light-dressing can sig-nificantly reduce the Coulomb-collision rates24, offering pathways forrelaxation rate control in devices. Experimental studies of Rabisplitting of exciton resonances within two-dimensional (2D) chargecarrier systems can directly confirm these predictions. Additionally,amplifying the resonant optical excitation access nonperturbativeexcitation regimes, leading to the emergence of new absorptionresonances and coherent gain phenomena beyond Rabi splitting25–28.High-quality transition metal dichalcogenides (TMDCs) or quantumwell (QW) structures serve as ideal platforms for studying thesephenomena in 2Dmany-body systems. Such studies transcend earlierinvestigations of Rabi splitting in microcavity systems, where light-matter interactions are enhanced at exciton resonances29–31, as well asat intersubband transitions28,32. More recently, Rabi splitting has beenobserved in TMDC monolayers across various cavity designs33–37.Beyond resonant optical excitation, Rabi splitting of the 1s excitonalso occurs when driven with lower energies in the range of theexciton binding energies38–42.In this work, we demonstrate and compare the Rabi splittingdynamics of exciton resonances in twoprototypical 2D semiconductorheterostructures—a TMDC single monolayer and conventional (Ga,In)As QWs—under controlled excitation conditions at cryogenic tem-peratures of 6 K. These systems represent two fundamentally differentregimes of light-matter interaction, allowing us to disentangle how theinterplay between Coulomb interactions and optical driving governsRabi dynamics in condensedmatter systems. The key distinction lies inthe vastly different exciton binding energies relative to the observedRabi energies in each system, as well as in the applied excitationschemes. These parameters determine the physical nature and fun-damental processes as identified by a rigorous microscopic many-body theory: on the one hand, the TMDC monolayer with stronglyboundexcitons in co-linear excitation resulting in amultitude ofmany-particle Coulomb correlations represents the one pole, where Cou-lomb interaction dominates completely over the light-matter interac-tion. On the other hand, the (Ga,In)As QW with less strongly boundexcitons in co-circular excitation, where only a fraction of Coulombcorrelations are possible, represents the other pole, where Coulomband light-matter interaction are of similar magnitude. This contrastallows us to explore and interpret the fundamentally different non-linear features observed in each system such as sublinear splitting,spectral sidebands, and coherent gain within a unified theoreticalframework.ResultsTMDCs featuring large exciton binding energies, strong light-mattercoupling accompanied by comparatively narrow linewidths are amongthe clearest realizations of an interacting two-level exciton gas. Inparticular, MoSe2 exhibits a singular, bright 1s exciton as the lowestenergy transition with binding energies of >100 meV. Figure 1a illus-trates the linear response (black line) andnonlinear absorption spectrameasured in transmission geometry of the MoSe2 monolayer encap-sulated in hBN and placed on a diamond substrate.The exciton transition is excited close to resonance at 1.638 eVwith a full width at half maximum (FWHM) of 2.2 meV, ~1.7 meV abovethe center of the exciton resonance, but still well within the 1s excitonabsorption line. The 850 fs long pump pulse and the white-lightsupercontinuum probe pulse are polarized co-linearly with respect toeach other. Figure 1a depicts results for a time delay between pumpand probe of −90 fs; the field maxima in time of pump and probeoverlap on the sample at time zero. Energy densities of 8 μJ/cm2 andabove invoke a splitting of the exciton resonance in the nonlinearabsorption spectrum. An exemplary time evolution of the excitonresonance splitting for an energy density of 16 μJ/cm2 is depicted inFig. 1b on the left. The unperturbed exciton resonance is observed fornegative time delays. It splits into two initially fairly symmetric bran-ches as the time delays approach zero. Eventually, these branches re-merge into a single absorption peak for time delays exceeding 0.5 ps.The absorption peak is then broadened and attenuated due to theexcitations in the system.The spectral splitting during temporal overlap is readily explainedusing a description based on an exciton Bloch equation-of-motionapproach43. This description accounts for 1s coherent exciton transi-tions (Eq. (6) in the Supplemental Material (SM)) and incoherentoccupations (Eq. (14) in the SM) as well as many-particle Coulombcorrelations. The latter account for four-particle Coulomb correla-tions, i.e. spin-like and spin-unlike biexcitons (Eq. (15) in the SM), andFig. 1 | Ultrafast nonlinear absorption dynamics in monolayer MoSe2: experi-ment and simulation. a Nonlinear absorption spectra of the MoSe2 monolayersample (left) for co-linear polarization geometries at a nominal pump-probe timedelay of −90 fs, shown for various excitation fluences. The linear absorptionspectrum (black) and the spectral profile of the optical pump pulse (orange) areincluded as references. Corresponding spectra from numerical simulations aredisplayed on the right. b 2D false-color plots of the nonlinear absorption as func-tion of time delay for an excitation energy density of 16 μJ/cm2 (left) and thecorresponding simulated results (right). E1s marks the 1s exciton energy.Article https://doi.org/10.1038/s41467-025-63098-7Nature Communications |         (2025) 16:8109 2www.nature.com/naturecommunicationssix-particle Coulomb correlations, i.e. spin-like and spin-unlike exci-ton-biexciton transitions (Eq. (16) and Eq. (17) in the SM), which sig-nificantly impact the nonlinear dynamics16,44–48. The details areprovided in the Supplemental Material. The model takes into accountoptical field-induced blocking, Coulomb-mediated excitation-induceddephasing, excitation-induced energy shifts and formation of inco-herent occupations due to optical interaction as well as exciton-phonon coupling. All excitonic couplingmatrix elements involving themany-body effects responsible for the nonlinearities in optically exci-ted semiconductors are calculatedmicroscopically; only the 1s excitonenergy, the transition dipole moment and a residual nonradiativebroadening unrelated to phonons are adjusted to experimental linearabsorption spectra.Results from the theoretical calculations for theMoSe2monolayerare shown in the right-hand panels of Fig. 1a, b. In the TMDC case, theexciton binding energy Eb significantly exceeds the Rabi energyℏΩ = dcv ⋅ E, where dcv represents the transition dipole moment and Ethe optical field. At the applied pump powers, the ratio ℏΩ/Eb ~10−3remains negligible. Additionally, co-linear excitation leads to the for-mation of spin-unlike biexcitons and exciton-biexciton transitions, asobserved in the absorption in Fig. 1 at ~1.60 eV and in the transientdifferential absorption spectra in Fig. 4. Here, a well-defined peak~30meV below the excitonic resonance appears, whose oscillatorstrength increases with increasing pump fluence. In this regime, thestrong Coulomb interaction between quasiparticles in MoSe2 dom-inates over the light-matter interaction, dictating the observed spec-tral features and governing the nonlinear response.Next, we study the scenario where the light-matter interaction isof a similar order of magnitude as the Coulomb interaction of thequasiparticles. The prototypical structure realizing these conditionsare (Ga,In)AsQWs in co-circular excitation geometry. These havemuchweaker bound excitons compared to the MoSe2 monolayer, i.e.,ℏΩ/Eb ~10−1 at the applied pump fluences and feature no spin-unlikebiexcitons and exciton-biexciton correlations. Varying the time delaybetween the optical pump and probe pulses provides insights into thedynamics of the nonlinear light-matter interaction, as depicted inFig. 2a. The inset illustrates the linear absorption spectrum of themultiple quantum well (MQW) sample, revealing its narrow 1s excitonabsorption peak at 1.467 eV, which is selectively excited by the opticalpulse centered at 1.468 eV. At an energy density of 4 μJ/cm2, the exci-ton resonance remains undisturbed for negative time delays, wherethe probe pulse arrives before the excitation pulse. However, as thetime delay approaches zero, causing temporal overlap between the1.3 ps-long excitation pulse and the short probe pulse, the excitonresonance begins to split into two distinct absorption peaks. Withincreasing temporal overlap, the probe pulse experiences a highereffective driving photondensity, leading tomore pronounced splittingand the eventual emergence of additional absorption features. Thisnonlinear response is consistently observed across different excitationpulses of similar duration and spectral range, as long as the exciton isexcited near resonance. Notably, significant optical gain emergesbetween these absorption peaks, even at amoderate energy density of4 μJ/cm2. This coherent gain feature49 is evident in Fig. 2a as a deeppurple region around 1.469 eV, near the 1s exciton resonance in thelinear absorption spectrum. Alongside the spectral Rabi splitting and acoherent gain, the associated Rabi oscillations manifest in the timedomain. These oscillations are particularly pronounced for the high-energy branchdue to its favorable scattering conditions, which reducethe Coulomb collision rates24. The Rabi oscillations can be analyzedquantitatively by examining the time traces spectrally averaged acrossthe gain region, cf. Fig. 2b. A clear single Rabi flop is found at 11 μJ/cm2while 42 μJ/cm2 yield nearly three full Rabi cycles, well-resolved intheory and experiment.DiscussionNext, we explain the key experimental features in Figs. 1 and 2: (a) Rabioscillations, (b) coherent gain, (c) spectral shifts and splitting and (d)bound biexciton formation for different values of ℏΩ/Eb and differentexcitation geometries, i.e. for dominating Coulombmany-body effectsin co-linearly excited TMDCs as well as for moderate Coulomb inter-action and comparably more pronounced light-matter coupling in co-circularly excited (Ga,In)As QWs.(a) Rabi oscillations. The microscopic model traces the temporalRabi oscillations back to oscillations of the total coherently excited 1sexciton density ∣P∣2 and incoherent 1s exciton density N.In the (Ga,In)As MQW, we focus the discussion on N, since itprovides the strongest contribution. N can be generated by exciton-phonon, exciton-light and exciton-exciton interaction. In the MQWcase, incoherent exciton formation via exciton-phonon scattering50,51 isnegligible, cf. Eq. (18). Rabi oscillations are solely determined by Pauli-blocking effects of incoherent excitonic occupations N in fourth orderof the optical field, cf. first and second line in Eq. (14) in the SM, sinceexciton-exciton interaction, cf. third and fourth line in Eq. (14) in theSM, is also of minor importance.Compared to theMQW, in themonolayerMoSe2, exciton-phonon,Pauli-blocking and exciton-exciton interactions scale differently. Here,in contrast to the (Ga,In)As MQW, optical blocking in the incoherentoccupations N is of minor importance, so that this formationFig. 2 | Coherent Rabi oscillations and gain dynamics in (Ga,In)As quantumwells: experiment and microscopic theory. a Left: 3D false-color representationof the nonlinear absorption for a high-quality (Ga,In)As MQW sample, revealingspectral Rabi splitting and a temporal beat corresponding to Rabi oscillations,particularly prominent in the higher-energy branch. The onset of negativeabsorption indicates coherent gain. The linear absorption (black) and the pumppulse spectrum (orange) are given in the inset. Right:Microscopically calculated 3Dplot of the nonlinear absorption as a function of time andphoton energy at a pumppower of 60 μW(16 μJ/cm2) and a pump detuning of 0.5meV above the 1s excitonenergy for a co-circular pump-probe polarization configuration. b Left: Transientsthrough the gain region at 1.469 eV highlighting Rabi oscillations for various opticalexcitation fluences under a co-circular pump-probe polarization configuration.Right: Corresponding results frommicroscopic calculations. The grey-shaded arearepresents the excitation pulse.Article https://doi.org/10.1038/s41467-025-63098-7Nature Communications |         (2025) 16:8109 3www.nature.com/naturecommunicationsmechanism is even outcompeted by exciton-phonon interaction at theapplied pump powers: The stronger Coulomb interaction reduces thePauli-blocking contribution in the incoherent excitonic occupationsN,Eq. (14) first line, cf. also Tab. SI and SII in the SM, since the excitonicwave functions are more spread out in q-space, and enhances theexciton-exciton interaction in the excitonic transitions P, cf. third andfourth line in Eq. (6) in the SM. Thus, the incoherent occupations N donot contribute to the Rabi-flopping dynamics. Similarly, the coherentlyexcited exciton density ∣P∣2 is especially decreased by the excitation-induced dephasing via the biexciton and exciton-biexciton continuum.The different scaling of these mechanisms result as a direct con-sequence of the stronger confinement in atomically thin TMDC, whichincreases exciton-exciton and exciton-phonon interaction comparedto exciton-light interaction. All in all, in the Coulomb-dominatedmonolayer MoSe2, no Rabi oscillations are observed.(b) Coherent gain. In the (Ga,In)As QW, the full spectro-temporaldynamics from our calculations, shown in Fig. 2a on the right, alsocapture the emergence of gain. Its attribution of a coherent natureoriginates from the two-pulse superposition in the blocking con-tribution (third term in the first line of Eq. (6) in the SM), whichtransfers parts of the pump-induced dynamics in the direction of theprobe pulse, i.e. it emerges only during the presence of the pumppulse. This ”wave-mixing-like” process is significantly different fromthe more common, incoherent gain due to inversion of an incoherentcharge-carrier population. The low excitation density of 16 μJ/cm2 andthe resulting absence of population inversion conclusively rule out anypossibility of incoherent gain. Occupation gratings due to Coulombinteraction (second line in Eq. (6) in the SM) and unbound spin-likeexciton-biexciton transitions (second term in the last line in Eq. (6) inthe SM)play aminor role, but do contribute to anoverall enhancementof the coherent gain. In contrast, in the MoSe2 monolayer, the strongCoulomb interaction, coupled with the presence of spin-unlike biex-citons and exciton-biexciton transitions due to co-linear excitation,effectively suppresses the coherent gain signatures at the appliedpump powers.(c) Energy shifts and splitting. The narrow linewidth of the MQWsample allows for a detailed quantitative analysis of the light-drivenspectral shifts and Rabi splitting using our many-body model. To thisend,we examine the individual spectral features inducedby theopticalexcitation. Figure 3a illustrates the measured nonlinear absorption forco-circularly polarizedpumpandprobe pulses at a timedelayof 300 fsfor increasing photon densities. The exciton resonance begins to splitinto two distinct absorption peaks at energy densities of 1.4 μJ/cm2. Inparticular, the low-energy branch, positioned at 1.4666 eV, maintainsits spectral position as the energy density increases, while the high-energy branch shifts from 1.4677 eV to 1.4727 eV, which it reaches atthe highest energy density of 99.0 μJ/cm2. Both branches have nar-rower linewidths than the linear 1s exciton resonance at low drivingphoton fluences. Moreover, additional absorption features emerge asthe excitation strength increases. A weak resonance appears at lowerenergies for pump energy densities of 2.8 μJ/cm2, subsequently shift-ing from 1.465 eV to 1.460 eV as the excitation is further increased.Another weak absorption peak arises between the two initial branchesat 1.468 eV for energy densities of 17.0 μJ/cm2, maintaining its spectralposition even under stronger driving conditions.Figure 3b traces the spectral positions of all observed absorptionpeaks relative to the 1s exciton resonance in the linear absorptionagainst the square root of the excitation power, i.e. the field amplitude.In this regime, an approximate analytical formula of the Rabi splittingfor vanishing exciton-exciton interaction V = 0 and vanishing pumpsaturation is derived (cf. Eq. (64) in the SM):E ± = E1s ±ffiffiffiffiffiffi2DpℏΩ1s, ð1Þwhere the Coulomb-enhanced blocking parameter reads: D=Pqφ3qPqφqwith the 1s excitonic wave function φq at relative momentum q. Eq. (1)is formally equivalent to the splitting observed in a classical 2-LS andtherefore shows a linear splitting in the Rabi energy. This linearsplitting behavior is observed in the experiment as well as in the fullsimulations, cf. Fig. 3b. We conclude that the (Ga,In)As MQW sampledisplays an effective, modified 2-LS response due to its relatively weakCoulomb interaction and the co-circular polarization conditions (nospin-unlike biexcitons or exciton-biexciton transitions). Note that inthe simulations, a weak saturation onset at higher fluences can beobserved, which is traced back to the onset of optical blocking of the 1sexciton density.Finally we consider the TMDCmonolayer case where the spectralshifts observed in Fig. 1 appear at first glance to be similar to thoseobserved in the QW sample (cf. Fig. 2). The low-energy resonanceremains fixed at ~1.633 eV, while the high-energy peak continuouslyshifts to higher energies with increasing excitation, as shown in Fig. 3bFig. 3 | Intensity-dependent Rabi splitting in (Ga,In)As and MoSe2: experimentandmicroscopic modeling. a Stacked nonlinear absorption spectra of the (Ga,In)As MQWmeasured at a 300 fs time delay between co-circular polarized pump andprobe pulses for various excitation densities. Rabi splitting is observed above anenergy density of 1.4 μJ/cm2, with additional absorption peaks emerging at higherpump intensities.b Energy positions of the absorptionpeaks fromboth experiment(triangles) and microscopic modeling (spheres), plotted against the square root ofthe pump power. Left: (Ga,In)As MQW (10 μW1=2 =̂ 28.3 μJ/cm2); Right: MoSe2Monolayer (10μW1=2 =̂ 157.2 μJ/cm2). The inset sketches the sample structure of theMoSe2 monolayer. The pulse area is defined as: Pulse area=R1�1 dtΩcvðtÞ andillustrates the light-matter interaction strength acting directly within the corre-sponding sample.Article https://doi.org/10.1038/s41467-025-63098-7Nature Communications |         (2025) 16:8109 4www.nature.com/naturecommunicationson the right, displaying a highly asymmetric splitting. Here, the initialsymmetric Rabi splitting into a repulsive (upper) and an attractive(lower) branch is significantly altered by the strong Coulomb interac-tion, which dominates over the optical interation (ℏΩ/Eb ~10−3): Theupper branch exhibits a strong Coulomb-dominated density-depen-dent blue shift, which is a common behavior of TMDCs52–54, while thelower branch remains relatively stable as it is more light-dominated(see SM for more details). Notably, a closer analysis reveals a slightsublinear increase of the splitting with the Rabi energy, even thoughthe optically excited exciton density is still far from optical saturation,which deviates from the effective 2-LS behavior, cf. Equation (1). In theMoSe2 monolayer case, where Coulomb interactions completelyovershadow optical interactions, a regime of vanishing Pauli-blockingemerges55,56 as long as the exciton density is well below the Motttransition57,58. Under such conditions, an approximate analyticalexpression for the Rabi splitting E± at zero detuning can be derived (cf.Eq. (62) in the SM):E ± = E1s ±ffiffiffi3p∣V ∣13∣ℏΩ1s∣23: ð2ÞHere, E1s is the exciton energy, V represents the effective exciton-exciton interaction, and ℏΩ1s =∑qφqℏΩ is the excitonic Rabi energy.This Hartree-Fock description already predicts a weak sublinearsplitting with respect to the Rabi energy ℏΩ. The analysis of themicroscopicmodel identifies theorigin of thisweak sublinear behaviorin a regime far from optical saturation: First, we note that the splittingitself originates from an occupation grating rather than a polarizationgrating, since, at zero pump-probe delay, i.e. maximal splitting, theoverall excitonic density is already dominated by incoherent occupa-tions. At that time, four-particle biexcitons (Eq. (15) in the SM) arealready decayed and six-particle exciton-biexciton transitions (Eq. (16)and Eq. (17) in the SM) are formed, since they possess a combinedcoherent (probed transitions) and incoherent (pump-induced exci-tonic occupation) source. Second, in co-linear excitation, next to spin-like exciton-biexciton transitions, additional spin-unlike exciton-biexciton transitions are induced, which are absent in co-circularexcitation. Due to their Coulomb-correlated intervalley nature, theycause a dynamic coupling between both K and K′ valleys, which causesan overall attenuation of the Rabi splitting.The qualitative validity of Eq. (2) results from the appearanceof intervalley exciton-biexciton transitions which overcompensatethe excitation-induced dephasing by amplifying coherent Coulombrenormalization effects in the probe-induced transition (cf. thesecond line in Eq. (6) in the SM). The experimentally observedlinewidth narrowing of the splitting peaks corroborates this inter-pretation: Compared to the 6.7 meV FWHM of the linear absorptionresonance, the lower-energy peak narrows to between 3.32 and 6.18meV,while the higher-energy peak exhibits a FWHM in the range of 3.11to 4.74 meV as the photon density increases. The simulations alsodisplay a linewidth narrowing at small to moderate photon densities.(d) Biexciton formation. An additional resonance on the low-energy side is clearly observed in co-linearmeasurements in theMoSe2monolayer, cf. Figures 1 and 4. Its spectral position is ~30meV belowthe excitonic resonance similar to other works, which report biexci-tonic binding energies in the range of 20–30meV59–62. Its spectralposition experiences a fluence-dependent red shift, which mirrors adensity-dependent repulsive interaction between the biexcitonic andthe excitonic resonance, as well as an increase in oscillator strength. Inco-circular excitation in a (Ga,In)As MQW, no biexcitonic resonanceappears. Here, only biexcitonic continua occur, since only oneelectron-heavy-hole spin configuration is optically addressed. Theobserved features in the experiments are well reproduced by themicroscopic theory, confirming the contributions of bound biexcitonsand the biexcitonic continuum.In summary, we present experimental evidence and theoreticalconfirmation for Rabi splitting of the 1s exciton resonances in (Ga,In)AsQWs and MoSe2 monolayers under close-to-resonant excitationconditions. The fundamentally different nature of both 2-LS, particu-larly regarding the interplay of Coulomb interactions and light-mattercoupling, in conjunction with specific excitation conditions, dictatesthe splitting behavior, occurrence of Rabi oscillations, and coherentgain: In (Ga,In)As MQWs with co-circular excitation and weakerCoulomb interaction, the Rabi splitting scales nearly linearly with theRabi energy and is accompanied by pronounced Rabi oscillationsand coherent optical gain. Conversely, in MoSe2 monolayers, with co-linear excitation and stronger Coulomb interaction, only sublinearRabi splitting is observed, with no evidence of Rabi oscillations orgain. Our findings are corroborated by a microscopic theory thatelucidates the physical mechanisms underlying these effects.These insights are pivotal for leveraging exciton-based coherent phe-nomena in next-generation ultrafast optoelectronic and switchingdevices.MethodsSample fabricationWe investigate two distinct types of samples. The first sample is aMoSe2 monolayer (ML) encapsulated in hBN and placed on a diamondsubstrate. This configuration promotes high optical quality, a narrowFig. 4 | Fluence-dependent biexciton shifts in MoSe2: experiment vs. theory.Comparison betweenexperimental (a) and theoretical (b)ΔαL spectraof theMoSe2ML as a functionof excitationfluence, ranging from low (blue) to high fluence (red).A fluence-dependent blueshift of the 1s exciton resonance and a redshift of theexcitation-induced biexciton absorption—highlighted in the zoomed-in panel—areobserved. These trends are well captured by the microscopic simulations.Article https://doi.org/10.1038/s41467-025-63098-7Nature Communications |         (2025) 16:8109 5www.nature.com/naturecommunicationsexciton linewidth, and improved environmental stability due to theprotective hBN encapsulation.The second sample is a type-I band alignmentmulti-quantumwellstructure. High-resolution X-ray diffraction (HRXRD) measurementsreveal that the Ga0.942In0.058As QW layers are 7.6 nm thick. These QWsare arranged in a stack of 10 layers, separated by barriers composed ofGaAs/GaAsP/GaAs, which are designed to provide strain compensa-tion. The barrier thickness is 28.6 nm. Atomic forcemicroscopy (AFM)measurements reveals a well-defined surface quality with a root-mean-square roughness of 0.3 nm. Additional details on the growth processand structural characterization are provided in Ref. 63.Both samples exhibit a pronounced and well-defined 1s excitonresonance in their linear absorption spectra, which can be resonantlyexcited using an optical pulse.Experimental detailsFor the investigation of the MQW sample, we utilize a regenerativeamplifier system operating at a 5 kHz repetition rate. This systemdelivers 50 fs pulses centered around 800 nm with a pulse energy of1.6 mJ. A small portion of the amplifier output generates a white-lightsupercontinuum in a 4mm thick sapphire-crystal, while the remainingoutput drives an optical parametric amplifier (OPA). TheOPA providestunable pulses at the desired central wavelength, which are thenspectrally narrowed using a pulse shaper. After shaping, the pulseshave aduration of 1345 fs. The shapedpulses are then focusedonto thesample to a spot size of 300 μm. In the white-light beam path, a wedgebeamsplitter is used to split the light into two beams: one is focuseddown to a 200 μm spot on the sample to probe the excitation-inducedabsorption changes, while the other serves as a reference pulse. Thesample is held at liquid helium temperatures in a cold-finger cryostat.After passing through the sample, the white-light supercontinuum andthe reference pulse are spectrally analyzed using an imaging spectro-meter equipped with a 600 lines/mm grating and an sCMOS camera(see Fig. 5). The sCMOS detector has 2160 lines that can be read outindividually, allowing multiple lines to be grouped into a region ofinterest (ROI). By imaging the white-light pulse transmitted throughthe sample and the reference pulse onto different lines of the detector,both spectra can be captured simultaneously and independently. Thissetup allows for the calculation of a transfer function (Tf) that convertsthe reference spectrum (Tref) into the spectrum of the pulse trans-mitted through the unexcited sample. The reference path thusprovides the spectrum of the unexcited sample at any time, compen-sating for fluctuations in the white-light spectrum, as the transferfunction remains stable over time. By introducing optical excitation,we can simultaneously measure the transmission through the excitedsample (TP) and the transmission through the unexcited sample (T0).At the beginning of each measurement, the photoluminescencebackground (TPl) and the scattered light background (TBg) in bothpaths are recorded. The differential absorption at each time step isthen calculated as follows:ΔαL= � ln TP � TPl� �= T f � T ref � TBg� �� �� �: ð3ÞThis method offers several advantages: the simultaneous measure-ment of the excited and unexcited sample transmissions minimizesdistortion from white-light fluctuations, and each time step can bemonitored in real time by simply reading out the spectrometer,resulting in a highly accurate, fast, and low-noisemethod to determinedifferential absorption. Additionally, this approach can also be used todetermine the linear absorption of the sample. To do this, we firstremove the sample from the beam path to establish the transferfunction that converts the reference path into the spectrum of thepulse transmitted through the sample holder. The sample is thenreintroduced, allowing us to measure the transmission through theunexcited sample and calculate the linear absorption. Adding thelinear absorption to the differential absorption then yields the totalabsorption of the excited sample.To measure the MoSe2 monolayer (see Fig. 6 for a brightfieldimage), we utilized an amplifier laser system operating at a 100 kHzrepetition rate with a central wavelength of 1030 nm. As before, aportion of the amplifier output is used to generate a white-lightsupercontinuum in a 4mm thick sapphire-crystal, while the remainingoutput drives anOPA. The output from the OPA is spectrally narrowedusing a combination of short-pass and long-pass filters. The shapedpulse is then focusedonto the sample to a spot size of 28.5μmtoexcitethe sample. The probing white-light supercontinuum is expanded to adiameter of 1 inch before being focused down to an ~5μm spot on thesample using a 1-inch lens with a 4 cm focal length. The transmittedlight is then focused into a spectrometer for spectral analysis. Tomonitor the spot position on the sample, the setup allows for imagingthe transmitted light onto a camera. By strongly attenuating the white-light probe and using additional background illumination, the probeposition on the monolayer sample can be precisely observed.For theMoSe2monolayermeasurements, we employ a systematicapproach using mechanical shutters in both the optical excitation andwhite-light probing paths to capture various spectra at each time step.This method allows us to record the light transmitted through theunexcited sample (T0), the spectrum of the excited sample (TP), abackground measurement (TBg), and the photoluminescence back-ground (TPl). These measurements enable us to calculate differentialabsorption via:ΔαL= � ln TP � TPl� �= T0 � TBg� �� �: ð4ÞSince the diamond substrate introduces light scattering that affectstransmission through the sample, directly measuring the linearabsorption by comparing the light transmitted through the samplewith that through the sample holder alone is not feasible. To addressthis challenge, we analyze the transmitted light through the sampleitself. The resulting spectrum shows a distinct dip corresponding tothe exciton resonance of the monolayer, as illustrated as gray line inFig. 7. Due to the high exciton binding energy in theMoSe2monolayer,the spectrum is dominated by the exciton resonance, which is furthercorroboratedby thedifferential absorptionmeasurements. To create areference spectrum, we fit a polynomial function to the transmittedlight spectrum, deliberately excluding the exciton dip. The polynomialFig. 5 | Schematic of the optical pump-probe setup with pulse shaper andreference path. Illustration of the optical pump—optical probe setup, featuring apulse shaper to tailor the excitation pulse and two probe paths: one directedthrough the sample and the other serving as a reference.Article https://doi.org/10.1038/s41467-025-63098-7Nature Communications |         (2025) 16:8109 6www.nature.com/naturecommunicationsfit and its boundaries are shown in Fig. 7. This polynomial fit serves asthe reference white-light spectrum, which we use to calculate thesample’s linear absorption viaαL= � lnT0T fit� �: ð5ÞThis approach provides the most accurate estimate of the sample’slinear absorption around the exciton resonance. By adding the dif-ferential absorption to the linear absorption, we obtain the nonlinearabsorption of the sample at each time step.TheoryWe account for optical light-matter and electron-electron, hole-hole andelectron-hole Coulomb interaction64 as well as Coulomb exchangeinteraction65,66 in second quantization, apply the unit operatormethod43,67, perform a cluster expansion68 with respect to electron-holepair operators and formulate the theory in the emerging correlatedexpectation values. The direct Coulomb interaction and optical inter-action are treated up to fourth order within the dynamics-controlledtruncation scheme (DCT)69, which is necessary to correctly capture theoptical formation of incoherent excitonic occupations. This is especiallyimportant in quantum wells at cryogenic temperatures, where the for-mation of incoherent excitonic occupations due to exciton-phononinteraction is negligible. Coulomb exchange interaction is included onlyup to third order DCT, since higher orders are expected to be negligible.Moreover, we include the formation of incoherent excitonic occupa-tions via exciton-phonon scattering50,51 on a second-order DCT level70.Density-dependent exciton-phonon interaction is neglected71,72. Sincewe are mainly interested in the dynamics within and shortly after theduration of the optical pump pulse, the optical formation of excitonicoccupations is much more important than a redistribution of momentadue to Coulomb scattering73,74. Therefore, we do not consider fullyincoherent excitonic Coulomb scattering.All in all, we consider correlated expectation values for up to sixparticles68, i.e., we account for one-, two- and three-exciton correla-tions: The excitonic transitions P = hvycic, the excitonic occupationsN = hcyvvycic, the biexcitonic correlationsB= hvycvycic and the exciton-biexciton correlations R= hcyvvycvycic. Here v/c(†) denote the valence/conduction band annihilation (creation) operators. All many-bodycorrelations are first established via the Heisenberg equations ofmotion in the electron-hole picture and then transformed into theexcitonic picture by an expansion in the solutions of the one-excitonSchrödinger equation (Wannier equation)43. The biexcitons B andexciton-biexciton transitionsR are further expanded in solutions of thetwo-exciton Schrödinger equation16,44. The subscript ”c” denotes thecorrelated part of the corresponding expectation value, where alllower order correlations have been removed. We consider only 1sexcitonic transitions P and occupations N, but include 1s, 2s and 3sstates in the calculation of the two-exciton states.Since the purpose of the theory is a qualitative understanding ofthe experiments, wemake the assumption NQ ≈ NδQ,0, which results inan effective momentum-independent formulation of the incoherentdynamics. A full momentum-dependent treatment of the incoherentexciton dynamics, which complicates the numerical evaluation of theexcitonic occupations N and especially the exciton-biexciton correla-tions R due to their partly incoherent nature, is beyond the scope ofthis work. A detailed description of the theory and the calculations canbe found in the SM.Fig. 7 | Linear absorption spectrum of monolayer MoSe2 derived from thetransmission spectrum. Transmission spectrum of the MoSe2 monolayer. Thetransmitted light (gray) shows a dip at the exciton resonance. The polynomial fit(red) excludes this dip and serves as the reference spectrum for calculating linearabsorption. The fit boundaries are marked in orange, and the calculated linearabsorption is shown in black.Fig. 6 | Brightfieldmicroscopy of the TMDC samplewithmagnified view of theMoSe2monolayer area.Brightfield image of the TMDC sample. The right side displaysan enlarged view of the monolayer MoSe2, marked with a red outline to indicate the specific region where measurements were taken.Article https://doi.org/10.1038/s41467-025-63098-7Nature Communications |         (2025) 16:8109 7www.nature.com/naturecommunicationsData availabilityThe datasets generated in this study have been deposited in the JLU-pub database https://doi.org/10.22029/jlupub-20064.Code availabilityCode pertaining to the theoretical model in this work will be madeavailable upon request to the corresponding author.References1. Kasevich, M. A. Coherence with atoms. Science 298, 1363–1368(2002).2. Shah, J. Ultrafast Spectroscopy of Semiconductors and Semi-conductor Nanostructures, Vol. 115 (Springer Science & BusinessMedia, 2013).3. Sommer, A. et al. Attosecond nonlinear polarization andlight–matter energy transfer in solids. Nature 534, 86–90 (2016).4. Karnieli, A., Rivera, N., Arie, A. & Kaminer, I. 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A.C. and M.C.acknowledge financial support by the DFG through Würzburg-DresdenCluster of Excellence on Complexity and Topology in Quantum Matterct.qmat (EXC 2147, Project-ID 390858490) and SFB 1277 (project B05,Project-ID: 314695032). K.W. and T.T. acknowledge support from theJSPS KAKENHI (Grant No. 21H05233 and 23H02052) and World PremierInternational Research Center Initiative (WPI), MEXT, Japan.Author contributionsA.K. and S.C. conceived the study. C.F., and K.V. designed and epitaxi-ally grew the quantum well sample. M.C. and A.C. exfoliated themonolayer sample, encapsulated it in h-BN and placed it on a diamondsubstrate. K.W. and T.T. provided the hBN. F.S., M.S., D.A., I.M., F.D. andS.C. set up and carried out the experiments and analysed the experi-mental data. H.M.,O.V., L.G., andA.K. developed themicroscopic theoryand carried out the numerical calculations. F.S., H.M.,M.S., A.K. andS.C.wrote the manuscript with contributions from all authors.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-025-63098-7.Correspondence and requests for materials should be addressed toMarkus Stein.Peer review information Nature Communications thanks Xiaolong Zhu,and the other, anonymous, reviewer(s) for their contribution to the peerreview of this work. 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If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2025Article https://doi.org/10.1038/s41467-025-63098-7Nature Communications |         (2025) 16:8109 10http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Distinct Rabi splitting in confined systems of MoSe2 monolayers and (Ga,In)As quantum wells Results Discussion Methods Sample fabrication Experimental details Theory Data availability Code availability References Acknowledgements Author contributions Funding Competing interests Additional information