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Aleksander Rodek, Thilo Hahn, James Howarth, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Marek Potemski, Piotr Kossacki, Daniel Wigger, Jacek Kasprzak

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[Controlled coherent-coupling and dynamics of exciton complexes in a MoSe<sub>2</sub> monolayer](https://mdr.nims.go.jp/datasets/2c8086ca-9d61-4b93-82dd-d5b6671f1bdf)

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Controlled coherent-coupling and dynamics of exciton complexes in a MoSe2 monolayer2D MaterialsPAPER • OPEN ACCESSControlled coherent-coupling and dynamics ofexciton complexes in a MoSe2 monolayerTo cite this article: Aleksander Rodek et al 2023 2D Mater. 10 025027 View the article online for updates and enhancements.You may also likeTwo-dimensional sub-wavelength atomlocalization in an electromagneticallyinduced transparency atomic systemJ. C. Wu and B. Q. Ai-(Invited, Digital Presentation) PhotocurrentDetection of Cooperative Exciton QuantumInterference in Nanocrystal Thin FilmsHirokazu Tahara and Yoshihiko Kanemitsu-Multidimensional coherent opticalspectroscopy of semiconductornanostructures: a reviewGaël Nardin-This content was downloaded from IP address 144.213.253.16 on 04/04/2023 at 06:50https://doi.org/10.1088/2053-1583/acc59a/article/10.1209/0295-5075/107/14002/article/10.1209/0295-5075/107/14002/article/10.1209/0295-5075/107/14002/article/10.1149/MA2022-0220922mtgabs/article/10.1149/MA2022-0220922mtgabs/article/10.1149/MA2022-0220922mtgabs/article/10.1149/MA2022-0220922mtgabs/article/10.1088/0268-1242/31/2/023001/article/10.1088/0268-1242/31/2/023001/article/10.1088/0268-1242/31/2/0230012D Mater. 10 (2023) 025027 https://doi.org/10.1088/2053-1583/acc59aOPEN ACCESSRECEIVED8 January 2023REVISED28 February 2023ACCEPTED FOR PUBLICATION20 March 2023PUBLISHED3 April 2023Original Content fromthis work may be usedunder the terms of theCreative CommonsAttribution 4.0 licence.Any further distributionof this work mustmaintain attribution tothe author(s) and the titleof the work, journalcitation and DOI.PAPERControlled coherent-coupling and dynamics of exciton complexesin a MoSe2 monolayerAleksander Rodek1,∗, Thilo Hahn2, James Howarth3, Takashi Taniguchi4, Kenji Watanabe5,Marek Potemski1,6,7, Piotr Kossacki1,∗, Daniel Wigger8 and Jacek Kasprzak1,9,10,∗1 Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland2 Institute of Solid State Theory, University of Münster, 48149 Münster, Germany3 National Graphene Institute, University of Manchester, Booth St E, Manchester, M13 9PL, United Kingdom4 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan5 Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan6 Laboratoire National des Champs Magnétiques Intenses, CNRS-UGA-UPS-INSA-EMFL, 25 Av. des Martyrs, 38042 Grenoble, France7 CENTERA Labs, Institute of High Pressure Physics, PAS, 01-142 Warsaw, Poland8 School of Physics, Trinity College Dublin, Dublin 2, Ireland9 Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France10 Walter Schottky Institut and TUM School of Natural Sciences, Technische Universität München, 85748 Garching, Germany∗ Authors to whom any correspondence should be addressed.E-mail: aleksander.rodek@fuw.edu.pl, piotr.kossacki@fuw.edu.pl and jacek.kasprzak@neel.cnrs.frKeywords: transition metal dichalcogenides, coherent nonlinear spectroscopy, ultrafast exciton dynamics, coherent coupling,two-dimensional spectroscopy, four-wave mixing, optical microscopy and imagingSupplementary material for this article is available onlineAbstractQuantifying and controlling the coherent dynamics and couplings of optically active excitations insolids is of paramount importance in fundamental research in condensed matter optics and fortheir prospective optoelectronic applications in quantum technologies. Here, we perform ultrafastcoherent nonlinear spectroscopy of a charge-tunable MoSe2 monolayer. The experiments showthat the homogeneous and inhomogeneous line width and the population decay of excitoncomplexes hosted by this material can be directly tuned by an applied gate bias, which governs theFermi level and therefore the free carrier density. By performing two-dimensional spectroscopy, wealso show that the same bias-tuning approach permits us to control the coherent coupling strengthbetween charged and neutral exciton complexes.1. IntroductionThe ability to isolate monolayer flakes of semicon-ducting transition-metal dichalcogenides (TMDs),such as MoSe2, and the discovery of their enhancedlight–matter interaction at the monolayer limit a dec-ade ago [1, 2] established a novel benchmark for semi-conductor optics. The combination of heavy effect-ive masses and spin-valley locking—both inherent inthe monolayer’s band structure—with a large out-of-plane dielectric contrast result in particularly strongCoulomb interactions among free carriers hosted byTMD monolayers [3, 4]. For the same reason, theexciton transitions display a non-hydrogenic excita-tion spectrum [5] and have large binding energies andoscillator strengths [6, 7], such that they dominatethe optical response of TMD materials even at roomtemperature.Further to that, the ability to stack different sortsof layered van der Waals materials, such as TMDs,hexagonal boron nitride (hBN), graphene or graph-ite, into van der Waals heterostructures [8, 9], facilit-ates the fabrication of optoelectronic devices [10–14].Sandwiching layered TMDs or other materialsbetween high quality hBN diminishes the electronicdisorder and protects them from degradation byambient agents. In general, the hBN-encapsulationprocedure reduces the exciton’s spectral inhomo-geneous broadening ℏσ, thus improving the opticalperformance of TMDs [15–18]. It even enables reach-ing the homogeneous limit for the exciton’s spectralline shape [19, 20]. This has been a milestone for© 2023 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/2053-1583/acc59ahttps://crossmark.crossref.org/dialog/?doi=10.1088/2053-1583/acc59a&domain=pdf&date_stamp=2023-4-3https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://orcid.org/0000-0002-0263-3122https://orcid.org/0000-0001-7434-9940https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-7558-1044https://orcid.org/0000-0003-4960-2603mailto:aleksander.rodek@fuw.edu.plmailto:piotr.kossacki@fuw.edu.plmailto:jacek.kasprzak@neel.cnrs.frhttps://doi.org/10.1088/2053-1583/acc59a2D Mater. 10 (2023) 025027 A Rodek et almonolayer spectroscopy, as it enhances the visibilityof phenomena previously blurred by different typesof inhomogeneity, with the appearance of a noveldestructive photon echo being an example [21].We are employing a four-wave mixing (FWM)spectroscopy approach to study the coherent opticalproperties of TMD exciton complexes. This methodis recently gaining more popularity for the investig-ation of layered semiconductors: Besides the funda-mental studies of the exciton coherence and popula-tion dynamics in different TMD systems [19, 22, 23]it was also used to probe intervalley scattering pro-cesses [24], valley coherence times [25], and mech-anisms of exciton broadening [26]. In particular itallowed for first observations of biexcitons [27, 28] orcoupling between different exciton states [28] in vari-ous TMDmaterials.Electron beam lithography and metal depositioncan further be employed to complement van derWaals heterostructures with electronic contacts andgates [29, 30]. In this way, one can inject carriersinto TMD monolayers, permitting us to control thefree electron density ne in our sample. This is cru-cial, not only for optoelectronic applications, but alsowhen exploring the fundamental physics of correlatedmany-body systems in solids; landmark examplesbeing the recent revelation of Wigner crystals in agate-tunable MoSe2 monolayer [31] or optical sens-ing of the quantum Hall effect in graphene [32].The presence of free carriers modifies theexcitonic optical response substantially [33, 34],similarly as was shown and partially explained forsemiconductor quantum wells [35–38]. Firstly, it issupposed that free carriers affect the relative absorp-tion (oscillator or effective dipole strength) and thepossible couplings between neutral excitons (X) and(negatively) charged excitons (X−), also called trions.Secondly, the exciton’s dephasing and therefore thespectral line shape should be sensitive to the electrondensity ne [39], in an analogous way as it dependson the total exciton density, through the mechan-isms called excitation induced dephasing [20, 40, 41],sometimes described by a local field effect [42, 43].Thirdly, with increasing ne the exciton complexesare screened more efficiently from intrinsic disorder,which in turn should reduce their inhomogeneousbroadening [3, 44], further impacting the excitons’radiative decay rates [45]. Here, we address the basicinterplay between free carriers ne, excitons X, andtrions X− by performing coherent nonlinear spec-troscopy [46, 47] of a gated and hBN-encapsulatedMoSe2 monolayer.2. Sample and experimentA microscopy image of our device is presentedin figure 1(a), where the MoSe2 monolayer flake(marked in red) is encapsulated between thin bot-tom (dashed white) and top (solid white) hBN films.The bottom graphite flake (dashed black) lies underthe hBN spacer, while the top few-layer grapheneflakes (solid black) are adjacent to the MoSe2 asshown in the layer schematic in the top right inset.The graphite and graphene operate as bottom andtop gates, respectively. Details regarding the fab-rication of our device are provided in the sup-porting information (SI). We have fabricated andinvestigated two samples of the same design, whichshow very similar results throughout the studiedproperties.The device permits us to control the free elec-tron density ne by applying a gate bias U via the goldbands, as shown in yellow in figure 1(a), connect-ing with both graphite and graphene layers acting aselectrodes. When increasing the bias and thereforene we observe modifications of the optical responseof the monolayer, both in the transition energy andthe spectral line shapes, as shown in the white-lightreflection contrast spectra presented in figure 1(b).We note that by varying ne we can alter the relat-ive spectral intensities of X and X−, as quantified in(c). We see that already the linear optical response, asobserved in reflectance, is tuned when increasing ne,as the oscillator strength shifts fromX to X−—wewillcome back to this point in the last section.To infer the coherent nonlinear response ofexcitons, we perform FWM microscopy, similarly asin our recent works [22]. We use three, co-linearlypolarized pulses of 150 femtosecond duration, herelabeled as E1,2,3, that can be tuned into resonancewith the excitonic transitions. The beams are focuseddown to a diffraction-limited spot on the sample’ssurface, placed in a helium flow cryostat, settingthe temperature at T= 8K for all experiments. Thecomplex-valued FWM signal, generated in a stand-ard, so-called photon echo, configuration is retrievedin reflectance by combining optical heterodyning andspectral interferometry with a reference pulse [48].The reference beam is focused through the sameobjective (NA= 0.65) below E1,2,3 and is reflectedfrom the neighboring hBN without MoSe2, as shownin figure 1(a) by the white dots.To investigate the exciton’s coherence dynamicswemeasure the FWMamplitude as a function of timedelay τ 12 between the first and the second arrivingpulses, while fixing τ 23. Conversely, the exciton’s pop-ulation dynamics is probed in FWM when varyingthe delay τ 23, while fixing τ 12. These pulse sequencesare depicted in figures 2(a) and 3(a), respectively. Wenote that for all experiments we keep the same excit-ation conditions of 0.3µW average power for eachof the beams, generating an exciton density of a few1010 cm−2 [20, 22]. To confirm the expected linearscalings of the FWMamplitudewith the pulse areas ofE1,2,3 for small powers, we checked the FWM powerdependence (see SI figure S1), also monitoring thatthe excitation induced dephasing can be neglected forthis range of exciton densities [20].22D Mater. 10 (2023) 025027 A Rodek et alFigure 1. Investigated MoSe2 device and optical characterization. (a) Optical image of the heterostructure. The individualcomponents are marked by colored lines and the white dots mark the positions of the laser beams for the FWMmeasurementsand the inset shows the layer structure of the sample. (b) Reflectance contrast spectrum of the sample as a function of the appliedvoltage showing the neutral (X) and negatively-charged (X−) exciton transitions. (c) Individual peak intensities X in blue and X−in red from (b) and their ratio in green. (d) Imaging of the time-integrated FWM at τ12 = 1.2 ps. The black area in the lower-leftcorner of the image is an experimental artefact generated when the reference beam impinges on the highly reflecting metalliccontact∼10 µm below the edge of the MoSe2 monolayer. The outline of the MoSe2 flake is marked in red and the line where thereference beam Eref crosses the bottom edge of the top hBN flake in white.To characterize the overall device, we set U=−0.5V and perform a FWM amplitude mappingacross the MoSe2 flake for τ12 = 1.2ps, by scanningthe objective’s position. The resulting time-integratedFWM image is presented in figure 1(d) (for detailedanalysis of this imaging see SI figures S2–S4). Thedominant signals reflect the third-order optical sus-ceptibility of the MoSe2 monolayer (red line). Note,that the generated FWM isweaker in the upper part ofthe flake, which is due to the reduced reflectance of thereference pulse Eref when impinging on the top hBN.This crossover between low andhigh FWMamplituderegions (marked in white) faithfully follows the shapeof the top hBN’s edge when Eref leaves the top hBN,while the other pulses E1,2,3 still probe the fully encap-sulated MoSe2 monolayer.In time-resolved FWM, a typical case presen-ted in figure 2(b), without introducing free carri-ers we observe a more or less pronounced photonecho [22, 23, 26], which is a consequence of inhomo-geneous broadening ℏσ. The latter induces rephasingof the signal for t= τ12, such that normally the echois aligned with the diagonal in the (t, τ12)-space, asdepicted as dashed diagonal in (b). However, for shortdelays τ12 < 0.5ps we spot deviations from the typ-ical echo. In this range the signal is retarded in time,such that the transient is not following the diagonal,and is instead curved, into a comet-like shape. Theecho distortion is readily reproduced, when takinginto account the local field effect [21], representing aneffective exciton–exciton interaction, and is describedin leading order in the local field coupling by|pFWM(t, τ12)| ∼Θ(τ12)Θ(t)te−β(t+τ12)− 12σ2(t−τ12)2+Θ(−τ12)Θ(t+ τ12)(t+ τ12)× e−β(t−τ12)− 12σ2(t−τ12)2. (1)The fitted (t, τ12)-dynamics is presented infigure 2(c), from which we extract homogen-eous β and inhomogeneous σ dephasing ratesand corresponding full-width at half-maxima32D Mater. 10 (2023) 025027 A Rodek et al(FWHM) spectral broadenings ℏβ = 0.86meVand 2√2 ln(2)ℏσ = ℏσ̃ = 3.9meV, respectively. Byinspecting spectrally-resolved FWM mappings, wereveal typical fluctuations of the exciton’s centralenergy (≈±5meV) and line widths (≈±1meV), asshown in figure S2 in the SI, attributed to varyingstrain generated during the sample assembly [49–51].We therefore fix the excitation spot for the entireexperiment, as marked in figure 1(a), to probean area of relatively small σ within the distri-bution. The same mapping is performed for thetime-resolved FWM signal, which allows to extracthomogeneous and inhomogeneous broadenings,summarized in SI figure S3. We find an expec-ted and pronounced correlation between σ andthe FWHM of the FWM spectra, as shown in SIfigure S4. There, we also find a correlation betweenthe exciton transition energy and the inhomogeneousbroadening.3. Coherence dynamics of neutral andcharged excitonsWe now proceed to the investigation of the excitonhomogeneous linewidth ℏβ depending on theinduced electron density ne. For this purpose, wecenter the excitation at the exciton energy, which forthe investigated area equals 1640 meV, and meas-ure the photon echo as a function of the gate biasU. Exemplary exciton coherence dynamics, i.e. time-integrated FWM amplitudes as a function of τ 12, areshown in figure 2(d) (blue dots). Due to the localfield, the measured dynamics deviate from simplecombined exponential and Gaussian decays accord-ing to equation (1). The corresponding fits are shownas pale blue lines and reproduce the experimentswell. The data clearly indicate a growth of the deph-asing rate when increasing the electron density (topto bottom). All fitted homogeneous broadenings arepresented in (f, blue filled circles) as function of thegate bias, which quantifies the impression of a strongrise of ℏβ with increasing ne. The fitted dynamics alsoyieldℏσ̃ as a function of ne (f, blue open circles). Note,that for the fits we used the entire signal dynamics inreal and delay time from which we also extract thetime-integrated data in figures 2(d) and (e). Interest-ingly, the inhomogeneous broadening is suppressedefficiently, when injecting more electrons. The FWMtransient thus evolves from the photon echo to free-induction decay like, as we explicitly show in SIfigure S5. This strikingly shows that the excitonsbecome less sensitive to the underlying disorder,which can be static due to the strain or dynamic viafluctuating charges. With increasing ne, the disorderthus gets screened more efficiently and the appliedvoltage can neutralize fluctuating charges [44], bothmechanisms making the system less inhomogeneous.We have previously found that the reduced inhomo-geneity goes hand-in-hand with a shortening of theexcitons’ radiative lifetime [19]. The small rise ofthe X’s inhomogeneous broadening at U > 4 V is anartefact from the weak optical response which res-ults in an increased uncertainty of the fit. This is alsoreflected by the significant increase of the error bars(shaded area).The second resonance, occurring at 1620 meV isthe charged exciton transition X−, which we select-ively address by centering the laser pulse spectrum tothis energy. We carry out the same routine as beforeto determine the charged exciton’s homogeneous andinhomogeneous broadening and their dependence onne. The results are presented in figures 2(e) and (f)in red colors. When comparing X’s and X−’s homo-geneous broadenings in (f), we observe that for thesame electron density the X broadening is signific-antly larger than that of X−. This finding is similar toexperiments performed on non-intentionally dopedsamples [25]. Following an intermediate drop withincreasing ne, we eventually also observe an increaseof the charged exciton’s homogeneous broadeningaccompanied by a decrease in inhomogeneous broad-ening (open red circles). The increased line width ofboth exciton complexes is attributed to the dephas-ing due to interactions with the Fermi-sea of elec-trons, similarly as in past studies on semiconductorquantum wells [39, 40].With this, we evidence the control the inhomo-geneous broadening and excitonic dephasings andthus the spectral line shape of the optical trans-itions, via a tunable gate bias introducing freecarriers into the MoSe2 monolayer. We distin-guish this line broadening mechanism from theones previously investigated in TMDs, i.e. phonon-induced [22, 23, 26, 49, 52] and excitation-induceddephasing [19, 20].At this point, we remark that the measured beha-vior of dephasings does not match with a simplepicture of the dipole (oscillator) strength transferfrom X to X− when increasing ne. Assuming thatthe dephasing was entirely governed by the radiativelifetime, a reduction for the X (an increase for X−)dipole strength should increase (reduce) the lifetimeand consequently the dephasing time. However, theopposite trend is observed in figure 2(f). Next to thesuppression of σ via screening, the second possiblecause for this behavior is the presence of other decaychannels that increasewith a growingne, whichwouldresult in an accelerated dephasing while remainingclose to a lifetime-limited condition. To shed light onthe dominating decay mechanism, we thus measureFWM as a function of τ 23, monitoring the coherentpopulation dynamics of X and X−.42D Mater. 10 (2023) 025027 A Rodek et alFigure 2. X and X− dephasing versus applied gate bias. (a) Scheme of the three-pulse FWM sequence probing coherencedynamics via τ 12. (b) Measured time-resolved FWM amplitude showing a clear photon echo. (c) Theoretical fit of (b) employingthe local field model with the fitted homogeneous and inhomogeneous dephasing rates β and σ as given in the plot. (d) and(e) Exemplary time-integrated FWM amplitude dynamics as a function of τ 12 for different gates biases. (d) For X and (e) for X−.(f) Homogeneous (filled circles) and inhomogeneous (open circles) FWHM line widths extracted from (d), (e) and time-resolveddata as a function of gates bias with X in blue and X− in red. The shaded areas show the uncertainties.4. Population dynamics of neutral andcharged excitonsIn figure 3(b) we present the time-integrated FWMsignals of X as a function of τ 23 for selected gate biases(amplitude as red and phase as blue dots). As theFWM response is measured in a coherent fashion,we can retrieve its amplitude and phase, improvingthe insight into the involved decay processes affectingthe excitonic populations that occur. A natural choiceto describe this dependence is to consider a coherentsuperposition of several exponential decays [19, 53]viaSFWM(τ23, t) = Aoff exp(iφoff)+ exp(iφdrt)×{Anr exp(iφnr −τ 223τ 20)+∑nAn[1+ erf(τ23τ0− τ02τn)]× exp(iφn +τ 202τ 2n− τ23τn)}(2)where t is the measurement time of the FWM signal,(Anr,φnr) are amplitude and phase of the two-photonabsorption, (An, τn,φn) are the amplitude, character-istic time, and phase of a given decay processes, φdr isthe phase drift during the measurement, (Aoff,φoff)are the amplitude and phase of the complex offset,and τ 0 is the pulse duration of the femto-second laserof around 150 fs.As exemplarily represented by the fitted palecurves in figure 3(b, top), we find that for gatebias values U⩽ 0, which corresponds to the charge-neutrality regime, i.e. no free electrons, the exciton’spopulation dynamics can be fitted with a bi-exponential decay with characteristic timescalesτ1 < 1ps and τ2 ≈ 7ps. We plot the extracted times-cales of the relaxation processes in figure 3(c). Thisresult is in agreement with previous studies [22, 53],with the faster component being attributed to theexciton’s decay, which can be due to several mechan-isms: radiative recombination, non-radiative scatter-ing into variousmomentum-dark states and localizedstates generated by the disorder, and finally X to X−conversion. The longer decay stems from the relax-ation of thermalized higher k-vector exciton states,which scatter back into the light-cone (i.e. stateswith momentum values |k|< nω/c) through non-radiative processes and subsequently contribute tothe FWM signal.52D Mater. 10 (2023) 025027 A Rodek et alFigure 3. X and X− population dynamics versus gate bias. (a) Scheme of the three-pulse FWM probing the population dynamics.(b) Exemplary FWM amplitude (red dots) and phase (blue dots) dynamics of X as a function of τ 23, with fitted curves in palecolors, according to equation (1). (c) Extracted characteristic decay times of the three identified relaxation channels as a functionof gate bias. The gray curve shows the blue data from figure 2(f) in the form TX2 = 1/β. (d) and (e) same as (b) and (c) but for X−.With increasing gate bias U and thus ne, thedescription of the observed population dynamicsrequires the introduction of a third, slower relaxa-tion process τ 3. Earlier FWMexperiments on samplesnaturally doped with electrons also indicate the pres-ence of this long-lived component and attribute it to adecaying population of excitons with spin-forbiddentransitions [53]. As the disorder is screenedmore effi-ciently with increasing ne, we tentatively suggest thatthe observed faster dynamics could be also due to aweaker exciton localization.The most pronounced change in the popula-tion dynamics, caused by the increase of ne, canbe observed during the first few picoseconds infigure 3(a). For positive U in figure 3(c), we observea rapid increase of the fastest relaxation rate. Thisbehavior, fortified by the shift of the FWM phase,occurs in the same voltage range as the previouslymentioned shortening of the exciton dephasing time(see blue dots in figure 2(f)). Interestingly, when com-paring the retrieved τ 1 values with TX2 /2= 1/(2β)(gray line in figure 3(c, bottom)), we see the sametrend with increasing bias. In particular, we findthat for large gate biases the measurements approachτ1 = TX2 /2 for U≈ 3V and therefore the special situ-ation of lifetime-limited dephasing. This result is alsoin line with the previous finding that the exciton losesnearly all inhomogeneous broadening for U> 3Vand therefore the dominant dephasing mechanismfor smaller U.In figures 3(d) and (e) we present analogous datafrom the measurements where the excitation energyis tuned to the charged exciton transition X−. In thiscase, we only observe two distinct relaxation pro-cesses [53, 54] contributing to the investigated rangeof τ 23. X− is the lowest energy state and—in con-trast to X—cannot relax to other excitonic states.Consequently, the initial decay of X− of around 6 ps,reflects its radiative and non-radiative recombina-tion. We note here also, that the simple extensionof the light-cone escape process as for the neutralexcitons is now not valid for X−. The free carrierleftover after the recombination allows for the addi-tional momentum transfer and recombination of X−from non-zero k-vectors [55, 56].5. Controlled coherent coupling betweenneutral and charged excitonsSo far, we have investigated the coherence dynam-ics of X and X− separately, by selectively addressingthe respective resonances. Conversely, when the twocomplexes are excited simultaneously, both excitonspecies coexist and interact with each other. To trigger62D Mater. 10 (2023) 025027 A Rodek et alFigure 4. Quantum beat of the X–X− Raman coherence revealed in the population dynamics. (a) Amplitude of the FWM spectraof X and X− in blue together with the amplitude of reference spectrum in red. (b) Spectral dynamics of the FWM amplitude as afunction of τ 23 showing a clear beating of X−. (c) Theoretical simulation of (b). (d) Dynamics of the X− single after spectralintegration over the respective peak. Experiment as green dots and simulation as pale line.Figure 5. Phase-referenced two-dimensional FWM spectroscopy of X and X−. (a) The pulse sequence used in the 2D FWMexperiment. τ 12 is scanned, while the other delay is set to τ23 = 0.13ps maximizing the coherent coupling. (b) Examples of 2DFWM spectra for different gate biases as given in the plots. (c) Integrated peak amplitudes of the four peak X, X, XX−, and X−X(marked in (b)) as function of gate bias. (d) Characteristic peak rations extracted from (c) as dots and from figure 1(c) as grayline. The shaded areas in (c) and (d) mark the uncertainty ranges.the interplay between X and X− and reveal theircoupling, we excite them in tandem, as shown spec-trally in figure 4(a) by the laser spectrum in red andthe FWM spectrum in blue. The measured spectrallyresolved population dynamics are presented in (b).Interestingly, for the initial delays τ23 < 0.5ps, theFWM displays a beating particularly pronounced onthe X− resonances. The first driving beam inducesthe Raman coherence between X and X− as depic-ted schematically in (b). Due to the spectral splittingδ between X and X−, the density grating generated bythe second beam oscillates with the period 2πℏ/δ ≈0.14ps. The FWM released by the probe thereforeshows the Raman quantum beat [57], as confirmedby the simulation in (c), and directly shown in (d)as time traces after spectrally integrating over the X−peak. This result indicates that X and X− are Raman-coupled, as they share a common ground state. Note,that the simulation (pale green line) only considers asingle exciton decay channel, while in the experiment(green dots) the interplay between different relaxa-tion paths leads to the rising signal for τ23 > 0.5ps(see figure 3).A beating of a similar origin is observed in thecoherence dynamics, probed by the τ 12-dependence,as shown in SI figure S6. When coherently coupled,the first-order absorptions of X and X−, created bythe leading pulse, are mutually converted into theFWM of X− and X, respectively, by the following twopulses, as schematically shown in figure 5(a). To pin-point this phenomenonwe perform two-dimensional(2D) FWM spectroscopy [58, 59]. In this approach,72D Mater. 10 (2023) 025027 A Rodek et aloriginating from nuclear magnetic resonance spec-troscopy, Fourier transformations are performedalong two time axes: the direct time axis and theindirect delay axis [60–62]. In our case, the trans-form along the direct axis is automatically performedby the optical spectrometer, yielding the FWM spec-tra. Conversely, the Fourier transform of the indir-ect axis has to be recovered from the τ 12-sequence.To track the FWM phase when varying τ 12, we applythe phase-referencing scheme [63], which overcomesthe need for an active phase-stabilization and permitsus to accurately perform the Fourier transform alongτ 12, yielding the energy axis ℏω12.A conclusive display for the coherent couplingbetween X and X− is presented in figure 5(b), show-ing 2D FWM amplitude spectra for selected gatevoltages as labeled in the pots. The delay betweenpulses 2 and 3 was chosen to τ23 = 0.13ps, whichlocates us in the coherent coupling range as demon-strated in the SI figure S7. We have checked thatfor τ23 > 1ps the X and X− coupling is present(see figure S8), although dominated by an inco-herent population transfer between the two com-plexes [26]. Conceptually, the 2D FWM spectra ofour V-shaped system consist of only four peaks. Forthe diagonal pair, labeled X− and X, detection energyand ℏω12 energy are identical: FWM emission arisesfrom the same absorption. Decisively, we clearlydetect off-diagonals, labeled XX− and X−X. Thismeans that the FWM of the charged exciton is alsodriven by the neutral exciton’s first-order absorption,and vice-versa, respectively. Such coherent couplingwas previously reported in semiconductor nanostruc-tures [48, 59, 64], including TMDs [28, 59, 65]. Here,thanks to the tunability of ne in this gated MoSe2,we obviously find that the coherent nonlinear opticalresponse of the X –X− complex can be controlledsimply by applying an external bias.The variation of the peak amplitudes in the 2Dspectrawith changing gate bias has twopotential reas-ons: (i) change of the dipole strengths of X and X−due to additional free charges, as already demon-strated in the reflectivity measurement in figure 1(b),(ii) the coherence transfer between both exciton spe-cies is affected by the free carriers. To disentanglethese two effects we need to quantify the strengthof the coherent coupling depending on ne. In panel(c) we plot the peaks’ integrated amplitudes versusthe gate bias, where we expectedly find that X (blue)clearly drops, while X− (red) builds up when increas-ing the carrier density. At the same time the off-diagonal peaks (green and yellow)—representing thecoupling—showno clear trend. Asmentioned before,the variation of the reflectivity spectrum upon anapplied gate bias indicates a change of the dipolestrengths of X and X− and therefore of the pulse areasθX and θX− applied to the two transitions. Note, thatwe do not consider any specific origin for the changeof the strength of optical response of X and X−. Now,in order to extract the strength of the X –X− couplingfound in the 2D spectra, we can extract a quantity thatdoes not depend on the pulse areas. To achieve this,we note that the amplitudes of diagonal peaks scale asθ3X and θ3X−, whereas the off-diagonals follow the scal-ing θ2XθX− for XX− and θXθ2X− for X−X. Plotting nowthe expression, XX−·X−X/(X·X−) in figure 5(d, top)the pulse areas and therefore the dipole strengths can-cel out. We see though, that this quantity increaseswith the electron density, indicating the increase ofthe coupling strength.Another insightful quantity that can be extractedfrom the reflectivity spectra and the 2D spectra is thesum of peak ratios. For the 2D spectra we calculateXX−/X+X−X/X−, which translates into pulse areasas θX−/θX + θX/θX− . To extract an equivalent quant-ity from the reflectivity spectra we need to calculate√RX−/√RX +√RX/√RX− from the peak intensit-ies RX and RX− . The two quantities are plotted (nor-malized to unity) in figure 5(d) as turquoise dotsfor 2D and gray line for reflectivity. The behavior ofboth curves is rather unspecific for negative and smallgate voltages, where X− is only weakly addressed ifpresent at all. Note, that in this range of bias val-ues the X− is very weak which is leading to the rel-atively large uncertainties (shaded areas). Therefore,the peak ratios in this range are not particularly sig-nificant and we do not expect to draw conclusionsfrom this bias range. However, the 2D spectra resultshows a clearly growing trend for increasing chargedensities, while the reflectivity results shrink. This dis-crepancy is a second hint that the variation of thedipole strengths with increasing gate bias alone, can-not fully explain the variation of the coherent coup-ling between X and X− observed in 2D FWM.6. Conclusions and outlookFWM spectroscopy methods are powerful tools thathave led in the recent years to a significant improve-ment of the understanding of the rich exciton physicsin TMDs. This has established the method as a ver-satile technique for studies of atomically thin mater-ials and TMD-based heterostructures as detailed inthe Introduction. However, so far a particular focuson the important issue of the free carrier influenceon the studied phenomena has beenmissing. A factorthat fundamentally affects the optical response ofTMDs, and governs even the most basic characterist-ics like the number and respective intensities of dif-ferent exciton states.Using FWM micro-spectroscopy, we have shownthat the homogeneous linewidth and population life-time of excitonic complexes hosted by a MoSe2-monolayer-based van der Waals heterostructure canbe tuned via the free electron density, which isinjected by a gate bias applied to our device. Withincreasing gate bias, the exciton’s inhomogeneousbroadening decreases, reflecting the screening of the82D Mater. 10 (2023) 025027 A Rodek et aldisorder via free the electron gas, which also increasesthe excitons’ radiative decay rates. Conversely, thehomogeneous broadening increases, which is attrib-uted to the combined increase of the radiative decayrate and the conversion rate of the neutral excitontowards the charged one. By exciting the neutral (X)and charged exciton (X−) simultaneously and prob-ing the population dynamics we demonstrated thatthe coherence between X and X− leads to a charac-teristic quantum beat in the FWM signal. By thenperforming 2D FWM spectroscopy for a variety ofapplied gate biases, we have further demonstratedthat the X –X− coherent coupling can be controlledby the gate voltage, and hence by the free electrondensity. Through considering specific peak ratios, wewere able to demonstrate that the change of coher-ent coupling can be disentangled from the variationof dipole strengths arising from the injection of freecarries. An increase of the coupling strength with necould be linked with the screening of disorder via theelectron gas. This illustrates the utility and versatil-ity of ultrafast nonlinear spectroscopy in investigatingoptical responses of excitonic systems, going beyondthe capabilities of linear methods.In future developments, by performing FWMwith spatially separated driving beams, while usingdevices hosting highly diffusive excitons, it will bepossible to achieve non-local coherent coupling ina 2D semiconductor. Combining this approach withtwo-color FWM spectroscopy would also permit toselectively address the coherence transfer betweenneutral and charged excitons. Our findings yieldexciting prospects for forthcoming investigations ofcoherent phenomena in the context of recent dis-coveries of strongly-correlated exciton phases insolids [31], optically probed quantumHall states [32],moiré superlattices [66], and magnetic 2D materi-als [67]. In practice, coherent nonlinear spectroscopycould be used to optically infer the dephasing pro-cesses of the Umklapp branches of TMDWigner crys-tals and fractional quantum Hall states in graphene.Data availability statementThe data that support the findings of this study areavailable upon reasonable request from the authors.AcknowledgmentsThis work was supported by the PolishNational Science Centre under decisions DEC-2020/39/B/ST3/03251. The Warsaw team (A R, PK and M P) acknowledges support from the ATO-MOPTO project (TEAM program of the Foundationfor Polish Science, co-financed by the EU withinthe ERDFund), CNRS via IRP 2D Materials, EUGraphene Flagship. M P acknowledges support bythe Foundation for Polish Science (MAB/2018/9Grant within the IRA Program financed by EUwithinSG OP Program). The Polish participation in theEuropean Magnetic Field Laboratory (EMFL) is sup-ported by the DIR/WK/2018/07 of MEiN of Poland.A R acknowledges support of this work from theDiamentowy Grant under decision DI2017008347 ofMEiN of Poland. K W and T T acknowledge supportfrom the Elemental Strategy Initiative conducted bythe MEXT, Japan, (Grant No. JPMXP0112101001),JSPS KAKENHI (Grant No. JP20H00354), and theCREST (JPMJCR15F3), JST. J H acknowledges sup-port from EPSRC doctoral prize fellowship. DW wassupported by the Science Foundation Ireland (SFI)under Grant 18/RP/6236. 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Introduction 2. Sample and experiment 3. Coherence dynamics of neutral and charged excitons 4. Population dynamics of neutral and charged excitons 5. Controlled coherent coupling between neutral and charged excitons 6. Conclusions and outlook References