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Haonan Wang, Heejun Kim, Duanfei Dong, Keisuke Shinokita, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Kazunari Matsuda

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[Quantum coherence and interference of a single moiré exciton in nano-fabricated twisted monolayer semiconductor heterobilayers](https://mdr.nims.go.jp/datasets/fedaecfc-102f-4883-9174-9411fad063a6)

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Quantum coherence and interference of a single moiré exciton in nano-fabricated twisted monolayer semiconductor heterobilayersArticle https://doi.org/10.1038/s41467-024-48623-4Quantum coherence and interference of asingle moiré exciton in nano-fabricatedtwisted monolayer semiconductorheterobilayersHaonan Wang1, Heejun Kim1, Duanfei Dong1, Keisuke Shinokita 1,Kenji Watanabe 2, Takashi Taniguchi 3 & Kazunari Matsuda 1The moiré potential serves as a periodic quantum confinement for opticallygenerated excitons, creating spatially ordered zero-dimensional quantumsystems. However, a broad emission spectrum resulting from inhomogeneityamong moiré potentials hinders the investigation of their intrinsic properties.In this study, we demonstrated a method for the optical observation ofquantum coherence and interference of a single moiré exciton in a twistedsemiconducting heterobilayer beyond the diffraction limit of light. Weobserved a single and sharp photoluminescence peak from a single moiréexciton following nanofabrication. Our findings revealed the extended dura-tion of quantum coherence in a single moiré exciton, persisting beyond 10 ps,and an accelerated decoherence process with increasing temperature andexcitation power density. Moreover, quantum interference experimentsrevealed the coupling between moiré excitons in different moiré potentialminima. The observed quantum coherence and interference of moiré excitonwill facilitate potential applications of moiré quantum systems in quantumtechnologies.A quantum two-level system has garnered considerable attention inrecent years due to its numerous potential applications in the fields ofphysics, such as quantum simulation, quantum computing, andquantum information processing1–5. The development of these sys-tems facilitates the construction and utilization of quantum bits(qubits), which serve as fundamental units for quantum computingand quantum information6–8. Resonant light–matter interactions, suchas Rabi oscillation9,10, Ramsey interference11, andHahn echoes12, enablethe manipulation of quantum two-level systems13,14 by generatingsuperposition states. However, the superposition states of qubitssuffer from the interaction and fluctuations from the environment,resulting in an accelerated decoherence process15. This decoherenceprocess imposes a temporal limitation on the precise manipulation ofquantum systems16–18, hindering their potential applications19,20.Hence, for platforms aiming to achieve qubits, a sufficiently longcoherence time is imperative21,22. Furthermore, it is also necessary tocontrol the interaction between quantum systems, since such inter-action not only modifies the quantum coherence within each indivi-dual system but also facilitates the formation of a coupled-quantumsystem. This introduces interference or entanglement23 between sys-tems, which is essential for the development of large-scale quantumdevices24–26.Recent progress in artificial van der Waals (vdW) structures,achieved by stacking atomically thin two-dimensional (2D) materials,Received: 7 October 2023Accepted: 2 May 2024Check for updates1Institute of Advanced Energy, Kyoto University, Uji, Kyoto 611-0011, Japan. 2Research Center for Electronic and Optical Materials, National Institute forMaterials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan. 3Research Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan. e-mail: matsuda@iae.kyoto-u.ac.jpNature Communications |         (2024) 15:4905 11234567890():,;1234567890():,;http://orcid.org/0000-0002-7752-3251http://orcid.org/0000-0002-7752-3251http://orcid.org/0000-0002-7752-3251http://orcid.org/0000-0002-7752-3251http://orcid.org/0000-0002-7752-3251http://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/0000-0002-3990-8484http://orcid.org/0000-0002-3990-8484http://orcid.org/0000-0002-3990-8484http://orcid.org/0000-0002-3990-8484http://orcid.org/0000-0002-3990-8484http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48623-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48623-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48623-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-48623-4&domain=pdfmailto:matsuda@iae.kyoto-u.ac.jphas opened up opportunities for designing novel quantumplatforms27–30. vdW heterobilayer assembled from monolayers ofsemiconducting transition metal dichalcogenides exhibited variousintriguing physical phenomena, including strongly correlated insu-lator phases31,32, superconductivity33, and novel ferromagnetism34.Moiré superlattices with varying atomic registries in vdW hetero-bilayers can be constructed using monolayer semiconducting transi-tionmetal dichalcogenideswith a small latticemismatchor twist angle.The resulting moiré superlattice leads to the formation of periodic,ordered potential traps, confining and spatially organizing opticallygenerated bound electron–hole pairs (excitons) into periodic arrays ofquantum two-level systems35–38. The trapped excitons in moirépotentials are expected to exhibit long quantum coherence due totheir limited degree of freedom as 0D quantum systems39,40; further-more, coupling interactions can be formed between spatially sepa-rated moiré potentials, leading to quantum interference of emittedphotons41. These results establish moiré exciton quantum systems asnot only a promising platform for achieving extended coherence butalso an effective tool for exploring interactions within or betweenquantum systems. However, experimentally, important informationon the quantum coherence and interference of moiré excitons in vdWheterobilayers remains unexplored. This is due to the overlapping ofmultiple emission peaks from moiré excitons in their inhomogen-eously broadened spectra, hindering these intrinsic insights within thediffraction limit of light. The experimental approaches of strain-induced exciton traps introduced by metallic nanopillars in hetero-bilayers have been previously reported, to observe spectra with dis-crete emission lines42,43; however, the intrinsic properties of the moiréexciton systemmight be inadvertently concealed due to the additionalstrain-induced effects.In this study, we provide a approach of quantum optics in semi-conducting twisted heterobilayer beyond the diffraction limit of light.A single and sharp optical signal from a single exciton trapped in themoiré superlattice was successfully demonstrated in MoSe2/WSe2heterobilayer, enabling the direct measurement of the first-ordercorrelation function of moiré exciton emission. The quantum coher-ence of a single moiré exciton within a potential was maintained forover 12 ps at a low temperature of 4 K, considerably longer than thecoherence of a 2D exciton in a monolayer semiconductor44–46. Quan-tum beats were experimentally observed with a period of approxi-mately 230 fs, indicating the coupling betweenmoiré excitons trappedin different potentials. Furthermore, this study discussed themechanism of quantum decoherence of a single moiré exciton statewith increasing temperature and excitation power mediated by themoiré exciton–phonon and exciton–exciton interaction.Results and discussionFigure 1a and b illustrate schematics of the concept using a nanos-tructure fabrication process, and experimental setup employed in thisstudy. Nanoscale fabrication using reactive-ion etching (RIE) enablesemissions from a single moiré exciton and the observation of itsquantum coherence in the MoSe2/WSe2 heterobilayer. During usualoptical measurements, obtaining clear spectrum frommoiré excitonicstates is difficult due to the inhomogeneity of the moiré potential,leading to ensemble-averaged and broadened emissions comprisingmultiple peaks. This issue arises because the focused laser light with aspot diameter of approximately 1.5μmis determinedby the diffractionlimit of light, which excites a large number of moiré potentials due tothemuch smaller spatial period ofmoiré potentials. To address this, inthe microfabrication process, we applied a nanoscale fabricationtechnique to reduce the optical excitation and detection area of theMoSe2/WSe2 heterobilayer with a nano-pillar structure. The nanofab-ricated heterobilayer, with a pillar size smaller than the wavelength oflight, will result in a reduced number of spectral peaks beyond thediffraction limit of light. Thus, this approach allows for the anticipatedobservation of emission from a single moiré potential, enabling therevelation of the quantum coherence of moiré excitons.The details of the RIE procedure for preparing the nanostructureare described in theMethods section47. Figure 1a displays the designedpattern for nanostructure fabrication. Black circular areas with a dia-meter of 2 μm, larger than the focused laser spot size, are exposed toreactive ions and subsequently etched. The inner white areas areshielded during the RIE process via electron beam resist, and theresultant pillar sizes (D) corresponding to the optically active size ofthe heterobilayer are designed to be 50, 100, 150, 200, and 500nm.Figure 1bpresents a typical scanning electronmicroscopy (SEM) imageof the fabricated nanostructure with an actual pillar size of 240nm inthe heterobilayer. Dotted circle at the center corresponds to the pillarregion in Fig. 1a, while the area between the outer and inner dottedcircles denotes the etched region. The SEM images of various pillarsizes are presented in Supplementary Fig. 3.Figure 1c displays the photoluminescence (PL) spectra of thenanostructure-fabricated MoSe2/WSe2 heterobilayer, measured at 4 Kwith varying pillar sizes. In the heterobilayer with a pillar size (D) of500 nm, the PL spectrum reveals an inhomogeneously broadenedensemble average of multiple peaks from large number of moiréexcitons. This finding agrees with previously reported results48. Thebroadened spectrum, characterized by a Gaussian distribution, arisesfrom the inhomogeneity of moiré potentials in the heterobilayer. Asthe pillar size decreases, the number of peaks in the spectra sig-nificantly decreases. Consequently, the PL spectrum of the hetero-bilayer with a 50nm pillar size reveals a singular emission peak from amoiré exciton, attributed to the reduced number of moiré potentialswithin the optical excitation and detection area determined by thepillar size. An additional series of PL spectra for different pillars in theheterobilayer are shown in Supplementary Fig. 2a, and the results aresimilar to those in Fig. 1c.Figure 1d shows the integrated PL intensity frommoiré exciton asa function of the actual pillar size. As the pillar size decreases, theintegrated intensity is reduced rapidly, particularly smaller sizes. Thisreduction is attributed to the Gaussian distribution of the excitationlaser profile, which results in the varying power densities across dif-ferent pillar sizes, as presented in Supplementary Fig. 2b. The inte-grated intensities are calibrated against the average laser intensity. Theresults demonstrate a lineardecrease in intensitywith pillar size, whichstrongly supports the reduction of the number of optically excited anddetectedmoiré potentials within the nanostructure-fabricatedMoSe2/WSe2 heterobilayer (Supplementary Fig. 2c).Power dependence of PL spectraFigure 2a presents a contour plot showing the excitation power-dependence of the normalized PL spectra in the range of0.8–3124W/cm2 in the MoSe2/WSe2 heterobilayer with D = 50 nm at4 K. The power densities are calibrated based on the pillar size andlaser spot size, as previously described. The spectral shape of the PLspectra varies with the excitation power density. Figure 2b displays thePL spectra of a moiré exciton normalized to the excitation powerdensity. At a low excitation power density of approximately 0.8W/cm2,a sharp peak from a moiré exciton is observed at 1.380 eV with alinewidthof600μeV, as definedby the spectral resolution in this setupcondition. As the excitation power density increases, the normalizedPL peak of a moiré exciton at 1.380 eV gradually decreases, and addi-tional spectral peaks emerge.Figure 2c illustrates the excitation power dependence of the PLspectra of a moiré exciton on an expanded energy scale to clearly seechanges in the spectra. At low excitation power densities, specificallybelow 7.8W/cm2, a single PL peak is observed at 1.380 eV because thenanostructure fabrication limits the observed number of moirépotentials. As the excitation power density increases, the primary PLpeak at 1.380 eV begins to broaden and exhibits saturation behaviorArticle https://doi.org/10.1038/s41467-024-48623-4Nature Communications |         (2024) 15:4905 2around 15.6W/cm2, which will be discussed later. Moreover, above15.6W/cm2, an additional PL peak appears at the higher energy side ofprimary PL peak at 1.382 eV.In order to assign the origin of spectral peaks, the PL intensity ofthe peak as a function of excitation power density is plotted in Fig. 2e.The PL intensity at 1.380 eV increases linearly and then graduallysaturates at approximately 15.6W/cm2, consistent with the spectralbehaviors observed in Fig. 2d. The saturation power density corre-sponds to a generated exciton density of 9 × 1011 cm−2, which is almostconsistentwith the estimatedmoiré potential density of 3 × 1012 cm−2 inthe MoSe2/WSe2 heterobilayer with a twist angle of 56.5° ± 0.3, asindicated by the SHG results shown in Supplementary Fig. 1b. More-over, the circularly and linearly polarized PL spectra are measured tofurther support the moiré exciton emission, as shown in Supplemen-tary Fig. 8 and Supplementary Fig. 9. The polar pattern of PL intensitiesdetected through linear polarizers as a function of the detection angleindicates that the emission state maintains C3v symmetry, which isexpected for a moiré exciton system49. Circular polarization–resolvedspectroscopy shows stronger co-polarized PL intensities, which is alsoconsistent with the results of the H-type stacking of theheterobilayer29. The above experimental results support the findingthat the primaryPLpeak at 1.380 eVoriginates from the recombinationof moiré exciton. The PL peak at 1.382 eV emerges after the saturationof moiré exciton emission above 15.6W/cm2, and the nonlinearincrease of PL intensity shows a square dependence on the excitationpower density in Fig. 2e, which suggests that the PL peak is attributedto the recombination ofmoiré biexciton confinedwithin the potential.Moreover, the blueshifted emission of moiré biexciton than that ofmoiré exciton due to dipolar repulsion between excitons and theenergy difference of 2meV corresponding to the binding energy ofmoiré biexciton are consistent with previously reported results50.Figure 2d displays the spectral linewidth of moiré exciton peak asa function of the excitation power density, derived from the Voigtfunction fitting procedure. The spectral linewidth exhibits a narrowvalue of 600μeV at lower excitation conditions below 10W/cm2,where the PL intensity increases linearly. Above a saturation powerdensity of 15.6W/cm2, the spectral linewidth of the moiré excitonpeak becomes increasing dependent on the excitation power density,Relative IntensityDelay time(d)(a) (b)500nmD = 240 nm10 μm 10 μm(c)0 1 2 3 405101520D2 (105 nm2)Integreted Intensity (a. u.)051015202530Calibrated Intensity (a. u.) 1.32 1.35 1.38 1.41 1.44012345Nor. Intensity (a. u.)Photon Energy (eV)50 nm100 nm500 nm200 nm150 nmFig. 1 | Nanoscale fabrication. a, b Schematic of the concept in this study.Nanoscale fabrication using reactive-ion etching enables to obtain emission from asinglemoiré exciton and the observation of its quantum coherencewithMichelsoninterferometer in MoSe2/WSe2 heterobilayer beyond the diffraction limit of light.Designed pattern for RIE (left) and the optical image of MoSe2/WSe2 heterobilayerwith anarrayofnanofabricated structures (right) arepresented in (a). The designedsizes of pillar used in the nanostructure fabrications are 50, 100, 150, 200, and500 nm. A SEM image of typical pillar is presented in (b). The inner dotted circleshows a pillar with a diameter of 240nm corresponding to the optical excitationand observation area of moiré potential. Optical spectra from the pillars areFourier-transformed to temporal interferograms using the Michelson inter-ferometer. c PL spectra of MoSe2/WSe2 heterobilayers with various pillar sizes at4 K.d Integrated PL intensity and calibrated intensity of heterobilayer as a functionof pillar size. Solid line in the image represents the guide line, wherein calibratedintensity is determined as integrated intensity/average laser power density. Thecalibrated intensity shows linearly dependence on D2, indicating the linearlydependence of peak numbers on the pillar size.Article https://doi.org/10.1038/s41467-024-48623-4Nature Communications |         (2024) 15:4905 3which suggest the influences of exciton density on the moiré excitoncoherence.PL spectral wandering and first-order correlation functionTo quantify intrinsic spectral broadening, we measured the time-dependent PL spectra in the MoSe2/WSe2 heterobilayer. Figure 3apresents the time evolution of the moiré exciton PL spectra, with eachspectrumaccumulatedover 30 s. Randomfluctuations in spectral peakpositions, known as, spectral wandering or spectral jittering, areclearly observed in the moiré exciton emission, which is a character-istic of 0D quantum systems51. Figure 3b traces the energy peak posi-tions derived from the data in Fig. 3a. The frequency distribution ofeach peak position is shown as a histogram in Fig. 3c. The histogramreveals that the energy positions during spectral wandering rangefrom 1.3755 to 1.3758 eV. The spectral wandering of the PL peak is alsoobserved across different pillars of the heterobilayer (refer to Sup-plementary Fig. 10 and Supplementary Fig. 16).To obtain the information on the quantum coherence, the first-order correlation function g(1)(τ) of emission signals from a moiréexciton ismeasured using aMichelson interferometer (SupplementaryFig. 11). Figure 3d presents the contourmapof interferometry of the PLspectra as a function of delay time at 4 K for an excitation powerdensity of 14W/cm2. The amplitude of the oscillation fringe betweenthe maximum and minimum intensities gradually decreases withincreasing delay time, indicating the process of decoherence, as pre-sented by the temporal interferogram in Fig. 3e. The visibility V(τ) iscalculated as follows:V ðτÞ= Imax � IminImax + Iminð1Þwhere Imax and Imin denote the maxima and minima intensitiesobtained from an oscillation period around a certain delay time in theinterferometry. The visibilityV(τ) (blue circle) as a function of the delaytime is plotted in Fig. 3f. The visibility as a function of delay time,corresponds to the Fourier transform of the emission spectrum, withthe convolution result of extrinsic inhomogeneous and intrinsicinhomogeneous linewidth, in form of Lorentz and Gaussian functions,respectively. Consequently, the delay time-dependent visibility inFig. 3f can be modeled by the product of exponential and Gaussianfunctions, as follows52,53:V ðτÞ= e�B1τ � e�B22τ2 ð2Þwhere B1 and B2 are parameters defined in the exponential and Gaus-sian functions, respectively. The fitted result using Eq. (2) with B1(a) (b) (c)(d) (e)1.36 1.38 1.40 1.420.81.63.16.27.815.6941567803124Photon Energy (eV)Power density  (W/cm2 )0.20.60.70.80.91Nor. Intensity (a. u.)1 10 1000.51.01.52.02.5FWHM (meV)Power Density (W/cm2) 1 10 100 1000110100 1.380 eV 1.382 eV Intensity (counts/s)Power Density (W/cm2)1.36 1.38 1.40 1.42050010001500200025003000Intensity (a. u.)Photon Energy (eV)0.8 W/cm21.6 W/cm23.1 W/cm27.8 W/cm215.6 W/cm294 W/cm2156 W/cm2780 W/cm23124 W/cm26.2 W/cm2MoSe2/WSe2 heteroT=4 K1.376 1.379 1.382 1.3850246810Nor. Intensity (a. u.)Photon Energy (eV)0.8 W/cm21.6 W/cm23.1 W/cm26.2 W/cm27.8 W/cm215.6 W/cm294 W/cm2156 W/cm2780 W/cm23124 W/cm2MoSe2/WSe2 heteroT=4 KFig. 2 | Power dependence of PL spectra. a Contour plot of the normalized PLspectra of MoSe2/WSe2 heterobilayer at 4 K for various discrete excitation powerdensities ranging from 0.8W/cm2 to 3124W/cm2. b Low-temperature PL spectra ofheterobilayer with a pillar diameter of 50nm for different excitation power den-sities. c PL spectra of heterobilayer in the expanded energy scale. d Spectrallinewidth of PL peak at 1.380 eV represented by full width at halfmaximum (FWHM)as a function of excitation power densities. e PL intensities of peaks at 1.380 and1.382 eV as a function of excitation power density. Red and black lines denote thelinear and square excitation power dependence.Article https://doi.org/10.1038/s41467-024-48623-4Nature Communications |         (2024) 15:4905 41.28 1.34 1.40 1.460.00.51.01.52.0Photon Energy (eV)Time (ps)70100130160190Intensity (counts)MoSe2/WSe2T = 4 K)b()a((c)(d)(e) (f)(g) (h)(i) (j)0.0 0.5 1.0 1.5 2.00.20.40.60.81VisibilityDelay time (ps)1.3753 1.3756 1.375902468101214Time (min)Peak position (eV)1.3753 1.3756 1.37590246810Peak position (eV)Frequency 0.0 0.5 1.0 1.5 2.0050100150Intensity (a.u.)Delay time (ps) 4 K 14 W/cm21.370 1.375 1.380123456789101112131415Photon Energy (eV)Time (min)0.10.20.40.60.8Nor.Intensity (a. u.)MoSe2/WSe2T = 4 K4 6 8 10 120246Coherence time (ps)Temperature (K)0.0 0.5 1.0 1.5 2.00.40.71Under 40 W/cm2 4 K 6 K 8 K10 K12 K VisibilityDelay time (ps)0.0 0.5 1.0 1.5 2.00.40.71 Under 4 K14 W/cm240 W/cm280 W/cm2 120 W/cm2200 W/cm2VisibilityDelay time (ps)0 50 100 150 20002468Coherence time (ps)Power density (W/cm2)Fig. 3 | PL spectral wandering and first-order correlation function. a Timeevolution of PL spectrum of moiré exciton peak at low temperature, with eachspectrum accumulated for 30 s. b Time-trace of energy peak position for eachspectrum derived from the contour plot. c Frequencies of energy peak positionsrepresented as a histogram.dContour plot of the first-order correlation functionofPL signals as a function of delay time, measured using the Michelson inter-ferometer. e Interferogram of the moiré exciton peak in the time domain at 4 K inthe excitation power condition of 14W/cm2. f Decay profile of visibility in theinterferogram, with solid curve representing the fitted result of the product of anexponential and aGaussian function.gTemperature dependenceof the visibility ofthe moiré exciton peak as a function of delay time. Solid curves indicate the fittingresults of the product of an exponential and Gaussian function. h Plots of theextracted coherence time of moiré exciton (T2) as a function of temperature.i Visibility of the moiré exciton peak as a function of delay time for differentexcitation power densities. j Plot of coherence time as a function of power density.Article https://doi.org/10.1038/s41467-024-48623-4Nature Communications |         (2024) 15:4905 5( = 0.14 ps−1) and B2 ( = 0.35 ps−1) well reproduce the experimentalresults of visibility as a function of delay time. In the spectral domain,the homogeneous and inhomogeneous broadening of the spectrumare described as 2ℏB1, and 4ℏffiffiffiffiffiffiffiln2pB2, where ℏ is the Planck constantdivided by 2π.The exponential decay rate of B1, in the visibility of interferometryis inversely related to the coherence time of the quantum state asT2 = 1B1. The coherence time (T2) of a quantum state is directly deter-mined by the exciton lifetime and pure dephasing time, as describedby the following equation1T2=12T 1+1T *2ð3Þwhere T1 represents the energy relaxation lifetime and T *2 denotes thepure dephasing time. The lifetime of the interlayer moiré exciton inMoSe2/WSe2 heterobilayer was measured using time-resolved PLspectroscopy, employing a time-correlated single-photon countingmethod (Supplementary Fig. 13a) from 4 to 14 K under the excitationpower density of 56W/cm2. The PL decay profiles of moiré excitonexhibit a longer decay time of several tenths of ns, which is consistentwith previously reported results48. The decay curves are modeled byexponential functions as I (t) = A1exp(−t/τ1) + A2exp(−t/τ2) +A3exp(−t/τ3), where A1, A2, and A3 as well as τ1, τ2, and τ3 are thecoefficients of amplitude and decay times. The fitting results yield thevalues of τ1 = 14 ns, τ2 = 100 ns, and τ3 = 700ns at 4 K. As temperatureincreases, the PL decay profiles become faster, and the obtained decaytimes are summarized in Supplementary Fig. 13b. The τ2 can beascribed to the decay lifetime of moiré exciton54. Given that τ2 is muchlonger compared to the measured coherence time T2, the populationrelaxation process hardly contributes to the dephasing process. As aresult, pure dephasing time of this position is evaluated to be 7.1 ps,corresponding to a homogeneous linewidth of 184 μeV.Supplementary Fig. 15 exhibits the interferograms of moiré exci-ton emission at various temperatures from 4 to 12 K. The decay ofinterferograms is progressively faster with increasing temperature.Figure 3g presents the temperature dependence of visibility, which arealso well fitted using Eq. (2) by altering the value of B1, as shown inFig. 3h. The evaluated coherence times of moiré exciton considerablydecreases from 5.3 to 2.2 ps, corresponding to an increase of thehomogeneous linewidth from 250μeV to 600μeV. The broadening ofthe intrinsic homogeneous linewidth with increasing temperatureaffects the PL linewidth, which is consistent with the results in Sup-plementary Fig. 5. The broadening of the homogeneous linewidth withincreasing temperature can be linearly modeled using Γ(T) = γ0 + γ’T,where γ0 is the residual homogeneous linewidth at zero temperature.The coherence time T2 can be described by T2 = 2ħ/(γ0 + γ’T) due to therelationship of Γhomo = 2ħ/T2. Solid line in Fig. 3h reproduces theexperimental results at various temperatures. According to the fittingresult, the value of the residual homogeneous linewidth is 83 μeV,corresponding to a coherence time of 16 ps at the zero-temperaturelimit, and the linear coefficient of γ’ as a function of temperature is43μeV/T. The temperature-dependent linear increase in homo-geneous linewidth implies that the decoherence of moiré exciton isdetermined by the interaction of low-energy acoustic-phonon modesin the heterobilayer.Moreover, the value of the broadening coefficientγ’ dominated by the strength of exciton–acoustic phonon interactions,is smaller than that of 2D exciton (60μeV/T) in a monolayersemiconductor55. This result implies that the exciton-phonon interac-tion is suppressed by the quantum confinement of moiré potential.Supplementary Fig. 14 presents the interferograms of moiréexciton emission at 4 K for different excitation power densities. Withincreasing excitation power density, the decay of the interferogramsincreases. Figure 3i presents the power density dependence of visibi-lity, which also fits well using Eq. (2) with changing the value of B1, asshown in Fig. 3j. The evaluated coherence times of moiré excitondepending on the excitationpower density significantly decrease from7.1 to 2.0 ps, corresponding to a homogeneous linewidth increasingfrom 180 to 660 μeV. The broadening of homogeneous linewidth withincreasing excitation power density were linearly fitted byΓ(P) = β0 + β’P, where β0 is the homogeneous linewidth under weakexcitation power limit. Thus, the coherence time can be modeled byT2 = 2ħ/(β0 + β’T), represented by the black solid line in Fig. 3j, whichindicates a residual homogeneous linewidth of 160μeV and coherencetime of 8.2 ps under zero exciton density. The value of the broadeningcoefficient β’ resulting from the exciton–exciton interaction is esti-mated to be 2.6 × 10−3 meVcm2/W, significantly lower than that pre-viously reported for monolayer WSe255. The reduced coefficientsuggests thatmoiré potential confinement, substantially decreases theinteractions between excitons56.Quantum beat and coupling between two moiré excitonsWe further investigated the quantum interference of moiré excitonsfrom the first-order correlation function of the corresponding signalsin another pillar of the heterobilayer. Figure 4a shows the results of thePL spectra of moiré excitons for various excitation power densities. Atthe lowest excitation power (0.8W/cm2), two peaks are observed at1.327 and 1.309 eV, respectively, which is indicated by the solid circleand rectangle. With increasing excitation power density, each peakshows saturationbehavior independently, andnewpeaks appear at thehigher energy side. Intensities of these two peaks are plotted as afunction of the excitation power density in Fig. 4b. The intensities ofeach peak are fitted with function I = P α, represented by the solid lines.These two peaks exhibit linear increases at low excitation powerdensities and reach saturation at different power densities, indicatinginterlayer excitons trapped in moiré potentials of different depths (M1and M2). The peak at 1.309 eV is associated with a deeper moirépotential (M2) compared to the peak at 1.327 eV (M1), which is causedby inhomogeneity within the optical excitation area.Figure 4c displays a contour plot of the first-order correlationfunction of the moiré exciton PL signals measured at an excitationpower density of 3.2W/cm2. Figure 4ddemonstrates the interferogramof the peak at 1.327 eV, where the integrated peak intensity (M1) isplotted against delay time. The envelope of the interferogram as afunction of delay time presented in Fig. 4e shows a coherence time of12 ps, corresponding to a homogeneous linewidth of 110μeV. In con-trast to the previous results, the interferogram in Fig. 4d presents adistinct beating pattern comprising multiple periods. The beatingperiod BM is estimated to be 230 ± 10 fs, corresponding to an energysplitting of 18.0 ± 0.8meV. According to the spectra in Fig. 4a, the peakpositions of M1 and M2 show an energy difference of 18meV, which iswell matched to the splitting energy evaluated from the beating peri-ods observed in the time domain. Because M1 and M2 are moiré exci-tons trapped in different moiré potential minima, it can be confirmedthat the beating signal comes from the coupling of moiré excitons (M1and M2). As indicated in Fig. 4f, the coupling between moiré excitons(M1 and M2) creates a four-level system, including the coherentlycoupled state M12, two excited states (M1 and M2) trapped in moirépotentials, and the shared ground state (G). The emissions from thecoupled state lead to quantum interference (quantum beat) in the M1interferogram. Quantum couplings across adjacent electronic systemshave been reported in quantum wells57, colloidal quantum dots58, andmolecules59,60. However, we demonstrate new features of coupled-quantum systems based on moiré excitons in a semiconducting het-erobilayer, which can be extended to manipulate multiple quantumstates in periodic moiré superlattices.In conclusion, we demonstrated a new nanofabrication strategybased on RIE for investigating quantum physics in a semiconductingtwisted MoSe2/WSe2 heterobilayer. A reduction in the number ofmoiré potentials in the optical excitation and detection beyond theArticle https://doi.org/10.1038/s41467-024-48623-4Nature Communications |         (2024) 15:4905 6diffraction limit of light was realized by this nanofabrication, enablingthe observation of optical signals from a single moiré exciton in thetrapped potential. A significant single and sharp PL peak from a singlemoiré exciton in a potential has been successfully demonstrated,which also shows the characteristic spectral wandering in the 0Dquantum system. This study explored the quantum coherence of asingle moiré exciton for different temperatures and excitation powerdensities, confirming that the acceleration of decoherence was due tointeraction of moiré exciton–acoustic phonon and moiréexciton–moiré exciton. Further, the long duration of quantum coher-ence observed in a single moiré exciton was revealed to be more than12 ps at a low temperature of 4 K, which is much longer than that of anexciton in a monolayer semiconductor. Furthermore, quantum beatswere observed in the interferogram of moiré exciton, proving theexistence of coupling between moiré excitons trapped in differentmoiré potentials. The long coherence of moiré exciton revealed in thisstudy offer potential applications of moiré quantum systems inquantum technologies.MethodsSample preparation and nanofabricationMonolayer (1 L) MoSe2, WSe2, and encapsulating h-BN layers wereprepared on SiO2/Si substrates via mechanical exfoliation of bulkcrystals. The layer number and thickness of MoSe2 and WSe2 wereobtained from optical images and PL spectra. The MoSe2/WSe2 het-erobilayer encapsulated by top and bottom h-BN was fabricated by apolydimethylsiloxane (PDMS)-based dry-transfer method. The top h-BN, monolayer WSe2, and MoSe2 were sequentially picked up by poly(methylmethacrylate) (PMMA)-coated PDMS stampanddroppedontothe bottomh-BNon a SiO2/Si substrate. The entire dry-transfer processwas performed in an N2-filled glove box. The fabricated sample wasimmersed after in acetone solution for removing residual PMMA.Electron beam lithography and selective reactive ion etching (RIE)using Ar gas were used for the fabrication of designed nanostructuresinMoSe2/WSe2 heterobilayer–encapsulating h-BN layers. The selectiveRIE using Ar gas was performed in the conditions of power (50W) andflow rate (40 s.c.c.m).Optical measurementA linearly polarized semiconductor laser (2.38 eV)was employed as theexcitation source for low-temperature PL measurements. A 50×objective lens with a numerical aperture of 0.67 was used for solelyfocusing on the excitation laser light on the surface and acquire opticalimages. The sample was positioned in a cryogen-free cryostat with atemperature ranging from 4K to room temperature. The emissionsignals were coupled with an optical fiber and detected using a spec-trometer and charge coupled device with a typical spectral resolutionof 0.6meV. A pulsed supercontinuum light source passed through abandpass filter with a photon energy of 1.7 eV, with a repetition rate of1MHz was used for the time-resolved PL measurement. The emissionsignals were transported through the optical fiber, and detection wasrealized via an Si avalanche photodiode using a time-correlated single-0.0 0.3 0.6 0.9 1.2 1.5 1.8100200300400500Intensity (a.u.)Delay time (ps)1.29 1.32 1.35 1.380200040006000800010000Intensity (a.u.)Photon Energy (eV)0.8 W/cm21.6 W/cm22.4 W/cm23.2 W/cm24.8 W/cm26.4 W/cm28 W/cm212.8 W/cm224 W/cm232 W/cm240 W/cm2BMα1 = 1α2 = 0.83M12M2M1G18 meV(a)(b)(c)(d)(e)(f)0.0 0.4 0.8 1.2 1.6 2.00.20.40.60.81Visibility Delay time (ps)1 3 6 30 500.1110Intensity (a. u.)Power (W/cm2)1.25 1.30 1.35 1.40 1.450.00.30.60.91.21.51.8Photon energy (eV)Delay time (ps)50607080Intensity (a. u.)Fig. 4 | Quantum beat and coupling between two moiré excitons. a Low-temperature PL spectra (4K) of a newly fabricated pillar with diameter of 50nm forvarying excitation powers. b PL intensities of peaks at 1.327 and 1.309 eV as afunction of excitation power density. c Contour plot of the first-order correlationfunction of PL signals obtained using theMichelson interferometer as a function ofdelay time at the same position under excitation power of 3.2W/cm2.d Interferogram of the peak at 1.327 eV, displaying a quantum beat signal with aperiod BM= 230 ± 10 fs. e Envelope of the interferogram and its fitting result.f Schematic of the coupling of moiré excitons from two different moiré potentialminima. M1 indicates the moiré exciton state at 1.327 eV, while M2 indicates themoiré exciton state at 1.309 eV. The period of beating signal reveals the energysplitting between two moiré excitons.Article https://doi.org/10.1038/s41467-024-48623-4Nature Communications |         (2024) 15:4905 7photon counting technique. A Michelson interferometer was used togather the first-order correlation function g1(τ) of the PL signals forcoherence measurement. The detailed optical setup is shown in Sup-plementary Fig. 11.Data availabilityData presented in this paper and the supplementary materials areavailable from the corresponding author upon request.References1. Zrenner, A. et al. Coherent properties of a two-level system basedon a quantum-dot photodiode. Nature 418, 612–614 (2002).2. Veldhorst, M. et al. A two-qubit logic gate in silicon. Nature 526,410–414 (2015).3. Monz, T. et al. 14-qubit entanglement: Creation and coherence.Phys. Rev. Lett. 106, 130506 (2011).4. Koong, ZheXian et al. 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Nano Lett. 20,2843–2848 (2020).AcknowledgementsThis work was supported by JSPS KAKENHI (Grant NumbersJP16H00910, JP16H06331, JP17H06786, JP19K14633, JP19K22142,JP20H05664, JP21H05232, JP21H05235, JP21H01012, JP21H05233, andJP22K18986), JST FOREST program (Grant Number JPMJFR213K), theKeihanshin Consortium for Fostering the Next Generation of GlobalLeaders in Research (K-CONNEX) established by the Human ResourceDevelopment Program for Science and Technology, MEXT, and theCollaboration Programof the Laboratory forComplex EnergyProcesses,Institute of Advanced Energy, Kyoto University. h-BN growth was sup-ported by JSPS KAKENHI, (Grant Number JP20H00354) and by MEXT(Grant Number JPMXP0112101001). We thank Mr. Yuki Okamura, Dr.Yukiko Yamada-Takamura, Dr. Kohei Aso, and Dr. Yoshifumi Oshima forthe help of TEM measurement, which was supported by AdvancedResearch Infrastructure for Materials and Nanotechnology in Japan(ARIM) of the Ministry of Education, Culture, Sports, Science and Tech-nology (MEXT), Proposal Number JPMXP1222JI0020.Author contributionsH.W., K.W., and T.T. contributed to the fabrication of samples studied inthis work. H.W., K.S., and K.M. designed the experiments, which wereperformed by H.W., H.K., and K.M. Data analysis was performed by H.W.and K.M. The draft was written by H.W., K.S., and K.M., with all authorscontributing to reviewing and editing. The project was supervisedby K.M.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-48623-4.Correspondence and requests for materials should be addressed toKazunari Matsuda.Peer review information Nature Communications thanks the anon-ymous reviewer(s) for their contribution to thepeer reviewof thiswork. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-48623-4Nature Communications |         (2024) 15:4905 9https://doi.org/10.1038/s41467-024-48623-4http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Quantum coherence and interference of a single moiré exciton in nano-fabricated twisted monolayer semiconductor heterobilayers Results and discussion Power dependence of PL spectra PL spectral wandering and first-order correlation function Quantum beat and coupling between two moiré excitons Methods Sample preparation and nanofabrication Optical measurement Data availability References Acknowledgements Author contributions Competing interests Additional information