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[Lingxiao Zhou](https://orcid.org/0009-0009-6048-9788), Bin Liu, Yuze Liu, [Yang Lu](https://orcid.org/0009-0003-9690-3916), [Qiuyang Li](https://orcid.org/0000-0002-8192-3960), Xin Xie, [Nathanial Lydick](https://orcid.org/0000-0002-5991-4240), Ruofan Hao, Chenxi Liu, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Yu-Hsun Chou, [Stephen R. Forrest](https://orcid.org/0000-0003-0131-1903), [Hui Deng](https://orcid.org/0000-0003-0629-3230)

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[Cavity Floquet engineering](https://mdr.nims.go.jp/datasets/48e79e0f-4617-4839-833f-8e2bb2c4f521)

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Cavity Floquet engineeringArticle https://doi.org/10.1038/s41467-024-52014-0Cavity Floquet engineeringLingxiao Zhou 1, Bin Liu2, Yuze Liu2, Yang Lu 2, Qiuyang Li 1, Xin Xie1,Nathanial Lydick 1, Ruofan Hao3, Chenxi Liu4, Kenji Watanabe 5,Takashi Taniguchi 6, Yu-Hsun Chou7,8, Stephen R. Forrest 1,2 &Hui Deng 1,2,3Floquet engineering is a promising tool to manipulate quantum systemscoherently. A well-known example is the optical Stark effect, which has beenused for optical trapping of atoms and breaking time-reversal symmetry insolids. However, as a coherent nonlinear optical effect, Floquet engineeringtypically requires high field intensities obtained in ultrafast pulses, severelylimiting its use. Here, we demonstrate using cavity engineering of the vacuummodes to achieve orders-of-magnitude enhancement of the effective Floquetfield, enabling Floquet effects at an extremely lowfluence of 450photons/μm2.At higher fluences, the cavity-enhanced Floquet effects lead to 50 meV spinand valley splitting of WSe2 excitons, corresponding to an enormous time-reversal breaking, non-Maxwellian magnetic field of over 200 T. Utilizing suchan optically controlled effective magnetic field, we demonstrate an ultrafast,picojoule chirality XOR gate. These results suggest that cavity-enhanced Flo-quet engineering may enable the creation of steady-state or quasi-equilibriumFloquet bands, strongly non-perturbative modifications of materials beyondthe reach of other means, and application of Floquet engineering to a widerange of materials and applications.Coherent electromagnetic waves form a periodic potential in time forelectric dipoles and modify electronic transitions, which is known asFloquet engineering1–3. By controlling the frequency and spatial-temporal mode of an off-resonant laser field, different Floquetpotentials can be formed to modify the spatial-temporal symmetry4–7,topology8–10, and energy landscape11–13 of electronic transitions,potentially rendering rich new phenomena and novel applications14,15.In most materials, however, Floquet effects are often overwhelmed byinhomogeneity, phonon-induced dephasing, and other dissipationchannels. Ultrafast lasers with high fluence are typically required toproduce significant Floquet effects. This has restricted the applicationof Floquet engineering to a fewmaterials and to transient phenomenathat can be difficult to model, understand, or use. To circumventcomplications of high-intensity lasers, vacuum fields in cavities havebeen explored in theory for Floquet engineering recently16,17. Alter-natively, cavities can also be designed tomodify the effective field of adriving laser and thus the Floquet effects18,19. Here we demonstrate theuse of an optical cavity to achieve two orders of magnitude enhance-ment of the effective fluence of a driving Floquet field, enabling anenormous non-Maxwellian magnetic field over 200T and an ultrafast,picojoule all-optical-chirality XOR gate. Our results also suggest that asufficiently large static effective magnetic field and steady-state Flo-quet engineering can be achieved with commonly availablecontinuous-wave lasers. The work opens doors to Floquet engineeringReceived: 6 August 2024Accepted: 21 August 2024Check for updates1Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, MI 48109-2122, USA. 2Department of Electrical Engineering and ComputerScience, University ofMichigan, 1301 Beal Avenue, AnnArbor,MI 48109-2122, USA. 3Applied Physics Program, University ofMichigan, 450Church Street, AnnArbor, MI 48109-2122, USA. 4Nuclear Engineering and Radiological Science, University of Michigan, 2355 Bonisteel Blvd, Ann Arbor, MI 48109-2122, USA.5Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 6Research Center forMaterials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 7Department of Photonics, National Cheng KungUniversity, Tainan, ROC, Taiwan. 8Academy of Innovative Semiconductor and Sustainable Manufacturing, National Cheng Kung University, Tainan, ROC,Taiwan. e-mail: dengh@umich.eduNature Communications |         (2024) 15:7782 11234567890():,;1234567890():,;http://orcid.org/0009-0009-6048-9788http://orcid.org/0009-0009-6048-9788http://orcid.org/0009-0009-6048-9788http://orcid.org/0009-0009-6048-9788http://orcid.org/0009-0009-6048-9788http://orcid.org/0009-0003-9690-3916http://orcid.org/0009-0003-9690-3916http://orcid.org/0009-0003-9690-3916http://orcid.org/0009-0003-9690-3916http://orcid.org/0009-0003-9690-3916http://orcid.org/0000-0002-8192-3960http://orcid.org/0000-0002-8192-3960http://orcid.org/0000-0002-8192-3960http://orcid.org/0000-0002-8192-3960http://orcid.org/0000-0002-8192-3960http://orcid.org/0000-0002-5991-4240http://orcid.org/0000-0002-5991-4240http://orcid.org/0000-0002-5991-4240http://orcid.org/0000-0002-5991-4240http://orcid.org/0000-0002-5991-4240http://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/0000-0003-0131-1903http://orcid.org/0000-0003-0131-1903http://orcid.org/0000-0003-0131-1903http://orcid.org/0000-0003-0131-1903http://orcid.org/0000-0003-0131-1903http://orcid.org/0000-0003-0629-3230http://orcid.org/0000-0003-0629-3230http://orcid.org/0000-0003-0629-3230http://orcid.org/0000-0003-0629-3230http://orcid.org/0000-0003-0629-3230http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52014-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52014-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52014-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-52014-0&domain=pdfmailto:dengh@umich.eduwww.nature.com/naturecommunicationsof a wide range of materials, frequency bands, and spatio-temporalmode structures.To demonstrate cavity-enhanced Floquet engineering, we con-sider the Floquet effects of a pulsed driving field on excitonic transi-tions (Fig. 1a), which is also known as the optical Stark effect (OSE)20,21.OSE has been used to create optical trapping potentials for ultracoldatomic gases, where even a small shift in resonance is pronouncedcompared to the narrow linewidths of atomic transitions22–24. Inmaterials,OSEhas been studied since the 1980s25,26 and,more recently,in transition metal dichalcogenide (TMD) monolayers4–6,27. Yet thechange of the resonances due to non-resonant OSE has been less thantheir linewidths, leaving the effects in the perturbative regime. Lasersresonantly driving discrete and continuum electronic transitions canproduce strong modifications to materials, yet absorption-inducedincoherent effects and multitudes of higher-order effects oftendominate13,28–30.In this work, we demonstrate coherent, non-resonant OSE wellbeyond theperturbative regimevia cavity enhancement.Wemeasure ashift of the exciton transition energy up to 46meV in a WSe2 mono-layer, many times the inhomogeneously broadened exciton linewidthof 10meV, and a valley splitting of 50meV, corresponding to a non-Maxwellian magnetic field of >200T31,32.ResultsThe systemOur system can be modeled by a four-level Hamiltonian in the fieldinteraction picture. As illustrated in Fig. 1a, we consider a laser fieldwith frequency ν and amplitude ε, coupled to the exciton transitionfrom theground state of thematerial ∣g�to the exciton state ∣Xi, with atransitionmatrix element μgX (Fig. 1a). The exciton energy is EX, and thelaser is red-detuned from EX by Δ = EX − hν. In the field interactionpicture, the exciton-field coupling leads to two virtual states ∣X � hν�and ∣g +hν�, coupled with ∣g�and ∣Xi respectively, with a couplingstrength μgX∣ε∣ (Fig. 1a). Using base vectors ∣g�,∣X � hν�,∣g +hν�and∣Xi, the system Hamiltonian is given by:H =0 μgX jεj 0 0μgX jεj Δ 0 00 0 EX � Δ μgX jεj0 0 μgX jεj EX0BBB@1CCCAð1ÞDiagonalizing the Hamiltonian gives the energies of the shifted energylevels ∣g 0� and ∣X 0�, and thus the dressed-exciton energy EX + ΔE,where the Stark shift ΔE is given by:ΔE = � Δ+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2 + 4μ2gX jεj2q: ð2ÞUnder the weak field approximation, μ2gX jεj2=Δ2 ≪ 1, the Stark shiftreduces to ΔE ffi 2μ2gX jεj2=Δ, which increases linearly with the fieldintensity ∣ε∣2. In a resonant cavity, the effective field intensity at thecentral antinode is enhancedby a factorηcav � jεj2=jεinj2 ∼F=π, whereF is the finesse of the cavity, and εin is the input field amplitude in freespace. The enhancement ηcav can be many orders of magnitude intypical microcavities.To achieve optimal cavity enhancement for a red-detuned pump,we design an asymmetric λ/2 cavity as illustrated in Fig. 1b. A higherquality factor would produce a stronger enhancement, but only forfrequencies within the cavity linewidth. Therefore, we match theresonance frequency and linewidth of the cavity with those of our 360-fs pulsed pump laser with 2.3meV linewidth, 67meV red detuningfrom theWSe2 A exciton resonance. The enhancement factor averagedover the pulse is calculated to be 145. To facilitate probing the excitontransition, the first side-bandminimum of the topmirror is positionedat the exciton frequency, while the bottommirror has a stopband thatcovers both the pump laser and exciton frequencies, as shown inFig. 1c. (See Methods for more details).To study the OSE of our device, we use an ultrafast pump pulseand measure the change induced by the pump via the reflectancecontrast (RC) spectra of a weak white-light probe (Fig. 1b). The pumpand probe pulses are both 360 fs in duration with variable relativedelay. Due to spin-valley locking in TMDs33–36, the chirality of the pumpand probe allows valley selectivity.Cavity enhancementWe first measure the cavity enhancement by comparing the pumplaser intensity required for the same optical Stark shift in the cavityversus on a TiO2/SiO2 mirror. The mirror is designed to have a high-reflectance stopband that covers both the cavity and exciton energies(see Supplement Materials). As shown in Fig. 2, similar blueshifts ofexciton absorption, ΔE, are clearly seen in the RC spectra when theWSe2 monolayer is either placed inside a cavity (Fig. 2a) or on amirroronly (Fig. 2b), with fitted ΔE = 2.58 ± 0.09meV and 2.52 ± 0.07meV,respectively. Blueshifts appear only when the pump and probe pulsesoverlap in time, which confirms that they result from the OSE. How-ever, the pump pulse fluence used is only 18 fJ/μm2 when in the cavityPump pulseProbe pulseSapphireSiO2SiNZnSMgF2hBN-WSe2-hBN-PMMA1.6 1.65 1.7 1.75 1.8 1.85Energy (ev)00.20.40.60.81Reflectance|Xk±|gk±+ hνσ+ Pump|Xk±- hν|gk±MgxεMgxεΔ++Δ++σ+ Probe σ- Probe|X*k+|g*k+|X*k-|g*k-Δ+-Δ+-⟩⟩⟩⟩⟩⟩⟩⟩a               b                      c|E|2200100 0Fig. 1 | Theprinciple and experimental systemof cavity-enhancedoptical Starkeffect (OSE). a Illustration of the red-detuned chiral OSE. The circularly polarizedpump leads to Floquet states (∣X� hν�and ∣g+hν�), which hybridize with theground and excited state of excitons ∣g�and ∣Xi, resulting in blueshifted dressedstates ∣g0� and ∣X0�. A weak probe pulse measures the shifted transition energy.b Schematic of the half-wavelength cavity with a monolayer WSe2 at the antinode.The simulated field distribution at the cavity resonance is plotted on the left side,showing a 200-fold enhancement of the resonant field at the antinode. Opticalmeasurements are performed through transparent Sapphire. cReflectance spectraof the SiN=SiO2 bottom distributed Bragg reflector (DBR) (blue) with a side-bandminimum at the exciton resonance, the ZnS/MgF2 top DBR (red) with highreflectance at the laser and exciton energies, and the complete cavity (black)showing cavity (1.67 eV) and exciton (1.74 eV) resonances. The solid/dashed curvesare the measured/simulated results, respectively.Article https://doi.org/10.1038/s41467-024-52014-0Nature Communications |         (2024) 15:7782 2www.nature.com/naturecommunications(Fig. 2a), compared to 460 fJ/μm2 when on the Distributed Braggreflector (DBR) (Fig. 2b). SinceΔE/Δ ≈2.5/67 ≈ 0.037≪ 1, the weak fieldapproximation is valid. Using ΔE ∝ ∣ε∣2, the cavity enhances the effec-tive field intensity by 26-fold compared to a DBR. Since the DBRalready provides a 3.4-fold enhancement of effective field intensityrelative to free space (see Supplement), we deduce a cavity enhance-ment factor ηcav = 88.Fitting the dependence of ΔE on pump intensity in the weakfield limit (P < 0.015GW/cm2) with Eq. (2), we obtain μ2gX �ηcav ≈4900± 100Debye2 (red dashed line in Fig. 2g). Comparing thisvalue with μ2gX of 59 Debye2 37 and 45 Debye 24 previously measured byabsorption and OSE, we obtain ηcav ≈83 to 109, consistent with themeasured value of 88.Pump fluence dependence of enhanced OSEThe large cavity enhancement allowsOSE in regimes thatwere difficultto access previously. As an example in the weak-field limit, Fig. 2cshows where the Stark shift remains clearly discernible down to anextremely lowpump fluence of 0.12 fJ/μm2, or about 450photons/μm2.The corresponding intensity is 0.33mW/μm2 during our 360 fs pulse.This suggests that the effect is achievable using a continuous-wave(CW) laser of only 5.6mWpower over 17 μm2. In contrast, in free space,a high power of about 500mW would be required for CW OSE.The cavity enhancement also allows us to reach an abnormallylarge coherent Stark shift beyond the weak-field approximation, evenwith a large pumpdetuning. Figure 2d shows an examplewhere a Starkshift of 46.3 ± 0.4meV ismeasured forΔ = 67meV. This is the highestreported coherent optical Stark shift in TMD.In Fig. 2g, we summarize the pump intensity dependence of thecavity-enhanced Floquet. The weak-field, linear regime extends froman extremely low pump fluence of 0.12 fJ/μm2 to about 40 fJ/μm2, withcorresponding Stark shifts of 0.008meV to about 4.6meV. Through-out this regime,weobserve the same26-fold enhancementof the Starkshift in the cavity (red circles in Fig. 2g) compared to the DBR (blackdiamonds), or equivalently 88-fold compared to free space (purpledashed line). For a pump fluence >100 fJ/μm2, the Stark shift deviatesfrom the predictions of Eq. (2) (red dashed line in Fig. 2g), and even-tually saturates to a maximum shift of about 50meV. We will discusslater possible causes of the saturation and mitigation strategies.Ultrahigh effective magnetic fieldThe cavity-enhanced Floquet effect can be used to create a non-Maxwellian magnetic field B* in the material based on the opticalselection rules of exciton-photon coupling. In TMDs, optical selectionrules lead to valley selectivity due to spin-valley locking. We use probepulses that are co- or cross-circularly polarized with the pump toseparately measure the OSE of the two valleys. As shown in Fig. 2d–f, a1200 fJ/μm2σ+ pump introduces a blueshift of 46.3 ± 0.4meV of theco-circular OSE, but a redshift of 3.8 ± 0.2meV for the cross-circularOSE. The latter has been attributed to many-body Coulomb1.72 1.74 1.76-101Time (ps)-0.5-0.4-0.3-0.210-5 10-4 10-3 10-2 10-1 10010-310-210-11001011020.25 2.5 25 250 2500Fluence (fJ/μm2)Intensity (GW/cm2)ΔE+ (meV)1.72 1.74 1.76-101Time (ps)-0.5-0.4-0.3-0.2-0.11.72 1.74 1.76Energy (ev)-101Time (ps)-505x10-31.72 1.74 1.76 1.78 1.8-101Time (ps)-0.3-0.25-0.2-0.15-0.1-0.0501.72 1.74 1.76 1.78 1.8-101Time (ps)-0.3-0.25-0.2-0.15-0.1-0.0501.72 1.74 1.76 1.78 1.8Energy (ev)-0.200.2R++-R+-ΔE+ (meV)Fluence (fJ/μm2)Intensity (GW/cm2)10-5 10-4 10-3 10-2 10-1 10010-310-210-11001011020.25 2.5 25 250 25000.030.3330300B (T)a           d                                      gb           e                 c           f                   h19.7 fJ/μm2σ+/σ+506 fJ/μm2σ+/σ+0.13 fJ/μm2σ+/σ+1315 fJ/μm2σ+/σ+1315 fJ/μm2σ+/σ-1315 fJ/μm225x 4x50 meVFig. 2 | Extreme Floquet engineering. a, b Co-circularly polarized probe reflec-tance spectra R++(t) at different delay time t relative to the pump. a Themonolayeris in a cavity, and pump fluence P = 18 fJ/μm2. b The monolayer is on a DBR, andP = 460 fJ/μm2. The dashed white curves are fitted exciton resonances. The sameamount of Stark shift of ~2.5meV is observedwhen the pump and probeoverlap intime, while the pump fluences differ by 26 times between the cavity and DBRdevices. cWeak pump, co-circular differential reflectance spectraΔR++(t)/R++(t) forthe cavity device with a pump intensity of 33 kW/cm2, or a fluence of 0.12 fJ/μm2.d Strong pump, co-circular reflectance spectra R++(t) of the cavity device with apump intensity of 330MW/cm2 or fluence of 1.2 pJ/μm2, showing a very largeblueshift at zero time delay. e Cross-circular probe R+−(t) under the same pump asin (d), showing a redshift at zero time delay. f Energy difference between K and K 0valley excitons when the same pump as in (d, e) is turned on (red) and off (blue) attime =0, showing effective Zeeman splitting of 50meV.gCo-circular Stark shift vs.the pump fluence and intensity for a monolayer in a cavity (red), on a DBR (black),and in free space (purple). Symbols are measurement results. The red and blackdashed lines are fits using Eq. (2). The red solid line is a fit including both two-photon absorption and cavity-shifting. The blue dashed line is a fit onlywith cavity-shifting. The purple dashed line is an extrapolation from DBR to vacuum.hDressing induced valley splitting vs. the pump fluence and Intensity for the cavitydevice. Symbols are measurement results. The black dashed line is modelprediction.Article https://doi.org/10.1038/s41467-024-52014-0Nature Communications |         (2024) 15:7782 3www.nature.com/naturecommunicationsinteractions, with biexciton as the dominant contribution38–40. There-fore, the 1200 fJ/μm2 pump pulse leads to a spin and valley splitting ofΔZ = 50meV in WSe2 exciton. Using the exciton g factor of 1.9 in anearly report41, the same amount of Zeeman splitting would require anexternal magnetic field of B ~ 455T. Using the g factor of 4.1 in morerecent work31 would give B ~211 T. Figure 2h shows the control of B* bypump fluence.Ultrafast low-power chirality XOR switchLastly, we show that the enhancedOSEmay facilitate energy-efficient all-optical computing. A switch, which functions as an XOR gate, is a criticalbuilding block for optical computing. A coherent optical switch has theadvantage of ultrafast speed andminimal pump-induced heating. Whilecoherent OSE can change the optical response of the system on anultrafast timescale, it has been impractical for switching when it hasrequired high-intensity lasers and produced only weak responses. In ourcavity device, however, a large optical Stark shift can be achievedusing alow-intensity pulse with minimal excitation of incoherent carriers.As an example, we evaluate our device as a chirality XOR gate. Achirality XOR is a logic gate with the additional chirality degree offreedom, where the output signal is logic 1/0 according to the chiralityof the two input beams (Fig. 3a). In TMDs, chirality is also locked withthe valley degree of freedom.With a 120 fJ/μm2 (2 pJ) pump, we obtaina valley splitting of 14.6 ± 0.3meV (~1.5 × linewidth) in the WSe2exciton. Figure 3b shows the corresponding change in normalizedreflectance dRσ = RCpump−on −RCpump−off, where σ = +( − ) presentsthe co- (cross-) circularly polarized probe. Since it is straightforward toadd a non-dispersive linear absorptive medium to uniformly reducethe minimum reflectance, we consider the extinction ratio of theswitch as the contrast between σ ± vs. the power noise δ:10log10ðjdR+ � dR�j=δÞ. From the variance of 500 pulses, we obtainδ ~ 1.1− 2.0%. The extinction ratio is shown in Fig. 3b (right axis), whichreaches about 15 dB at E ~ 1.75 eV. Furthermore, as a coherent switch,the switching has the same ultrafast bandwidth as the pump. Figure 3cshows the time-dependent modulation amplitude at E = 1.75 eV. Aswap of pump and probe polarization will complete the truth table ofanXORgate listed in Fig. 3a. Coherent switchingwith similar speed andpower has only been achieved using 6-mm LiNbO3 bulk crystals42. Theperformance of our switch and XOR gates can be further improved byreducing the inhomogeneously broadened exciton linewidth.DiscussionThe demonstrated time-reversal breaking, manifested in valley split-ting of 50meV, is already beyond the reach of presently availableMaxwellian magnetic fields. An intriguing question is how strong theeffect, or, how large an optical Stark shift ΔE, can be reached throughcavity-enhanced Floquet engineering.We consider two main factors that lead to the saturation of ΔE inour present experiment. First, the large shift in the exciton resonanceleads to a change in the dielectric constant of the monolayer, which inturn shifts the cavity resonance and reduces the enhancement factorηcav at the original cavity frequency. For our device, we calculate areduction of ηcav by up to 13% at the pump fluence used, shown by theblue dashed line in Fig. 2g (See Supplement). However, this is not anintrinsic limitation and can be mitigated by tuning the pump laserwavelength to match the shifted cavity resonance.Second, linear and higher-order absorption of the red-detunedpump becomes non-negligible at high pump intensities, leading toexcitations in the monolayer, which saturate the exciton transitiondipole moment μgX. This can be seen in Fig. 2d, e, where the excitonresonance shows reduced reflection contrast, as well as a small blue-shift after the pump. We measure the excitations generated by thepump through time-integrated photoluminescence (PL), which showsthat linear and two-photon absorption (TPA) are the dominant con-tributions, and TPA dominates above P ~ 0.2 GW/cm2 (see Supple-ment). Including both the shift of the cavity resonance and thesaturation of μgX, we achieve excellent agreement between the mea-sured and fitted pump intensity dependence of the optical Stark shift(the red symbols and red solid line in Fig. 2g).The above analysis suggests that the maximum shift can beincreased in a few ways. First, we can blueshift the pump to follow theblueshift of the cavity resonance, typically by less than 1meV adjust-ment. Second, linear absorption can be strongly suppressed bydecreasing the exciton linewidth toward the radiative limit, or byincreasing the red detuning,Δ, of the pump. Although the Stark shift isalso reduced inversely with Δ, it can be countered by a linear increasein pump intensity until TPA dominates. The main limitation comesfrom TPA, as its efficiency is approximately constant for detuningmuch less than half of the exciton resonance43.We evaluate possible maximum shifts using the fitted linear andTPA coefficients for our device. The exciton is modeled as a Lorentzoscillator, and the pump as a Fourier-limited pulse with a Gaussianspectrum (see Supplement for details). As shown in Fig. 4, we calculatea maximum optical Stark shift of ΔEmax ~ 48meV for our device at anoptimized pump red detuning Δopt ~ 49meV. Considering an excitonlinewidth of 1meV, which is close to the radiative limit, we obtainΔEmax ~ 74meV with Δopt ~ 22meV. If TPA can be suppressed tenfold,then Δopt ~ 41meV leads to ΔEmax ~ 131meV.1.72 1.73 1.74 1.75 1.76 1.77-0.6-0.4-0.200.20.40.6051015Energy (ev)dRσExtinction ratio (dB)-2 -1 0 1 2-0.6-0.4-0.200.2051015t (ps)dRσExtinction ratio (dB)Signal Control Output σ- σ- σ+ σ+ σ- σ+ σ- σ+0101a                           b                            cFig. 3 | A chirality all-optical switch. a The truth table of an XOR gate. Whencontrol is co-circular to the signal, a decrease in the reflected signal is definedas “0”output. When control is cross-circular to the signal, an increase in the reflectedsignal is defined as “1” output. b Left axis: The modulation amplitude of the probereflectance by a 120 fJ/μm2 (2pJ) pump at zero time delay: dR± =R±(on) − R±(off),where ± denotes the co- (blue) and cross-circularly (red) polarized probe. Rightaxis: The corresponding extinction ratioΘ = 10log10(∣dR+ − dR−∣/δ) (green). cTime-dependence of the modulation amplitudes for the co- (blue) and cross-polarized(red)probe and the corresponding extinction ratioΘ (green) at E = 1.75 eV, showingan ultrafast switch time of about 0.4 ps, reflecting the coherent nature of the Flo-quet effect.Article https://doi.org/10.1038/s41467-024-52014-0Nature Communications |         (2024) 15:7782 4www.nature.com/naturecommunicationsIn conclusion, we demonstrate near two orders of magnitudeenhancement of the effective Floquet field intensity in an asymmetriccavity compared to in free space. This approach has enabled a valleysplitting as large as 50meV that corresponds to time-reversal breakingby a 210 T magnetic field, a measurable Floquet effect with a pumpfluence as low as 0.12 fJ/μm2 or an average intensity of 33 kW/cm2, andan ultrafast coherent chirality XOR switch with 15 dB on/off switchingratio. These effects can be further enhanced by reducing the excitonlinewidth to the radiative limit and suppressing high-order nonlinearprocesses.Theory work suggests a corresponding enhancement of a Max-wellianmagnetic field via the inverse Faraday effect32. Futureworkmayidentify the optically-induced Maxwellian magnetic field, and mayclarify the nature of the effective non-Maxwellian magnetic field wehave measured here, such as its effects on the electronic band struc-tures and dielectric breakdownofmaterials in contrast to aMaxwellianmagnetic field.The demonstrated cavity-enhanced Floquet can be broadlyapplied to optically active materials of different spectral bands toinduce pseudo-magnetic fields inaccessible by other means, toenable CW Floquet engineering of new quantum phases44, and tofacilitate ultra-low-energy, ultrafast, all-optical switches andsensors45. In materials with stronger light-matter interactions, cav-ities have been explored as an alternative to laser driving for Floquetengineering16,17.MethodsComposition and fabrication of the deviceThe device consists of, from substrate to top, a bottomDBR, a WSe2monolayer encapsulated by hexagonal boron nitride (hBN) crystals,a polymethyl methacrylate (PMMA) spacer layer, and a top DBR.The bottom DBR comprising 10.5 pairs of SiN (~102 nm)/SiO2(~138 nm) and 70-nm top spacer SiO2 layers are grown on 500-μmthick Sapphire substrate by plasma enhanced chemical vapordeposition (PECVD). The monolayer WSe2 and two few-layer hBNflakes are mechanically exfoliated from bulk crystals. A poly-propylene carbonate (PPC) film combined with polycarbonate (PC)protrudes, and a polydimethylsiloxane (PDMS) stamp is used topick up the top hBN, WSe2 monolayer, and the bottom hBN under amicroscope. After all of the layers are picked up, the PPC film ismelted, stamped onto the bottom DBR, and then cleaned indichloromethane. After the monolayer is placed on the bottomDBR, a PMMA layer is spin-coated on top, followed by the transfer ofthe top DBR46. The top transferable DBR comprising 9.5 pairs of ZnS(~75 nm) and MgF2 (~123 nm) layers is grown on a silica substrate bythermal evaporation in a vacuum chamber with a base pressure of10−7 Torr. The transferable DBR allows us to form the cavity withprecise control of the cavity resonance without degrading the 2Dmaterials46.Circularly-polarized pump-probe spectroscopyA 360-fs pump pulse with tunable photon energy is generated from ahigh-power 1035-nm pulse at a 100-kHz repetition rate through anoptical parametric amplifier (OPA) and filtered to 2.3meV linewidthby tunable spectral filters. The probe pulse is a 1.2 ps chirped whitelight supercontinuum with a spectral range of 645–735 nm, gener-ated by focusing the 1035-nm residue pulse from the OPA onto aYAG crystal. We use a linear chirp correction for the probe. Amotorized delay stage controls the pump-probe time delay. Thepumpbeam is chopped at 30 Hz and focused to awaist size of 2.3 μmon the sample. The probe is focused on a slightly smaller beam at thecenter of the pump. The total fluence of the probe is 54.8 fJ/μm2; thefluence at the exciton resonance is about 0.6 fJ/μm2/nm, and there isno cavity enhancement at the exciton resonance. The probe-induced shift or saturation of the exciton resonance is observed.The pump-on/off intensities of the reflected probe are recorded by aspectrometer, synchronized with the chopper at double its fre-quency (60 Hz). The circular polarizations of the pump and probeare independently controlled by quarter-wave and half-wave platesand linear polarizers.We probe the valley-selective OSE via circularly polarized pump-probe spectroscopy. Due to TMDspin-valley lock-in33–36, a co-circularlypolarized pump and probe will dress and detect the exciton in thesame valley, while cross-circular polarization means dressing K=K0valley but detecting K0=K valley.TMM simulationWe use the transfer matrix method (TMM) to calculate the device’sreflection spectrum and the electric field distribution of the cavitydevice. The simulated refractive index of the materials:SiO2, SiN,MgF2, ZnS, TiO2 are 1.47, 2.00, 2.32, 1.42, and 2.53,respectively.Data availabilitySource data generated in this study is available at the repositoryDeepBlue under https://doi.org/10.7302/f6xv-d389.References1. Autler, S. H. & Townes, C. H. Stark effect in rapidly varying fields.Phys. Rev. 100, 703 (1955).2. Jaynes, E. & Cummings, F. 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High quality factor microcavity for Van der Waalssemiconductor polaritons using a transferrable mirror. Adv. Opt.Mater. 11, 2201440 (2023).AcknowledgementsThe authors acknowledge fruitful discussionswith RobertoMerlin. L.Z.,Y.L., Q.L., X.X., N.L., R.H., C.L., and H.D. acknowledge the support bythe Army Research Office under Awards W911NF-17-1-0312, the AirForce Office of Scientific Research under Awards FA2386-21-1-4066,the National Science Foundation under Awards DMR 2132470, theOffice of Naval Research under Awards N00014-21-1-2770, and theGordon andBettyMoore Foundation underGrantGBMF10694. B.L. andS.R.F. acknowledge the support by the Army Research Office underAwards W911NF-17-1-0312. Y.-H.C acknowledges the support of thePilot Directions for NSTCGrant for the Einstein Program, funded by theNational Science and Technology Council (NSTC) under grant numberNSTC 112-2636-M-006-004. K.W. and T.T. acknowledge support fromthe JSPS KAKENHI (Grant Numbers 20H00354 and 23H02052) andWorld Premier International Research Center Initiative (WPI),MEXT, Japan.Author contributionsL.Z. and H.D. conceived and designed the research. L.Z. performed themeasurements, simulation and data analysis. B.L. and Y.-H.C. fabricatedthe DBRs. Y.Liu, Q.L., X.X., and Y. Lu assisted in the measurements. N.L.assisted in the simulation. R.H. and C.L. assisted in sample preparation.K.W. and T.T. provided the hBN single crystal. H.D. and S.R.F. supervisedthe project. L.Z. and H.D. wrote the manuscript. All authors read andcommented on the manuscript.Competing interestsThe authors declare no competing interests.Article https://doi.org/10.1038/s41467-024-52014-0Nature Communications |         (2024) 15:7782 6https://doi.org/10.1038/s42254-023-00681-1https://doi.org/10.1038/s42254-023-00681-1www.nature.com/naturecommunicationsAdditional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-52014-0.Correspondence and requests for materials should be addressed toHui Deng.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/s41467-024-52014-0Nature Communications |         (2024) 15:7782 7https://doi.org/10.1038/s41467-024-52014-0http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Cavity Floquet engineering Results The system Cavity enhancement Pump fluence dependence of enhanced OSE Ultrahigh effective magnetic field Ultrafast low-power chirality XOR switch Discussion Methods Composition and fabrication of the device Circularly-polarized pump-probe spectroscopy TMM simulation Data availability References Acknowledgements Author contributions Competing interests Additional information