Real-Time Observation of Spin-Resolved Plexcitons Between Metal Plasmons and Excitons of WS2

Real-time observation of spin-resolved plexcitons between metal plasmons and excitons of WS2

R. K. Chowdhury,1 P. K. Datta,1 S. N. B. Bhaktha,1 and S. K. Ray*1, 2 1Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India 2S. N. Bose National Centre for Basic Sciences, Kolkata - 700106, India Strong light-matter interactions between resonantly coupled metal plasmons and spin decoupled bright excitons from two dimensional (2D) transition metal dichalcogenides (TMDs) can produce discrete spin-resolved exciton-plasmon polariton (plexciton). A few eﬀorts have been made to per- ceive the spin induced exciton-polaritons in nanocavities at cryogenic conditions, however, successful realization of spin-resolved plexciton in time-domain is still lacking. Here, we are able to identify both the spin-resolved plexcitons discretely at room temperature and investigate their ultrafast temporal dynamics in size-tunable Au−WS2 hybrid nanostructures using femtosecond pump-probe spectroscopy technique. Furthermore, we attribute that zero detuning between the excitons and plasmons is achieved at ∼7.0 ps along with transient Rabi-splitting energy exceeding ∼250 meV for both the spin-plexcitons, validating the strong-coupling conditions of polariton formation. Re- alization of these novel spin-plexcitons in the metal-TMDs platform is, therefore, interesting for both fundamental understanding and their possible futuristic applications in quantum photonics operating at room temperature.

I. INTRODUCTION P . Therefore, the strong coupling can only be achieved, if more dipoles in TMDs are oriented along out-of-plane direction. A few-layer TMDs is a possible pathway in- Hybrid nanostructures interconnecting fundamental stead of a monolayer to achieve stronger coupling, as light-matter interactions of dissimilar components pro- proposed by Kleemann et al.29, which eventually lead vide an innovative platform for designing futuristic pho- to the generation of plexcitons. In general, plexcitons 1,2 3 tonic devices like polariton nanolasers , spin-switches , are tracked down by observing the anti-crossing behav- 4 digital data storage for quantum computing , single pho- ior of polariton dispersion curves and formation of two 5 6 ton transistor and metamaterials . Even though prac- energetically well-separated hybridized peaks governed tical realization of exciton-plasmon polariton devices is by Rabi-splitting, a measure of coupling-strength30. In- still underdeveloped, interesting experimental outcomes deed, several reports show that plexcitonic states can be have been reported recently on plexcitonic hybrids com- achieved by the fine tuning of oscillator strength and posed of metallic nanostructures and cavities interact- spectral linewidth of plasmons and excitons, via linear ing with molecular excitons in organic semiconductors and angle-resolved spectroscopy techniques that required 7–15 and dyes or semiconducting excitons in 2D transition ultrahigh optical pumping (≥103 W/cm2)9,10,31,32. Al- 16–21 metal dichalcogenides (TMDs) . Indeed, the supe- ternatively, the ongoing quest is to tune exciton-plasmon rior coupling between optical field and surface-plasmons overlap via two-step pump-probe spectroscopy technique makes the metal nanostructures appealing for strong that can separately create exciton (X0) and plasmon (P ) 22,23 light-matter interactions . On the other hand, ex- in a time-resolved process, resulting in the generation of citons in TMDs (MoS2, WS2) are potentially attractive plexcitons11,27,28. due to their massive exciton binding energy (∼0.5 eV) Here, we report individual ultrafast detection of both and spin-orbit coupled photon energy selectiveness re- the spin-resolved plexcitons (X0 − P and X0 − P ) by sulting in alike but independent spin-locked bright ex- A B controlling the dimensions (∼30 and 90 nm) of self- citons (X0 and X0 ) in the same system24. Thus, the A B assembled gold nanoislands on layered WS (AuNI − coupling between spin resolved excitons and plasmons 2 WS ) fabricated on different substrates, such that their (P) can offer unexpected properties depending upon their 2 plasmonic resonance perfectly overlaps with individual coupling-strength. In general, surface-enhanced phenom- spin-resolved excitons of WS . Moreover, a strong out- ena like Purcell effect, Fano resonances are dominant in 2 of-plane dipolar interaction is attributed between pump- the weak to intermediate-coupling regime25,26, whereas arXiv:1909.05290v1 [cond-mat.mes-hall] 11 Sep 2019 induced excitons and probe-induced plasmons in time- the strong-coupling occurs at ultrafast timescale (faster domain. It results in remarkable transient Rabi-splitting than electron relaxation) which can lead to the formation energies as high as Ω (X0 −P ) ∼250 meV and Ω (X0 − of spin-plexcitons (X0 − P and X0 − P ) in metal-TMDs R A R B A B P ) ∼270 meV, thus providing new insights of these novel hybrid nanostructures11,27,28, hitherto unexplored. spin induced plexcitonic hybrids. Interestingly, excitons in monolayer TMDs are highly confined along in-plane directions, whereas metal nanostructures usually trap the optical field in perpendicular to the layer planes29. This misalignment of effective dipole- II. RESULTS AND DISCUSSION dipole interactions lead to poor coupling between X0 and 2

A. Individual overlap of excitons and plasmons

To provide independent experimental evidence of both the spin-resolved plexcitons, we have fabricated scalable prototypes of AuNI − WS2 hybrid nanostructures on glass and Si substrates with controlled AuNI dimensions of ∼30 and 90 nm (see details in Materials and Meth- ods). Scanning electron micrographs and atomic force micrographs confirm the size-distribution, roughness and surface profile of Au nanoislands on WS2 nanosheets, whereas Auger electron spectroscopy maps reveal the compositional homogeneity of the fabricated hybrids as shown in Fig.1(a)-(f) (also see Fig. S1, Supplemen- tary Information). Here, we have chosen Au nanostruc- FIG. 1. (Color online) Scanning electron micrographs of Au tures, a well-known non-reactive stable plasmonic sys- nanoislands on layered WS2 resulting in AuNI − WS2 hy- tem with efficient and wide spectral tunability (500-700 brid nanostructures for (a) ∼30 nm and (b) ∼90 nm diam- nm) in the desired wavelength regime by controlling their eter Au nanoislands and (c and d) Auger electron mapping size and shape. To detect both the spin-plexcitons inde- images of the corresponding hybrid samples showing composi- pendently, we have optimized the size of self-assembled tional homogeneity. Here, yellow color in figure (c) and blue, red and green colors in figure (d) denote S, W , Au and Si Au nanoislands to be of ∼90 and ∼30 nm in diameter, elements, respectively confirming a uniform composition of because of their characteristic plasmon resonance wave- samples. Here, scale bar is 500 nm for figure (a and b). (e length coinciding individually with the respective spin- and f) are the atomic force micrographs of ∼30 and ∼90 nm 0 0 coupled bright excitons (XA and XB) of WS2, as shown AuNI − WS2 to confirm the presence of nanoislands on WS2 in the steady state optical absorption spectra (Fig.1(g)). layers. The root mean square surface roughness values are 8.8 It is important to mention that we measured the steady- nm and 23.8 nm for ∼30 and ∼90 nm AuNI − WS2 samples, state spectra using a PerkinElmer spectrometer equipped respectively. (g) Steady-state absorption spectra of AuNI with a halogen lamp as the excitation source of broad- (∼30 and 90 nm diameter) and bare WS2 film on the glass 2 substrate, where both the excitons of WS2 individually match band UV-Visible illumination (∼0.1 W/cm ). The inten- 0 with the plasmonic modes of AuNI (blue dashed line for XA sity is thus insufficient to generate exciton-plasmon po- 0 and red dashed line for XB ) for different sizes. laritons (plexcitons) in the AuNI − WS2 hybrid system using our steady-state UV − vis absorption set-up31,32. sient spectra (Fig.2(e) and (g)) exhibit distinct transient Rabi-splitting features for both the ∼30 and 90 nm B. Transient Rabi-splitting in Au-WS2 hybrid AuNI − WS2 hybrids in comparison with the sharp ex- samples 0 0 citonic peaks (XA and XB) in pristine WS2 (Fig.2(d) and (f)), validating the individual spin-plexciton forma- The detailed time-resolved spin-plexciton dynamics of tion. Here, both the reflection and absorption modes these fabricated nanostructures have been monitored have been used, as in case of ∆A, the excited state ab- through an ultrafast (femtosecond) helicity controlled 0 sorption of AuNI overlapped and suppressed the XB ex- pump-probe spectroscopy technique at room tempera- citons, whereas for ∆R, X0 excitons are submerged due ture (300 K), as shown in Fig.2(a). An optical paramet- A 2 to the presence of huge pump induced reflection of AuNI ric amplified pump beam (405 nm, ∼ 30µJ/cm ), gener- beyond 550 nm (Fig. S2, Supplementary Information). ated from a T i-sapphire laser (808 nm), has been used for 0 0 the prior generation of both XA (1.97 eV) and XB (2.34 eV) excitons from fixed k-valley in layered WS2 (see details in Materials and Methods). Following the pump, C. Spin-resolved plexcitons a time-delayed (∆τ) broadband (1.65-2.75 eV) probe pulse has been used to record the transient absorbance Ideally, the plexcitonic resonance should occur due (∆A = Aon − Aoff ) or reflectance (∆R = Ron − Roff ) to the single excitonic dipolar overlap with the plas- of AuNI − WS2 hybrids fabricated on different large- monic vacuum field. In our case, we have designed the scale substrates (glass for ∆A or Si for ∆R), as shown metal nanostructures via dewetting of Au film with vari- in Fig.2(b)-(c). Here, Aon or Ron and Aoff or Roff rep- able thickness that results in self-assembled plasmonic resent the corresponding absorption or reflected probe hot-spots among Au nanoislands, such that the opti- spectrum in the presence and absence of the pump, re- cal field can be trapped more efficiently33, as compared spectively. We now discuss the time-resolved response to an isolated single metallic absorber. Thus, these of the as-fabricated WS2 samples and ∼30 nm and ∼90 quasi-continuous plasmonic modes originated from self- nm AuNI − WS2 hybrid nanostructures by measuring assembled AuNI hot-spots and pump-induced hot exci- ∆R and ∆A, as shown in Fig.2(d)-(g). The tran- tons of WS2 bring the system to a classical limit, where 3

FIG. 3. (Color online) Contour map for (a) transient absorption of ∼90 nm AuNI − WS2 hybrid nanostructures on glass 0 substrate to detect XA −P plexcitons and (b) transient reﬂec- tion of ∼30 nm AuNI − WS2 hybrid nanostructures on Si 0 substrate to detect XB − P plexcitons. Both the contour maps clearly show the transient hybridized mode splitting (E+ and E−) for corresponding spin-plexcitons, where the 0 0 position of excitons (XA and XB ) are denoted with blue and red dashed lines, respectively. The transient Rabi-splitting is further elaborated in ﬁgure (c-d) where individual temporal 0 0 dynamics of both the plexcitons (XA − P and XB − P ) is shown from 5-to-20 ps.

FIG. 2. (Color online) (a) Schematic illustration of the non- ter the zero-delay that implies a larger Rabi-splitting en- collinear transient absorption/reﬂection (∆A/∆R) setup used ergy (ΩR) value. The temporal dynamics of these hy- for the study. ∆τ is the time delay between the pump and bridized modes (E+ and E−) and their time-resolved probe, where the probe lags behind the pump for positive linewidth modulation have been investigated further as delay. Figure (b-c) is the corresponding photographs of the shown in Fig.3(c)-(d). Relatively broader probed ensem- wafer-scaled ∼30 nm and 90 nm AuNI −WS2 hybrid samples + − fabricated on Si (1×1cm2) and glass (2×2cm2) substrates for ble spectral linewidths of the E and E modes point transient measurements. Figure (d-e) and (f-g) are the corre- towards a higher dephasing time limit for both the plex- sponding ∆R and ∆A spectra of the WS2 and AuNI − WS2 citons. hybrid samples fabricated on Si and glass substrates, showing the transient build-up of Rabi-splittings at the respective exciton-plasmon overlap. D. Time-resolved anti-crossing

To further investigate these peculiar transient charac- normal hybridized mode splitting or Rabi-splitting can teristics (see details in Materials and Methods), we con- 0 11,34 be realized at exciton-plasmon (X − P ) resonance . sider a system consisting of two oscillating dipoles (X0 This splitting results in the formation of two distinct and P ) with associated dipole moments (µX and µP ) coupled polariton energy branches (E+ and E−) inde- interacting with an applied optical ﬁeld, which is again 0 0 pendently for both the XA − P and XB − P plexcitons, coupled with one another via the resultant local electro- 34 as shown in Fig.3. Previously Wang et al. suggested magnetic ﬁeld, as shown in Fig.4(a). In AuNI − WS2 that the hybridized mode splitting with an energy ΩR, hybrids, the similar interaction between the plasmons of 0 0 occurs due to the periodic energy transfer between plas- AuNI and the excitons (XA and XB) of WS2 is shown in mons and excitons with a time period of ∼ 2π/ΩR. This Fig.4(b), which leads to two new hybrid modes according indicates that one may be able to track down the Rabi- to the two-state model9. Now, the upper branch mode oscillation with our current transient spectroscopy set-up (E+ band) and the lower branch mode (E− band) exhibit where instrument response time is of 150 fs, only if the a typical anti-crossing behavior as shown in Fig.4(c), for 0 0 Rabi-splitting energy is below ∼150 meV. However, in both XA − P and XB − P plexcitons. These resultant our case, both the ∼90 and ∼30 nm AuNI−WS2 hybrids hybrid states have both excitonic and plasmonic charac- (Fig.3(a)-(b)) achieve a transient Rabi-splitting just af- ters when two un-hybridized components are close to the 4

0 0 the spin decoupled excitons (XA and XB) are already known in WS2, we can calculate the Rabi-splitting energy by directly solving E+ and E−. The extracted individual time-resolved Rabi-splitting energies are found to 0 0 be 269±26 meV and 246±7 meV for XA −P and XB −P plexcitons, respectively, by fitting the transient peak positions of E+ and E− with Eq. 1. The uncertainty value 0 for the normal mode splitting energy is greater for XA−P 0 compared to XB−P , as this energy uncertainty is directly related with the delay-dependent broadening of the transient spectra. The size-distribution of AuNI for ∼90 nm hybrid (X0 −P ) is broader than ∼30 nm hybrid (X0 −P ) FIG. 4. (Color online) (a) Schematic representation of plex- A B citon using the two-state model consisting two oscillating as shown in Fig. S1(e-f) of Supplementary Information dipoles exciton and plasmon. The exciton (yellow sphere) which introduces relatively higher uncertainty in the de- and plasmon (green sphere) are coupled together through a termination of spectral positions of the plexcitonic hybrid ± hypothetical spring (black dashed line) which are interacting modes (E ). Furthermore, if we consider the coupling under a strong electromagnetic field. (b) An artistic repre- losses in the surrounding medium, the ΩR get modified sentation of two-step pump-probe scheme which is equivalent q (∆ −∆ )2 as, (Ω ) ≈ 2 g2 − X0 P . Following this, to the two-state model to achieve the spin-resolved plexcitons R withloss 4 0 in AuNI − WS2 hybrid nanostructures. (c) Typical transient we have extracted both the excitonic linewidth (∆X ) + anti-crossing behavior of the hybridized energy branches (E as ∼100 meV and plasmonic linewidth (∆P ) as ∼150 − 0 0 and E ) for XA − P and XB − P plexcitons in AuNI − WS2 meV within ≤10 ps time-limit. These energies (∆X0 and hybrid nanostructures, as predicted in the two-state model. ∆P ) are much less than the calculated (ΩR)withloss (∼200 X0 and X0 are denoted as orange and purple dashed lines, 0 0 A B meV) for both XA −P and XB −P while considering the whereas the navy blue line represents delay dependent blue- losses. This validates the strong-coupling criterion of the shifted plasmons. Individual anti-crossings have been demon- 0 0 excitons and plasmons. Surprisingly, we discover the zero strated for both the XA −P and XB −P spin-plexcitons. The detuning (i.e. δ = 0) timescale between both the exci- zero detuning (δ = 0) or the normal mode splitting is observed 0 0 at the probe delay of 7.0 ps and the transient Rabi-splitting tons (XA and XB) and plasmons to be close to 7.0 ps, which exactly matches with the fast decay components energies (ΩR) at zero detuning as 269±26 meV (black arrow) 0 0 of both the plexcitons found experimentally and shown and 246±7 meV (red arrow) for XA − P and XB − P plexcitons, respectively. in Fig. S4, Supplementary Information. As the Auger scattering takes place typically in this timescale (sub- ps to few-ps) as reported earlier in the previous studies 38,39 resonance. Whereas far-off from resonance, the hybrid on WS2 , the hot-exciton cooling process may play a states are principally equivalent to the additive spectra crucial role to achieve this time-resolved zero detuning of of uncoupled P and X0. Here, the time-dependent anti- both the plexcitons. crossing originate due to the probe-delay dependent blue- shift of the plasmon resonance in AuNI35–37 (Fig. S3, Supplementary Information). The variation in the en- E. Origin of transient Rabi-splitting ergy amplitudes for both the branches (E+ and E−) is defined as12,29,33: To understand the definite origin of these robust Rabi- splitting energies, we evoke the idea of plasmonic hot- spots in Au nanoislands and pump induced exciton pop- r 2 + − ± 1 2 δ ulations in WS2. In the plexcitonic limit, E and E can E = (EP + EX ) ± (g + ) (1) 2 4 be written as E0 − g and E0 + g, where E0 = EP = EX is the resonance energy. A detailed calculation suggests where EP and EX are the corresponding energy of 0 that the following condition needs to be satisfied to real- the plasmons (P ) and excitons (X ), δ = EP − EX 9,11 0 ize the strong coupling between the X0 and P : is the detuning energy of P and X , g = ΩR/2 = 1 p + − 2 2 2 (E − E ) − δ is the coupling rate of plexcitons. 2 r In the two-state model, we have considered both the plas- D e NX ΩR or 2g ≈ ( ) µX (2) mons and excitons to be ideal dipole oscillators within ω0 me VP our considerable ultrafast time-domain (≤10 ps), by ne- glecting the incoherent cross-damping term to simplify where D is the dipolar coupling factor ∝ 1/(diame- 3 the model. However, the asymmetric anti-crossing be- ter of AuNI) , ω0 is the resonance frequency, e is elec- + − havior (Fig.4(c)) of E and E for both the plexcitons tronic charge, me is the effective mass of the electron, 0 0 (XA − P and XB − P ) indicates the possibility of non- µX is exciton dipole moment, NX is the effective number 0 ideal photon exchange between X and P for these hy- of excitons in the system and VP is the plasmon mode brids. Now, as the individual spectral positions of both volume. During our transient measurements, the pump 5

fluence has been maintained around ∼30 µJ/cm2 with spin-coated at 1,500 rotation per minute onto glass (2×2 a pump photon energy of 3 eV throughout the experi- cm2) or silicon (1×1 cm2) substrates at 90 0C under an ments to avoid any pump induced many-body effects and inert condition. Thereafter, optically thick gold films spectral linewidth modulation due to hot-exciton popu- with variable thickness (controlled through precursor 38 lation in WS2 . Thus, the effective number of both percentage) were deposited on as-fabricated WS2 layers 0 0 −6 the excitons (XA and XB) in the system always reaches using thermal evaporation at ∼10 mbar chamber pres- to a certain population right before the occurrence of sure. Following this, the samples were annealed for 3 probe excitation. On the other hand, plasmonic hot- hours at 300 0C constant temperature under an inert spots can trap sufficient number of probe-photons to cre- (Ar) atmosphere, resulting in the desired size of gold ate quasi-continuum of plasmons. As a result of this, nanoislands (AuNI) − WS2 hybrid nanostructures as 0 0 both the plexcitons (XA −P and XB −P ) can be accom- shown via the step-by-step fabrication schematic in Fig. plished facilely via strong dipolar coupling among pre- S5 of the Supplementary Information. The fabricated excited pump-induced excitons and probe-induced plas- samples can act as wafer-scaled emergent prototypes for mons. Thus, we conclude that our system reaches a tem- exploring the transient dynamics of spin-resolved plexci- porally stable strong coupling regime at a probe delay of tonic modes. The sample fabricated on glass (∼90 nm ∼7.0 ps, where the normal mode splitting energy exceeds AuNI − WS2) was used for the transient absorption and ∼250 meV for both the spin decoupled plexcitons. that on a Si substrate (∼30 nm AuNI − WS2) was used for transient reflection measurements.

III. CONCLUSIONS

In summary, our report reveals individual time-resolved generation and detection of both the spin-locked plexcitons in metal-TMDs hybrids. We attribute strong cou- B. Transient measurements set-up pling condition between plasmons and spin-resolved excitons in size-tunable AuNI − WS2 hybrid nanostructures through a two-step excitation process that dis- Ultrafast transient absorption/reflection spectra of plays clear transient anti-crossing behavior for both the AuNI − WS2 on glass/silicon was recorded with a com- spin-resolved plexcitons, in agreement with the results mercially available transient spectrometer (Newport) us- of two-state model. A remarkably robust Rabi-splitting ing 50 fs T i: Sapphire mode locked amplifier (Libra-He, energy (∼250 meV) and comparatively higher zero de- Coherent) with 808 nm center wavelength and a repeti- tuning time (∼7.0 ps) are realized for both the spin- tion rate of 1 kHz with 3 mJ average pulse energy. Princi- resolved plexcitons of AuNI −WS2 hybrids. We thereby ple part (70%) of the fundamental beam (808 nm) was fed strongly believe that our results suggest new possibilities into an optical parametric amplifier (TOPAS prime, Co- to study the ultrafast behavior of the spin-plexcitons in herent), which generated horizontally polarized 405 nm time-domain and may be an interesting route to explore (∼3 eV) pump beam that converts into right circularly possible spin-polaritonic device aspects at room temper- polarized output by passing through a broadband (400- ature in future. 800 nm) achromatic quarter-wave plate (Thorlabs). The minor part (30%) of the fundamental beam was focused on to a CaF crystal, which generated a stable white IV. ACKNOWLEDGMENTS 2 light continuum (400-750 nm) as a probe spectrum. This probe spectrum passed through an optical delay channel Authors acknowledge SDGRI-UPM project of IIT using a motorized translational stage (ILS-LM, Newport) Kharagpur for necessary equipment support in Ultra- in order to probe the system at different times, after fast Science Lab, Department of Physics, IIT Kharagpur. the pump excitation was employed to perturb the sys- RKC thanks Sayantan Bhattacharya and Manobina Kar- tem. Pump and probe beam spot-sizes were maintained makar for their inputs during transient measurements. at 1.0 mm and ∼100 µm, respectively which were used for our non-collinear pump-probe measurement setup. This kind of arrangement provided a pump energy fluence of V. APPENDIX: MATERIALS AND METHODS ∼30 µJ/cm2 with the pump:probe fluence ratio of 300:1. A spectrometer equipped with a fiber coupled linear Si A. Sample fabrication photodiode array (MS-260i, Oriel Instrument) was used to detect the differential probe spectra. Differential ab- Quasi-continuous layered WS2 films (typically ∼10 nm sorbance (∆A)/reflectance (∆R) spectra were measured thick) were fabricated using freshly exfoliated mono- by modulating the pump beam at 500 Hz with an optical to-few layer WS2 (Sigma-Aldrich) ink homogeneously chopper, which was ultimately fed to the spectrometer. dispersed in dimethylformamide (DMF , 99.9%, Sigma- Here, this ∆A or ∆R are measured in terms of mOD Aldrich) as reported earlier in our previous work40, and which is the change in optical density×10−3. 6

C. Two-state model distance between dipoles. Solving these two equations, we can calculate both the dipole moments (µX and µP ) that is useful to calculate the absorption cross-sections In our AuNI − WS2 hybrid nanostructures, self- 9,42 assembled Au nanoislands produce strong hot-spots on (σ) and exciton-plasmon coupling energy (g) . top of WS layers. As a result, only a small fraction of Now, to calculate the hybrid mode energies of the up- 2 per and lower branches (E±), two-state model is incorpo- WS2 experiences a strong near-ﬁeld coupling with quasi- continuous plasmonic hot-spots, which results in the for- rated, as shown in Fig.4(b). g(= ΩR/2) is the exciton- mation of plexcitons. plasmon coupling energy which is essential to achieve the strong-coupling condition of plexcitons. Now, one Following Thomas et al.9, we have considered the cou- can write the dipolar interaction in terms of eigen value pled harmonic oscillator model where we assume two equation, dipoles (one associated with excitons (X0) and the other one for plasmons (P )) interacting strongly under an EX g Hµi = µi (5) applied local electromagnetic ﬁeld (EM) (as shown in g EP Fig.4(a)). The equation of interaction can be written as9,33, Thus, the eigen values of the Hamiltonian matrix is equivalent to the hybrid mode energies of the plexcitonic system (E±) as shown in the Eq. 1. In Eq. 5, we have 2 not considered the damping term (γi) that will modify µ¨X + γX µ˙ X + ωX µX = AX [Ecos(ωt) + DµP ] (3) the Eq. 1 as9,

r 2 2 ± 1 2 δ ~ 2 2 E = (EP + EX ) ± g + − (γP − γX ) µ¨P + γP µ˙ P + ωP µP = AP [Ecos(ωt) + DµX ] (4) 2 4 16 ~ − j (γP + γX ) (6) where µX and µP are the dipole moments of the exci- 4 ton and plasmon respectively, ωi (= Ei/~) are the cor- √ ~γi responding resonance frequency, Ai is related with the when we replace Ei → Ei−j 2 ,(i = X and P , j = −1) oscillator strengths, γi the damping constant of the os- in Eq. 5. As our system validates the strong coupling ~|γP −γX | cillators (here, i = X and P ), Ecos(ωt) the driving local conditions of plexcitons, i.e., g 4 , Eq. 6 will EM ﬁeld and D ∝ 1/r3, where r is the center-to-center thus reduce to Eq. 1.

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