Multistable excitonic Stark effect
MMultistable excitonic Stark effect
Ying Xiong, Mark S. Rudner, and Justin C.W. Song
1, 3 Division of Physics and Applied Physics, Nanyang Technological University, Singapore 637371 Center for Quantum Devices and Niels Bohr International Academy,Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark Institute of High Performance Computing, A*STAR, Singapore, 138632
The optical Stark effect is a tell-tale signature of coherent light-matter interaction in excitonicsystems, wherein an irradiating light beam tunes exciton transition frequencies. Here we show that,when excitons are placed in a nanophotonic cavity, the excitonic Stark effect can become highlynonlinear, exhibiting multi-valued and hysteretic Stark shifts that depend on the history of theirradiating light. This multistable Stark effect (MSE) arises from feedback between the cavity modeoccupation and excitonic population, mediated by the Stark-induced mutual tuning of the cavity andexcitonic resonances. Strikingly, the MSE manifests even for very dilute exciton concentrations andcan yield discontinuous Stark shift jumps of order meV. We expect that the MSE can be realizedin readily available transition metal dichalcogenide excitonic systems placed in planar photoniccavities, at modest pump intensities. This phenomenon can provide new means to engineer coupledstates of light and matter that can persist even in the single exciton limit.
Strong light-matter interaction can provide a versa-tile platform for dynamically controlling quantum mat-ter [1]. A striking example is the excitonic optical Starkeffect [2–5]: off-resonant irradiation of an excitonic sys-tem, with frequency below the exciton transition energy,continuously blue-shifts the exciton transition to higherfrequencies as the light intensity increases [4–8]. In con-trast to the fixed Rabi splitting found for polaritons, thatis independent of the intensity of light [9–11], the opticalStark effect is linear in the irradiation intensity. Thisdependence grants on-demand tunability of excitonicproperties. In transition metal dichalcogenides (TMDs),Stark shifts are furthermore sensitive to light polariza-tion, thereby enabling direct control over the valley exci-tons necessary for valley opto-electronics [8, 12, 13].Here we propose that the optical Stark effect can takeon a markedly different character when an excitonic sys-tem is placed in a nanophotonic cavity (Fig. 1 inset). Inthis setting, the Stark shift becomes a dynamical vari-able, with the cavity field taking on the role of the irra-diating field that shifts the excitonic levels. In particular,when the excitonic and cavity modes are simultaneouslypumped, the optical Stark effect can become multistable,exhibiting a hysteretic Stark shift that depends on thehistory of the optical drive. As we explain below and in-dicate in Fig. 1, this multistable Stark effect (MSE) arisesdue to a Stark-induced mutual tuning: the excitonic tran-sition frequency (right panel) is sensitive to the cavitymode occupation, while the cavity resonance (left panel)is sensitive to the exciton population. When applied ex-citon and cavity driving fields are detuned from theirrespective bare transition frequencies, the mutual tun-ing sets up a feedback that shifts the exciton and cavitytransition frequencies into resonance with their respec-tive drives (dashed to solid lines, Fig. 1). This feedbackleads to a highly non-linear, and multistable Stark effect.The MSE features discontinuous transitions between
FIG. 1: Mutual tuning of exciton (right panel) and cavity(left panel) transitions induced by the optical Stark effect,wherein the exciton transition frequency is sensitive to thecavity mode occupation (and vice versa). When the excitonsand the cavity mode are simultaneously pumped (downwardarrows indicate the corresponding pump frequencies), the ex-citon and cavity transitions can shift into resonance withtheir drives (from dashed to solid curves). These population-induced shifts generate the feedback loop that gives rise tothe multistable Stark effect (MSE). (inset) A two-dimensionalexcitonic material such as a transition metal dichalcogenidecan be readily layered on top of a planar nanophotonic cavityformed by a photonic crystal defect to achieve the conditionsfor realizing the MSE. multiple distinct steady states of the combined cavity-exciton system, and can exhibit large discontinuous Starkshift jumps of order meV. Indeed, we find that the exci-ton population can take on multiple steady state values(Fig. 2a) with a hysteretic behavior that is controlled bya weak cavity drive far-detuned from the original exci-ton resonance. Further, the magnitude of the Stark shiftjump from one stable state to another can be directlytuned by the drive that pumps the excitonic population.These mechanisms provide in-situ means of tailoring theswitching behavior in the exciton/cavity system. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p We expect that the MSE can be realized in TMDs (e.g.,WS ) on currently available high quality factor planarphotonic cavities [14–17] (Fig. 1 inset), even at low op-tical drive strengths of several to tens of kW / cm . Thisplatform provides new means of constructing hysteretic,nonlinearly coupled states of light and matter that can,in principle, persist even at the single exciton limit. Stark-induced mutual tuning and nonlinearity.—
Thekey to achieving the MSE is the nonlinearity mediated bystrong coupling between cavity photon modes and exci-tons. As we now explain, this nonlinearity in the cavity-exciton system can arise directly through the Stark effect.As a simple and clear illustration of the MSE, we first fo-cus on a single localized excitonic mode interacting witha single cavity mode (of a single polarization). We willdiscuss the MSE for delocalized excitons in an extended2D excitonic layer later in the text.We model the localized exciton mode as a simple two-level system with bare resonance angular frequency ν (0) ;we denote the ground state (no exciton) by | P = 0 (cid:105) ,and the excited state (exciton present) by | P = 1 (cid:105) . Thecavity photon mode has angular frequency ω (0) . In thedispersive limit, the dynamics of the system are describedby the Hamiltonian H = H X + H + H int with (setting (cid:126) = 1 here and throughout, unless otherwise stated): H X = ν (0) ˆ P , H = ω (0) a † a, H int = V a † a ˆ P , (1)where a † is the creation operator for the cavity photonmode, and ˆ P = s z + / P | P (cid:105) = P | P (cid:105) , where is the 2 × s z = σ z /
2, where σ z is the third Pauli matrix.The last term in Eq. (1) encodes a dispersive coupling, V , between the excitons and cavity photons, that is validfor V (cid:28) | ν (0) − ω (0) | , ω (0) , ν (0) . In this limit, the magni-tude of V can be controlled directly through engineeringof the microcavity mode profile and its detuning from theexciton resonance, ∆ = ν (0) − ω (0) . Throughout this workwe will consider ∆ >
0, which ensures that
V >
0; fora derivation of the dispersive coupling V and parameterestimates for a TMD/cavity system, see SupplementaryInformation ( SI ).Crucially, through the dispersive coupling, both theexciton and cavity photon resonances are mutually de-pendent on the other’s occupation. For a state with m cavity photons present, and excitonic state P = { , } ,the (cavity-dressed) exciton and (exciton-dressed) cavityphoton resonance angular frequencies, ˜ ν ( m ) and ˜ ω ( P ),respectively, are given by:˜ ν ( m ) = ν (0) + V m, ˜ ω ( P ) = ω (0) + V P. (2)The excitonic resonance ˜ ν ( m ) experiences a blue shiftaway from its bare resonance frequency that is propor-tional to the photon number in the cavity – the opticalStark effect [3, 4, 6, 8]. We characterize this by the ex-citonic Stark shift: δE ≡ ˜ ν ( m ) − ν (0) = V m . Similarly,
FIG. 2: A single excitonic emitter coupled to a cavity candisplay bistable and hysteretic steady-states of the (a) excitonand (b) cavity photon populations (reflected in the Stark shift, δE ). The steady states are obtained by solving Eqs. (8) and(9); the thick solid lines indicate the stable solutions, andthe thin dashed lines indicate the unstable solutions, see SI .Illustrative dimensionless parameters used: F X /γ = 2, V /γ =0 . ν d − ν (0) = 7 γ , ω d − ˜ ω (0) = 1 . κ and γ/κ = 10. the cavity photon resonance frequency ˜ ω ( P ) also dependson the occupation of the excitonic state, shifting as P changes. The mutual tuning of exciton and cavity pho-ton transitions exhibited in Eq. (2) provides a naturalmeans of feedback, and as we now discuss, gives rise tononlinear dynamical phenomena in the system. Multistable Stark effect and cavity-exciton steadystates.—
To demonstrate the MSE, we consider anexciton-photon microcavity system with laser drives atangular frequencies ν d and ω d . These fields pump theexcitonic and cavity photon modes, respectively. Thisselectivity can be achieved by choosing ν d and ω d to beslightly detuned from ˜ ν (0) and ˜ ω (0), respectively, withtheir individual detunings much smaller than ∆.In the presence of these laser driving fields, the Hamil-tonian becomes H ( t ) = H + H ( d ) X ( t ) + H ( d )0 ( t ), with H ( d ) X ( t ) = F X (cid:0) e − iν d t σ + + h.c. ) ,H ( d )0 ( t ) = F e − iω d t a † + h.c. ) , (3)where F X and F are the drive amplitudes, and σ + =( σ x + iσ y ) /
2, where σ x,y are the x, y Pauli matrices. Inanticipation of making a rotating wave approximation be-low, we have discarded counter-rotating terms in Eq. (3).To explicitly demonstrate the MSE, we track the ex-citon and cavity photon populations in the driven sys-tem in the presence of Markovian dissipation that ac-counts for exciton relaxation (recombination) and cav-ity photon loss. As a first step, we transform into aframe that co-rotates with the drives, using U ( t ) =exp ( − iω d t a † a − iν d t ˆ P ). In the rotating frame, thesystem evolves according to the (static) Hamiltonian˜ H = ˜ H X + ˜ H + ˜ H int , with ˜ H X = ( ν (0) − ν d ) ˆ P + F X σ x /2,and ˜ H = ( ω (0) − ω d ) a † a + F ( a † + a ) /
2. The interaction˜ H int = V a † a ˆ P does not change under the transformation.Using the rotating-frame Hamiltonian ˜ H , we take thedensity matrix of the composite exciton and cavity sys-tem (in the rotating frame), ˜ ρ ( t ), to evolve according tothe master equation ∂ t ˜ ρ = i [˜ ρ ( t ) , ˜ H + ˜ H X + ˜ H int ] + I X [˜ ρ ( t )] + I [˜ ρ ( t )] , (4)where I X [˜ ρ ] = γ (2 σ − ˜ ρσ + − σ + σ − ˜ ρ − ˜ ρσ + σ − ) accountsfor recombination of the exciton, with rate γ , and I [˜ ρ ] = κ (2 a ˜ ρa † − a † a ˜ ρ − ˜ ρa † a ) describes losses in the microcav-ity photon mode with rate κ . The interaction ˜ H int (cid:54) = 0couples the cavity and exciton subsystems by the mutualtuning of their transition frequencies as in Eq. (2).While Eq. (4) can generically encode a variety of com-plex dynamical regimes of the composite system, as wenow discuss, a large separation in the cavity and excitondecay timescales enables direct evaluation of the MSEsteady states (cf. Ref.[18] for general discussion). In-deed, the regime wherein the excitonic system relaxesfar faster than the cavity photon system can be read-ily achieved in many exciton-cavity setups, see estimatebelow. Physically, this separation of timescales meansthat the reduced density matrix of the excitonic system˜ ρ X ( t ) ≡ Tr ˜ ρ ( t ) rapidly reaches a quasistationary stateover a time that is short compared with the character-istic evolution timescale of the cavity photon; here Tr [Tr X ] denotes the partial trace over photonic [excitonic]degrees of freedom. On the timescale of excitonic relax-ation, the cavity state ˜ ρ ( t ) ≡ Tr X ˜ ρ ( t ) can be treatedas quasistatic, allowing the formation of an excitonicsteady state that depends parametrically on ˜ ρ . On thetimescale that the cavity state ˜ ρ ( t ) evolves, ˜ ρ X ( t ) main-tains a quasistationary state that adiabatically followsthe slow evolution of ˜ ρ ( t ).Using this separation of timescales, in describing thetime evolution of the exciton and cavity photons weadopt a mean-field decoupling [18] of Eq. (4) by replac-ing the cavity-exciton coupling by its mean-field aver-ages Tr (˜ ρ ( t ) ˜ H int ) → V (cid:104) m ( t ) (cid:105) ˆ P and Tr X (˜ ρ ( t ) ˜ H int ) → V a † a (cid:104) ˆ P ( t ) (cid:105) , where (cid:104) ˆ P ( t ) (cid:105) ≡ Tr[ ˆ P ˜ ρ X ( t )] and m ( t ) ≡ Tr[ a † a ˜ ρ ( t )]. This mean-field decoupling is justified inthe semiclassical regime where the photon number inthe cavity is large and fluctuations are small [18]. Withthis mean-field decoupling, the (rotating frame) excitonand cavity density matrices ˜ ρ X ( t ) and ˜ ρ ( t ), respectively,evolve according to: ∂ t ˜ ρ X ( t ) = i (cid:104) ˜ ρ X ( t ) , ˜ H X + V (cid:104) m ( t ) (cid:105) s (cid:105) + I X [˜ ρ X ( t )] , (5) ∂ t ˜ ρ ( t ) = i (cid:104) ˜ ρ ( t ) , ˜ H + V a † a (cid:104) ˆ P ( t ) (cid:105) (cid:105) + I [˜ ρ ( t )] . (6)The exciton population dynamics can be obtained bydirectly evaluating the elements of ˜ ρ X ( t ) in Eq. (5)to obtain effective Bloch equations. Writing (cid:104) s i ( t ) (cid:105) ≡ Tr [ s i ˜ ρ X ( t )] where s i = σ i / i = x, y, z , and notingTr [˜ ρ X ( t )] = 1, we obtain ∂ t (cid:104) s x ( t ) (cid:105) = δν ( t ) (cid:104) s y ( t ) (cid:105) − γ (cid:104) s x ( t ) (cid:105) ,∂ t (cid:104) s y ( t ) (cid:105) = − F X (cid:104) s z ( t ) (cid:105) − δν ( t ) (cid:104) s x ( t ) (cid:105) − γ (cid:104) s y ( t ) (cid:105) ,∂ t (cid:104) s z ( t ) (cid:105) = F X (cid:104) s y ( t ) (cid:105) − γ ( (cid:104) s z ( t ) (cid:105) + 1 / , (7) where δν ( t ) = ν d − ˜ ν [ (cid:104) m ( t ) (cid:105) ]. We solve for the exci-tonic (quasi)-steady state by setting the three equationsabove equal to zero, and assuming that the cavity modeoccupation (cid:104) m ( t ) (cid:105) = m is fixed. We thus obtain the(quasi)-steady-state population of the excitonic mode asa function of the cavity occupation, m : P ( m ) = (cid:104) s z (cid:105) + 12 = F X / F X + 2 (cid:104) γ + (cid:0) ν d − ˜ ν ( m ) (cid:1) (cid:105) , (8)where (cid:104) s z (cid:105) is the time independent steady state solutionof Eq. (7). As evident from Eq. (8), the steady state ex-citonic population depends both on the excitonic drivestrength, F X , and parametrically on the cavity popula-tion through the stark-shifted exciton resonance, ˜ ν ( m ).The steady state cavity population can be obtainedheuristically by first considering the familiar expressionfor the average population of a driven cavity mode witha fixed resonance frequency, ω : ¯ m = ( F / / { κ + ( ω d − ω ) } . Due to the Stark-induced mutual tuning describedabove, the cavity resonance frequency changes with theexciton population, see Eq. (2). As a result, we replace ω → ˜ ω [ P ( m = ¯ m )] to yield a self-consistency relation forthe cavity mode population:¯ m = ( F / / { κ + (cid:0) ω d − ˜ ω [ P ( ¯ m )] (cid:1) } . (9)We note that this heuristically-obtained self-consistencyrelation agrees with results obtained through carefulanalysis of the evolution of the full density matrix ofthe joint system [42], in the regime κ/γ (cid:28)
1, and V /γ (cid:28) κ [18]. The steady state cavity occupation thusdepends on the steady state exciton population throughits mutually-tuned cavity transition in Eq. (2).We now explicitly exhibit the multistability describedby Eqs. (8) and (9). Choosing drive frequencies slightlyblue detuned from the bare exciton and cavity resonances(see Fig. 2 and caption for parameter values), Eqs. (8)and (9) yield multiple solutions for P as a function of F (for all other parameters held fixed in this regime). Thesemultiple steady states arise from the MSE, as evidencedby the jumps of the Stark shift δE (on the order of theexciton decay rate γ ) displayed in Fig. 2b.Within the bistable regime, two distinct stable steady-state solutions for P and δE exist for the same drive pa-rameters (solid lines). This enables a hysteretic behaviorof the excitonic system that depends on the history ofthe optical drive. Indeed, as F increases from zero (for-ward sweep), P (as well as ¯ m ) jump to the upper branchof solutions (upward arrow) at a forward threshold am-plitude. However, when F is then decreased (reversesweep), both P (and δE ) jump to the lower branch of so-lutions (downward arrow) at a distinct reverse thresholdamplitude. This hysteresis enables the system to operateas an optically-controlled “exciton switch” with “off” and“on” states as the lower and upper branches. Strikingly,this excitonic hysteresis occurs even for a single excitonicmode, in sharp contrast to other nonlinearities inducedat high exciton density [33]. MSE in transition metal dichalcogenides.—
Having ex-hibited the MSE mechanism for a single excitonic emit-ter, we now discuss the MSE in readily available two-dimensional excitonic systems. A natural class of candi-date materials are the atomically thin TMDs, which pos-sess room temperature stable excitons and large Starkeffects [8, 12, 13], and can be easily integrated with pla-nar photonic crystal cavities, as in the inset of Fig. 1.Here we will focus on zero center of mass momentum(COMM) excitons in a single valley, where excitons obeycircular polarization selection rules [19–24]: by drivingthe TMD with circularly polarized light of fixed handed-ness with frequency close to the exciton resonance, onlyexcitons in the corresponding valley will be excited.To describe the MSE in TMDs, we consider an ex-tended TMD layer placed on top of a photonic cavity, seeFig. 1 inset. We first note that the TMD excitonic modeat ν (0) can have a large effective degeneracy N . Thisdegeneracy accounts for excitons at distinct exciton cen-ter of mass spatial coordinates; these degenerate excitonemitters can form plane wave superpositions that leadto delocalized excitonic modes [25–27]. Importantly, themodes with zero COMM interact coherently (in phase)with the same cavity photonic mode [25–27] (with a wave-length of a few hundred nanometers); similarly, for exci-ton pumping fields that have large wavelengths of orderseveral hundred nanometers, multiple excitonic emitterscan be driven in phase with each other. As such, in de-scribing the TMD layer excitonic-cavity system, we re-place ˆ P → ˆ P tot = (cid:80) j ˆ P j in the Hamiltonian Eq. (1), aswell as σ + , − → s + , − tot = (cid:80) j σ + , − j in Eq. (3), where thesum over j runs over each of the j = 1 , . . . , N degenerateexcitonic emitters. Similarly, ˜ ω ( P tot ) → ˜ ω = ω (0) + V P tot in Eq. (2) where P tot = 0 , , , · · · are eigenvalues of ˆ P tot .Since all the emitters interact with the same cavity pho-ton mode, ˜ ν ( m ) in Eq. (2) remains unchanged.We follow a similar procedure and use the separationof timescales as discussed above for tracking the exci-ton and cavity photon populations (see SI for full de-tails). In so doing, we take a spin-coherent-state ansatzso that the dynamics of the multiple emitter system canbe analyzed in terms of the dynamics of a giant spin s tot = s x tot ˆ x + s y tot ˆ y + s z tot ˆ z , where s x,y,z tot = (cid:80) i s x,y,zi issummed over the excitonic emitters. For fixed cavityoccupation m we obtain the steady-state exciton popu-lation in the extended system (see SI ) as P tot = N F X / F X + 2(Γ + [ ν d − ν ( m )] ) , (10)where Γ is the exciton recombination rate (for the zeroCOMM excitons) in the extended TMD system; we note,parenthetically, that this rate can be estimated from the FIG. 3: (a) MSE shift δE of the excitonic system in a mono-layer TMD coupled to a cavity obtained from Eq. (9) and (10)displaying multiple steady states (bistable [low exciton drive]and then tristable [larger exciton drive]) and discontinuous δE jumps. Panel (b) shows line cut of δE as a function of ex-citon drive at a fixed cavity drive of 25 kW/cm [as indicatedby the blue line in (a)]. Panel (c) displays δE as a functionof cavity drive at a fixed exciton drive of 0 . [asindicated by the orange line in (a)]. Solid lines indicate sta-ble solutions; whereas dashed lines indicate unstable states.Here we used parameters: Γ = 1 meV, ν d − ν (0) = 4 meV,dispersive coupling V = 0 . κ = 0 . N = 1000and ω d − ω (0) = 0 . SI for detailed estimates anddiscussion of parameters. recombination rate γ of a single localized exciton emitteras Γ ∼ N γ [25–27]. In obtaining Eq. (10) we have takena large degeneracy N (cid:29) N can be largeand can range from N ∼ − [25]; this arisesfrom the large number of excitonic modes that can in-teract coherently within a single wavelength of either thecavity photon mode or the exciton drive [25–27]. Anestimate of N can be obtained from the ratio of themode area of the photonic mode (the square of its wave-length) and the effective size of an exciton (the squareof its Bohr radius) [27]. Further, recombination timesfor zero COMM excitons in typical monolayer TMDscan range from Γ − ∼ . κ (cid:28) Γ, justifying the separation of time scales and mean-field decoupling approach we have used to describe theMSE. Lastly, strong light-matter interaction in mono-layer TMDs [8, 12] can lead to sizeable values of disper-sive coupling V ≈ . − . SI for a detailedestimate. In the plots we have chosen V = 0 . and photoniccrystal cavities, discontinuous jumps in the excitonicStark shift can be readily achieved by moderate cavityand exciton drive intensities of order kW/cm .Interestingly, distinct regimes of multistability can beaccessed; at low exciton drive strength a bistable MSEmanifests (as cavity drive is swept) whereas larger ex-citon drives display tristabilities (see Fig. 3a,b). Indeed,the MSE displays hysteretic behavior as either exciton orcavity drives are swept, with Fig. 3b,c displaying sizeablediscontinuous δE of order meV. We note that togetherwith multistable δE shown in Fig. 3, the exciton popula-tion P tot similarly exhibits multistability and hysteresis(see also Fig. 2). While we have focused on the MSE andits concomitant excitonic multistability, multiple stablestates of the cavity mode (so-called “optical multistabil-ity,” characterized by distinct steady state values of ¯ m )can also arise via the MSE. (Note that in Fig. 2b, δE isdirectly proportional to the cavity photon occupation.)This effect is similar to dispersive optical multistabilityin highly nonlinear optical media [35–40], and may pro-vide new means for controlling optical states.From a fundamental perspective, the MSE arises fromthe fact that the excitonic Stark effect is an inescapableconsequence of the partial fermionic nature of excitons [6]– a property that is present even in dilute exciton sys-tems. Indeed, we find in Fig. 3 that a MSE manifestsfor a steady state excitonic population on the order of P tot ∼
1, see SI . This indicates that the MSE occurseven as approximately one exciton is excited in the en-tire photonic cavity (corresponding to a low exciton den-sity of order 10 cm − ). This distinguishes MSE andits associated nonlinear phenomena from other types ofmultistable behavior, e.g., optical bistabilities that orig-inate from exciton-exciton interactions that typically re-quire a large density of excitons to enable bistable behav-ior [33, 34]. Perhaps most exciting is how MSE-inducedhysteresis in the exciton population as a function of op-tical drive yields jumps in P tot of order unity; these mayprovide controllable means of selectively exciting/de-exciting a single exciton as well as controlling its emis-sion.J.C.W.S. acknowledges support from the National Re-search Foundation (NRF), Singapore under its NRF fel-lowship programme award number NRF-NRFF2016-05,the Ministry of Education, Singapore under its MOEAcRF Tier 3 Award MOE2018-T3-1-002, and a Nanyang Technological University start-up grant (NTU-SUG).M.R. gratefully acknowledges the support of the VillumFoundation, and the European Research Council (ERC)under the European Union Horizon 2020 Research andInnovation Programme (Grant Agreement No. 678862). [1] D. N. Basov, R. D. Averitt, and D. Hsieh, “Towards prop-erties on demand in quantum materials.” Nat. Mater. ,1077-1088 (2017).[2] A. Mysyrowicz, D. Hulin, A. Antonetti, A. Migus, W.T.Masselink, H. 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SUPPLEMENTARY INFORMATION FOR “MULTISTABLE EXCITONIC STARK EFFECT”Optical Stark effect
For the convenience of the reader, in this section we review the optical Stark effect and how it arises from the light-matter interaction between an exciton and light. In so doing, we concentrate on a single localized exciton emitterand its interaction with a single cavity mode. As discussed in the main text, generalization to multiple emitters isstraightforward. For illustration, we begin with a Jaynes-Cummings model H JC = ν bare σ z / ω bare a † a + g σ − a † + σ + a ) , (S1)where σ x,y,z are Pauli matrices, σ ± = ( σ x ± iσ y ) /
2, and g = M E is the coupling constant that captures the dipoleinteraction between the exciton mode (described by its dipole moment M ) and the cavity mode (described by theamplitude of the electric field E ). Here ν bare and ω bare are the bare frequencies of the exciton and cavity moderesonances in the absence of any interaction, i.e., g = 0. The interaction – i.e., the third term of Eq. (S1) – isresponsible for the optical Stark effect. To see this we perform a canonical transform T † H JC T with T = exp S ; here S = − g ( aσ + − a † σ − ) / (2∆). The transformed Hamiltonian is given by T † H JC T = H JC + [ H JC , S ] + 12! [[ H JC , S ] , S ] + 13! [[[ H JC , S ] , S ] , S ] + · · · . (S2)We note that [ ν bare σ z / ω bare a † a, S ] = − g ( σ − a † + σ + a ) /
2. By taking the dispersive limit g (cid:28) ∆, the transformedHamiltonian up to the first order in g/ ∆ is T † H JC T = ω bare a † a + ν bare σ z / V a † aσ z / V σ z / V / O ( V ) , = ( ω bare − V / a † a + ( ν bare + V /
2) ˆ P + V a † a ˆ P − ν bare / O ( V ) , (S3)where V = g / (2∆) is the dispersive coupling constant and ˆ P = σ z / /
2. We note that the last term, − ν bare / Stability analysis of the steady state solutions
In the plots in the main text, we have shown the multiple branches of steady state solutions that appear at agiven set of drive parameters. In these plots, solid lines denote the stable solutions while dashed lines denote unstablesolutions. The stability of each steady state solution can be ascertained by examining the time-evolution of the excitonoccupation factor ∂ t (cid:104) ˆ P ( t ) (cid:105) = ∂ t (cid:104) s z ( t ) (cid:105) in Eq. (7). In particular, taking a small deviation δP from the steady statesolution P and computing ∂ t (cid:104) ˆ P ( t ) (cid:105)| P + δP , we find a solution is stable if sgn (cid:16) ∂ t (cid:104) ˆP(t) (cid:105)| P+ δ P (cid:17) = − sgn( δ P); i.e., thesolution is stable if, after the system is slightly displaced from the steady state, it time evolves back to the originalsteady state solution (cid:104) ˆ P ( t ) (cid:105) = P . Steady state exciton population with multiple emitters
In this section, we describe the steady state solution for the MSE with multiple emitters. As in the main text,we replace ˆ P → ˆ P tot = (cid:80) i ˆ P i = (cid:80) i ( s zi + 1 /
2) in the Hamiltonian, Eq. (1), as well as σ + , − → s + , − tot = (cid:80) i σ + , − i inEq. (3), where the sum runs over the degenerate excitonic emitters. Similarly, ˜ ω ( P ) → ˜ ω = ω (0) + V P tot in Eq. (2)where P tot = 0 , , , · · · are eigenvalues of ˆ P tot . Since all the emitters interact with the same cavity photon mode,˜ ν ( m ) in Eq. (2) remains unchanged. For simplicity of notation, we have defined the spin operators s = σ /
2, where σ = ( σ x , σ y , σ z ) is the vector of Pauli matrices.Similar to our approach described for a single excitonic emitter in the main text, we exploit a separation of timescales between cavity and exciton systems that is easily achieved in readily available systems (see discussion in maintext). In this fashion, we can mean-field decouple the excitonic and cavity degrees of freedom. As a result the excitonicsystem evolves according to ∂ t ˜ ρ X, tot ( t ) = i [˜ ρ X, tot ( t ) , − δν ( t ) ˆ P tot + F X s x tot ] + I X [˜ ρ X, tot ( t )] , (S4)where δν ( t ) = ν d − ˜ ν [ (cid:104) m ( t ) (cid:105) ], and the dissipator for the multiple emitters reads I X [˜ ρ X, tot ] = γ (cid:2) s − tot ˜ ρ X, tot s +tot − s +tot s − tot ˜ ρ X, tot − ˜ ρ X, tot s +tot s − tot (cid:3) . (S5)As a sanity check, noting that [˜ ρ X, tot , ] = 0, we find that probability is conserved: ∂ t Tr(˜ ρ X, tot ) = 0.We can use Eq. (S4) to obtain the equation of motions governing the “Bloch” (giant spin) dynamics of the multipleemitters. Here we treat the collection of multiple emitters as a giant spin with magnitude S = N /
2. When the spin ispointing down such that (cid:104) s z tot (cid:105) = −S , no excitons are excited. When (cid:104) s z tot (cid:105) = S , all the excitonic modes are excited.In this work we focus on the regime far from saturation, where the exciton density remains low.Writing (cid:104) s i tot ( t ) (cid:105) = Tr[ s i tot ˜ ρ X, tot ( t )] for i = x, y, z , we find ∂ t (cid:104) s x tot ( t ) (cid:105) = δν ( t ) (cid:104) s y tot ( t ) (cid:105) − γ (cid:104) s x tot ( t ) (cid:105) + γ (cid:104){ s z tot , s x tot }(cid:105) ,∂ t (cid:104) s y tot ( t ) (cid:105) = − δν ( t ) (cid:104) s x tot ( t ) (cid:105) − F X (cid:104) s z tot ( t ) (cid:105) − γ (cid:104) s y tot ( t ) (cid:105) + γ (cid:104){ s z tot , s y tot }(cid:105) ,∂ t (cid:104) s z tot ( t ) (cid:105) = F X (cid:104) s y tot ( t ) (cid:105) − γ (cid:2) (cid:104) s (cid:105) − (cid:104) [ s z tot ] (cid:105) + (cid:104) s z tot ( t ) (cid:105) (cid:3) , (S6)where (cid:104) s (cid:105) = (cid:104) [ s x tot ] + [ s y tot ] + [ s z tot ] (cid:105) = S ( S + 1) and { a, b } = ab + ba is the anti-commutator. In obtaining Eq. (S6)we have cycled the operators in the trace, recalling that Tr( (cid:98) O A (cid:98) O B (cid:98) O C ) = Tr( (cid:98) O C (cid:98) O A (cid:98) O B ) = Tr( (cid:98) O B (cid:98) O C (cid:98) O A ), as well asnoted the identity [ s i tot , s j tot ] = i(cid:15) ijk s k tot , where (cid:15) ijk is the Levi-Civita symbol. Spin coherent state ansatz
As we now discuss, the dynamics of the multiple emitters can be understood as the dynamics of a giant spinˆ s tot = s x tot ˆ x + s y tot ˆ y + s z tot ˆ z with a spin magnitude S . Here S = N / S = 1 / { s z , s y } = { s x , s y } = { s z , s x } = 0; similarly, (cid:104) s (cid:105) − (cid:104) ( s z ) (cid:105) = 3 / − / /
2. Using these identities, we find Eq. (S6) reduces to Eq. (7) of the main text. Howeverwhen multiple emitters are involved, { s z tot , s x tot } does not necessarily vanish.Key to analyzing Eq. (S6) is obtaining (closed) forms for the terms (cid:104){ s z tot , s y tot }(cid:105) , (cid:104){ s z tot , s x tot }(cid:105) , and (cid:104) s z (cid:105) in terms ofthe giant spin expectations values (cid:104) s i tot (cid:105) . In order to do so and find the steady-state solutions of Eq. (S6), we adopta spin coherent state ansatz ˜ ρ X, tot = | ψ ˆ n (cid:105)(cid:104) ψ ˆ n | , s tot · ˆ n | ψ ˆ n (cid:105) = S| ψ ˆ n (cid:105) , (S7)with the giant spin pointing in the direction ˆ n :ˆ n = 1 S (cid:16) (cid:104) s x tot (cid:105) , (cid:104) s y tot (cid:105) , (cid:104) s z tot (cid:105) (cid:17) . (S8)In using this coherent state, it is useful to note that expectation values of various operators (e.g., s z tot s x tot ) in thiscoherent state can be obtained by applying a suitable rotation to the operator and evaluating the expectation valuein a well-known reference state (e.g., | ψ ˆ z (cid:105) ). For example, S = (cid:104) ψ ˆ n | s tot · ˆ n | ψ ˆ n (cid:105) = (cid:104) ψ ˆ z | U (cid:0) s tot · ˆ n (cid:1) U † | ψ ˆ z (cid:105) , U (cid:0) s tot · ˆ n (cid:1) U † = s z tot , U | ψ ˆ n (cid:105) = | ψ ˆ z (cid:105) , (S9)where U is unitary operator that rotates the spin (determined by ˆ n ).Using this coherent state ansatz, we obtain (cid:104) [ s z tot ] (cid:105) = s ⊥ / (cid:0) S (cid:1) + (cid:104) s z tot (cid:105) , (cid:104){ s z tot , s x tot }(cid:105) = −(cid:104) s z tot (cid:105)(cid:104) s x tot (cid:105) / S + 2 (cid:104) s z tot (cid:105)(cid:104) s x tot (cid:105) , (cid:104){ s z tot , s y tot }(cid:105) = −(cid:104) s z tot (cid:105)(cid:104) s y tot (cid:105) / S + 2 (cid:104) s z tot (cid:105)(cid:104) s y tot (cid:105) , (S10)where s ⊥ = ( (cid:104) s x tot (cid:105) + (cid:104) s y tot (cid:105) ) / . In obtaining Eq. (S10) we have used the identity (cid:104) ψ ˆ z | s x tot s z tot | ψ ˆ z (cid:105) = (cid:104) ψ ˆ z | s y tot s z tot | ψ ˆ z (cid:105) =0. Further we have noted that (cid:104) ψ ˆ z | s x tot s y tot | ψ ˆ z (cid:105) = −(cid:104) ψ ˆ z | s y tot s x tot | ψ ˆ z (cid:105) . This latter identity can be discerned from notingthat in s x tot s y tot = (cid:80) ij s xi s yj only the terms where i = j yield non-zero values, and that when i = j , we have { s x , s y } = 0.Substituting Eq. (S10) into Eq. (S6) we get ∂ t (cid:104) s x tot ( t ) (cid:105) = δν ( t ) (cid:104) s y tot ( t ) (cid:105) − γ (cid:104) s x tot ( t ) (cid:105) (cid:2) (cid:104) s z tot (cid:105) / S − (cid:104) s z tot (cid:105) (cid:3) ,∂ t (cid:104) s y tot ( t ) (cid:105) = − δν ( t ) (cid:104) s x tot ( t ) (cid:105) − F X (cid:104) s z tot ( t ) (cid:105) − γ (cid:104) s y tot ( t ) (cid:105) (cid:2) (cid:104) s z tot (cid:105) / S − (cid:104) s z tot (cid:105) (cid:3) ∂ t (cid:104) s z tot ( t ) (cid:105) = F X (cid:104) s y tot ( t ) (cid:105) − γ (cid:2) S ( S + 1) − s ⊥ / (2 S ) − (cid:104) s z tot (cid:105) + (cid:104) s z tot ( t ) (cid:105) (cid:3) . (S11)In the absence of a drive, F X = 0, and at steady state, the excitonic system yields its equilibrium value with (cid:104) s z tot (cid:105) eq = −S (similarly, (cid:104) s x tot (cid:105) eq = (cid:104) s y tot (cid:105) eq = 0). Interestingly, Eq. (S11) is a non-linear equation in (cid:104) s z tot (cid:105) . Suchnonlinearities become important only when the deviation of (cid:104) s z tot (cid:105) away from its equilibrium (no excitation) value islarge (i.e., comparable to (cid:104) s z tot (cid:105) eq ), due to saturation of the exciton resonance.We now analyze the steady state population of the excitons by first taking ˜ ν [ (cid:104) m ( t ) (cid:105) ] → ˜ ν ( m ) for a fixed m in thesame way as discussed in the main text. In the low-excitation regime where (cid:104) s z tot (cid:105) = −S + P tot for P tot (cid:28) S , we cansafely linearize Eq. (S11) in P tot . Solving for P tot at steady state in Eq. (S11), we obtain P tot = S F X F X + 2(Γ + [ ν d − ν ( m )] ) , (S12)where Γ ∼ S γ is the recombination rate for excitons in the entire multiple emitter system. In obtaining Eq. (S12),we have used that s ⊥ / S (cid:28) P tot . Writing S = N / Alternative derivation of exciton population in the low-density limit: Holstein-Primakoff transformation
In this section, we provide an alternative derivation of the steady state exciton population (in the case of multipledegenerate emitters) in the low-density limit. In so doing, we note that the dynamics of the exciton population can beunderstood from analyzing the evolution of a large spin with magnitude
S (cid:29)
1. In the large S limit, the excitationsof this system can be analyzed in a bosonic framework, using a Holstein-Primakoff transformation. To begin, we write s +tot → ( √ S ) b † (cid:104) − b † b S (cid:105) / , s − tot → ( √ S ) (cid:104) − b † b S (cid:105) / b, (S13)where b † is a bosonic creation operator so that [ b, b † ] = 1. Using Eq. (S13) we can readily verify2 s z tot = [ s +tot , s − tot ] = 2 S (cid:104) b † (cid:0) − b † b/ (2 S ) (cid:1) / , (cid:0) − b † b/ (2 S ) (cid:1) / b (cid:105) = 2( b † b − S ) . (S14)Note that Eq. (S14) is exact; the factors of [1 − b † b S ] / must be included in order to obtain the b † b dependence on theright-hand side (which will prove to be essential below).Noting that b † b counts the number of excitons created [see Eq. (S14)] we express the state where P tot excitons havebeen excited as | P tot (cid:105) ∝ (cid:2) b † (cid:3) P tot | (cid:105) boson , P tot < S , (S15)where the vacuum state (no excitons present) corresponds to the giant spin pointing downwards ( (cid:104) s z tot (cid:105) = −S ).In the co-rotating frame of the drive, we find the Hamiltonian that describes the excitons is˜ H X, tot = (˜ ν − ν d ) ˆ P tot + F X s x tot HP −→ (˜ ν − ν d ) b † b + F X ( √ S )( b + b † ) / , (S16)where we have recalled that ˆ P tot = s z tot + (cid:80) i / ≈ s z tot + S and s x tot = ( s +tot + s − tot ) /
2. Here we have estimated N = 2 S ;the arrow labeled HP denotes the Holstein-Primakoff transformation. In obtaining the last term of Eq. (S16) we havetaken the limit (cid:104) b † b (cid:105) (cid:28) S so that s +tot ≈ ( √ S ) b † and s − tot ≈ ( √ S ) b .Similarly, we find the exciton-cavity interaction reads as˜ H int , tot = V a † a ˆ P tot HP −→ V a † ab † b. (S17)This expression captures the nonlinear interaction between the exciton degrees of freedom (characterized by b ’s)and the cavity photons (characterized by a ’s). Note that this interaction is distinct from the bilinear exciton-cavityinteraction typically discussed for the polariton effect. The nonlinearity arises due to the underlying fermionic natureof the electrons and holes that make up the excitons [6]. The role of the fermionic Pauli exclusion is reflected inthe square root factors in the relation between the collective exciton creation operator s +tot and the bosonic creationoperator b † (and similarly for s − tot and b ), as well as the b † b dependence of [ s +tot , s − tot ] as shown in Eq. (S14).In a similar fashion to that described in the main text and above, we exploit a separation of time scales betweenthe exciton relaxation and cavity relaxation time scales and write the equation of motion of the excitonic excitationsas ∂ t ˜ ρ HP X, tot ( t ) = i [˜ ρ HP X, tot ( t ) , (˜ ν − ν d ) b † b + F X (cid:112) S / b + b † )] + I HP X [˜ ρ HP X, tot ( t )] , (S18)0where I HP X [˜ ρ ] = Γ(2 b ˜ ρb † − b † b ˜ ρ − ˜ ρb † b ), with Γ capturing a phenomenological decay rate of the excitonic mode.The steady state solution to Eq. (S18) can be readily obtained using a coherent state ansatz ˜ ρ tot , HP X = | β (cid:105)(cid:104) β | where b | β (cid:105) = β | β (cid:105) and β ∈ C . By direct calculation we verify that the coherent state ansatz indeed yields a steady statesolution of Eq. (6) for fixed oscillator excitation m , with β = ( − iF X (cid:112) S / / (Γ − i ( ν d − i ˜ ν [ m ])). The correspondingsteady state exciton population is given by P tot ≈ (cid:104) b † b (cid:105) = ( S F X / / { Γ + (cid:0) ν d − ˜ ν [ m ] (cid:1) } . (S19)As a consistency check, we note that in the limit F X (cid:28) Γ , ν d , ˜ ν , Eq. (S12) reduces to Eq. (S19). Estimate of parameter values for MSE in a TMD photonic cavity
In this section, we estimate the values of the parameters used in the main text to achieve the MSE in a transitionmetal dichacolgenide (TMD) sample placed in a photonic crystal cavity. For the purposes of the estimates below, wewill restore (cid:126) so that (cid:126) ω, (cid:126) ν as well as (cid:126) κ, (cid:126) Γ are energies.We consider a cavity with a high quality factor Q = ω (0) /κ and compressed effective mode volume V mode = α ( λ/n ) ,where κ is the cavity mode linewidth, ω (0) is the bare cavity resonance frequency, λ is the free space wavelength ofthe photons, n is the refractive index of the cavity, and α is a proportionality constant that depends on the geometryof the cavity. In photonic crystal cavities, α values in the range 0 . ∼ ∼ [14–17]. Here for the purposes of a simple demonstration, we choose α = 0 .
05 and Q = 20000.Similarly, we will consider an exciton resonance at (cid:126) ν (0) = 2 eV and set the cavity mode resonance frequency, ω (0) ,to be red-detuned away from it (as discussed below, we choose a detuning of 43 meV). These parameters correspondto cavity linewidth of (cid:126) κ ≈ . n ≈
3, we obtain an effective modevolume of V mode ≈ . × nm . TMD Stark shift and exciton-cavity interaction
Here we estimate the value for the dispersive coupling V for a TMD material placed on the photonic crystal cavitydescribed in the main text. First, we note that excitons in monolayer TMDs obey circularly polarized optical selectionrules wherein the excitons around K ( K (cid:48) ) valleys primarily interact with photons of left (right) circular polarization.This valley-circularly polarized light selectivity justifies our use of single flavor of exciton modes (e.g., excitons in the K valley) interacting with a single cavity mode (e.g., left hand circularly polarized cavity mode) in the main text.Indeed, this selectivity is well evidenced in experiment. It not only manifests in selective excitation of excitons in thevalleys (by using circularly polarized light), but also yields a valley Stark effect wherein light of left (right) circularpolarization only shifts the exciton resonances of K ( K (cid:48) ) excitons [8, 12, 13]. For a left-hand circularly polarized field E ( t ) = E (ˆ x cos ωt + ˆ y sin ωt ), excitons in the K valley experience a Stark shift [8, 12, 13]Stark shift = δE = ( M E ) / ∝ E / ∆ , (S20)where ∆ = (cid:126) ( ν (0) − ω ) and M is a material dependent dipole matrix element for the circularly polarized field.In Ref. [8], an optical Stark shift in a single valley of monolayer WS was measured. Ref. [8] reported that a peakStark shift value δE expt = 18 meV was induced by a circularly polarized laser pulse with fluence of 120 µ J/cm andpulse width 160 fs. The detuning between the exciton and laser pulse was ∆ expt = 180 meV. Recalling that theintensity of a circularly polarized field E ( t ) (see above) is I = (cid:15) c | E expt0 | , we estimate peak E expt0 = 5 . × V/min Ref. [8], where (cid:15) is the vacuum permittivity, and c is the speed of light. Here we have estimated peak intensity asfluence / pulse width.The value of V (Stark shift per circularly polarized photon in the cavity) for our TMD in a photonic cavity setupcan be estimated by comparing with the above experiments on the Stark effect in WS [8]. To do so, we first note thatthe electric field amplitude of the circularly polarized cavity mode can be estimated as (cid:126) ω (0) = V mode (cid:15) (cid:15) effcav [ E (0)cav ] ,where (cid:15) eff c is the effective relative permittivity of the cavity. Taking V mode from above, and (cid:15) effcav = 10, we estimatean electric field strength of the circularly polarized cavity mode as E (0)cav = 2 . × V/m. Comparing the cavitymode electric field amplitude with that used in the experiment above [8] and specifying an exciton-cavity detuning∆ cav = 43 meV, we obtain V (Stark shift per circularly polarized photon in the cavity) as V = ( E (0)cav /E expt (cid:1) ∆ cav / ∆ expt × δE expt ≈ . , (S21)1where we have used the proportionality relation in latter part of Eq. (S20). While we have chosen a specific value of∆ cav which yields V ≈ . V ∼ . − . cav .We note that V in the cavity system can also be estimated directly from the dipole matrix element for the circularlypolarized field acting on a TMD (in our case, e.g., WS ). Using Eq. (S20) and the experimental parameters fromRef. [8] discussed above, we find M ≈ . M is within a factorof unity from that obtained from a simple theoretical estimate [8] M theoretical ≈
56 Debye. For the exciton-cavityphoton system, we recall that the exciton-cavity photon interaction in Eq. (S1) is g = M E (0)cav ≈ .
15 meV. Using V = g / cav from Eq. (S3) we obtain V ≈ . Exciton drive strength, F X In this section, we connect the exciton driving field intensity (in kW/cm ) to the F X driving strength (in meV) usedin the main text. The strength of the exciton drive F X in Eq. (3) of the main text can be obtained in the same fashionas that described above via F X = M E d X , where E d X is the amplitude of a circularly polarized driving electric field E drive X ( t ) = E d X (cos ν d t ˆ x + sin ν d t ˆ y ). The intensity of the (exciton) driving field is given by I X = (cid:15) (cid:15) eff c ( E d X ) , where (cid:15) eff is an effective relative permittivity that depends on the geometry of the incident irradiation. For illustration, wehave taken (cid:15) eff = 1. For a driving intensity of I X = 0 . − . , we estimate a circularly polarized drivingelectric field strength as E d X = 0 . − . × V/m. Using the value of M above we obtain F X ≈ . − .
072 meV.We note parenthetically that these values of drive electric field are far smaller (two orders of magnitude smaller) thanthe electric field amplitude of the cavity mode discussed above.
Exciton recombination rate
The recombination rate of excitons in TMDs is typically in the range of 0 . ∼ several ps at cryogenic temperatures[28–32]. To illustrate MSE, we used (cid:126) Γ = 1 meV.
Cavity drive strength, F In this subsection, we relate the incident cavity drive power P to the cavity drive parameter F used in the maintext. To do so we first briefly review classical coupled mode theory [41]. In a cavity with cavity eigenmodes e i ( r ),the electric field profile in the cavity can be described by E cavity ( r , t ) = Re (cid:2) (cid:80) i C i ( t ) e i ( r ) (cid:3) ; here i indexes the cavityeigenmodes. C i ( t ) is a time-varying amplitude that describes the degree to which each of the cavity eigenmodes areoccupied over time. According to coupled mode theory, the classical cavity mode fields coupled to an external driveevolve according to [41] dC i ( t ) dt = − iω i C i ( t ) − κC i ( t ) − iξf + i ( t ) , (S22)where f + i ( t ) is the amplitude of the incident drive field that couples to the cavity mode i . Here ω i is the resonancefrequency of cavity mode i , κ is the decay rate of the cavity mode, and ξ = √ κ represents the strength of thecavity-incident driving field coupling [41].We now focus on a planar photonic cavity where there are two degenerate modes in the plane, which we label e x and e y , with common frequency ω x = ω y = ω . Here e x and e y are cavity modes that are polarized in the x and y directions. We will also assume that the system is isotropic in-plane so that κ x = κ y = κ . Further, we focus oncavities that primarily decay radiatively, consistent with a high quality factor cavity. We remark that, following thestandard convention [41], the cavity modes profiles e x,y ( r ) can be normalized in such a way that U ( t ) = (cid:88) i | C i ( t ) | = | C x ( t ) | + | C y ( t ) | , P = (cid:88) i | f + i ( t ) | = | f + x ( t ) | + | f + y ( t ) | , (S23)where U ( t ) is the energy in the cavity and P is the incident driving power. To describe the driving of the cavity by acircularly polarized beam we will use f + x ( t ) = f cos ω d t and f + y ( t ) = f sin ω d t .We now turn to the quantum mechanical description of the driven cavity: H cav = (cid:126) ω ( a † x a x + a † y a y ) + F √ a † x + a x ) cos ω d t + F √ a † y + a y ) sin ω d t, (S24)2where a x,y are annihilation operators for the x, y polarization cavity photon modes (in modes e x,y ), and the lasttwo terms describe a circularly polarized driving with amplitude F . The latter terms describe the full Rabi-typedrive-cavity coupling, see below for discussion. Here we have dropped the vacuum energy of the cavity mode as itacts as a constant energy off-set; including it does not alter our conclusions/estimate for F below.In the same fashion as described in the main text, the evolution of the cavity can be described by the masterequation ∂ t ρ cav ( t ) = i (cid:126) [ ρ cav ( t ) , H cav ] − I [ ρ cav ( t )] , (S25)where I ( ρ ) = κ (cid:2) a † x a x ρ − a x ρa † x + ρa † x a x + ( x → y ) (cid:3) tracks the decay of the mode.In order to track the evolution of the amplitudes above, we define the mode amplitude operators (cid:98) C i = C (0) i a i , for i = x, y ). Following Eq. (S23) above, we have U = (cid:80) i (cid:104) (cid:98) C † i (cid:98) C i (cid:105) = | C (0) x | (cid:104) a † x a x (cid:105) + | C (0) y | (cid:104) a † y a y (cid:105) . This energy mustcorrespond to the energy of the cavity obtained directly from the first term of Eq. (S24): (cid:126) ω (cid:104) a † x a x + a † y a y (cid:105) . As aresult, we obtain C (0) = | C (0) x | = | C (0) y | = √ (cid:126) ω .The temporal evolution of the cavity amplitude (cid:104) (cid:98) C i ( t ) (cid:105) = Tr (cid:2) (cid:98) C i ρ cav ( t ) (cid:3) can be described by multiplying (cid:98) C i intoEq. (S25) and taking the trace. Recalling the commutator identities [ a † a, a ] = − a , [ a, a † ] = 1, and cyclically permutingthe operators, we obtain d (cid:104) (cid:98) C x ( t ) (cid:105) dt = − iω (cid:104) (cid:98) C x ( t ) (cid:105) − κ (cid:104) (cid:98) C x ( t ) (cid:105) − i F C (0) √ (cid:126) cos ω d t, d (cid:104) (cid:98) C y ( t ) (cid:105) dt = − iω (cid:104) (cid:98) C y ( t ) (cid:105) − κ (cid:104) (cid:98) C y ( t ) (cid:105) − i F C (0) √ (cid:126) sin ω d t. (S26)Relating (cid:104) (cid:98) C i ( t ) (cid:105) with C i ( t ) in Eq. (S22), we see that Eq. (S26) directly corresponds to Eq. (S22). This yields theassociation F C (0) / √ (cid:126) = f ξ . As a result, we find that the value of F can be directly estimated from the inputdriving power F = √ (cid:126) f ξ/A (0) = 2 (cid:112) (cid:126) P /Q ≈ (cid:112) (cid:126) IL /Q, (S27)where in the last line we have estimated the incident driving power from the laser intensity P = IL where L is theincident area of the cavity; here I is the laser intensity. The incident area can be estimated from the mode volume V mode = L . For the parameters for mode volume chosen, we arrive at L ≈
78 nm.We remark that Eq. (S24) is the full Rabi Hamiltonian for a cavity mode driven by an external circularly polarizedfield. We now re-write Eq. (S24) into the basis of circularly polarized operators a L = ( a x − ia y ) / √ a R =( a x + ia y ) / √ H cav = (cid:126) ω ( a † L a L + a † R a R ) + F √ a † x + a x ) (cid:0) e iω d t + e − iω d t (cid:1) + F √ i ( a † y + a y ) (cid:0) e iω d t − e − iω d t (cid:1) = (cid:126) ω ( a † L a L + a † R a R ) + F √ (cid:0) a † x − ia † y + a x − ia y (cid:1) e iω d t + F √ (cid:0) a † x + ia † y + a x + ia y (cid:1) e − iω d t = (cid:126) ω ( a † L a L + a † R a R ) + F (cid:16) a L e iω d t + a † L e − iω d t (cid:17) + F (cid:16) a † R e iω d t + a R e − iω d t (cid:17) . (S28)This formulation allow us to focus on the component with a single circular polarization by discarding the counter-rotating component in the rotating wave approximation.We observe that in the Heisenberg picture, ∂ t a L,R = i/ (cid:126) [ H cav , a L,R ], the operator a L,R evolves as e − iωt . Similarly,the operator a † L,R evolves as e iωt . Thus, the right-handed circularly polarized component (cid:16) a † R e iω d t + a R e − iω d t (cid:17) rotatesabout twice as fast as ω when moving to the rotating frame (as employed in the main text). Therefore, within therotating wave approximation, we can discard the right-handed component and the resulting Hamiltonian is consistentwith Eq. (3) in the main text. Multistable P tot in a TMD photonic cavity In this section, we plot the multi-stable steady state excitonic population as a function of exciton and cavity drives inFig. S1. This displays how the average steady state P tot takes on small values on the order of P tot ∼
1. This indicatesthat the MSE occurs even as approximately one exciton is excited in the entire photonic cavity (corresponding to alow exciton density of order 10 cm − ). Further, we find that jumps in the exciton population (see Fig. S1b) can betuned to be of order unity.3 FIG. S1: Exciton population P tottot