Nonclassical Light from Exciton Interactions in a Two-Dimensional Quantum Mirror
NNonclassical Light from Exciton Interactions in a Two-Dimensional Quantum Mirror
Valentin Walther,
1, 2, 3, ∗ Lida Zhang, Susanne F. Yelin, and Thomas Pohl ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark
Excitons in a semiconductor monolayer form a collective resonance that can reflect resonant lightwith extraordinarily high efficiency. Here, we investigate the nonlinear optical properties of suchatomistically thin mirrors and show that finite-range interactions between excitons can lead to thegeneration of highly non-classical light. We describe two scenarios, in which optical nonlinearitiesarise either from direct photon coupling to excitons in excited Rydberg states or from resonanttwo-photon excitation of Rydberg excitons with finite-range interactions. The latter case yieldsconditions of electromagnetically induced transparency and thereby provides an efficient mechanismfor single-photon switching between high transmission and reflectance of the monolayer, with atunable dynamical timescale of the emerging photon-photon interactions. Remarkably, it turnsout that the resulting high degree of photon correlations remains virtually unaffected by Rydberg-state decoherence, in excess of non-radiative decoherence observed for ground-state excitons in two-dimensional semiconductors. This robustness to imperfections suggests a promising new approachto quantum photonics at the level of individual photons.
I. INTRODUCTION
The ability to couple light and excitons in a semicon-ducting material is foundational to the tremendous devel-opments in solid-state optics and nanophotonics research[1, 2]. Exploring the regime of quantum photonics, inwhich synthetic interactions between photons generatequantum states of light, remains an exciting scientificchallenge, since the optical nonlinearity that underliessuch interactions is weak in most materials. Remarkableadvances have been made by coupling single quantumdots to photonic waveguides [3] or by reaching strongexciton-photon coupling in semiconductor microcavities[4]. The latter has revealed a rich phenomenology of non-linear wave phenomena [5–12], while first indications ofweak photon correlations have been found only recentlyin experiments [13, 14].A potential approach to address this challenge and en-hance optical nonlinearities in semiconductors is to useexcited states of excitons [15]. Here, the increased po-larizability of excitons in excited Rydberg states leadsto greatly enhanced interactions [16–18] that can evenbe large enough to inhibit photon coupling to multipleexcitons within mesoscopic distances [16]. As success-fully demonstrated in experiments with atoms [19–21],this Rydberg blockade mechanism indeed leads to siz-able nonlinearities that can be sufficiently large to induceinteractions and strong correlations between individualphotons. Experiments with Rydberg excitons in Cu Osemiconductors [22–26] have found evidence for Rydbergblockade over distances of up to 5 µ m [22, 27, 28], andindeed suggest a substantial enhancement of optical non-linearities at higher exciton excitation levels [28, 29]. Yet,nonlinear effects have thus far been confined to the do- ∗ Electronic address: [email protected] b) c) a) FIG. 1: a) A quantum light field ˆ R in impinges on a two-dimensional semiconductor. Under strong driving, its largestfraction is transmitted into ˆ R out , while well-separated pho-tons are back-reflected into ˆ L out . b) Atomistic model of amonolayered transition metal dichalcogenide, where a layer oftransition metal atoms (green) is sandwiched between chalco-gen atoms (yellow). c) The exciton resonance is described bybosonic operators ˆ P ( r ), coupled to the laser field at a couplingstrength g with a detuning ∆. This coupling produces a nat-ural decay rate γ = g /c , which can be elevated by additionaldecay and dephasing ¯ γ . main of classical optics, largely due to the overall weakexciton-photon coupling [22] in these systems.Strong light-matter interactions, on the other hand,are possible in a new class of two-dimensional semi-conductors, monolayer transition metal dichalcogenides(TMDCs), that has emerged in recent years and offers a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b new perspectives for the manipulation of light, owing toits promising electrooptical properties [30, 31]. Excitonsin this material feature extraordinarily strong coupling tolight [2], and its in-plane translationally invariance ren-ders this coupling highly mode selective. This yields avery effective mirror [32–34], where a single layer of thematerial can reflect light with more that 80% efficiency,limited only by lattice defects and other non-radiativedecay mechanisms. While the nonlinearities of TMDCmonolayers due to heat diffusion [34] or collisional inter-actions between ground-state excitons [35–38] are gener-ally small, the possibility to realize a controllable mirrorat the smallest possible scales offers exciting perspectivesfor electro-optics applications [39–41] and optomechanics[42].Here, we explore the combination of finite-rangeinteractions between excitons in excited states andhighly coherent light-matter interactions possible in two-dimensional semiconductors. We analyze the resultingphotonic nonlinearities in this system by solving the cor-related quantum many-body dynamics of laser-driven ex-citons with strongly interacting excited states. We iden-tify conditions that afford a mapping to an isolated sat-urable emitter, which provides an intuitive understand-ing of the generated photon-photon correlations in thetransmitted and reflected light. Remarkably, we find asizeable antibunching of transmitted photons for surpris-ingly large dissipation of the interacting Rydberg state,allowing decay rates that can approach and even exceedmeasured linewidths of ground-state excitons in TMDCmonolayers. This robust mechanism for effective photon-photon interactions together with the extraordinary op-toelectronic properties of TMDC monolayers [43] offersa promising outlook for the exploration of quantum pho-tonics applications at the nanoscale. II. TWO-DIMENSIONAL EXCITONS COUPLEDTO LIGHT
Transition metal dichalcogenides are a class of materi-als whose chemical composition MX contains a transi-tion metal (M) such as Mo or W, and chalcogen atoms(X) such as S, Se, or Te [44]. Much like graphene, thesematerials can be isolated into individual monolayers witha hexagonal structure as illustrated in Fig. 1. However,unlike graphene, their monolayers can be direct semicon-ductors [45, 46] and exhibit sizable bandgaps of ∼ . E → ( r , t ) and ˆ E ← ( r , t ) that denote the slowly-varyingelectric-field envelope of the electromagnetic field andyield the photon density operators, ˆ E †→ ˆ E → and ˆ E †← ˆ E ← ,in each mode. In free space, the fields propagate withthe speed of light c , as described by the paraxial waveequation [52]. They couple to the two-dimensional ex-citons in the semiconductor, described by the bosonicoperator ˆ P ( r ⊥ , t ). Here, r ⊥ denotes the two-dimensionalcoordinate within the plane of the TMDC, and we willfrom now on denote three-dimensional spatial argumentsas ˆ E ← ( r ⊥ , z, t ), choosing the z -axis to be orthogonal tothe semiconductor plane and parallel to the light prop-agation axis. Within the rotating wave approximationand neglecting transverse beam diffraction, the Hamilto-nian describing the excitons, the light modes and theircoupling can then be written asˆ H = − ∆ (cid:90) d r ⊥ ˆ P † ( r ⊥ ) ˆ P ( r ⊥ )+ g (cid:90) d r ⊥ ˆ P † ( r ⊥ )[ ˆ E → ( r ⊥ ,
0) + ˆ E ← ( r ⊥ , − ic (cid:90) d r ˆ E †→ ( r ⊥ , z ) ∂ z ˆ E → ( r ⊥ , z )+ ic (cid:90) d r ˆ E †← ( r ⊥ , z ) ∂ z ˆ E ← ( r ⊥ , z ) , (1)where g denotes the light-matter coupling strength. Thisyields the following Heisenberg propagation equations forthe photon dynamics ∂ t ˆ E → ( r ⊥ , z, t ) = − c∂ z ˆ E → ( r ⊥ , z, t ) − ig ˆ P ( r ⊥ , t ) δ ( z ) , (2) ∂ t ˆ E ← ( r ⊥ , z, t ) = c∂ z ˆ E ← ( r ⊥ , z, t ) − ig ˆ P ( r ⊥ , t ) δ ( z ) , (3)which can readily be solved. As illustrated in Fig.1a,one can define the outgoing photon fields ˆ R out ( r ⊥ , t ) ≡ ˆ E → ( r ⊥ , L, t ) and ˆ L out ( r ⊥ , t ) ≡ ˆ E ← ( r ⊥ , − L, t ) that prop-agate away from the mirror to its left and right at adistance L , respectively. Defining equivalent expressions,ˆ R in ( r ⊥ , t ) ≡ ˆ E → ( r ⊥ , − L, t ) and ˆ L in ( r ⊥ , t ) ≡ ˆ E ← ( r ⊥ , L, t ),for the input fields, and letting the distance L →
0, thesolution of Eqs. (2) and (3) yields a simple set of input-output relations [32, 53]ˆ R out ( r ⊥ , t ) = ˆ R in ( r ⊥ , t ) − i gc ˆ P ( r ⊥ , t ) (4)ˆ L out ( r ⊥ , t ) = − i gc ˆ P ( r ⊥ , t ) (5)as well as the total field at the position of the semicon-ductorˆ E → ( r ⊥ , , t ) + ˆ E ← ( r ⊥ , , t ) = ˆ R in ( r ⊥ , t ) − i gc ˆ P ( r ⊥ , t ) , (6)where we assumed that the system is only driven fromthe left, such that ˆ L in can be omitted. The Heisenbergequation for the exciton operator governed by the Hami-tonian (1) together with Eq. (6) gives ∂ t ˆ P ( r ⊥ , t ) = − ig ˆ R in ( r ⊥ , t ) + ( i ∆ − γ ) ˆ P ( r ⊥ , t ) . (7)where γ = g c is the rate of radiative decay of the ex-citon into the forward and backward propagating pho-ton modes. Under realistic conditions, defects and non-radiative processes lead to additional dissipation and inparticular cause scattering into other modes. Experi-ments show [33, 34] that such additional losses can bewell accounted for by an phenomenological decay con-stant ¯ γ , such that the exciton dynamics can be describedby ∂ t ˆ P ( r ⊥ , t ) = − ig ˆ R in ( r ⊥ , t ) − Γ2 ˆ P ( r ⊥ , t ) . (8)with an effective complex decay rate Γ = (2 γ + ¯ γ − i ∆).In the following, we will assume a typical experimen-tal situation [14] in which the reflected and transmittedphotons are detected in the same transverse mode as theincident field. Denoting this detection mode by E ( r ⊥ ),one can project into this mode and define new operatorsˆ O = (cid:90) ˆ O ( r ⊥ ) E ∗ ( r ⊥ )d r ⊥ , (9)that describe the occupation of the transverse detectionand input mode E ( r ⊥ ), with (cid:82) | E ( r ⊥ ) | d r ⊥ = 1. In thisway, I out = (cid:104) ˆ L † out ˆ L out (cid:105) describes the linear density of out-going detected photons, while (cid:104)P † P(cid:105) count the numberof excited excitons in the spatial mode E . The equationsof motion then take the simple formˆ R out ( t ) = ˆ R in ( t ) − i gc ˆ P ( t ) , (10)ˆ L out ( t ) = − i gc ˆ P ( t ) , (11) ∂ t ˆ P ( t ) = − ig ˆ R in ( t ) − Γ2 ˆ P ( t ) . (12)Solving this simple set of linear equations in Fourier spacegives the reflection and transmission spectrum R ( ω ) = − g c (Γ + 2 iω ) (13) T ( ω ) = 1 − g c (Γ + 2 iω ) (14)of the TMDC. For resonant cw-driving (∆ = ω = 0), thereflection coefficient R (0) = − (1 + ¯ γ γ ) − ≈ − ¯ γ γ islimited only by non-radiative losses. The strong exciton-photon coupling of TMDC monolayers can render ra-diative processes dominant ( γ > ¯ γ ), which has madeit possible to realize reflection coefficients of more than80% [33, 34]. Under such conditions, residual trans-mission with T (0) = ¯ γ/ (2 γ + ¯ γ ) and photon losses 1 − | R (0) | − | T (0) | = 4 γ ¯ γ/ (2 γ + ¯ γ ) are greatly sup-pressed.The phase, φ ( ω ) of the complex reflection coefficient R ( ω ) = | R ( ω ) | e iφ ( ω ) contains information about thephoton-exciton interaction dynamics. For a long inputpulse with a spectral width well below 2 γ + ¯ γ , the re-flected light ˆ L out ( t ) ≈ R (0) ˆ R in ( t − ∆ τ ) (15)has a pulse delay of∆ τ = − dφdω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 = 2(¯ γ + 2 γ )4∆ + (¯ γ + 2 γ ) . (16)We can interpret this time as the characteristic timethe photon interacts with the monolayer and is trans-ferred to an excitonic excitation. For a perfect material(¯ γ = 0) under resonant driving (∆ = 0), the delay timeor photon-interaction time is expectedly given by the ra-diative lifetime ∆ τ = 1 /γ of the exciton.We can use this time to scale the equations of motionby introducing dimensionless times γt → t and lengths( γ/c ) r → r . This yields a simple set of linear equationsˆ R out ( t ) = R in ( t ) − i ˆ P ( t ) , (17)ˆ L out ( t ) = − i ˆ P ( t ) , (18) ∂ t ˆ P ( t ) = − i R in ( t ) − Γ2 ˆ P ( t ) , (19)that relate the dimensionless output fields R out → gc R out and L out → gc L out to the incident light field R in → gc R in with only two remaining parameters ¯ γ/γ and ∆ /γ thatdetermine the dimensionless width ˜Γ = Γ /γ . III. CLASSICAL OPTICAL NONLINEARITIESDUE TO FINITE-RANGE EXCITONINTERACTIONS
Nonlinear optical processes can alter the above behav-ior, whereby the absorption of one photon can affect theoptical response of the system to additional photons andbreak the conditions that otherwise lead to perfect reflec-tion. Possible mechanisms that have been investigatedinclude lattice heating due to photon absorption [34],phonon coupling and exciton-exciton interactions [32].The latter usually give rise to relatively weak nonlinear-ities, due to the typically short range of exciton interac-tions. This length scale is, however, enhanced for exci-tons in excited states [15, 17], as observed for TMDCs in[54–56]. Very highly excited states of excitons have beenobserved in bulk Cu O semiconductors [22] and foundto generate large optical nonlinearities due to an exci-tation blockade of multiple Rydberg excitons [28]. Sucha blockade is caused by the van der Waals interaction U ( r ) = C /r between excitons. The van der Waals co-efficient C ∼ n increases rapidly with the principalquantum number n [16]. The resulting energy shift ofexciton-pair states can inhibit the excitation of multi-ple excitons within a blockade radius R bl , at distances r for which U ( r ) exceeds linewidth of the excitation pro-cess. Experimental evidence for the increase of the block-ade radius with n and actual values of R bl ∼
25 nm at n = 2 in TMDCs [56] suggests that blockade radii closeto R bl ∼ µ m may be achieved for n = 10 [57].Interactions are included in the exciton propagationequation (8) according to ∂ t ˆ P ( r ⊥ ) = − i ˆ R in ( r ⊥ ) − ˜Γ2 ˆ P ( r ⊥ ) (20) − iγ − (cid:90) d r (cid:48)⊥ U ( | r ⊥ − r (cid:48)⊥ | ) ˆ P † ( r (cid:48)⊥ ) ˆ P ( r (cid:48)⊥ ) ˆ P ( r ⊥ ) , and lead to a correlated exciton dynamics that typicallyprevents a simple analytical solution as in the previoussection. In order to gain some more intuitive insightsinto the resulting optical response, we first analyze theclassical-optics limit for a coherent input field with a con-stant amplitude R in = (cid:104) ˆ R in (cid:105) , where (cid:104) ˆ R in ˆ P (cid:105) = R in (cid:104) ˆ P (cid:105) , (cid:104) ˆ R in ˆ P † ˆ P (cid:105) = R in (cid:104) ˆ P † ˆ P (cid:105) , etc. Under these assumptions,Eq. (20) results in an infinite hierarchy of equations forproducts of exciton operators, which can be truncated atthe lowest nonlinear order [15] to obtain an exact descrip-tion of the third-order susceptibility χ (3) of the reflectedfield as L out ( r ⊥ ) = χ (1) R in ( r ⊥ )+ (cid:90) d r (cid:48)⊥ χ (3) ( | r ⊥ − r (cid:48)⊥ | ) | R in ( r (cid:48)⊥ ) | R in ( r ⊥ ) , (21)where χ (1) = R (0) corresponds to the linear reflection co-efficient discussed in the preceding section and the non-linear susceptibility is given by χ (3) ( r ⊥ − r (cid:48)⊥ ) = 16˜Γ | ˜Γ | iU ( | r ⊥ − r (cid:48)⊥ | )Γ + iU ( | r ⊥ − r (cid:48)⊥ | ) . (22)The third-order susceptibility χ (3) ( r ) acts as an effec-tive interaction potential for two photons and its specificform affords a simple interpretation. At large distances, r , for which the exciton interaction potential U ( r ) (cid:28) | Γ | remains well below the linewidth, the nonlinear kernelscales as χ (3) ( r ) ∼ U ( r ) and directly reflects the exci-ton interaction in this perturbative regime. However,in the opposite limit U ( r ) > | Γ | for distances r < R bl within the blockade radius, the susceptibility approachesa constant χ (3) ≈ | ˜Γ | = − χ (1) / | ˜Γ | that reduces theoverall reflection due to the blocking of multi-photon re-flection at distances below R bl . The blockade radius R bl = | C / Γ | / follows directly from the denomina-tor in Eq. (22). Its magnitude relative to the waist ofthe input beam determines the extent of nonlinear ef-fects. To be specific, we chose a Gaussian mode profile E ( r ⊥ ) = e − r ⊥ / σ / ( √ πσ ) and consider cw-driving withan amplitude R in ( r ⊥ ) = R in E ( r ⊥ ). Upon projecting -1 beam radius p σ/R bl F ( σ / R b l , ) a) 10 -1 beam radius p σ/R bl g ( ) ( ) b) FIG. 2: Reflection properties for various beam sizes. a) Non-linear reflection, as characterized by the dimensionless func-tion F , reaches a maximal plateau value when σ < R bl andfalls off for larger beam radii, shown in terms of the function F defined in Eq. (23) for resonant driving ( ϕ = 0). The solidline represents the real part (nonlinear reflection), the dashedline represents the imaginary part (nonlinear refraction). b)The equal-time correlation function g (2) of reflected light isstrongly suppressed for small beams radii. onto this mode and carrying out some of the integralsin Eq. (21), one finds for the reflected field L out = − R in + 16˜Γ | ˜Γ | |R in | R in F (cid:18) σR bl , ϕ (cid:19) , where we have defined the function F (cid:18) σR bl , ϕ (cid:19) = (cid:18) R bl σ (cid:19) (cid:90) ∞ ± ix e iϕ e − (cid:16) x R bl √ σ (cid:17) x d x. (23)and ϕ = arg(˜Γ) denotes the phase of the complexlinewidth, which vanishes for resonant excitation. Thedifferent signs correspond to repulsive (+) and attrac-tive ( − ) exciton interactions, but for simplicity we focushere on the repulsive case. The function F determinesthe nonlinear reflection and gives a particularly simpleexpression R = − |R in | F (cid:18) σR bl , (cid:19) . (24)for an ideal monolayer (¯ γ = 0) under resonant driving.Fig. 2(a) shows its dependence on the beam radius √ σ . For large values σ (cid:29) R bl , multiple excitons can beindependently created within the illuminated area ∼ σ ,which reduces the effects of the nonlinearity and, hence,requires higher light intensities, |R in | , to alter the reflec-tion. In this regime, the interactions also affect the shapeof the output mode [see Eq. (21)], and therefore transferpopulation out of the incident mode E ( r ⊥ ). This is re-flected in the imaginary part of F , which increases withdecreasing σ as the effect of the nonlinearity becomesstronger. Eventually, however, nonlinear losses decreaseagain once the beam size decreases below the blockaderadius R bl . In this regime, incident photons only probethe constant inner part of χ (3) ≈ − χ (1) / | ˜Γ | , whichtherefore leads to shape-preserving reflection accordingto Eq. (21). Concurrently, the nonlinear reflection coef-ficient saturates to its maximal value, since a single ab-sorbed photon inhibits the reflection for any additionalphotons in this full-blockade limit σ < R bl . Such strongphoton-photon interactions can also give rise to photoniccorrelations, as we shall discuss in the following. IV. QUANTUM STATES OF LIGHT
Correlations between reflected photons can be quanti-fied by the two-time correlation function g (2)refl . ( τ ) = (cid:104) ˆ L † out ( t ) ˆ L † out ( t + τ ) ˆ L out ( t + τ ) ˆ L out ( t ) (cid:105)(cid:104) ˆ L † out ( t ) ˆ L out ( t ) (cid:105)(cid:104) ˆ L † out ( t + τ ) ˆ L out ( t + τ ) (cid:105) , (25)which only depends on the time difference between suc-cessive photon detection events once the system hasreached its long-time steady state ( t → ∞ ) under cw-driving. Using Eq. (18), g (2)refl . ( τ ) can be related directlyto temporal correlations of the generated excitons. Inparticular, we can obtain equal-time correlations ( τ = 0)from operator products of ˆ P ( t ), using the truncation ap-proach outlined in the previous section. As shown inFig. 2(b), the obtained dependence of g (2)refl . ( τ ) on thewidths of the incident laser beam shows similar behav-ior as discussed above for the nonlinear reflection co-efficient. For large values of σ > R bl , the incidentlight can simultaneously excite multiple excitons at dis-tances r > R bl , which facilitate the simultaneous reflec-tion of multiple photons in the transverse mode E ( r ⊥ )such that g (2)refl . (0) >
0. Eventually, the correlation func-tion approaches 1 with increasing waist of the incidentbeam, as multiple excitons can be excited unimpededlyfor σ (cid:29) R bl and, therefore, give rise to uncorrelated pho-ton reflection. However, in the opposite limit of σ < R bl ,the incident light in the driving mode E ( r ⊥ ) can onlygenerate a single exciton at a time while any furtherexcitation is blocked by the interaction. As a conse-quence, a single reflected photon effectively blocks reflec-tion of any further light, which passes the monolayer un-affected. The resulting quantum mirror, thus acts as anefficient single-photon filter that generates anti-bunchedlight with g (2)refl . (0) = 0, as shown in Fig. 2(b).In this strong-blockade limit, in which the interaction U ( r ) exceeds all other energy scales across the illumi-nated area, one can simplify Eq. (20) and describe the exciton dynamics by ∂ t ˆ P = − i R in − ˜Γ2 ˆ P − i ˜ U ˆ P † ˆ P ˆ P , (26)in terms of an effective interaction potential ˜ U = const . that extends over the entire array. Taking the subsequentlimit ˜ U → ∞ suppresses all contributions from multipleexcitons, and we, for simplicity, consider resonant exci-tation and neglect additional broadening. This permitstruncation of the hierarchy and an adiabatic eliminationof the corresponding interaction terms, which leads to aclosed set of propagation equations ∂ t (cid:104) ˆ P(cid:105) = − i R in − ˜Γ2 (cid:104) ˆ P(cid:105) + 2 i R in (cid:104) ˆ P † ˆ P(cid:105) , (27) ∂ t (cid:104) ˆ P † (cid:105) = i R in − ˜Γ ∗ (cid:104) ˆ P † (cid:105) − iR in (cid:104) ˆ P † ˆ P(cid:105) , (28) ∂ t (cid:104) ˆ P † ˆ P(cid:105) = i R in ( (cid:104) ˆ P(cid:105) − (cid:104) ˆ P † (cid:105) ) − ˜Γ + ˜Γ ∗ (cid:104) ˆ P † ˆ P(cid:105) (29)for the excitons. This simple set of equations describesthe dynamics of an effective spin-1 / S z = ˆ P † ˆ P − / S x = ( ˆ P + ˆ P † ) /
2. Sim-ilar to so-called Rydberg super-atoms [58], in which theRydberg blockade of an atomic ensemble enables strongphoton interactions with a single collective atomic excita-tion [59], the present setting thus realizes strong couplingbetween individual photons in a single photonic mode toa single effective saturable exciton.On resonance and without additional decay (∆ = ¯ γ =0), the steady-state expectation values (cid:104) ˆ P(cid:105) = − i R in R (cid:104) ˆ P † ˆ P(cid:105) = R R (30)yield the nonlinear reflection L out = −R in / (1 + 2 R ) = −R in + 2 R + O ( R ), in agreement with the third-order result Eq. (24) of the previous section. Note that (cid:104) ˆ P † ˆ P(cid:105) (cid:54) = (cid:104) ˆ P † (cid:105)(cid:104) ˆ P(cid:105) , which indicates the emergence ofphoton-photon correlations down to the lowest nonlin-ear order in the driving field R in . The photon correlationfunction is readily obtained from Eqs. (27)-(29) using thequantum regression theorem, giving the known result g (2)refl . ( τ ) = e − τ − (cid:16) τ √ κ (cid:17) √ κ − cosh (cid:18) √ κ τ (cid:19) + 1(31)for a driven two-level system [60], where the constant κ = 1 − R is determined by the driving field inten-sity R . For weak fields ( κ > g (2) ( τ ) → γ (Fig. 3). At higher incident intensities for which κ < g (2)refl . ( τ ) undergoes damped oscillations with a frequency ∼ √− κ . While the damping time scale is set by the ra-diative decay rate γ , the oscillation frequency increases as FIG. 3: Photon correlations in reflection and transmission. a)Photon correlations expressed in g (2)refl . ( τ ) for R in = 0 . R in = 5 (red) show strong antibunching and a periodicstructure of minima and maxima. b) The transmitted light atthe same parameters is strongly bunched under weak drivingbut exhibits only small oscillations around the long-time limitfor strong driving. ∼ R in , reflecting the coherence of the underlying single-body Rabi oscillations in the limit of strong driving [60].Most importantly, the outgoing light exhibits completeanti-bunching regardless of the driving intensity due tothe interaction blockade of simultaneous reflection, asdiscussed above.This picture is confirmed by the correlation function g (2)trans . ( τ ) = − e − τ ( κ − √ κ (cid:20) ( κ + 3) sinh (cid:18) √ κτ (cid:19) +( κ − √ κ cosh (cid:18) √ κτ (cid:19)(cid:21) + 1 (32)of the transmitted light, described by ˆ R out . At low in-tensities, the transmitted light is strongly bunched, with g (2)trans . (0) diverging as ∼ / (4 R ). Since transmissionthrough an otherwise perfectly reflecting monolayer isonly possible via exciton-exciton interactions, photonscan only be transmitted simultaneously, leading to thelarge antibunching displayed in Fig.3(b). However, sincea single generated exciton blocks reflection of all subse-quently incident photons the mirror saturates for high in-tensities and largely transmits the incident coherent fieldsuch that g (2)trans . (0) ≈ / R quickly approaches unitywith increasing driving intensity. Under strong coherent driving, the nonlinear mono-layer, therefore, transmits coherent radiation with weakcorrelations, while its reflected light is a highly non-classical train of antibunched single-mode photons. V. TWO-PHOTON DRIVING ANDELECTROMAGNETICALLY INDUCEDTRANSPARENCY
While the interaction between excitons is enhanced forexcited states, their coupling to light ( g ) tends to weakenwith increasing principal quantum number n . A stronglight matter coupling can, however, be maintained byusing an additional control laser field. More specifically,this can be achieved via a two-photon coupling of two dis-tinct exciton states as illustrated in Fig. 4. Hereby, theincident probe field with amplitude R in generates exci-tons described by the bosonic field ˆ P ( r ⊥ ), as introducedabove, while the control laser couples these excitons to ahigher lying excited state with Rabi frequency Ω. Denot-ing the bosonic field operator for these Rydberg excitonsby ˆ S , this adds the following Hamiltonianˆ H c = − δ (cid:90) d r ⊥ ˆ S † ( r ⊥ ) ˆ S ( r ⊥ ) (33)Ω (cid:90) d r ⊥ (cid:16) ˆ P † ( r ⊥ ) ˆ S ( r ⊥ ) + ˆ S † ( r ⊥ ) ˆ P ( r ⊥ ) (cid:17) to the light-matter Hamiltonian introduced in Eq. (1),where δ denotes the total detuning of the two-photontransition to the excited Rydberg state.Following a similar calculation as in section II, we nowobtain for the transmission spectrum of the monolayer T ( ω ) = 1 + 2 i ( δ − ω )2Ω + (2 ω − i ˜Γ)( δ − ω ) . (34) a) b) FIG. 4: Photon dynamics using electromagnetically inducedtransparency. a) The semiconductor is transparent under con-ditions of weak driving. Stronger driving breaks EIT andleads to strong reflection off the exciton resonance. The re-sulting individual photons emerge in reflection with a timeseparation of γ/ Ω . b) To establish EIT, the exciton reso-nance is coupled via a second laser to a high-lying Rydbergstate, described by ˆ S , at Rabi frequency Ω. The upper stateis quasi-stable, limited only by a small total decay rate γ ryd . While this coincides with Es. (14) for Ω = 0, a finite con-trol field leads to a vanishing reflection coefficient on two-photon resonance, δ = ω = 0. This is a direct manifesta-tion of electromagnetically induced transparency (EIT)[61, 62], as has been observed in a range of driven three-level systems [63–65], and can be traced back to the es-tablishment of a dark steady state that does not containexcitons in low-lying states ( ˆ P ). From Eq. (34) we obtaina simple expression ∆ τ = 1Ω , (35)for the group delay of resonantly transmitted photons,which can now be controlled by the Rabi frequency Ωand extends well beyond the values given by Eq. (16)for the two-level mirror discussed above. This delay ∆ τ corresponds to the characteristic times for which a trans-mitted photon is transferred into a Rydberg exciton and,therefore, directly affects the dynamics of the optical non-linearity. Neglecting the comparably weak interactionsbetween the low lying exciton states, the two-photon res-onant dynamics ( δ = 0) of the excitons is now describedby the coupled equations ∂ t ˆ P = − i R in − ˜Γ2 ˆ P − i Ω ˆ S (36) ∂ t ˆ S = − i Ω ˆ
P − i ˜ U ˆ S † ˆ S ˆ S , (37)where we take the limit ˜ U → ∞ to obtain the interactionblockade of multiple Rydberg excitons within the illumi-nated area of the monolayer. The situation is, however,more complex than in the previous section, since the in-teraction does not confine the number of excitons in lowlying states. Thus, one has to solve the driven and corre-lated many-body dynamics of multiple excitons coupledto their strongly interacting excited state. Starting fromEqs. (17), (36), and (37), this can be expressed in an infi-nite hierarchy of equations for operator products for thetwo types of excitons ( ˆ P and ˆ S ) along with the transmit-ted photon field ( ˆ R out ). Defining the correlators A n,q = (cid:104) ( ˆ R † out ) n ˆ S † ˆ R q out (cid:105) , (38) B n,q = (cid:104) ( ˆ R † out ) n ˆ S ˆ R q out (cid:105) , (39) C n,q = (cid:104) ( ˆ R † out ) n ˆ S † ˆ S ˆ R q out (cid:105) , (40) D n,q = (cid:104) ( ˆ R † out ) n ˆ R q out , (cid:105) (41)this hierarchy can be written in closed form as ∂ t A n,q = − ( n + q ) A n,q + Ω D n +1 ,q − Ω R in D n,q (42)+ 2Ω R in C n,q − C n +1 ,q − q Ω C n,q − ∂ t B n,q = − ( n + q ) B n,q + Ω D n,q +1 − Ω R in D n,q (43)+ 2Ω R in C n,q − C n,q +1 − n Ω C n − ,q ∂ t C n,q = − ( n + q ) C n,q − Ω R in ( A n,q + B n,q ) (44)+ Ω B n +1 ,q + Ω A n,q +1 a)b) FIG. 5: Correlations in transmitted light under conditions ofEIT. a) Numerical solution of Eqs. (42)-(45) shows slowingoscillations for descreasing Ω. b) Fluctuations intensify bothin frequency and in number with growing R in . ∂ t D n,q = − ( n + q ) D n,q − n Ω A n − ,q (45) − q Ω B n,q − , where D , = 1 and ∆ = ¯ γ = 0 has been assumed for sim-plicity. For any finite input power |R in | , the solution ofthis set of equations converges for sufficiently large co-efficient matrices A n,q , B n,q , C n,q , and D n,q . We canthus calculate the steady-state expectation values anduse the quantum regression theorem to determine thetwo-photon correlation functions from Eqs. (42)-(45) fora finite set of equations with n, q < ν max , and subse-quently verify convergence of the result with respect to ν max .As shown in Fig. 5, the transmitted light is strongly an-tibunched and the photon correlation function g (2)trans . ( τ )exhibits damped oscillations at finite times. This re-versed response as compared to the previously discussedcase with a single exciton state, is readily understood bynoting that the linear mirror is now completely transmis-sive, instead of being fully reflective. As a single photonis transmitted through the mirror it generates a Rydbergexciton for a time t ∼ ∆ τ , which blocks EIT for any otherphotons. More specifically, by preventing the formationof the EIT dark state, the Rydberg-exciton blockade ex-poses the strong photon coupling to the low-lying excitonstates, which leads to high reflection and thereby reducesthe simultaneous transmission of multiple photons.Interestingly, we find that the degree of antibunching, ℛ in = Ω = ℛ in = Ω = ℛ in = Ω = τ g t r a n s . ( ) FIG. 6: Comparison of exact solution with adiabatic elimina-tion. After large differences for small τ , agreement is generallygood. g (2)trans . (0), depends on the amplitude R in of the drivingfield as well as the control Rabi frequency Ω. We cananalyze this behavior more systematically by first con-sidering the limit of weak control fields, Ω (cid:28)
1, in whichwe can adiabatically eliminate the dynamics of the inter-mediate states. Neglecting the time derivative in Eq. (36)gives ˆ P = − i ˜Γ R in − i Ω˜Γ ˆ S , (46)and substitution into Eq. (37) yields a single equationfor the Rydberg exciton that can once again be mappedonto an effective spin-1/2 system. This makes it possibleto obtain exact expressions for the steady-state excitondensity (cid:104) ˆ S † ˆ S(cid:105) = R R + Ω , (47)and for the two-photon correlation of the transmittedlight g (2)trans . ( τ ) = 1 − e − τ √ κ (cid:20)
3Ω sinh (cid:18) √ κ Ω2 τ (cid:19) (48)+ √ κ cosh (cid:18) √ κ Ω2 τ (cid:19)(cid:21) , where we have set ∆ = 0 for simplicity and where κ = Ω − R . We see that the additional control-fieldcoupling now makes it possible to independently tune thecharacteristic correlation time ∼ Ω − and the oscillationfrequency Ω √− κ , by varying the control and probe fieldamplitudes, Ω and R in . In particular, the weak-field limitΩ (cid:28) a)b) FIG. 7: Effects of Rydberg decay γ ryd on light correlations.a) Examples of the correlation function for different γ ryd at R in = 1 and Ω = 0 .
3. b) Zero-time antibunching in thetransmitted photons tends to decrease with γ ryd and R in butremains at remarkably high levels even in the presence of largeRydberg decay. not too strong intensities of both applied laser fields,Ω , R in <
1. We can gain a better understanding of theobserved deviations at short times by considering theperturbative solution of Eqs. (42-45) for small drivingstrengths, R in (cid:28)
1. The obtained Rydberg-exciton den-sity (cid:104) ˆ S † ˆ S(cid:105) = 1Ω R + 2 (cid:0) Ω − Ω − (cid:1) Ω (Ω + 1) R + O ( R ) (49)establishes Ω (cid:28) g (2)trans . (0) = Ω (1 + Ω ) (50) − (cid:0) Ω (cid:0) − − − (cid:1)(cid:1) (cid:16) (Ω + 1) (Ω + 2) (Ω + 24Ω + 12) (cid:17) R → Ω + 2 R , (51)shows that weak control and probe field amplitudes Ωand R in indeed permit generating strongly antibunchedlight as predicted in the adiabatic limit.A final important factor is the linewidth of the excitedexciton state. While the radiative Rydberg-state cou-pling is known to decrease with increasing principal quan-tum number, the influence of defects and non-radiativedecay processes might remain substantial and limit thelinewidth γ ryd of the excited state. We can investigatesuch dissipation effects on the resulting photon correla-tions by including a decay term, − γ ryd ˆ S , in Eq. (37). InFig.7a we show the obtained two-photon correlation ofthe transmitted light for different values of γ ryd . Sur-prisingly, the short-time behavior of the photon corre-lations remains virtually unaffected by excited-state de-coherence, even for values of γ ryd = 0 . γ that are al-ready twice as large as the non-radiative linewidths thathave been measured for ground-state excitons in TMDCmonolayers [33, 34]. As shown if Fig.7b, it turns out thatthe excited-state decay rate can be substantially largerthan this value and still retain significant antibunching ofthe transmitted light. This surprising level of robustnessof the generated photon correlations against Rydberg-state broadening can be understood intuitively from thefact that – even in the presence of additional broad-ening – the control-field coupling to the excited statewill always lower the otherwise near-perfect reflectivitygenerated on the lower exciton transition, driven by theprobe field R in . Hence, the interaction blockade of theexcited-state excitons can still provide an efficient nonlin-ear switching mechanism of the monolayer transmissionand generate strong photon correlations despite substan-tial excited-state broadening that may exceed the decayrate of the low-lying exciton state. Provided that the in-teraction blockade remains effective, the asymptotic so-lution of the equal-time correlation function ( γ ryd (cid:29) , Ωand R in (cid:28) g (2)trans . (0) ≈ − γ ryd + O ( γ − ) , (52)suggests that much larger decay rates on the order of γ ryd ∼ γ still permit the generation of correlated, non-classical light with photon antibunching well below cur-rent values of g (2) (0) ∼ .
95 in semiconductor microcav-ities [13, 14].
VI. CONCLUSION
In this work, we have elucidated the effects of finite-range exciton interactions on the optical properties ofatomistically thin mirrors formed by two-dimensionalsemiconductors. Remarkably, this combination turnedout to permit an exact solution of the underlying many-body problem of interacting excitons coupled to quantumlight fields. This stands in marked contrast to equiv-alent bulk systems [28] or cavity settings [66], where atheoretical description [67–72] beyond the few-photon orsemiclassical limits remains a formidable numerical chal-lenge. We have made use of this property to investigatethe properties of the scattered light and showed that theinteraction-induced nonlinear reflection and transmissionof the semiconductor can generate highly nonclassical states of light. We have proposed a two-photon cou-pling scheme that permits exploiting the strong photon-coupling to ground-state excitons, while a classical con-trol beam is used to efficiently couple these excitons to aninteracting excited state. By realizing conditions of elec-tromagnetically induced transparency, this approach pro-vides an efficient nonlinear switching mechanism betweenhigh transmission and high reflection and, thereby, toconvert a coherent input field into strongly antibunchedphotons. Importantly, the proposed two-photon schemeis surprisingly robust against unavoidable line broaden-ing of the excited Rydberg state, which offers a promisingoutlook on future experiments.While experiments on pristine samples of Cu O [22],have already revealed high-lying Rydberg states withstrong interactions [16] and sizable nonlinear signals [28],equivalent observations for two-dimensional excitons arecurrently limited to lower lying states. Measurements onmonolayer TMDCs have observed excited states of exci-tons [54–57] with assigned principal quantum numbers ofup to n = 11 and found signatures for the enhancementof the induced optical nonlinearities and exciton inter-action range with increasing principal quantum number[56]. A recently measured blockade radius of ∼
25 nm at n = 2 in WSe monolayers [56] suggests blockade radiiof ∼ µ m for moderate principal quantum numbers of n ∼ ...
10. Small beam waists of ∼ µ m are possi-ble with optical fibers [73] and the present calculationsdemonstrate that significant antibunching below previousmeasurements [13, 14] should still be possible for block-ade radii that are 4-6 times smaller than this value, whilefabricated masks and electrostatic gate control [2, 74]may be used to isolate small excitation spots that enablecomplete interaction blockade for even smaller principalquantum numbers.Such coupling of focused in- and outgoing light viaproximate fibre ends also permits creating high-qualityoptical resonators [75] that can lead to transverse con-finement of optical modes, which was recently shown togenerate observable photon correlations with GaAs quan-tum wells [13, 14]. The combination of strong mode con-finement, the mode-selective photon coupling of TMDCmonolayers in an optical resonator [76–78] and the non-linear mechanisms described in this article, thus presentsa promising approach to quantum nonlinear optics in thesolid-state that remains to be explored in future work.Here, the remarkable electro-optical properties of TMDCmonolayers open up a number of interesting possibili-ties. Their strong spin-orbit coupling, for example, leadsto valley-dependent photon coupling with tunable po-larization selection [47]. This, in turn, can generate apolarization-dependent nonlinearity that may give rise topolarization entanglement and offers an enabling mech-anism for generating and manipulating more complexquantum states of light.0 VII. ACKNOWLEDGMENTS
We thank Nikola ˇSibali´c, Nikolaj Sommer Jørgensenand Trond Andersen for useful discussions. This workhas been supported by the EU through the H2020-FETOPEN Grant No. 800942640378 (ErBeStA), by theDFG through the SPP1929, by the Carlsberg Foundationthrough the Semper Ardens Research Project QCooL,by the DNRF through the Center for Complex Quan-tum Systems (Grant agreement no.: DNRF156), and bythe NSF through a grant for the Institute for Theoreti- cal Atomic, Molecular, and Optical Physics at HarvardUniversity and the Smithsonian Astrophysical Observa-tory. SFY would like to thank for support from the NSFthrough the CUA PFC (context of Rydberg 2D arrays)and from the DOE through group grant DE-SC0020115(for applications to excitons).
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