sign in sign up
<<
Home , Polariton, Polaron

PHYSICAL REVIEW X 10, 021011 (2020) Featured in

Interacting -

Li Bing Tan,1,* Ovidiu Cotlet,1 Andrea Bergschneider ,1 Richard Schmidt ,2,3 Patrick Back,1 Yuya Shimazaki,1 † Martin Kroner,1 and Ataç İmamoğlu1, 1Institute for Electronics, ETH Zürich, CH-8093 Zürich, Switzerland 2Max Planck Institute of Quantum Optics, 85748 Garching, Germany 3Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 Münich, Germany

(Received 21 October 2019; revised manuscript received 19 January 2020; accepted 19 February 2020; published 15 April 2020)

Two-dimensional provide an ideal platform for exploration of linear and physics, primarily due to large exciton binding energy and strong - coupling. These features, however, generically imply reduced exciton-exciton interactions, hindering the realization of active optical devices such as lasers or parametric oscillators. Here, we show that electrical injection of itinerant into monolayer molybdenum diselenide allows us to overcome this limitation: dynamical screening of exciton-polaritons by electrons leads to the formation of new termed polaron-polaritons that exhibit unexpectedly strong interactions as well as optical amplification by Bose-enhanced polaron- scattering. To measure the nonlinear optical response, we carry out time- resolved pump-probe measurements and observe polaron-polariton interaction enhancement by a factor of 50 (0.5 μeV μm2) as compared to exciton-polaritons. Concurrently, we measure a spectrally integrated transmission gain of the probe field of ≳2 stemming from stimulated scattering of polaron-polaritons. We show theoretically that the nonequilibrium nature of optically excited quasiparticles favors a previously unexplored interaction mechanism stemming from a -space filling in the screening cloud, which provides an accurate explanation of the strong repulsive interactions observed experimentally. Our findings show that itinerant electron-exciton interactions provide an invaluable tool for electronic manipulation of optical properties, demonstrate a new mechanism for dramatically enhancing polariton-polariton interactions, and pave the way for realization of nonequilibrium polariton condensates.

DOI: 10.1103/PhysRevX.10.021011 Subject Areas: , Quantum Physics, Physics

I. INTRODUCTION or holes [6], and the realization of atomically thin mirrors [7,8]. However, nonlinear optical properties, crucial Monolayer transition metal dichalcogenides (TMDs) for the realization of novel photonic devices ranging from combine strong light-matter coupling and a large number parametric oscillators to exciton-polariton lasers, have been of degrees of freedom with which to manipulate . largely missing in the reported studies. This is at a first Because of the strong Coulomb interaction, glance not surprising since nonlinearities typically scale as constitute the elementary optical excitation and dominate the exciton Bohr radius and hence are suppressed in TMD optical spectra in TMDs. Many novel features of linear monolayers where excitons are strongly bound [6,9,10],as optical properties of TMD excitons have been demon- compared to quasi-2D materials such as gallium arsenide strated, including the optical control of the valley degrees of (GaAs) quantum wells. Consequently, an outstanding freedom using the helicity of the excitation light [1,2] or challenge for the field is to find ways to enhance the external magnetic fields [3–5], electrical control of the exciton-exciton interaction strength, or more precisely, its optical spectrum through injection of itinerant electrons ratio to the exciton radiative decay rate. Theoretical proposals to date include the exploitation of large inter- actions between Rydberg excitons with three-level driving *[email protected] schemes [11] and the reduction of the exciton radiative † [email protected] decay rate by placing a monolayer in a high quality-factor (Q) cavity [12] or, more elegantly, by placing a monolayer Published by the American Physical Society under the terms of at λ=2 distance away from a mirror [13]. However, experi- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to ments have so far only confirmed that exciton-exciton the author(s) and the published article’s title, journal citation, interactions in neutral TMD monolayers are more than an and DOI. order of magnitude weaker than those in GaAs [8,14].

2160-3308=20=10(2)=021011(17) 021011-1 Published by the American Physical Society LI BING TAN et al. PHYS. REV. X 10, 021011 (2020)

In this work, we demonstrate that the presence of itinerant the polaron-LP state, this scattering process is Bose electrons dramatically enhances nonlinear optical properties enhanced [34,35], leading to the observed gain. of TMD monolayers. In an earlier work, we investigated the After describing the experimental setup and reviewing linear optics of doped monolayer MoSe2 and identified the linear optical properties in the presence of nonpertur- exciton- as the relevant quasiparticles [15–17].In bative coupling between excitons and the cavity mode as TMD monolayers, the exciton has an ultralarge binding well as excitons and electrons in Sec. II, we present energy that dominates over the electron the experimental signatures unveiled in nonlinear spectros- and Fermi energy. In consequence, the exciton can be copy, namely the enhancement of nonlinearities (Sec. III) considered as a robust quantum impurity interacting with and polariton amplification in the presence of itinerant a fermionic bath. The itinerant electrons dynamically screen electrons as compared to the charge neutral regime the exciton to form new branches—the attrac- (Sec. IV). We then present the theoretical model and tive and repulsive polaron—each with a renormalized mass calculations of the scattering rates due to residual inter- and energy [15,16]. A simple description of the polaron as a actions in Sec. V. Finally, in Sec. VI, we discuss avenues for superposition of a bare exciton and an exciton dressed with a further work. single electron-hole pair [18] was sufficient to accurately predict the observed in linear spectroscopy. We II. ELEMENTARY OPTICAL EXCITATIONS OF remark that optical excitations in the presence of a 2DEG MoSe AN ELECTRON-DOPED MONOLAYER 2 were previously analyzed both theoretically and experimen- EMBEDDED IN A MICROCAVITY tally, where the attractive polaron was termed a tetron [19] or a -hole pair [20,21]. Our experiments are based on a van der Waals hetero- Here, we use nonlinear spectroscopy to investigate the structure consisting of a MoSe2 monolayer semiconductor residual quasiparticle interactions in the strong cavity- embedded between 2 hexagonal boron nitride (h-BN) coupling regime [22–26] where elementary optical excita- flakes. A top graphene layer allows us to control the tions are polaron-polaritons [15]. We find their effective electron density [Fig. 1(a)]. We place the heterostructure in polariton-polariton interaction strength to exceed that of a tunable zero-dimensional open cavity [36] consisting of a their undressed counterparts by up to a factor of ∼50, and flat distributed-Bragg-reflector- (DBR) coated transparent we demonstrate an unexpected but unequivocal amplifica- substrate and a concave DBR-coated fiber facet [Fig. 1(b)] tion of polaron-polaritons accompanying the interaction- (Q ≃ 1800). All experiments are carried out at induced blueshifts, with gain factors ≳2. helium temperature (see the Appendix A). The enhanced nonlinearities can be attributed to residual Figures 1(c) and 1(d) show the white light transmission interactions between the polaron quasiparticles. Using a spectrum measured for vanishing electron density ¼ −30 ¼ 5 wave function technique, we show how the measured (Vg V) and finite electron density (Vg V), repulsive interaction shift in the presence of itinerant respectively, in the strong-coupling regime as a function electrons can be understood in terms of a phase-space of the cavity length. In both cases, the measured spectra filling (PSF) effect. Here, the strong correlations between exhibit avoided crossings associated with the formation of pump-generated polaritons and electrons, that are associ- half-light, half-matter quasiparticles termed polaritons ¼ −30 ated with the formation of a polaron dressing cloud, lead to [15,37]. In the charge neutral regime at Vg V, an effective depletion of the electronic medium. Thus, we observe an exciton-polariton normal mode splitting additional polaritons created by the probe cannot be of 14 meV. When the monolayer is electron doped ¼ 5 screened with maximal efficiency, resulting in an increase (Vg V), dynamical screening of the excitons by elec- of their energy. Already at the lowest order, our approach trons dramatically alters the nature of elementary optical leads to a remarkable agreement between theory and excitations, leading to the appearance of attractive and experiment: we attribute this in part to the zero-dimensional repulsive exciton-polaron resonances. nature of the cavity mode which ensures that the lowest- To analyze how the exciton-electron interaction modifies energy optically excited state of the coupled system— the ground state of the cavity-coupled system, we start from ¼ þ the lower-polaron-polariton (polaron-LP)—is effectively the Hamiltonian H Hxe Hcav, which can be written as gapped from the higher-lying continuum of polaron states, X X thereby suppressing higher-order interaction processes. ¼ ω † þ ϵ † Hxe kxkxk kekek Residual interactions between polarons and electrons are k k X also responsible for the observed polariton amplification v † † þ 0 ð Þ [27–31]. A fraction of pump-generated polaritons contrib- A xkþqxkek0−qek ; 1 k k0 q ute to the creation of a high-momentum polaron reservoir. ; ; These polarons then relax into the polaron-LP state [32,33] X X ¼ ν † þ Ωð † þ Þ ð Þ by generating additional electron-hole pair excitations in Hcav kckck ckxk H:c: : 2 the electron system. In the presence of a seed population in k k

021011-2 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020)

(a) (b) Here, xk, ek, and ck are the annihilation operators of the exciton, electron, and cavity of momentum k, 2 2 respectively, with ωk ¼jkj =ð2mxÞ, ϵk ¼jkj =ð2meÞ, and 2 νk ¼jkj =ð2mcÞþΔ their energy dispersions. The detun- ing Δ between cavity photons and the excitons is controlled by changing the cavity length. For numerics we use the electron mass me ¼ 0.6m0, the exciton mass mx ¼ 2me, −5 while the cavity mass mc ≈ 10 m0. (c) (d) Note that in the absence of doping, the first term in Eq. (1) describes excitons as elementary excitations in the MoSe2 that exhibit weak mutual residual interactions which we have chosen to neglect here. Including the second and third term takes into account the itinerant electrons and their effective interaction v with excitons when the monolayer is capacitively doped, which we model here as a contact interaction and A is defined as the (e) (f) quantization area. Equation (2) describes the effect of the cavity mode: the first term takes into account the cavity energy and the second term, which is proportional to Ω, denotes the exciton-cavity coupling. The contact interaction v between excitons and electrons is regularized by a UV cutoff Λ. Physically, Λ can be related to the inverse Bohr radius of the exciton. However, assuming that the exciton Bohr radius is the smallest length scale in the problem, one may take Λ → ∞ at the end of the (g) calculation. Since any attractive interaction supports a in 2D, the constant v can be related to the binding energy of the exciton-electron bound state known as the trion: 1 X 1 v−1 ¼ − ; ð3Þ A E þ ωk þ ϵk jkj<Λ T

where ET denotes the trion binding energy. It has been shown that the eigenstates of the interacting FIG. 1. Experimental setup and characterization of the sample. polariton-electron system can be accurately described using (a) Contrast-enhanced optical microscope image of the heterostruc- the Chevy ansatz [15]: ture. The h-BN, graphene, and MoSe2 layers are indicated by dotted lines. (b) Schematic of the sample inside the cavity. The hetero- † jΨpi¼apjΦi structure sits on a flat mirror facing a fiber mirror which together form a cavity. The h-BN and graphene layer thicknesses are X ¼ ϕc † þ ϕ † þ ϕ † † jΦi ð Þ chosen such that the MoSe2 lies in an antinode while the graphene pck pxp pkqxpþq−kekeq ; 4 lies close to a node of the cavity electric field. Applying a gate kq voltage between MoSe2 and graphene provides electron density control. The cavity mode can be tuned in situ by a z-axis nano- where jΦi denotes the ground state of the Fermi sea and the positioner which controls the cavity length. The substrate is mounted cavity. Note that we introduced the polaron-polariton on x-andy-axis nanopositioners which allow for in-plane trans- † creation operator ap, which obeys commutation relations lation. (c),(d) White light transmission spectrum for gate voltages V ¼ −30 VandV ¼ 5 V, respectively. Nonperturbative cou- that are approximately bosonic. The attractive polaron g g jϕ j2 ∼ pling between the cavity mode and optical resonances show up as oscillator strength is given by p and grows as EF anticrossings in the transmission spectrum. The faint lines originate for low doping ðEF

021011-3 LI BING TAN et al. PHYS. REV. X 10, 021011 (2020) depicted in Fig. 1(e), stems predominantly from interfer- In Figs. 2(b) and 2(c), we show the normalized trans- ence effects and indicates that polariton line broadening is mission spectra of the probe pulse around the exciton-LP primarily due to cavity losses [12]. However, the excess and polaron-LP resonances respectively: here, we compare broadening of the polaron-UP as compared to the polaron- the transmission for τ < 0 (see dashed lines) as well as for LP hints toward the presence of residual polaron inter- τ ¼ 2.3 ps and τ ¼ 2.7 ps (see lines), the time delays actions. In the following section, we describe experiments at which the maximum blueshift of the exciton-LP and that probe the transient changes in the transmission polaron-LP are observed. From these data, we extract Δ — spectrum due to the evolution of a polaron subject to these ELP the magnitude of the maximal LP resonance energy residual interactions. shift relative to its value for negative time delay. By We remark that due to the valley selection rules, a σþ- monitoring this differential energy shift, we isolate the (σ−-) polarized photon will excite an exciton in the K (K0) anharmonicity of the polariton resonance arising from valley [see Fig. 1(g)]. When the monolayer MoSe2 hosts a interactions with the pump-field injected polaron-polaritons 2DEG with a Fermi energy of ∼3.3 meV, only the lower- from cumulative changes in the optical response stemming energy -split conduction band in each valley is popu- from multiple pulse excitations that last for timescales lated. Then, a K (K0) valley exciton gets dressed by exceeding the pulse repetition period. In Fig. 2(d), we plot 0 Δ electron-hole pairs formed from the Fermi sea of the K ELP as a function of the polariton density (see Appendix C (K) valley to form an intervalley polaron described by for a description of the method used to determine the Eq. (4). The valley degree of freedom does not play an polariton density). We find a dramatic enhancement of important role in the discussion of Sec. III but is relevant the polariton-polariton interaction strength by a factor ∼50 in Sec. IV. in the presence of itinerant electrons. Specifically, in the case pol 2 of the polaron-LP, gpp ¼ 0.5 μeV μm , while for the exciton- exc ¼ 0 01 μ μ 2 III. ENHANCED POLARITON NONLINEARITIES LP, gpp . eV m . Our experimentally measured pol DUE TO RESIDUAL QUASIPARTICLE value for gpp agrees with the theoretically expected value INTERACTIONS discussed in Sec. V. The bare exciton-exciton interaction exc ¼ 0 08 μ μ 2 The time-resolved pump-probe spectroscopy setup is strength extracted from this result is gxx . eV m ,in shown in Fig. 6. We use a spectrally narrow pump field agreement with earlier measurements [8,14]. (1 meV) with pulse duration of τ ¼ 2.62 0.01 ps to For excitons, it is well understood that at low densities, pump nonlinearities stem mainly from Coulomb interaction. The inject a majority polariton population up to as high as effective exciton-exciton interaction then scales with the n ¼ 2 × 1012 cm−2. The probe field is spectrally broad LP exciton Bohr radius a , which is small in TMDs compared (12 meV) with a pulse duration of τ ¼ 0.3 0.1 ps and B probe to GaAs-based semiconductors. In contrast, for excitons that it injects a minority polariton population in a linear orthogo- form polarons in the presence of an electronic environment, nal polarization unless stated otherwise. We monitor the the effective interaction has two main contributions: PSF pump-induced changes of the probe transmission as a stemming from the finite electron number inside the cavity τ τ ¼ 0 function of time delay . Zero time delay ( ) is defined area and the exchange of an electron-hole pair between two with respect to the leading edge of the pump pulse; i.e., the polarons. Here, we focus on the former since we expect the probe pulse impinges on the sample concurrently with the latter to be negligible in the nonequilibrium case which is leading edge of the pump. We expect the probe transmission relevant for the description of short-lived polaritons in our τ 0 (i) to be unperturbed by the pump for < , (ii) to show a experiments (see Sec. VB). The second term of Eq. (4) tells nonlinear response due to the presence of pump-injected us the effective number of electron-hole pairs used from the 0 τ ≲ ðτ þ τ Þ coherent polaritons for < pol pump , and (iii) to be Fermi sea to dress a single exciton impurity. As the number modified due to interactions with a pump-induced longer- of impurities is increased, each successive exciton gets lifetime high-momentum (incoherent) polaron population dressed by an increasingly depleted Fermi reservoir. This τ ≳ ðτ þ τ Þ for pol pump . depletion of the electron reservoir has two main conse- Figure 2(a) shows a typical pump-probe measurement of quences: (1) effective blueshift of the resonance, as not as the probe transmission as a function of τ. Two distinctive many electrons can participate in the dressing, and (2) reduc- features are observed: a strong blueshift of the polaron-LP tion of the quasiparticle weight. The second effect shows up energy and the amplification of the probe transmission. as a reduced oscillator strength that results in a reduction of We discuss the former in this section and the latter in the the normal mode splitting as well as a blueshift of the LP. next section (Sec. IV). In the first set of experiments, Consistent with the two described processes, we indeed we compare changes to the probe transmission in two experimentally observe both a blueshift of the polaron-LP cases: when the pump field is tuned into resonance with and a reduced splitting. ¼ 0 Δ the exciton-LP (ne ) and the polaron-LP transition We emphasize that the saturation behavior of ELP is ¼ 8 1011 −2 (ne × cm , see Appendix D). The schematic is observed as a function of the polariton density nLP rather shown in the inset of Fig. 2(a). than the incident pump intensity. We estimate nLP in a

021011-4 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020)

(a) (b) (c)

(d) (e)

FIG. 2. Blueshift of the lower polariton due to interactions. (a) A typical time delay scan of the probe transmission spectrum from a linear cross-polarized pump-probe experiment. In this case, the polaron-lower-polariton (LP) is resonantly pumped. Inset: An illustration of the energy level scheme of the upper and lower polariton (UP and LP) branches and the pump and probe fields. The pump field is spectrally narrow and tuned in resonance to the LP while the probe field is spectrally broad and covers both branches. Yellow dashed lines show the evolution of UP and LP resonance wavelengths as a function of time delay. (b) Normalized probe transmission ¼ 8 0 4 1011 −2 spectrum of the exciton-LP under resonant pumping for various polariton densities nLP , 2.1, and . × cm . An offset has been added for clarity. Solid lines are the probe transmission spectrum measured at τ ¼ 2.7 ps. Dashed lines are the average τ < 0 probe transmission spectrum obtained by taking the mean of the transmitted signal at a given wavelength over 5 different negative time delay ¼ 5 7 0 4 1011 −2 values. (c) Analogous to (b) but for the polaron-LP at polariton densities nLP . , 2.6, and . × cm . (d) Energy shift of the exciton- and polaron-LP at τ ¼ 2.7 ps measured with respect to τ < 0 as a function of pump power. Blue shaded area indicates the regime where polariton density and electron density are comparable in the system. The error bars are too small to be visible in the plot. For example, for the points indicated by the red arrows, the error bars are 4 and 9 μeV for the energy shift of the polaron and the exciton, respectively. (e) Evolution of the LP energy shift as a function of time delay. There is a contribution from coherent polaritons at short timescales and one from incoherent polarons at longer timescales. Black dashed line indicates the arrival of the pump pulse as determined in an independent measurement (see Appendix B). consistent way (see Appendix C) to reflect the changes in while the exciton-LP for τ < 0 shows a significant amount the detuning between the lower polariton resonance and the of blueshift as the excitation power is increased, the pump energy as the intensity is changed. In doing so, we polaron-LP resonance energy remains largely unchanged. account for the change in polariton injection efficiency and The observed blueshift of the exciton resonance at τ < 0 Δ remark that the observed saturation of ELP for the could originate from optical doping effects due to strong polaron-polaritons as compared to the exciton-polaritons pump laser illumination, effectively turning the exciton into cannot be due to the energy shifts of the polariton a repulsive polaron. In contrast, in the presence of itinerant resonance. Rather, we attribute the saturation of the electrons, we do not expect to observe significant optical polaron-polariton energy shift to the breakdown of the doping since the carrier density in this regime is fixed by polaron picture as the increasing exciton density becomes the applied voltage. comparable to the electron density [indicated by the blue In Fig. 2(e), we turn our attention to the time delay ≳ 2 1011 −2 Δ Δ region of Fig. 2(d)]. For nLP × cm , ELP dependence of ELP. Two timescales can be seen to continues to increase but with a gentler slope. In this limit, dominate the behavior. The resonance first blueshifts we are dealing with a Bose-Fermi mixture consisting of monotonically as it follows the excitation pulse shape degenerate electrons and polaritons and we expect the and reaches its peak at τ ∼ 2.7 ps before it starts to decay, simple Fermi-polaron model to be invalid. consistent with the evolution of coherent polariton pop- We note that signatures of aforementioned cumulative ulation resonantly injected by the pump. Then, a second long-timescale changes on polariton spectra are visible in mechanism which builds up over ∼12 ps takes over that the data at τ < 0 [dashed lines in Fig. 2(b)]. Remarkably, eventually decays over a much longer time. The buildup of

021011-5 LI BING TAN et al. PHYS. REV. X 10, 021011 (2020) the second peak grows noticeably more prominent as power is increased. We tentatively attribute this additional blue- shift to the feeding of high-momentum polaron states from (a) higher-energy optical excitations that are either generated by two-photon absorption or by Auger processes [38]. Since high-momentum polarons contribute to PSF as well, they also lead to a blueshift of the polaron-LP resonance. In this scenario, the rise time of the blueshift should be determined by the feeding time from higher-energy states populated by two-photon absorption and/or the Auger recombination rate, while the decay indicates the loss rate (b) (c) of these high-momentum polarons. For the exciton-LP, there is no clear evidence of an incoherent contribution which is likely due to the absence of itinerant carriers (see Appendix F).

IV. POLARITON GAIN A second striking feature in Fig. 2(a) is the enhancement of the polaron-LP transmission for 2 < τ < 6 ps as com- (d) (e) pared to τ < 0 transmission. The magnitude of the increased transmission cannot be explained by a simple change in the cavity-polaron detuning as discussed in the previous section, but rather suggests an amplification of polaritons. In Fig. 3(a), we show example line cuts for the exciton and polaron-polariton spectrum for two different τ contrasted with the typical spectrum at τ < 0. We define the net transmission gain at every τ by the ratio of the integral FIG. 3. Polariton amplification observed in transmission. under the shaded regions to the average integral at τ < 0 (a) Solid lines show the probe transmission spectrum under (area under the dotted lines). A simple change in the cavity resonant pumping of the exciton lower-polariton and polaron-LP detuning would not lead to net gain deviating from 1 since at different time delays τ. Dashed lines show the average τ < 0 the cavity content of the two polariton branches must probe transmission spectrum obtained by taking the mean of the always be unity. Therefore, net spectrally integrated gain transmitted signal at a given wavelength over 5 different negative exceeding unity suggests an amplification of the probe time delay values. (b) Evolution of the total gain for polaron-LP field. In semiconductor-microcavity systems, optical gain as a function of τ. Gain for a given τ is calculated by dividing can be observed due to parametric scattering of polaritons the integrated transmission under both UP and LP branches when pumping conditions are fine-tuned such that pump, by the average integral taken at τ < 0. Black dashed line indicates signal, and idler conserve energy and momentum [35]. the arrival of the pump pulse as determined in an independent However, the significant gain we observe for the polaron- measurement (see Appendix B). Black solid line shows the expected time evolution of the coherent LP population injected by LP in Fig. 3(b) considerably outlives the coherent LP the pump. (c) Dependence of the peak gain on the density of LPs population (black solid line), suggesting a contribution injected by the pump. Blue shaded area indicates the regime from a reservoir of long-lived incoherent polarons, and where polariton density and electron density are comparable in rules out the possibility of a coherent process as the sole the system. (d) Time delay scan of the probe transmission mechanism. spectrum for resonant pumping of the polaron-UP. (e) Time There are a few possibilities as to how an incoherent evolution of integrated gain corresponding to (d). Inset: The polaron population can be generated. Firstly, due to the probe transmission spectrum at τ ¼ 1 and 3 ps indicated by small normal mode splitting of the polaron-polariton, there yellow dashed lines in (d). is finite overlap of the LP with high-momentum polaron states which are not coupled to the cavity. Polarons can be both energy and momentum conservation. We find such created directly in high-momentum states due to the processes to be consistent with the second rise of the presence of disorder. Secondly, as we argued in the blueshift in Fig. 2(e) which grows in prominence relative to previous section, Auger recombination [38] or relaxation the first peak as pump power is increased. These high- following direct two-photon absorption can efficiently momentum polarons can scatter electron-hole pairs from populate high-momentum attractive polaron states. These the Fermi sea of electrons to relax their momentum and states are immune to radiative decay since they lie outside energy into the polaron-LP state. Bose stimulation of this the light cone and cannot recombine radiatively and satisfy process due to a seed population from the probe then leads

021011-6 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020) to gain. We verified that the gain factors observed for all pump powers are independent of the incident probe power. (a) (b) (c) In Fig. 3(c), we plot the power dependence of the peak gain for the exciton- and polaron-polariton. The saturation behavior of the gain for the polaron-polariton resembles that of the blueshift: it first increases sharply and then continues with a gentler slope as nLP approaches ne. The maximum gain for the polaron-polariton is ∼1.8 while for the exciton-polariton it remains constant at 1.1 for all n . FIG. 4. Evolution of lower-polariton gain as a function of time LP delay τ for various pump-probe polarization configurations. LP We remark that the mechanism underlying weak exciton- gain for a given τ is calculated by taking the ratio of the integral polariton gain and its saturation behavior remain unclear. under the LP branch at τ to the average integral taken at τ < 0. In order to understand the polaron-LP gain mechanism, we (a) LP gain for copolarized or cross-polarized probe are plotted studied gain under UP pumping since the population of high- when resonantly pumping the polaron-UP with a circularly momentum polarons is expected to be more efficient when polarized laser. (b) Analogous to (a) but for linear polarization. pumping the UP. This is because, while the UP can relax to (c) The polaron-LP is resonantly pumped with either linearly or lower-energy, high-momentum polaron states via disorder or circularly polarized laser while the probe is always cross-polarized. generation of additional Fermi sea electron-hole pairs, there 10 are no available states for the LP to relax into (for T< K), The polarization dependence and the long-lived nature of since it is the lowest-energy optical excitation. In Fig. 3(d), the gain lends strong evidence for the significance of an τ we plot the probe transmission spectrum at different . Gain is incoherent but valley-preserving reservoir population. We observed to start from the UP wavelength and redshifts with τ also note that similar polarization-dependent behavior is . However, as compared to pumping the LP, the polaron- observed when pumping the LP as shown in Fig. 4(c). polariton splitting collapses into a single amplified mode [inset of Fig. 3(e)], which at even longer time delays evolves V. THEORETICAL MODEL into the UP resonance [Fig. 3(d)]. It is not obvious if this is due to a nonuniform gain spectrum or a breakdown of the We now turn to the discussion of the theoretical model strong-coupling regime. The peak gain when pumping the that underlies the interpretation of our experimental results. UP was found to be ∼1.8 as compared to 1.5 when pumping In order to model the observed phenomena theoretically, the LP for the same pump power [Fig. 3(e)]. we use a wave function approach based on the Chevy Pumping the UP provides the advantage that both ansatz given by Eq. (4). The interaction between the copolarized and cross-polarized pump-probe experiments polarons described by this ansatz arises in our model from can be done since pump photons can be filtered spectrally the symmetrization of two polaron wave functions that instead of using polarization suppression. The polarization accounts for the underlying fermionic nature of the polaron degree of freedom is important due to valley-dependent dressing cloud. Physically this accounts for the PSF and the optical selection rules in monolayer TMDs: circularly related depletion of the electronic medium. As outlined polarized photons excite polarons where excitons in a below, in a similar fashion, we calculate the residual single valley are dressed by electrons in the opposite valley interactions between polarons and electrons that lead to (as discussed at the end of Sec. II). Therefore, we expect the decay of high-momentum polarons to the polaron-LP that pump-generated reservoir polarons can only be stimu- state, which could explain the observed gain. lated by probe-generated LPs in the same valley. To verify Our experiments probe polariton-polariton interactions this, we pump the UP with circularly polarized photons and in the nonequilibrium limit where the radiative lifetime of probe with cross- and co-circularly polarized pulses. optical excitations is comparable to or shorter than the Because of spectral filtering, we limit the integration area electronic timescales, such as the disorder-induced lifetime. to the LP branch when monitoring the gain. Consequently, Since this is intrinsically a nonequilibrium problem, we changes in the cavity detuning can lead to gain values have to consider interactions between polaron states that deviating from unity. Indeed, for circular polarization, the are not necessarily the lowest-energy optically excited ’ cross-polarized probe gain was found to be negligible as many-body state. While a full Keldysh Green s function compared to its copolarized counterpart [see Fig. 4(a)]. approach is a method of choice to analyze such non- On the other hand, the copolarized and cross-polarized equilibrium problems in the presence of dissipation, we probe gain are identical in linear polarization. This suggests show here that a wave function approach leads to remark- that the gain process conserves the valley population but ably good agreement between theory and experiment. not the valley coherence, which is consistent with the proposed mechanism. The entanglement of polarons with A. Fermi polaron-polaritons the electrons during the scattering process leads to a loss The coefficients of the polaron-polariton wave function c of the phase relation between the valley populations. given in Eq. (4), ϕp, ϕp, and ϕpkq, are determined by the

021011-7 LI BING TAN et al. PHYS. REV. X 10, 021011 (2020)

† minimization of hΦjapðH − EpÞapjΦi, where the polaron polaron; we emphasize that we explicitly introduce the area energy Ep is a Lagrange multiplier ensuring the normali- factor in this expression to ensure that the interaction U has zation of the wave function. Performing this minimization units of energy times length squared, as expected for a we obtain the following expression for the Fermi-polaron momentum-space interaction. To calculate E2 we need quasiparticle energy: to determine the two-polaron wave function. To first approximation the two-polaron wave function addressed † † Ω2 by the probe pulse is given by a a jΦi (properly normal- ¼ ν þ ð Þ 0 0 Ep p − ω − Σ ðp Þ : 5 ized). Higher-order contributions to the interaction can Ep p xe ;Ep appear from the coupling to higher-energy states such as † † This equation shows that the hybridization with the ak≠0a−kjΦi (k ≠ 0) or states of two polarons and a number excitonic degrees of freedom results in a self-energy for of electrons. Since these states have higher energy, they lead Σ the photon. The exciton, in turn, acquires a self-energy xe to attractive interactions within perturbation theory. The due to its interactions with the electron. These interactions experimental evidence in our case shows that the interaction are renormalized from the contact potential v and deter- between polaron-polaritons is effectively repulsive, sug- mined by the electron-exciton scattering T matrix that gesting that first-order contributions dominate over higher- accounts for effects of the finite electron density: order interaction terms. A reason for this is the ultralow polariton mass, which ensures a large kinetic energy cost in 1 X 1 Tðp; ωÞ−1 ¼ v−1 − ; ð6Þ coupling to nonzero-momentum states. A ω − ϵk − ωp−k In this approximation, the strength of interaction jkj>kF between two polaron-LPs is given by where p and ω denote the total momentum and energy of † † the exciton and the electron. The self-energy is expressed in U hΦja0a0Ha0a0jΦi ¼ − 2E0: ð11Þ T A † † terms of the matrix as hΦja0a0a0a0jΦi

1 X † † Σ ðp ωÞ¼ ðp þ q ω þ ϵ Þ ð Þ Here, the overlap hΦja0a0a a jΦi accounts for the nor- xe ; A T ; q : 7 0 0 jqj

B. Residual interactions between polarons These sums have direct physical interpretations. To For a sufficiently strong laser pump pulse, a finite polaron illustrate this, consider an attractive polaron at some density is generated and interactions between the quasipar- position in the sample (the zero-momentum polaron state ticles become important. To assess their strength, we focus is a superposition of such states). The exciton attracts the here on two polarons of zero total momentum. The inter- surrounding electrons, forming a dressing cloud which has action U is given by U=A ¼ E2 − 2E0, where E2 is the a size roughly given by the trion Bohr radius aT, in the limit energy of a system with two polarons of zero total momen- of low electron density. This also depletes locally the Fermi tum and E0 denotes the energy of a single zero-momentum sea, in a radius roughly given by the interelectron distance

021011-8 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020)

−1 ≫ A kF aT. Consider now adding a second polaron in the To understand the various terms in U= , it is helpful to same zero-momentum state. The depletion of the Fermi sea recognize that a single polaron is formed from the super- † † created by the first exciton prohibits the formation of position between a bare photon c0jΦi, an exciton x0jΦi, and another polaron in this region, reducing this overlap. an excitonP entangled with a neutral excitation in the Fermi This effect is contained in the (negative) hole exchange † † sea kq ϕkqxq−kekeqjΦi. The strength of the hybridization I contribution h, which reduces the overlap by a factor of the between the latter two states relies on the exciton’s ability to −2 A order of kF = corresponding to the probability that the −1 create electron-hole pair excitations in the Fermi sea. which two excitons are roughly within kF of each other. This necessarily depends on the number of electrons in the depletion argument does not take into account the electron Fermi sea. density increase in the immediate vicinity of the exciton, However, when the two excitons are close to each within a radius of roughly a . This correction is taken into T Pother, and one of the excitons is in the dressed state account by the electron and electron þ hole exchange † † kq ϕkqxq−kekeqjΦi while the other exciton is in the contributions Ie and Ieh, which are therefore of the order † 2 jΦi of a =A. Hence, in the limit of low electron densities, the state x0 , then it will be more difficult for the second T P † † hole exchange contribution will dominate the other con- exciton to evolve into the state kq ϕkqxq−kekeqjΦi, 2 tributions roughly by a factor of 1=ðkFaTÞ . To simplify because some of the electrons in the Fermi sea are the derivation, we therefore neglect the smaller contribu- already correlated with the first exciton. This fact is tions and focus only on the interaction effects due to captured by the first term in the square brackets. On hole exchange from now on. With this approximation, we the other hand, if also the second exciton is in the P † † obtain state kq ϕkqxq−kekeqjΦi, this state will have slightly higher energy because of the Pauli blocking between † † hΦja0a0a0a0jΦi ≈ 2ð1 þ IhÞ: ð16Þ the dressing clouds of the two excitons. This repulsion mechanism is captured by the last two terms, which contain Substituting this back into Eq. (11), we obtain the exchange corrections coming from the exciton-electron (second term) interaction and exciton-hole interaction (third term). U 1 † † ≈ hΦja0a0Ha0a0jΦið1 − IhÞ − 2E0 ð17Þ One can check that the last two terms go to zero as A 2 2 Λ → ∞. Indeed, since jϕkqj ∝ 1=jkj for large momenta, 1 one can check that the first sum in the square brackets ≈ hΦj † †jΦi − 2 − 2 ð Þ diverges logarithmically with Λ, while the others remain 2 a0a0Ha0a0 E0 E0Ih; 18 constant. Therefore, only the first term can compensate the logarithmic decrease of the interaction strength v as ≪ 1 where we used the fact that Ih and in the second line Λ → ∞ 1 † † , and the interaction is given by we replaced 2 hΦja0a0Ha0a0jΦi ≈ 2E0 in the term propor- tional to Ih, since corrections to this approximation are of X X U 2v 2 higher order. ≈− ϕ ϕ jϕ 0 j ð Þ A A kq k q : 20 The remaining expectation value can be evaluated by kq k0 applying Wick’s theorem once again. In this calculation, the terms −2E0 and −2E0Ih cancel all the intrapolaron direct 1 † † This expression allows us to make the connection to and exchange terms that appear in 2 hΦja0a0Ha0a0jΦi. Thus, in agreement with the expectation, only the interpo- the intuitive picture of phase-space filling and the laron interaction terms will contribute to the interaction U. related polaron-induced depletion of the electronic envi- ronment. Indeed, the last summation in Eq. (20)P quantifies Furthermore, the direct interpolaron interaction is zero due 2 jϕ 0 j ¼ to number conservation. Keeping only the hole-exchange the depletion of the Fermi sea,P since k0 k q † † 2 terms yields 1 − hΦja0cqcqa0jΦi. Therefore, v k0 jϕk0qj denotes the change in the amplitude for creating an electron-hole X U 2v 2 pair with momenta k, q due to the presence of the attractive ≈− ϕ ϕjϕ 0 j A A kq k q polaron at zero momentum, again, making explicit the kk0q effect of a depletion-induced interaction. X X 2 We can rewrite the above interaction in terms of the þ ϕkqϕk0q0 ϕkq0 ϕk0þq−q0;q − ϕkq0 ϕkqjϕk0qj : kqk0q0 kk0q0q T matrix using the definition for the polaron wave −1 functionP introduced in Eq. (10). Using ∂T ðp; ωÞ=∂ω ¼ ð Þ 2 19 ð1=AÞ 1=ðω − ωp−k − ϵkÞ , we obtain jkj>kF

021011-9 LI BING TAN et al. PHYS. REV. X 10, 021011 (2020) ϕ4 X ∂ 2ðq ωÞ polaron interactions increase as we decrease the Fermi U ¼ T ; ð Þ A A ∂ω : 21 energy, because they arise due to the Pauli blocking jqj ω¼E0þϵq

C. Polaron-electron interaction As we argued in Sec. IV, the observed probe gain originates from the residual interactions between polarons and the Fermi sea. For simplicity, we focus here on the gain in polaron-LP when the polaron-UP branch is pumped. We envision that the relaxation of the excited polarons to FIG. 5. (a) Interaction strength of exciton-polarons in the absence of cavity coupling. The black dashed line is a guide to polaron-LP state takes place in two sequential steps. In the eye to illustrate the Fermi energy probed experimentally, i.e., the first step, excitations in the polaron-UP branch decay EF=ET ¼ 3.3=25. (b) Residual interactions between polaron- into finite-momentum polarons by generating Fermi sea polaritons. The thicker blue line denotes the interactions of electron-hole pairs, to form a long-lived reservoir of finite- Fermi polaron-polaritons as a function of Fermi energy when momentum polarons. In the second step, these polarons Ω ¼ 0.2ET , corresponding to the experimental value. The green decay into the polaron-LP state by generating an additional Ω ¼ 0 5 Ω ¼ 1 and red solid lines correspond to . ET and ET , electron-hole pair. This second process can be stimulated respectively. For this plot we choose the detuning of the cavity by a finite population in the final state, leading to net Δ ¼ þ Ω2 E0 =ET ,whereE0 denotes the energy of the attractive transmission gain for the probe field. Because of the exciton-polaron, to ensure that the polaron-polariton normal- complexity of the problem, we will only attempt to mode splitting vanishes at zero Fermi energy. The interaction obtain an order-of-magnitude estimate of the gain, by strength of exciton-polarons (scaled by a factor of 0.25) in the absence of cavity coupling has been replotted here for com- calculating the rates for the two decay processes mentioned parison. The enhancement of interactions with increasing above. cavity-polaron coupling Ω is evident. The black dashed line The residual interactions between polarons and the is a guide to the eye to illustrate the Fermi energy probed Fermi sea can be estimated in an analogous way to the experimentally, i.e., EF=ET ¼ 3.3=25. Energies are in units previous section. To this end we evaluate the matrix 2 of ℏ =me. element:

021011-10 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020)

X † † Spkq v hΦj jΦi¼ ð þ þ ϵ − ϵ Þþ ϕ ϕ þ ϕ 0 0 ϕ 0 0 eqekapþq−kHap 2 Ep Epþq−k k q A p pþq−k pk q pþq−k;k ;q k0q0 v X X 1 X 1 X − ϕ 0 ϕ 0 0 þ ϕ 0 ϕ 0 0 − ϕ ϕ 0 0 − ϕ 0 ϕ A pk q pþq−k;k q pkq pþq−k;k q 2 pkq pþq−k;k q 2 pk q pþq−k k0q0 k0q0 k0q0 k0 1 X − ϕ 0 ϕ ð Þ 2 pkq pþq−k : 22 q0

In the above, the term proportional to Spkq ≡ jpj ∼ kF=2.5, implying that most polarons in the reservoir † † hΦjeqekapþq−kapjΦi arises because we are working with will have this momentum. a nonorthogonal basis and does not represent an interaction We can also calculate the decay rate from a finite- term. From the other terms, we focus only on the ones that momentum polaron into a lower-polaron-polariton: remain finite in the UV limit Λ → ∞, i.e., the second, third, X 2 Vp pþq q ΓL ¼ 2π ; ; δð − − ϵ þ ϵ Þ ð Þ and fourth terms in the last parentheses. In the low doping p 2 E0 Ep pþq q : 26 q A limit kFaT ≪ 1, relevant for the description of our experi- 1 ð Þ ment, the fourth term dominates by a factor of = kFaT ,so L 2 Evaluating the above, we obtain Γ 2 5 ≈ 1 μeV μm =A. we keep only this term, denoted by Vpkq in the following kF= . Assuming that the finite-momentum polaron reservoir has a analysis. It is given by ¼ 1011 −2 ¼ X population density between n0 cm and n0 v 12 −2 L L 10 cm , we obtain a decay rate ℏΓ ≈ N0Γ 2 5 ¼ Vpkq ≈ ϕpk0qϕpþq−k ð23Þ kF= . 2 2 k0 1 μeV μm × n0, which gives a value between 1 and 10 meV. Assuming a polariton lifetime of ℏ=1 meV, our ¼ ϕ ϕ ðp þ q þ ϵ Þ ð Þ p pþq−kT ;Ep q ; 24 simple calculation would therefore predict a net gain at ’ polaron-LP exceeding unity, which agrees well with the which agrees with previous calculations using Green s experimentally observed gain. functions [48]. This matrix element describes the amplitude for the emission of an electron-hole pair with momenta k and q by a polaron of momentum p. We emphasize that VI. CONCLUSION the above matrix element can also describe the residual inter- In this work, we have explored polariton-polariton action between the upper polaron-polariton and the Fermi interactions and Bose-enhanced scattering of polaron- ϕ sea, if we replace the coefficients and energy Ep with the polaritons in a charge-tunable MoSe2 monolayer embedded corresponding values for the upper polaron-polariton. in a zero-dimensional optical cavity. We found polariton- To calculate the decay rate from the upper polaron- polariton interactions to be enhanced by a factor of 50 when polariton into finite-momentum polaron states, we use the monolayer is electron doped as compared to the charge- Fermi’s golden rule: neutral regime. This dramatic enhancement originates from the restriction of the oscillator strength of polarons formed X X X VU 2 ΓU ¼ ΓU ≡ 2π 0;q−p;q within an optical spot with a fixed electron density. p 2 p p q A Intuitively, this enhancement is a consequence of the rearrangement of the polaron dressing cloud to accommo- δð U − − ϵ þ ϵ Þθð − jqjÞθðjq − pj − Þ × E0 Ep q−p q kF kF ; date a larger number of optically injected impurities. In an ð25Þ undoped semicondutor, exciton-exciton interaction scales linearly with the Bohr radius (aB), implying that strong EU where 0 denotes the energy of the excited upper polaron- light-matter coupling (Ω ∝ 1=aB) is normally associated polariton state and we used the superscript U to emphasize with weak nonlinearity. In stark contrast, the new mecha- that the interaction V should be evaluated using the nism we identify leads to stronger polariton interactions for coefficients and the energy of the upper polaron-polariton. stronger light-matter coupling (Fig. 4). This dependence In the above, Γp denotes the decay rate into the polaron suggests that, against expectations, strong exciton binding state of momentum p by emission of electron-hole pairs of may even help to increase polariton-polariton interactions total momentum −p. Choosing the same parameters as in in electron-doped samples. the previous section, we obtain ℏΓU ≈ 2 meV. Since the Our work further highlights the importance of time radiative lifetime of polaritons is about ℏ=ð1 meVÞ,we resolution when observing interaction-induced effects on conclude that pumping the upper polariton will create polariton spectra. In particular, the measurements revealed a sizable reservoir of finite-momentum Fermi polarons. spectral features that survive up to timescales far outliving We remark that Γp is strongly peaked at momenta the coherent polaritons. This observation sheds light on the

021011-11 LI BING TAN et al. PHYS. REV. X 10, 021011 (2020) possible existence of an incoherent reservoir which con- with a distributed Bragg reflector (DBR) coating: ten tributes to the blueshift of the polaron-LP resonance that alternating layers of NB2O5 and SiO2. This constitutes exists for timescales >10 ps. Residual polaron interactions the planar mirror of the zero-dimensional cavity. The DBR also provide an efficient relaxation channel from these was designed to provide >99.3% reflectivity between 680– high-momentum reservoir states into the lower polariton 800 nm with an intensity maximum at the DBR surface. The mode, which when stimulated by probe-field injected MoSe2 and graphene flakes were contacted with Au on a polaritons results in optical gain. Our findings suggest that 5 nm Ti sticking layer. The concave mirror was prepared by injection of itinerant electrons into a monolayer TMD could ablating a dimple of radius of curvature 30 μm onto a single- overcome the relaxation bottleneck and enable the reali- mode fiber facet which was then coated with an identical zation of a polariton laser [34,49]. DBR as the flat substrate. After fabrication of the sample, the The signatures revealed by nonlinear spectroscopy graphene layer was found to be torn and could not provide in our work represent an exciting new realm of polaron reliable doping. Thus, an additional graphene layer was physics: prior results in this field are often well described placed on top which restored gate tunability in the sample. by the Chevy ansatz which models many-body dressing The planar mirror substrate is mounted on two nano- in terms of a single electron-hole pair. However, an accurate positioners which provide in-plane spatial degrees of description of interactions that arise from the reorganization freedom. The fiber mirror is mounted on one nanoposi- of impurity screening by the bath will require a description tioner which provides an out-of-plane degree of freedom. of higher-order exciton-electron correlations. Furthermore, our experiments have also made a first APPENDIX B: OPTICAL MEASUREMENTS venture into the exciting regime of degenerate Bose-Fermi mixtures where the polariton density becomes comparable The optical setup is illustrated in Fig. 6. The sample sits to the electron density. Addressing open questions such as in a vacuum-pumped environment which is filled with the electron density dependence of the onset as well as 20 mbar of He exchange at room temperature. It is then magnitude of the saturation behavior in the observed immersed in a liquid He bath at 4.2 K for all optical blueshift and gain will provide insight to guide theoretical measurements. understanding. While such investigations are limited in the current sample due to the normal mode splitting of the 1. Pulse preparation polaron-polaritons rapidly diminishing when electron den- The output from a Ti:sapphire femtosecond pulsed laser sity is tuned, enhancing the cavity quality factor from with 76 MHz repetition rate is split into two arms: pump ∼1000 to 10 000 should allow a wider range of electron and probe. The pump arm is spectrally filtered with a 4f densities to be explored. pulse shaping setup with a single grating. The 12 meV The data that support the findings of this paper are broad pulse is first spectrally dispersed using a transmission available in the ETH Research Collection [50]. grating, and its spectrum selected with an adjustable slit aperture in front of a mirror and then recombined using ACKNOWLEDGMENTS the same grating. The probe pulse is spectrally unfiltered. Its The authors gratefully acknowledge many enlightening discussions with Eugene Demler. We also thank Jacqueline Bloch, Georg Bruun, Jerome Faist, Meinrad Sidler, Sina Zeytinoglu, Thibault Chervy, and Ido Schwartz for useful discussions. This work is supported by a European Research Council (ERC) Advanced investigator grant (POLTDES), and by a grant from Swiss National Science Foundation (SNSF).

Note added.—Recently, a closely related work appeared [51].

APPENDIX A: SAMPLE FABRICATION AND SETUP FIG. 6. A schematic of the pump-probe setup. From right to The monolayer MoSe2, graphene, and two h-BN flakes left: The output of a mode-locked Ti:sapphire laser (76 MHz were mechanically exfoliated onto SiO2 substrates. Then repetition rate) is split into two arms: pump and probe. The pump they were stacked by picking up the graphene, top h-BN, is spectrally filtered using a pulse shaper setup. The probe pulse is MoSe2, and the bottom h-BN, in that order, using a sent through an optical delay line to control the time delay τ with polycarbonate sacrificial layer on a polydimethylsiloxane respect to the pump pulse and a pulse compressor to compensate stamp. The stack is deposited onto a fused silica substrate for linear in the optical fibers.

021011-12 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020)

fiber polarization controllers and a polarizing beam splitter (a) (b) (PBS) in the detection setup, we can project the signal onto any two orthogonal polarizations. In a typical cross- polarized pump-probe measurement scheme, we measure the signal from the PBS arm that is cross-polarized with respect to the pump field using the spectrometer and, as a function of τ, we obtain the following: (i) the transmission spectrum when only the pump is turned on and (ii) the transmission spectrum when both pump and probe are turned on. In (i), we measure the cross-polarized polariton (c) (d) emission induced exclusively by the pump field and subtract this from (ii) in order to investigate how the transmission of the probe is influenced in the presence of the pump and as a function of τ.

3. Measurement of the zero delay between pump and probe After mixing the pump and probe pulses with a beam splitter, the output from one port is sent to the cavity sample FIG. 7. (a),(b) A typical spectrum of the pump and probe pulse, and the output from the second port is sent to a photodetector respectively, after traveling through the optical fibers. (c),(d) In- which measures the interfered signal. We note that the pump terferometric autocorrelation signal of the pump and probe, and probe travel through a common path after the beam respectively. Black lines are Gaussian fits to the data from which splitter and, therefore, we can determine the zero delay the linewidths (a),(b) and pulse duration (c),(d) are extracted. between them as the point about which the interfered signal is symmetrical. Since this measurement is not performed optical path length difference with respect to the pump pulse in situ but typically done right before or after the relevant is controlled by adjusting the position of a retroreflector. In pump-probe measurements, we cannot rule out the possibil- this way, the probe can be made to arrive with a variable time ity of slight drifts in the true zero-delay position over time. delay τ with respect to the pump. Both pump and probe pulses are then guided by optical fibers to the excitation arm of the transmission microscope. To avoid unwanted non- APPENDIX C: POLARITON linear effects in the fibers, we attenuate the laser powers to DENSITY ESTIMATION <1 mW before coupling them into the fibers. We check the We measure the reflection contrast of the bare cavity spectrum of the pulse after traveling through the fiber in mode of η ∼ 0.13 between resonant and off-resonant order to ensure such nonlinear effects are not present for the conditions. This gives an estimate of the efficiency of powers we are interested in [Figs. 7(a) and 7(b)]. In addition, the coupling into the cavity. The density of intracavity there are also unwanted linear effects such as group velocity polaritons per pulse is then given by ðηI ε=Aℏωf Þ, dispersion. Since the probe pulse is ∼12 meV broad, it is pump rep where f is the repetition rate of the pulse train, ε is the affected more significantly than the pump pulse (∼1 meV). rep spectral overlap of the pump and the lower polariton We compensate for the dispersion using a single grating branch, A is the area of the excitation spot, and ℏω is pulse compressor. The pump and probe pulse durations are the photon energy. We are interested in the polariton then measured using an interferometric autocorrelation setup τ density within the polariton lifetime pol, which can be in collinear geometry [Figs. 7(c) and 7(d)]. The uncertainty ðη ε ℏω Þ ðτ τ þ τ Þ of the pulse durations arise from the possible deviations of written as Ipump =A frep · pol= pulse pol . εðΔ Þ the amount of dispersive elements (i.e., fiber lengths) The spectral overlap cav is in fact a function of the incorporated in the autocorrelation measurement setup from detuning between the cavity and the polaron (or exciton) the pump-probe experimental setup even as this was already which determines the cavity content C and the resonance of taken into consideration in designing the former. We note the lower polariton (or the mode being pumped) ELP. Therefore, it is important to take into account the changes that while we achieved significant compression of the probe, ε it remains not transform limited. in due to blueshifts of the polaron resonance when calculating the polariton density. We want to compute ε τ during the pump-pulse illumination, i.e., at max instead of 2. Pump-probe measurement at τ < 0. However, when there is significant gain, it After the pulses are prepared, the pump and probe are becomes difficult to determine the cavity content of the coupled into the cavity via the free-space accessible side lower polariton from the area of the transmitted signal. The and the transmission is detected through the fiber. Using method we use is the following.

021011-13 LI BING TAN et al. PHYS. REV. X 10, 021011 (2020)

(1) From the polariton transmission data at τ < 0, where (a) (b) there is no gain, we extract C and ELP and calculate Ecav using the experimentally determined value of ℏΩ and the expression for the Hopfield coefficient: 1 jCj2 ¼ ; ðC1Þ ELP−Ecav 2 1 þð ℏΩ Þ where ℏΩ is the oscillator strength of the polaron. τ (2) hen we extract EUP and ELP from the max data where gain is observed. By taking the sum and difference of the two quantities and assuming that the cavity length remains unchanged when excited by the pump, we can ℏΩ0 0 (c) (d) find the altered oscillator strength and Epol using qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E þ E 1 ¼ cav pol ð − Þ2 þ 4jℏΩj2 EUP;LP 2 2 Ecav Epol : ðC2Þ τ (3) Then the cavity content at max can be calculated using Eq. (C1).

APPENDIX D: ELECTRON DENSITY ESTIMATION FIG. 8. (a) Transmission T spectrum of the repulsive polaron- polariton as a function of V at a bare cavity mode energy of We use a capacitive model to estimate the electron g ℏω ¼ 1643 meV. (b) Black dots with green error bars: Rabi density. The capacitance per unit area between the top c splitting extracted from Lorentzian fits to the transmission spec- gate and the sample is given by trum. The red dashed line is a fit of the model detailed in the text to ℏω ¼ 1644 1 −1 the data. (c),(d) Analogous to (a),(b) for c meV. C ¼ t þ ð Þ 2 ; D1 A ϵ0ϵ e DðE Þ h-BN F ð Þ grep EF is the oscillator strength of the repulsive polaron, where the first and second terms are the geometric and 1 which is ∼g0(1 − ðE =E Þ) for small E , E refers to the t ¼ð88 5Þ 2 F T F T quantum capacitances, respectively. nm is the trion binding energy, which we take to be 25 meV. h ϵ ¼ 3 5 0 5 thickness of the -BN flake, h-BN . . is the static Figures 8(a) and 8(c) show the measured transmission h m ¼ 0 5m dielectric constant of -BN, . e is the effective spectrum of the repulsive polaron-polariton for two differ- DðE Þ electron mass in the conduction band, and F is the ent cavity detunings as a function of V . It is observed E E > 0 g density of electronic states at Fermi energy F.For F , [shown in Figs. 8(b) and 8(d)] that there are two distinct one can neglect the quantum capacitance and write the Ω Ω regimes for the Rabi splitting rep. rep reacts much less Fermi energy as a function of applied gate voltage Vg as sensitively to Vg when increasing it from −6 to ∼1 V, after πℏ2ϵ ϵ h-BN 0 which it starts to decrease sharply. We attribute this E ðV Þ¼ ðV0 − V Þ: ðD2Þ F g tem g apparent slow increase of EF as a filling of localized states located in the midgap region resulting in a slight decrease In order to calculate the Fermi energy EF and therefore Ω of rep, which we can represent with a heuristic linear the electron density ne from the applied gate voltage Vg, Ω model. On the other hand, the behavior of rep is governed we first need to determine the voltage V0 at which we begin by Eq. (D3) as soon as itinerant electrons start to populate to dope the monolayer with itinerant electrons. To that end, the conduction band (for V >V0). In our fits, all param- we measured the transmission spectrum of the repulsive g eters in Eq. (D3) were fixed except for V0, which remained polaron-polariton as a function of gate voltage Vg.For Ω a fit parameter. We determine V0 to be 1 V. This implies that small EF (i.e., Vg

021011-14 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020)

[1] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of Valley Polarization in Monolayer MoS2 by Optical Helicity, Nat. Nanotechnol. 7, 494 (2012). [2] H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Valley Polarization in MoS2 Monolayers by Optical Pumping, Nat. Nanotechnol. 7, 490 (2012). [3] A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke, A. Kis, and A. Imamoğlu, Valley Zeeman Effect in Elementary Optical Excitations of Monolayer WSe2, Nat. Phys. 11, 141 (2015). [4] G. Aivazian, Z. Gong, A. M. Jones, R.-L. Chu, J. Yan, D. G. Mandrus, C. Zhang, D. Cobden, W. Yao, and X. Xu, Magnetic Control of Valley Pseudospin in Monolayer FIG. 9. Evolution of the LP energy shift as a function of time WSe2, Nat. Phys. 11, 148 (2015). ń delay for different polariton densities. [5] T. Smole ski, M. Goryca, M. Koperski, C. Faugeras, T. Kazimierczuk, A. Bogucki, K. Nogajewski, P. Kossacki, and M. Potemski, Tuning Valley Polarization in a WSe2 Monolayer with a Tiny Magnetic Field, Phys. Rev. X 6, function of τ in Fig. 9. We do not observe clear 021024 (2016). evidence of effects occurring after the decay of coherent [6] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. polaritons. Seyler, W. Yao, and X. Xu, Valleytronics in 2D Materials, Nat. Rev. Mater. 1, 16055 (2016). [7] P. Back, S. Zeytinoglu, A. Ijaz, M. Kroner, and A. Imamoglu, APPENDIX F: LONG-TIMESCALE Realization of an Electrically Tunable Narrow-Bandwidth PUMP-PROBE Atomically Thin Mirror Using Monolayer MoSe2, Phys. Rev. Lett. 120, 037401 (2018). In the pump-probe measurements where we reso- [8] G. Scuri, Y. Zhou, A. A. High, D. S. Wild, C. Shu, K. nantly pump the attractive polaron-LP and the attractive De Greve, L. A. Jauregui, T. Taniguchi, K. Watanabe, P. polaron-UP resonantly in Figs. 2 and 3, we consistently Kim, M. D. Lukin, and H. Park, Large Excitonic Reflectivity observed a shift of the oscillator strength from the of Monolayer MoSe2 Encapsulated in Hexagonal Boron attractive polaron-LP to the UP resonance that lasted Nitride, Phys. Rev. Lett. 120, 037402 (2018). long after the gain in the transmission was over. We [9] X. Xu, W. Yao, D. Xiao, and T. F. Heinz, Spin and conducted follow-up measurements for long time-delay Pseudospins in Layered Transition Metal Dichalcogenides, scans (up to 400 ps) while pumping the attractive Nat. Phys. 10, 343 (2014). polaron-UP. We find the timescale for the recovery of [10] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Colloquium: Excitons the initial conditions to be ∼300 ps (see Fig. 10). in Atomically Thin Transition Metal Dichalcogenides, Rev. Mod. Phys. 90, 021001 (2018). [11] V. Walther, R. Johne, and T. Pohl, Giant Optical Nonlinearities from Rydberg Excitons in Semiconductor Microcavities, Nat. Commun. 9, 1309 (2018). [12] S. Zeytinoglu and A. Imamoglu, Interaction-Induced Photon Blockade Using an Atomically Thin Mirror Embedded in a Microcavity, Phys. Rev. A 98, 051801(R) (2018). [13] D. S. Wild, E. Shahmoon, S. F. Yelin, and M. D. Lukin, Quantum Nonlinear Optics in Atomically Thin Materials, Phys. Rev. Lett. 121, 123606 (2018). [14] F. Barachati, A. Fieramosca, S. Hafezian, J. Gu, B. Chakraborty, D. Ballarini, L. Martinu, V. Menon, D. Sanvitto, and S. Kena-Cohen, Interacting Polariton Fluids in a Monolayer of Tungsten Disulfide, Nat. Nanotechnol. 13, 906 (2018). [15] M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, E. Demler, and A. Imamoglu, Fermi Polaron- Polaritons in Charge-Tunable Atomically Thin Semicon- FIG. 10. Transmission (T) spectrum of the attractive polaron- ductors, Nat. Phys. 13, 255 (2017). polariton when resonantly pumping the UP (indicated by black [16] D. K. Efimkin and A. H. MacDonald, Many-Body Theory of dashed line) as a function of time delay τ from −10 to 400 ps at Trion Absorption Features in Two-Dimensional Semicon- ¼ð8 1Þ 1011 −2 ¼ 0 07 ne × cm and Ipump . mW. ductors, Phys. Rev. B 95, 035417 (2017).

021011-15 LI BING TAN et al. PHYS. REV. X 10, 021011 (2020)

[17] Y.-C. Chang, S.-Y. Shiau, and M. Combescot, Crossover [32] G. M. Bruun and P. Massignan, Decay of Polarons and from Trion-Hole Complex to Exciton-Polaron in n-Doped in a Strongly Polarized , Phys. Rev. Two-Dimensional Semiconductor Quantum Wells, Phys. Lett. 105, 020403 (2010). Rev. B 98, 235203 (2018). [33] R. Schmidt and T. Enss, Excitation Spectra and rf Response [18] F. Chevy, Universal Phase Diagram of a Strongly Interact- Near the Polaron-to- Transition from the Func- ing Fermi Gas with Unbalanced Spin Populations, Phys. tional Renormalization Group, Phys. Rev. A 83, 063620 Rev. A 74, 063628 (2006). (2011). [19] R. A. Suris, Optical Properties of 2D Systems with Inter- [34] A. Imamoglu, R. J. Ram, S. Pau, and Y. Yamamoto, acting Electrons (Springer, New York, 2003), pp. 111–124. Nonequilibrium Condensates and Lasers without Inversion: [20] R. Rapaport, E. Cohen, A. Ron, E. Linder, and L. N. Exciton-Polariton Lasers, Phys. Rev. A 53, 4250 (1996). Pfeiffer, Negatively Charged Polaritons in a Semiconductor [35] P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. S. Microcavity, Phys. Rev. B 63, 235310 (2001). Skolnick, D. M. Whittaker, and J. S. Roberts, Angle-Resonant [21] F. Pulizzi, D. Sanvitto, P. C. M. Christianen, A. J. Shields, Stimulated Polariton Amplifier, Phys. Rev. Lett. 84, 1547 S. N. Holmes, M. Y. Simmons, D. A. Ritchie, M. Pepper, (2000). and J. C. Maan, Optical Imaging of Trion Diffusion and [36] B. Besga, C. Vaneph, J. Reichel, J. Esteve, A. Reinhard, J. Drift in GaAs Quantum Wells, Phys. Rev. B 68, 205304 Miguel Sanchez, A. Imamoglu, and T. Volz, Polariton (2003). Boxes in a Tunable Fiber Cavity, Phys. Rev. Applied 3, [22] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, 014008 (2015). Observation of the Coupled Exciton-Photon Mode Splitting [37] C. Schneider, M. M. Glazov, T. Korn, S. Höfling, and B. in a Semiconductor Quantum Microcavity, Phys. Rev. Lett. Urbaszek, Two-Dimensional Semiconductors in the Regime 69, 3314 (1992). of Strong Light-Matter Coupling, Nat. Commun. 9, 2695 [23] H. Deng, H. Haug, and Y. Yamamoto, Exciton-Polariton (2018). Bose-Einstein Condensation, Rev. Mod. Phys. 82, 1489 [38] B. Han, C. Robert, E. Courtade, M. Manca, S. Shree, T. (2010). Amand, P. Renucci, T. Taniguchi, K. Watanabe, X. Marie, [24] I. Carusotto and C. Ciuti, Quantum Fluids of Light, Rev. L. E. Golub, M. M. Glazov, and B. Urbaszek, Exciton States Mod. Phys. 85, 299 (2013). in Monolayer MoSe2 and MoTe2 Probed by Upconversion [25] X. Liu, T. Galfsky, Z. Sun, F. Xia, E.-c. Lin, Y.-H. Lee, Spectroscopy, Phys. Rev. X 8, 031073 (2018). S. K´ena-Cohen, and V. M. Menon, Strong Light–Matter [39] Z. Yu and C. J. Pethick, Induced Interactions in Dilute Coupling in Two-Dimensional Atomic , Nat. Pho- Atomic and Liquid Helium Mixtures, Phys. Rev. A tonics 9, 30EP (2015). 85, 063616 (2012). [26] S. Dufferwiel, S. Schwarz, F. Withers, A. A. P. Trichet, F. Li, [40] D. H. Santamore and E. Timmermans, -Mediated M. Sich, O. Del Pozo-Zamudio, C. Clark, A. Nalitov, D. D. Interactions in a Dilute Bose-Einstein Condensate, Phys. Solnyshkov, G. Malpuech, K. S. Novoselov, J. M. Smith, Rev. A 78, 013619 (2008). M. S. Skolnick, D. N. Krizhanovskii, and A. I. Tartakovskii, [41] H. Hu, B. C. Mulkerin, J. Wang, and X.-J. Liu, Attractive Exciton–Polaritons in van der Waals Heterostructures Fermi Polarons at Nonzero Temperatures with a Finite Embedded in Tunable Microcavities, Nat. Commun. 6, Impurity Concentration, Phys. Rev. A 98, 013626 (2018). 8579 (2015). [42] A. Camacho-Guardian and G. M. Bruun, Landau Effective [27] G. Malpuech, A. Kavokin, A. Di Carlo, and J. J. Baumberg, Interaction between Quasiparticles in a Bose-Einstein Polariton Lasing by Exciton-Electron Scattering in Condensate, Phys. Rev. X 8, 031042 (2018). Semiconductor Microcavities, Phys. Rev. B 65, 153310 [43] H. Tajima and S. Uchino, Many Fermi Polarons at Nonzero (2002). Temperature, New J. Phys. 20, 073048 (2018). [28] P. G. Lagoudakis, M. D. Martin, J. J. Baumberg, A. Qarry, [44] B. J. DeSalvo, K. Patel, G. Cai, and C. Chin, Observation of E. Cohen, and L. N. Pfeiffer, Electron-Polariton Scattering Fermion-Mediated Interactions between Bosonic , in Semiconductor Microcavities, Phys. Rev. Lett. 90, Nature (London) 568, 61 (2019). 206401 (2003). [45] M. A. Ruderman and C. Kittel, Indirect Exchange Coupling [29] M. Perrin, P. Senellart, A. Lemaitre, and J. Bloch, Polariton of Nuclear Magnetic Moments by Conduction Electrons, Relaxation in Semiconductor Microcavities: Efficiency of Phys. Rev. 96, 99 (1954). Electron-Polariton Scattering, Phys. Rev. B 72, 075340 [46] T. Kasuya, A Theory of Metallic Ferro- and Antiferromag- (2005). netism on Zener’s Model, Prog. Theor. Phys. 16,45 [30] A. I. Tartakovskii, D. N. Krizhanovskii, G. Malpuech, (1956). M. Emam-Ismail, A. V. Chernenko, A. V. Kavokin, V. D. [47] K. Yosida, Magnetic Properties of Cu-Mn Alloys, Phys. Kulakovskii, M. S. Skolnick, and J. S. Roberts, Giant Rev. 106, 893 (1957). Enhancement of Polariton Relaxation in Semiconductor [48] O. Cotleţ, F. Pientka, R. Schmidt, . Zarand, and E. Microcavities by Polariton-Free Carrier Interaction: Demler, and A. Imamoglu, Transport of Neutral Optical Experimental Evidence and Theory, Phys. Rev. B 67, Excitations Using Electric Fields, Phys. Rev. X 9, 041019 165302 (2003). (2019). [31] A. Qarry, G. Ramon, R. Rapaport, E. Cohen, A. Ron, A. [49] M. Waldherr, N. Lundt, M. Klaas, S. Betzold, M. Mann, E. Linder, and L. N. Pfeiffer, Nonlinear Emission due Wurdack, V. Baumann, E. Estrecho, A. Nalitov, to Electron-Polariton Scattering in a Semiconductor Micro- E. Cherotchenko, Cai, E. A. Ostrovskaya, A. V. Kavokin, cavity, Phys. Rev. B 67, 115320 (2003). S. Tongay, S. Klembt, S. Hofling, and C. Schneider,

021011-16 INTERACTING POLARON-POLARITONS PHYS. REV. X 10, 021011 (2020)

Observation of Bosonic Condensation in a Hybrid Mono- [51] R. P. A. Emmanuele, M. Sich, O. Kyriienko, V.Shahnazaryan, layer MoSe2-GaAs Microcavity, Nat. Commun. 9, 3286 F. Withers, A. Catanzaro, P. M. Walker, F. A. Benimetskiy, (2018). M. S. Skolnick, A. I. Tartakovskii, I. A. Shelykh, and D. N. [50] L. B. Tan, https://www.research-collection.ethz.ch/handle/ Krizhanovskii, Highly Nonlinear Trion-Polaritons in a Mono- 20.500.11850/401640. layer Semiconductor, arXiv:1910.14636.

021011-17