Polaron-Polaritons in the Integer and Fractional Quantum Hall Regimes
Sylvain Ravets, Patrick Knüppel, Stefan Faelt, Martin Kroner, Werner Wegscheider, Atac Imamoglu
PPolaron-Polaritons in the Integer and Fractional Quantum Hall Regimes
Sylvain Ravets, Patrick Kn¨uppel, Stefan Faelt,
1, 2
Ovidiu Cotlet, Martin Kroner, Werner Wegscheider, and Atac Imamoglu Institute of Quantum Electroncis, ETH Z¨urich,CH-8093 Z¨urich, Switzerland Solid State Physics Laboratory, ETH Z¨urich,CH-8093 Z¨urich, Switzerland
Elementary quasi-particles in a two dimensional electron system can be described as exciton-polarons sinceelectron-exciton interactions ensures dressing of excitons by Fermi-sea electron-hole pair excitations. A relevantopen question is the modification of this description when the electrons occupy flat-bands and electron-electroninteractions become prominent. Here, we perform cavity spectroscopy of a two dimensional electron systemin the strong-coupling regime where polariton resonances carry signatures of strongly correlated quantum Hallphases. By measuring the evolution of the polariton splitting under an external magnetic field, we demonstratethe modification of electron-exciton interactions that we associate with phase space filling at integer fillingfactors and polaron dressing at fractional filling factors. The observed non-linear behavior shows great promisefor enhancing polariton-polariton interactions.
PACS numbers: 71.36.+c, 73.43.Fj
Strong coupling of excitons in a semiconductor quantumwell (QW) to a microcavity mode leads to formation of quasi-particles called cavity exciton polaritons [1]. Polaritons haveplayed a central role in the investigation of nonequilibriumcondensation and superfluidity of photonic excitations [2, 3].While polaritons acquire a finite nonlinearity due to their ex-citon character, interactions between polaritons in undopedQWs are not strong enough for realizing strongly interactingphotonic systems [4].Two-dimensional electron systems (2DES) evolving inlarge magnetic fields, in contrast, are a fertile ground formany-body physics due to prominence of electron-electroninteractions. Formation of skyrmion excitations in the vicin-ity of filling factor ν = 1 is a consequence of such interac-tions. More spectacularly, electron correlations lead to theformation of fractional quantum Hall (FQH) states where theground state exhibits topological order [5–7]. Moreover, ithas been proposed that a sub-class of FQH states exhibit non-abelian quasi-particles which can be used to implement topo-logical quantum computation [8]. The nature of optical ex-citations of a 2DES have also recently generated lots of in-terest: experimental and theoretical [9, 10] studies in tran-sition metal dichalcogenide (TMD) monolayers have estab-lished that these excitations should be described in the frame-work of the Fermi polaron problem, as a collective excitationresulting from exciton-electron interactions [11–14]. In thiscontext, optical excitation of an exciton leads to generationof an electron screening cloud that results in formation of alower energy attractive exciton-polaron [9, 10]. In this work,we report corresponding signatures in GaAs, where energyscales are known to differ significantly compared to TMDmonolayers [15, 16], due to a particularly small binding en-ergy of the bound-molecular trion state. More importantly,our work presents experimental signatures of polaron forma-tion at nonzero magnetic fields where electrons are confinedto the lowest Landau level.It has recently been demonstrated that embedding a 2DES FIG. 1. Observation of polaron-polaritons in a GaAs QW. (a) Wecouple a 2DES to the optical mode of a microcavity composed of twodistributed Bragg reflectors (DBRs). In the strong coupling regime,the lowest energy eigenstates of the coupled system are the lower po-lariton (LP) and the upper polariton (UP). (b) Reflectivity spectrumof the system as we tune the cavity frequency. inside a microcavity realizes an alternate method for probingquantum Hall (QH) states [17]. In the strong coupling regime,polariton excitations are sensitive to elementary properties ofthe many-body ground state, such as spin-polarization and in-compressibility due to their part-exciton character. In contrastto bare excitons though, polaritons are immune to decoher-ence processes such as phonon or impurity scattering due totheir ultra-light mass, ensuring that they are delocalized. Con- a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p FIG. 2. Cavity spectroscopy of the system in the fractional quantum Hall regime, as we tune B for fixed E cav (white dashed line). Measurementperformed with (a) σ − and (b) σ + polarized light. (c) Relevant energy levels and optical transitions around ν = 1 . (d) Polariton splittingmeasured in σ − (blue) and σ + (red) polarizations. sequently, the energy resolution achievable in polariton-basedspectroscopy is only limited by the polariton decay rate dueto mirror losses, which can be on the order of 20 mK in state-of-the-art microcavities [18]. In the present work, we alsodemonstrate a new feature of cavity-polariton-spectroscopy ofFQH states: by adjusting the separation distance between the2DES and the doping layers we substantially reduce unwantedlight-induced variations of the 2DES electron density n e [19].Our structure consists of a 2DES in a 20 nm modu-lation doped GaAs QW, embedded at the center of a λ Al . Ga . As microcavity [19]. The front (back) dis-tributed Bragg reflector (DBR) is composed of 19 (25) pairsof AlAs / Al . Ga . As layers leading to the measured qual-ity factor Q (cid:39) (5 . ± . × for the cavity. The QWfeatures a double-sided silicon δ -doping with a set-back dis-tance of λ/ above and below the center of the cavity. Frommagneto-transport measurements [20], we estimate the 2DESelectron density n e (cid:39) . × cm − and the mobility µ (cid:39) . × cm V − s − . We deliberately choose a rela-tively low n e to access the physics of the lowest Landau levelin the range of magnetic fields currently available on our ex-perimental setup ( | B | ≤ ). The relatively high µ ensuresthat we can probe FQH physics.We perform polarization-resolved spectroscopy of the2DES using an infrared light emitting diode centered around820 nm. We shine excitation light onto the sample placed ina dilution refrigerator with a 30 mK base temperature. An as-pheric lens ( NA = 0 . ) collects the light reflected off thesample, which is analyzed using a spectrometer. Earlier stud-ies on the optics of 2DES have shown extreme sensitivity of n e to optical power [17, 21–23]. Increasing the optical powernot only changes n e , but also causes qualitative changes in thereflectivity spectrum [24], which is detrimental to the studyof fragile QH states. These unwanted effects are attributedto photoexcitation of DX centers in Si-doped Al x Ga − x As with x > . [25]. We minimize light-induced variations of n e by keeping x < . in the structure, and more importantly by placing the dopants in 10 nm GaAs ( x = 0 ) doping quan-tum wells (DQW). Further, we locate the DQWs in nodes ofthe electric field inside the cavity [19], which minimizes theintracavity light intensity at the position of the dopants.We first carry out cavity spectroscopy of the 2DES (seeFig. 1a) when B = 0 T . Fig. 1(b) shows the reflectivity spec-trum as a function of E cav . In contrast to undoped QW struc-tures [1], we observe coupling to three exciton-like resonancesas we scan E cav . For the lowest energy anticrossing, we mea-sure a normal mode splitting of g = 2 . ± .
01 meV . Since g is larger than the bare-cavity linewidth γ c (cid:39) ± µ eV ,the system is in the strong coupling regime of cavity-QEDand the elementary excitations should be characterized ascavity-polaritons [1]. Since the cavity-exciton coupling inthis system is comparable to energy level splittings of thethree exciton-like resonances, the polariton modes observedin the reflection spectrum can only be described as a su-perposition of all underlying resonances (see Fig. 1a). Weidentify the lowest energy exciton-like resonance observedin Fig. 1(b) as the heavy-hole attractive polaron ( X hhatt ) – aheavy-hole exciton dressed by Fermi sea electron-hole pairexcitations [9]. Since the attractive polaron resonance is as-sociated with the bound-molecular singlet trion channel, itwas previously referred to as “trion mode” [26, 27]. We as-sign the middle-energy excitonic resonance to the heavy holerepulsive-polaron ( X hhrep ) [9, 28]. The magnitude of the split-ting of this mode from the attractive polaron ( . meV) is afactor of 2 larger than the bare trion binding energy and isfully consistent with its identification as the repulsive polaronbranch. Finally, we tentatively identify the highest energy ex-citonic mode to the light-hole exciton [39].Next, we analyze the B (cid:54) = 0 case where the electrons areconfined to the lowest Landau Level (LL), with filling factor ν [5]. To explore the interplay between quantum Hall statesand polaritonic excitations, we tune E cav to ensure that thecavity mode dressed by nonperturbative coupling to higher en-ergy excitonic modes is resonant with X hhatt . Since the lowest FIG. 3. Cavity spectroscopy of the system in the fractional quantum Hall regime, as we tune B for fixed E cav (white dashed line). Measurementperformed with (a) σ − and (b) σ + polarized light. (c) Relevant energy levels and optical transitions around ν = 2 / . (d) Polariton splittingaround ν = 2 / measured in σ − (blue) and σ + (red) polarizations. (e) Lower polariton ( A LP ) and upper polariton ( A UP ) peak areas in σ − (blue) and σ + (red) polarizations (arbitrary units). energy polariton has predominantly X hhatt character, the spinstate of the optically generated electron is determined by thephoton polarization [17]: left-hand circularly polarized light σ − probes transitions to the lower electron Zeeman spin sub-band ( |↑(cid:105) ) and right-hand circularly polarized light σ + probestransitions to the upper electron Zeeman spin subband ( |↓(cid:105) ).Consequently, the observed spectral signatures are stronglydependent on how the electrons are arranged in the LLs i.e.on the spin-polarization of the different ground states of the2DES [22, 23].Figure 2(a-b) shows the white light reflection spectrum asa function of ν , varied by scanning B . Here, we tuned E cav close to resonance with the |↑(cid:105) -transition of lowest Landaulevel LL0 at ν = 1 . The most striking feature is the collapse of g σ − around B = 1 .
31 T , concurrent with the enhancement of g σ + . We associate this feature with the ν = 1 QH state, in ex-cellent agreement with the value of the electron density mea-sured independently. The observed behavior is a direct conse-quence of the high degree of spin-polarization of the QH fer-romagnet at ν = 1 : for a fully polarized state, g σ − is expectedto collapse due to the fact that all |↑(cid:105) -electron states are occu-pied. Phase space filling thus prevents optical excitation of anelectron to that level, and therefore the oscillator strength forthat transition collapses (see Fig. 2(c)). Concurrently, all |↓(cid:105) -electron states are free and g σ + increases due to the increasednumber of available states [30, 31]. Fig. 2(d) shows g σ − and g σ + extracted from fits of the reflection spectra. From this, wecalculate the spin-polarization S z (cid:39) ( g σ + − g σ − ) / ( g σ + + g σ − ) at ν = 1 [17, 23]. We obtain S z (cid:39)
70 % [19], suggesting thatfull polarization is not achieved at ν = 1 , contrary to whatis expected for the quantum Hall ferromagnet. In our low n e sample, incomplete polarization may arise due to disorder andreduced screening of impurity potentials [32]. Furthermorethe cyclotron frequency is comparable to the exciton bindingenergy, ensuring that exciton formation has a sizable contribu-tion from higher LLs. As a consequence, our measurementsonly yield a lower-bound on S z , and we do not expect full cancellation of g σ − at ν = 1 . We finally observe a rapid,symmetric depolarization on both sides of ν = 1 which iscompatible with formation of many-body spin excitations inthe ground state (skyrmions and anti-skyrmions) [30–34] as aconsequence of the competition between Coulomb and Zee-man energies. Finally, coupling to ν > integer QH statesis also visible in Fig. 2(a-b) as variations of the lower polari-ton energies vs ν as a consequence of phase space-filling [19].We emphasize that these spectral features are robust againstincreased optical powers, which demonstrates that our samplestructure provides, through “cavity protection” of the 2DES, aunique platform for optical studies of QH physics [19].We investigate FQH states by scanning B to up to 5 T foran increased value of E cav as shown in Fig. 3(a-b). Increasing B reduces ν , thus leading to absorption in a partially filledlowest LL [35–37]. Cavity coupling to several FQH states isobserved in Fig. 2 and Fig. 3 as a ν -dependent normal modesplitting in both polarizations. Such spectral signatures areparticularly striking when ν reaches the fractional values ν =1 / , 2/5, 2/3 and 5/3. We observe that g σ − and g σ + differsignificantly at ν = 1 / , 2/5 and 5/3, which shows that thesefractional QH states experience sizable spin-polarization [19].On the contrary, g σ − (cid:39) g σ + at ν = 2 / shows that this state isnot polarized, as expected for samples with n e in the range ofthe one studied here. Increasing n e should allow us to probethe phase transition from an unpolarized to a polarized / -state [17, 38].We now focus on filling factor ν = 2 / , see Fig. 3(a-b). In stark contrast with the integer QH states, both |↑(cid:105) and |↓(cid:105) states are available and phase-space filling only plays amarginal role here. Fig. 3(d) shows polariton splittings g σ − and g σ + extracted from fits of the reflectivity spectra. Onestriking feature is that the collapse of g σ − around ν = 2 / is not accompanied with an appreciable increase in g σ + , con-trary to what was observed for ν = 1 . Because the LLs arepartially filled, the mechanism leading to modification of thepolariton splitting is indeed modified.We argue that the decrease g σ − for a spin-polarized state isdue to the polaron nature of optical excitations that are acces-sible when promoting an electron into the |↑(cid:105) -state with σ − -polarized light. For a fully polarized state, all electrons are inthe same |↑(cid:105) -state and there are no electrons in the |↓(cid:105) -state.Since the oscillator strength of the σ − singlet X hhatt is propor-tional to the density of |↓(cid:105) electrons, perfect spin polarizationwould lead to vanishing cavity coupling. In contrast, promo-tion of an electron in the |↓(cid:105) -state with σ + -polarized light al-ways leads to formation of a singlet polaron excitation withelectrons available in the |↑(cid:105) -state, and the polariton splittingis only marginally modified. Fig. 3(e) plots the evolution ofthe polariton peak areas around ν = 2 / . The decrease in po-lariton splitting in σ − -polarization is accompanied with a loss(gain) of weight of the lower (upper) polariton. This obser-vation is fully consistent with a reduced cavity-polaron cou-pling strength and a finite detuning between the bare polaronand cavity resonances, ensuring that the lower (upper) polari-ton has predominantly polaron (cavity) character at ν = 2 / .The absence of a similar oscillator strength transfer in σ + -polarization on resonance further supports the interpretationof our data in terms of inhibition of σ − polaron-dressing by |↓(cid:105) -electrons at ν = 2 / .Finally, we address the question of the modification of thepolaron-polariton effective mass in the vicinity of ν = 2 / .We use a NA = 0 . lens to excite a broad range of in-planemomenta k (cid:107) using the same broadband light emitting diode.A low NA lens couples the reflected light into a fiber, whichenables angle selective measurements. The dispersion relationin Fig. 4(a) at ν = 2 / clearly shows the anticrossings with X hhatt and X hhrep as pointed out already in Fig. 1(b). We fit aparabola to the lower polariton dispersion at ν = 2 / (dashedorange line) and compare it, in Fig. 4(b), to the dispersionsmeasured at filling factors slightly above (green) and below(blue). Strikingly, we find an increase of the effective mass m ∗ at ν = 2 / (orange) by a factor of (cid:39) ± compared to ν = 0 . (green) and ν = 0 . (blue) [ ? ]. This observationillustrates further the strong reduction in the oscillator strengthof the attractive-polaron resonance which reduces the cavity-character and enhances m ∗ .We emphasize that theory of exciton-polarons has been pre-viously developed for excitons interacting with a 2DES inthe limit B = 0 [9, 10]. A quantitative modeling of ourexperiment requires extending prior theoretical work to thecase of screening of excitons by electrons occupying a sin-gle LL: a significant advance in this direction was the re-cent development of the theory of exciton-polarons in thelimit of strong magnetic fields but without taking into accountelectron-electron interactions leading to FQH states [28]. Ourwork focused on the singlet channel which plays a prominentrole in the limit of moderate magnetic fields ( B ≤ . T) usedin our experiments. Yet, we expect triplet channels to play akey role in determining the full polariton spectrum, particu-larly at higher B-fields relevant for samples with higher elec-tron density. A more challenging problem is exciton-electroninteractions in the vicinity of FQH states: polaron-polariton
FIG. 4. Polaron-polariton dispersion around ν = 2 / . (a) Cavityspectroscopy exactly at ν = 0 . for different in-plane momenta k (cid:107) .The flat reflection signal observable between the lower and upper po-laritons around k = 0 µ m − is an experimental artefact, stemmingfrom an etalon effect in the detection path. (b) Energy of the lowerpolaron-polariton for small k (cid:107) at filling factor ν = 0 . (green), ν = 0 . (orange) and ν = 0 . (blue). Dashed lines show parabolicfits to the lower polariton energies. formation in this limit may be described using polariton dress-ing by fractionally charged quasi-particle-hole pairs [40]. Thelatter problem is related to identification of signatures of in-compressibility of the many-body ground state in the polaritonexcitation spectrum.On the technical side, we demonstrate that cavity elec-trodynamics is an invaluable platform to probe fragile frac-tional states. This could potentially enable optical manip-ulation of anyonic quasi-particles associated with strongly-correlated phases. Furthermore, increasing the quality factorof the cavity could further enhance the sensitivity of our mea-surements [18]. Finally, the observed filling factor-dependentpolariton splitting could be particularly useful to engineersingle-photon non-linearities between polaritons and allowfor experimental realization of strongly correlated driven-dissipative photonic systems in arrays of semiconductor mi-crocavities [4].The Authors acknowledge many useful discussions withHadis Abbaspour, Valentin Goblot, Wolf Wuester and SinaZeytinoglu. This work was supported by NCCR QuantumPhotonics (NCCR QP), an ETH Fellowship (S. R.), and anERC Advanced investigator grant (POLTDES). [1] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Ob-servation of the coupled exciton-photon mode splitting in asemiconductor quantum microcavity,” Phys. Rev. Lett. , 3314(1992).[2] H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton bose-einstein condensation,” Rev. Mod. Phys. , 1489 (2010). [3] I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod.Phys. , 299 (2013).[4] A. Amo and J. Bloch, “Exciton-polaritons in lattices: A non-linear photonic simulator,” Comptes Rendus Physique , 934(2016).[5] R. Prange, M. Cage, K. Klitzing, S. Girvin, A. Chang, F. Dun-can, M. Haldane, R. Laughlin, A. Pruisken, and D. Thou-less, The Quantum Hall Effect , Graduate Texts in Contempo-rary Physics (Springer New York, 1989).[6] S. Das Sarma and A. Pinczuk,
Perspectives in Quantum HallEffects: Novel Quantum Liquids in Low-dimensional Semi-conductor Structures , A Wiley-Interscience publication (Wiley,1997).[7] S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, “Con-tinuous quantum phase transitions,” Rev. Mod. Phys. , 315(1997).[8] C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma, “Non-abelian anyons and topological quantumcomputation,” Rev. Mod. Phys. , 1083 (2008).[9] M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kro-ner, E. Demler, and A. Imamoglu, “Fermi polaron-polaritons incharge-tunable atomically thin semiconductors,” Nat Phys ,255 (2017).[10] D. K. Efimkin and A. H. MacDonald, “Many-Body Theory ofTrion Absorption Features in Two-Dimensional Semiconduc-tors,” arXiv:1609.06329 (2016).[11] A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwierlein,“Observation of fermi polarons in a tunable fermi liquid of ul-tracold atoms,” Phys. Rev. Lett. , 230402 (2009).[12] M. Koschorreck, D. Pertot, E. Vogt, B. Frohlich, M. Feld, andM. Kohl, “Attractive and repulsive fermi polarons in two dimen-sions,” Nature , 619 (2012).[13] R. Schmidt, T. Enss, V. Pietil¨a, and E. Demler, “Fermi polaronsin two dimensions,” Phys. Rev. A , 021602 (2012).[14] P. Massignan, M. Zaccanti, and G. M. Bruun, “Polarons,dressed molecules and itinerant ferromagnetism in ultracoldfermi gases,” Reports on Progress in Physics , 034401 (2014).[15] A. V. Koudinov, C. Kehl, A. V. Rodina, J. Geurts, D. Wolverson,and G. Karczewski, “Suris tetrons: Possible spectroscopic evi-dence for four-particle optical excitations of a two-dimensionalelectron gas,” Phys. Rev. Lett. , 147402 (2014).[16] R. A. Suris, “Correlation between trion and hole in fermi dis-tribution in process of trion photo-excitation in doped qws,” in Optical Properties of 2D Systems with Interacting Electrons ,(Springer Netherlands, Dordrecht, 2003) pp. 111–124.[17] S. Smolka, W. Wuester, F. Haupt, S. Faelt, W. Wegscheider, andA. Imamoglu, “Cavity quantum electrodynamics with many-body states of a two-dimensional electron gas,” Science ,332 (2014).[18] M. Steger, G. Liu, B. Nelsen, C. Gautham, D. W. Snoke,R. Balili, L. Pfeiffer, and K. West, “Long-range ballistic mo-tion and coherent flow of long-lifetime polaritons,” Phys. Rev.B , 235314 (2013).[19] See Supplemental Material at [URL will be inserted by pub-lisher] for additional information about the sample structure,the light sensitivity of n e and the measurements of S z .[20] D. W. Koon and C. J. Knickerbocker, “What do you measurewhen you measure resistivity?” Review of Scientific Instru-ments , 207 (1992).[21] I. V. Kukushkin, K. von Klitzing, K. Ploog, V. E. Kirpichev,and B. N. Shepel, “Reduction of the electron density in gaas- al x ga − x as single heterojunctions by continuous photoexcita-tion,” Phys. Rev. B , 4179 (1989).[22] B. Goldberg, D. Heiman, A. Pinczuk, L. Pfeiffer, and K. West, “Magneto-optics in the integer and fractional quantum hall andelectron solid regimes,” Surface Science , 9 (1992).[23] J. G. Groshaus, P. Plochocka-Polack, M. Rappaport, V. Uman-sky, I. Bar-Joseph, B. S. Dennis, L. N. Pfeiffer, K. W. West,Y. Gallais, and A. Pinczuk, “Absorption in the fractional quan-tum hall regime: Trion dichroism and spin polarization,” Phys.Rev. Lett. , 156803 (2007).[24] W. W¨uster, Cavity quantum electrodynamics with many-bodystates of a two-dimensional electron system , Ph.D. thesis, ETH-Z¨urich (2015).[25] T. Ihn,
Semiconductor Nanostructures: Quantum States andElectronic Transport (OUP Oxford, 2010).[26] A. Esser, R. Zimmermann, and E. Runge, “Theory of TrionSpectra in Semiconductor Nanostructures,” Physica Status So-lidi B Basic Research , 317 (2001).[27] I. Bar-Joseph, “Trions in gaas quantum wells,” SemiconductorScience and Technology , R29 (2005).[28] D. K. Efimkin and A. H. MacDonald, “Many-body theory oftrion absorption in a strong magnetic field,” arXiv:1707.05845(2017).[39] The absence of attractive and repulsive polaron branches asso-ciated with the light-hole exciton may stem from the absence ofa bound light-hole trion, or from the lower oscillator strengthand the larger broadening of these resonances.[30] E. H. Aifer, B. B. Goldberg, and D. A. Broido, “Evidenceof skyrmion excitations about ν = 1 in n -modulation-dopedsingle quantum wells by interband optical transmission,” Phys.Rev. Lett. , 680 (1996).[31] J. G. Groshaus, V. Umansky, H. Shtrikman, Y. Levinson, andI. Bar-Joseph, “Absorption spectrum around ν = 1 : Evidencefor a small-size skyrmion,” Phys. Rev. Lett. , 096802 (2004).[32] M. Manfra, B. Goldberg, L. Pfeiffer, and K. West, “Optical de-termination of the spin polarization of a quantum hall ferromag-net,” Physica E: Low-dimensional Systems and Nanostructures , 28 (1997).[33] V. Zhitomirsky, R. Chughtai, R. Nicholas, and M. Henini,“Spin polarization of 2d electrons in the quantum hall ferro-magnet: evidence for a partially polarized state around fillingfactor one,” Physica E: Low-dimensional Systems and Nanos-tructures , 12 (2002).[34] P. Plochocka, J. M. Schneider, D. K. Maude, M. Potemski,M. Rappaport, V. Umansky, I. Bar-Joseph, J. G. Groshaus,Y. Gallais, and A. Pinczuk, “Optical absorption to probe thequantum hall ferromagnet at filling factor ν = 1 ,” Phys. Rev.Lett. , 126806 (2009).[35] B. B. Goldberg, D. Heiman, A. Pinczuk, L. Pfeiffer, andK. West, “Optical investigations of the integer and fractionalquantum hall effects: Energy plateaus, intensity minima, andline splitting in band-gap emission,” Phys. Rev. Lett. , 641(1990).[36] G. Yusa, H. Shtrikman, and I. Bar-Joseph, “Charged excitons inthe fractional quantum hall regime,” Phys. Rev. Lett. , 216402(2001).[37] M. Byszewski, B. Chwalisz, D. K. Maude, M. L. Sadowski,M. Potemski, T. Saku, Y. Hirayama, S. Studenikin, D. G. Aust-ing, A. S. Sachrajda, and P. Hawrylak, “Optical probing ofcomposite fermions in a two-dimensional electron gas,” NatPhys , 239 (2006).[38] J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, and K. W. West,“Evidence for a spin transition in the ν =2/3 fractional quantumhall effect,” Phys. Rev. B , 7910 (1990).[39] Note that the estimation of the lower polariton mass at ν = 2 / is rendered difficult by the low curvature of the parabola.[40] F. Grusdt, N. Y. Yao, D. Abanin, M. Fleischhauer, and E. Dem- ler, “Interferometric measurements of many-body topologicalinvariants using mobile impurities,” Nature Communications7