Reports × 1011 cm−2 (9). Sample B exhibits an order of magnitude higher mobility of μ = 2.5 × 106 cm2(V s)−1 at ne = 2.3 × 1011 cm−2. Figure 1D shows the differential reflectivity (dR) (9) spectra of sample A (blue) as the electron density ne in the 2DEG is varied from ne < 2 × 1010 cm−2 to ne = 8 × 1010 cm−2. The cavity mode Stephan Smolka,1* Wolf Wuester,1,2* Florian Haupt,1 Stefan Faelt,1,2 is visible as a dR peak at 1521 meV. Changing ne modifies the nature of Werner Wegscheider,2 Ataç Imamoglu1† elementary optical excitations: at low 1 Institute of Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland. 2Solid State Physics Laboratory, electron densities (ne < 2 × 1010 cm−2) ETH Zurich, 8093 Zurich, Switzerland. the QW spectrum is dominated by the 0 heavy-hole exciton X hh peak. The *These authors contributed equally to this work. 0 weak resonance on the red side of X hh †Corresponding author. E-mail: [email protected] stems from the excitation of a bound state of two electrons and one hole (X−, Light-matter interaction has played a central role in understanding as well as trion). For 2 × 1010 cm−2 ≤ ne < 8 ×1010 engineering new states of matter. Reversible coupling of excitons and photons cm−2, the trion transition emerges as the enabled groundbreaking results in condensation and superfluidity of dominant feature in the spectrum. Innonequilibrium quasiparticles with a photonic component. We investigate such creasing n e leads to an asymmetry in cavity-polaritons in the presence of a high-mobility two-dimensional electron gas, the X− lineshape and a blue-shift of its exhibiting strongly correlated phases. When the cavity is on resonance with the center-frequency (10). Finally, for ne > Fermi level, we observe novel many-body physics associated with a dynamical hole 8 × 1010 cm−2, the absorption spectrum scattering potential. In finite magnetic fields, polaritons show unique signatures of is dominated by an asymmetric resointeger and fractional quantum Hall ground states. Our results lay the groundwork nance at the Fermi edge (11). Photolufor probing nonequilibrium dynamics of quantum Hall states and exploiting the minescence (PL) spectra (orange), electron density dependence of polariton splitting to obtain ultrastrong optical obtained after tuning the cavity mode to nonlinearities. 1531 meV, is shown with dR in Fig. 1D. For ne > 6 × 1010 cm−2 the PL spectrum (12) exhibits a double-peaked Strong electric-dipole coupling of excitons in an intrinsic semiconductor quantum well (QW) to microcavity photons leads to the formation of structure with a total width that agrees well with EF, estimated using mixed light-matter quasiparticles called cavity-polaritons (1, 2). In con- independently determined ne (9). We first address the confluence of Fermi-edge singularity and nontrast to excitons, optical excitations from a two-dimensional electron gas (2DEG) show many-body effects associated with the abrupt turn-on of a perturbative light-matter coupling (Fig. 2). In stark contrast to earlier valence-band hole scattering potential. The resulting absorption spec- work on edge singularity physics, our system exhibits a dynamical hole trum exhibits a power-law divergence, stemming from final-state inter- scattering potential due to vacuum Rabi oscillations in the final state of action effects leading to orthogonal initial and final many-body wave- the optical transition, and a finite hole recoil induced modification of the functions, and is referred to as Fermi-edge singularity (3). Despite initial cavity-2DEG coupling. We note that prior work - based on a fixed ne attempts (4–6), the nature of strong coupling of a Fermi-edge singularity, 2DEG embedded in a microcavity - demonstrated normal-mode splitting rather than an exciton, to a microcavity mode remained largely unex- at a relatively high temperature of 2 K and used single-particle transfermatrix calculations to obtain a fit to the data (5). We show on the other plored. We investigate a 2DEG in a modulation-doped GaAs QW embedded hand that only by measuring the polariton dispersion as well as the ne, in a distributed Bragg reflector (DBR) micro-cavity (Fig. 1A). We use mobility and temperature dependence of the vacuum Rabi coupling, it is this unique system to analyze two distinct problems: in the absence of an possible to develop a consistent picture of the underlying many-body external magnetic field (Bz = 0), optical excitations from a 2DEG are physics. To demonstrate nonperturbative light-matter coupling, we fix ne to simultaneously subject to Coulomb interactions and strong cavity coupling, which lead to the formation of Fermi-edge polaritons - a many- 1.2 × 1011 cm−2 in sample A and tune the cavity mode across the absorpbody excitation with a dynamical valence-hole scattering potential (7). tion edge at EF. When the cavity frequency Ec is resonant with EF (Fig. In the Bz ≠ 0 limit, we establish cavity quantum electrodynamics (QED) 2A), we observe normal-mode splitting of 2g = (2.0 ±0.1) meV, with g as a new paradigm for investigating integer and fractional quantum Hall denoting the dipole-coupling strength. Because 2g is larger than the bare (QH) states (8) where information about a strongly correlated electronic cavity linewidth Γc = 1 meV, it is concluded that the system is in the ground state is transcribed onto the reflection and transmission ampli- strong coupling regime of cavity-QED where the elementary optical excitations could be described as trion/Fermi-edge polaritons (5, 13). In tudes of a single cavity photon. Figure 1B (left panel) depicts the dispersion of the relevant electron- contrast to the exciton-polariton excitations observed in undoped QWs ic states when Bz = 0. The Fermi energy EF is tuned by applying a gate coupled to microcavities (1), in our case the upper polariton is substanvoltage (Vg) between a p-doped GaAs top layer and the 2DEG. When Bz tially broader than the lower polariton. This broadening and the asym≠ 0, the electronic states form discrete Landau levels (LL) (Fig. 1B, right metry stem from the shake-up of the Fermi sea that accompanies bound panel). Each (spin-resolved) LL has a degeneracy e Bz/h; the filling fac- trion or Fermi-edge resonance formation (6). The dispersion of Fermi-edge polaritons is shown in Fig. 2B. The tor ν = ne h/(e Bz) determines the number of filled LLs at a given electron density ne. We studied two different samples with gate-tunable ne: sam- data are obtained by measuring angle-resolved dR spectra (14) for a ple A exhibits an electron mobility of μ = 2 × 105 cm2(V s)−1 at ne = 1.0 cavity detuning that yields minimal normal-mode splitting at normal
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Cavity quantum electrodynamics with many-body states of a twodimensional electron gas
incidence (Fig. 2A). We observe two split branches each with an ultrasmall effective polariton mass which is approximately twice as large as that of the bare cavity-photon mass (9). In order to study the density and mobility dependence of the Fermiedge polaritons, we recorded the polariton excitations versus detuning from the Fermi-edge at ne = 1.6 × 1011 cm−2 on the high mobility sample B (Fig. 2C). The data shows a significantly smaller normal mode splitting as compared to sample A (Fig. 2A) which is a consequence of weaker hole confinement due to higher ne and mobility. A detailed study of the normal-mode splitting at resonance as a function of ne is presented in Fig. 2D showing a clear decrease from 2g = (2.0 ± 0.1) meV to 2g = (1.3 ± 0.2) meV as the density is increased from ne = 0.8 × 1011 cm−2 to ne = 1.6 × 1011 cm−2 (9). The scattering of an electron with momentum ∼kF on the Fermi surface is accompanied by a hole momentum kick, leading to a hole recoil energy Er (15). A larger hole recoil partially inhibits the formation of the electron screening cloud as ne(kF) is increased resulting in the observed reduction of the normal-mode splitting. Remarkably, polariton excitations depend strongly on temperature as well: when we compare the ne = 0.9 × 1011 cm−2 dR spectrum at T = 0.2 K with the one at T = 4 K (Fig. 2D, inset), we observe that the normalmode splitting disappears at T = 4 K. This observation is striking since the corresponding splitting at T = 0.2K (2g = 1.3 ± 0.2meV) is larger than the thermal energy ET = 345 μeV at T = 4 K. Our experimental findings point to a scenario where the nonperturbative cavity-QW coupling leads to the formation of Fermi-edge polaritons for ne > 8 × 1010 cm−2. In contrast to the Fermi-edge singularity associated with screening of a localized hole by the Fermi sea (4, 6), these new many-body optical excitations are delocalized as demonstrated by the polariton dispersion (Fig. 2B). A key requirement for the observation of Fermi-edge polaritons is that the cavity-QW coupling strength exceeds Er: in this limit, the enhancement of the 2DEG-cavity coupling by optical excitations with energy ≥ EF + Er ensures strong coupling, albeit with reduced normal-mode splitting. This conclusion is supported by the reduction of normal mode splitting (Fig. 2D) with increasing ne, or equivalently, increasing Er observed in the high-mobility sample B. Conversely, the persistence of polariton splitting implies that the hole population in the QW exhibits Rabi oscillations, leading to a time-dependent scattering potential. In contrast to the Bz = 0 limit where the initial state of the optical transition is a Fermi sea without interaction-induced correlations, the electronic states for Bz ≠ 0 in the absence of photonic excitations are QH states. We analyze polariton excitations out of these many-body states by performing polarization-resolved dR measurements. Since the absorption of right-hand (left-hand) circularly-polarized σ+(σ−) light probes the transition from a heavy-hole ⇑
( ⇓ ) state to a spin-down
↓ (spin-up
↑ ) electronic LL state, the observed spectrum is highly sensitive to spin polarization-dependent phase-space-filling. When the magnetic field is increased, the filling factor ν is reduced, allowing optical absorption into LLs that are no longer (completely) filled. In Fig. 3, A and B, the cavity resonance on sample B is tuned to E = 1523.8 meV. In the low field range (Bz = 0 − 2 T) absorption into higher lying empty LLs (ν ≥ 2), enabled by the weak off-resonant coupling of these resonances to the cavity mode, is visible. The reflection spectrum from the lowest LL (LL0) at ne = 1.4 × 1011 cm−2 (Fig. 3, A and B) shows that a finite polariton splitting emerges at Bz ∼ 3 T (ν ∼ 2). To study the resonant interaction between the cavity mode and the optical excitations of QH states, we fixed Bz = 3 T, tuned the cavity to resonance with the LL0 spin-up (σ−) transition at ν = 2 and varied ne through Vg. Fig. 3, C and D, show that for both light polarizations only the bare cavity resonance is visible for ν ≥ 2, since phasespace-filling blocks all 2DEG optical excitations at Ec. In contrast, a
normal-mode splitting appears for ν < 2; the splitting grows monotonically as ne is reduced, since the number of available final states is thereby increased. We remark that the normal-mode splitting of σ+ polarization starts at higher ν than the one of σ−, suggesting that 2DEG electrons remain spin-polarized for 1.7 ≤ ν < 2.0. Unlike the Bz = 0 case, the normal-mode splitting at Bz ≠ 0 is comparable to that of a heavy-hole exciton polariton in the limit ne → 0, pointing to a reduced screening of electron-hole attraction. The dominant optical excitations into LL0 for 1 < ν < 2 can be described as heavy-hole singlet trions (16), as the strongest binding energy and the highest optical oscillator strength would be obtained for a complex of two electrons of opposite spin in the same electronic orbital m of size am = eBz and a heavy-hole with a strong spatial overlap. Such a trion could have an absorption strength comparable to that obtained for an exciton since the magnetic length am is comparable to the exciton Bohr radius for Bz ∼ 5T. By increasing the magnetic field to Bz = 4.9 T we explore polariton excitations around ν = 1. In Fig. 3, E and F, we see a drastic change in the optical response: while the normal-mode splitting collapses for σ− light, it increases sharply for σ+. This feature shows the emergence of a high degree of spin polarization, which we associate with the ν = 1 integer QH state (17, 18). In case of full spin polarization the oscillator strength (∝ g2) of the σ+ polariton excitations is maximum since no ↓ states are occupied. For ν = 1, the degree of spin polarization,
(
P =− g σ2+ g σ2−
) (g
2 σ+
)
+ g σ2− , is determined from the measured mini-
mum normal-mode splittings and found to be P = 90 ± 10%. The error arises primarily from the measurement uncertainty of the individual splittings and simultaneous coupling to higher lying optical transitions. As we move away to both sides from ν = 1 by changing either Vg or Bz, we observe a symmetric decrease (increase) of the normal-mode splitting for the σ+ (σ−) polariton excitations which we attribute to spindepolarization due to skyrmion excitations (19). In order to investigate fractional QH states we tuned Ec into resonance with the ↓ -transition of LL0 at ν = 2/3. Fig. 4, A and B, present dR spectra at Bz = 6 T as a function of ne: for ν ≅ 2/3, we observe a small decrease (increase) in the normal-mode splitting for the σ− (σ+) polariton modes, which points to enhanced spin polarization associated with the ν = 2/3 fractional QH state (20, 21). The small change in the splitting stems from the sizeable degree of spin polarization that persists for ∀ν < 1 and the fact that even for complete spin polarization at ν = 2/3 one third of the states in the spin-up LL0 are available for excitonic excitations. To highlight the variations in the normal-mode splitting, we plot in Fig. 4C the energy difference ∆ELP between the dR peaks of the σ+ and σ− lower polariton branches as a function of ν at Bz = 6 T (blue circles): a local maximum in ∆ELP around the fractional QH state ν = 2/3 signifies spin polarization (22–25). To confirm that the peak in ∆ELP at ν = 2/3 does not have an auxiliary origin, we spatially tune the cavity mode to Eν = 1 (red squares in Fig. 4C) and find that the local maximum in ∆ELP at ν = 2/3 persists. It has been previously established that a phase transition from an unpolarized ν = 2/3 to a polarized ν = 2/3 QH ground state takes place at a critical magnetic field Bc that depends strongly on the effective width of the electron wave function (26, 27). Strikingly, at a lower magnetic field of Bz = 4.9 T (Figs. 3, E and F), we do not observe a local maximum in ∆ELP at ν = 2/3 (Fig. 4C), suggesting that the excess spin polarization is lost: this observation is consistent with a phase transition from an unpolarized to a polarized ν = 2/3 state. Our Bz = 0 experiments reveal a novel cavity excitation induced strongly correlated state which requires a new theoretical framework describing a mobile 2DEG impurity in a cavity-QED setting. On the
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other hand, Bz ≠ 0 experiments establish cavity-QED as an invaluable spectroscopic tool for investigating and manipulating properties of many-body states in a 2DEG. By increasing the cavity lifetime it should be possible to enhance the measurement sensitivity and to observe the incompressibility induced modification of the polariton splitting. Semiintegrated cavity structures not only allow for a factor of 20 increase in lifetime, but also enable tuning the electric-dipole coupling strength (28). More intriguing possibilities raised by our observations include observation of nonequilibrium dynamics following an optical quench into a fractional QH state. Finally, the system we consider in the Bz = 0 limit could also be viewed as a new nonlinear optical system where ne of the 2DEG can be used to change the effective Bohr radius and consequently the strength of polariton-polariton interactions (29). References and Notes 1. C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314–3317 (1992). Medline doi:10.1103/PhysRevLett.69.3314 2. H. Deng, H. Haug, Y. Yamamoto, Exciton-polariton Bose-Einstein condensation. Rev. Mod. Phys. 82, 1489–1537 (2010). doi:10.1103/RevModPhys.82.1489 3. K. Ohtaka, Y. Tanabe, Theory of the soft-x-ray edge problem in simple metals: Historical survey and recent developments. Rev. Mod. Phys. 62, 929–991 (1990). doi:10.1103/RevModPhys.62.929 4. N. S. Averkiev, M. M. Glazov, Light-matter interaction in doped microcavities. Phys. Rev. B 76, 045320 (2007). doi:10.1103/PhysRevB.76.045320 5. A. Gabbay, Y. Preezant, E. Cohen, B. M. Ashkinadze, L. N. Pfeiffer, Fermi edge polaritons in a microcavity containing a high density two-dimensional electron gas. Phys. Rev. Lett. 99, 157402 (2007). Medline doi:10.1103/PhysRevLett.99.157402 6. M. Baeten, M. Wouters, (available at http://arxiv.org/abs/1404.2048). 7. B. Sbierski, M. Hanl, A. Weichselbaum, H. E. Türeci, M. Goldstein, L. I. Glazman, J. von Delft, A. Imamoğlu, Proposed Rabi-Kondo correlated state in a laser-driven semiconductor quantum dot. Phys. Rev. Lett. 111, 157402 (2013). Medline doi:10.1103/PhysRevLett.111.157402 8. S. Das Sarma, A. Pinczuk, (eds), Perspectives on Quantum Hall Effects (Wiley, New York, 1996). 9. See supplementary materials on Science Online. 10. B. R. Bennett, R. A. Soref, J. A. del Alamo, Carrier-induced change in refractive index of InP, GaAs and InGaAsP. IEEE J. Quantum Electron. 26, 113–122 (1990). doi:10.1109/3.44924 11. P. W. Anderson, Infrared Catastrophe in Fermi Gases with Local Scattering Potentials. Phys. Rev. Lett. 18, 1049–1051 (1967). doi:10.1103/PhysRevLett.18.1049 12. M. S. Skolnick, J. M. Rorison, K. J. Nash, D. J. Mowbray, P. R. Tapster, S. J. Bass, A. D. Pitt, Observation of a many-body edge singularity in quantum well luminescence spectra. Phys. Rev. Lett. 58, 2130–2133 (1987). Medline doi:10.1103/PhysRevLett.58.2130 13. M. Perrin, P. Senellart, A. Lemaître, J. Bloch, Polariton relaxation in semiconductor microcavities: Efficiency of electron-polariton scattering. Phys. Rev. B 72, 075340 (2005). doi:10.1103/PhysRevB.72.075340 14. R. Houdré, C. Weisbuch, R. P. Stanley, U. Oesterle, P. Pellandini, M. Ilegems, Measurement of cavity-polariton dispersion curve from angle resolved photoluminescence experiments. Phys. Rev. Lett. 73, 2043–2046 (1994). Medline doi:10.1103/PhysRevLett.73.2043 15. A. Rosch, T. Kopp, Heavy Particle in a d-Dimensional Fermionic Bath: A Strong Coupling Approach. Phys. Rev. Lett. 75, 1988–1991 (1995). Medline doi:10.1103/PhysRevLett.75.1988 16. I. Bar-Joseph, Trions in GaAs quantum wells. Semicond. Sci. Technol. 20, R29–R39 (2005). doi:10.1088/0268-1242/20/6/R01 17. P. Plochocka, J. M. Schneider, D. K. Maude, M. Potemski, M. Rappaport, V. Umansky, I. Bar-Joseph, J. G. Groshaus, Y. Gallais, A. Pinczuk, Optical absorption to probe the quantum Hall ferromagnet at filling factor nu=1. Phys. Rev. Lett. 102, 126806 (2009). Medline doi:10.1103/PhysRevLett.102.126806 18. L. Tiemann, G. Gamez, N. Kumada, K. Muraki, Unraveling the spin polarization of the ν = 5/2 fractional quantum Hall state. Science 335, 828– 831 (2012). Medline doi:10.1126/science.1216697
19. E. H. Aifer, B. B. Goldberg, D. A. Broido, Evidence of Skyrmion excitations about nu =1 in n-modulation-doped single quantum wells by interband optical transmission. Phys. Rev. Lett. 76, 680–683 (1996). Medline doi:10.1103/PhysRevLett.76.680 20. B. I. Halperin, Hel. Phys. Acta 56, 75–102 (1983). 21. J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, K. W. West, Evidence for a spin transition in the ν=2/3 fractional quantum Hall effect. Phys. Rev. B 41, 7910– 7913 (1990). doi:10.1103/PhysRevB.41.7910 22. I. V. Kukushkin, K. von Klitzing, K. Eberl, Spin Polarization of Composite Fermions: Measurements of the Fermi Energy. Phys. Rev. Lett. 82, 3665– 3668 (1999). doi:10.1103/PhysRevLett.82.3665 23. J. H. Smet, R. A. Deutschmann, F. Ertl, W. Wegscheider, G. Abstreiter, K. von Klitzing, Gate-voltage control of spin interactions between electrons and nuclei in a semiconductor. Nature 415, 281–286 (2002). Medline doi:10.1038/415281a 24. B. Verdene, J. Martin, G. Gamez, J. Smet, K. von Klitzing, D. Mahalu, D. Schuh, G. Abstreiter, A. Yacoby, Microscopic manifestation of the spin phase transition at filling factor 2/3. Nat. Phys. 3, 392–396 (2007). doi:10.1038/nphys588 25. J. Hayakawa, K. Muraki, G. Yusa, Real-space imaging of fractional quantum Hall liquids. Nat. Nanotechnol. 8, 31–35 (2013). Medline doi:10.1038/nnano.2012.209 26. T. Chakraborty, Spin-reversed ground state and energy gap in the fractional quantum Hall effect. Surf. Sci. 229, 16–20 (1990). doi:10.1016/00396028(90)90821-O 27. N. Kumada, D. Terasawa, Y. Shimoda, H. Azuhata, A. Sawada, Z. F. Ezawa, K. Muraki, T. Saku, Y. Hirayama, Phase diagram of interacting composite fermions in the bilayer nu=2/3 quantum hall effect. Phys. Rev. Lett. 89, 116802 (2002). Medline doi:10.1103/PhysRevLett.89.116802 28. B. Besga et al., (available at http://arxiv.org/abs/1312.0819). 29. I. Carusotto, C. Ciuti, Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013). doi:10.1103/RevModPhys.85.299 30. C. Latta, F. Haupt, M. Hanl, A. Weichselbaum, M. Claassen, W. Wuester, P. Fallahi, S. Faelt, L. Glazman, J. von Delft, H. E. Türeci, A. Imamoglu, Quantum quench of Kondo correlations in optical absorption. Nature 474, 627–630 (2011). Medline doi:10.1038/nature10204 31. F. Haupt, S. Smolka, M. Hanl, W. Wüster, J. Miguel-Sanchez, A. Weichselbaum, J. von Delft, A. Imamoglu, Nonequilibrium dynamics in an optical transition from a neutral quantum dot to a correlated many-body state. Phys. Rev. B 88, 161304 (2013). doi:10.1103/PhysRevB.88.161304 32. S. Pau, G. Björk, J. Jacobson, H. Cao, Y. Yamamoto, Microcavity excitonpolariton splitting in the linear regime. Phys. Rev. B 51, 14437–14447 (1995). doi:10.1103/PhysRevB.51.14437 33. D. Snoke, Spontaneous Bose coherence of excitons and polaritons. Science 298, 1368–1372 (2002). Medline doi:10.1126/science.1078082 34. I. V. Kukushkin, K. von Klitzing, K. Ploog, V. E. Kirpichev, B. N. Shepel, Reduction of the electron density in GaAs-AlxGa1-xAs single heterojunctions by continuous photoexcitation. Phys. Rev. B 40, 4179–4182 (1989). doi:10.1103/PhysRevB.40.4179 35. A. J. Shields, J. L. Osborne, M. Y. Simmons, D. A. Ritchie, M. Pepper, Comparison of optical and transport measurements of electron densities in quantum wells. Semicond. Sci. Technol. 11, 890–896 (1996). doi:10.1088/0268-1242/11/6/007 Acknowledgments: The Authors acknowledge insightful discussions with L. Glazman, M. Goldstein, A. Rosch, A. Srivastava and J. von Delft. This work is supported by NCCR Quantum Photonics (NCCR QP), research instrument of the Swiss National Science Foundation (SNSF). Supplementary Materials www.sciencemag.org/content/science.1258595/DC1 Materials and Methods Supplementary Text Figs. S1 to S8 References (30–35) 10 July 2014; accepted 17 September 2014 Published online 3 October 2014 10.1126/science.1258595
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Fig. 1. Cavity QED in a two-dimensional electron system. (A) The sample consists of a modulation doped 20 nm GaAs QW inside a DBR microcavity. (B) left: Dispersion of the conduction and valence bands at Bz = 0 T; EG and EF denote the band gap and Fermi energy. Right: For Bz ≠ 0 the bands split into discrete Landau energy levels; electron-hole pairs
−
, can be excited with σ (σ ) polarized +
light into the corresponding (electron) spin-split Landau levels (LLO). (C) Magnetotransport 11 −2 measurements of the resistivity ρxx and ρxy of wafer B at an electron density of ne = 2.3 × 10 cm . (D) dR (blue) and photoluminescence (orange) spectra of sample A for different electron densities at T = 0.2 K. The spectra are vertically shifted for clarity. The uncertainty in the estimated electron density for the 10 −2 10 −2 plots with ne ≥ 4 × 10 cm is ∼ 2 × 10 cm .
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Fig. 2. Experimental signatures of Fermi-edge polaritons. (A) dR spectra taken from sample A as a function of cavity detuning δ from the Fermi-edge at T = 0.2 K 11 −2 and ne = 1.2 × 10 cm . (B) Fermi-edge polariton energy dispersion obtained at T = 11 −2 1.6 K, ne = 1.2 × 10 cm and δ = 0. The dashed straight line depicts the Fermi energy. The blue (red) color-coded area displays low (high) dR signal. (C) dR spectra measured on sample B versus cavity detuning at T = 0.2 K and ne = 1.6 × 11 −2 10 cm . (D) dR spectra for different densities from sample B at δ = 0 and T = 0.2 11 −2 11 −2 11 K for ne=0.8 × 10 cm (black), ne = 1.2 × 10 cm (green) and ne = 1.6 × 10 −2 cm (red). The spectra shift to higher energies as the density increases due to the increasing EF. (Inset) dR spectra for two different temperatures of T = 0.2 K (red) 11 −2 and T = 4 K (black) at δ = 0. In both cases we have ne = 0.9 × 10 cm .
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Fig. 3. Integer QH polariton excitations. (A and B) Polarization-resolved dR measurements versus magnetic field of sample B at a constant cavity 11 −2 energy of Ec = 1523.8 meV and an electron density of ne = 1.4 × 10 cm (Vg = −4 V). (C and D) Polarization-resolved dR measurements versus Vg around ν = 2 when the cavity energy is on resonance with the lowest Landau level (Bz = 3 T). The blue (red) color-code represents low (high) dR signal. + − σ (σ ) corresponds to right-hand (left-hand) circularly polarized light. (E and F) Polarization-resolved dR measurements as a function of the filling factor + obtained by changing Vg around ν = 1 at a magnetic field of Bz = 4.9 T for σ − in (E) and σ -polarized light in (F). The spectral white light power in panels (A −1 and B) is P ≈ 15 pW nm . In panels (C) to (F) we used a spectral power P ≈ −1 0.2 pW nm ; the plotted data are an average over 150 measurements.
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Fig. 4. Fractional QH polariton excitations. (A and B) Polarization-resolved dR measurements versus Vg performed on sample B when the cavity energy is on resonance with the lowest Landau level (Bz = 6 T). The color-code is the same as in − + − + Fig. 3. (C) The energy difference, ∆ELP = E(LPσ ) − E(LPσ ), of the σ and σ lower polariton branches versus filling factor for different magnetic fields and resonance conditions. Blue circles (Bz = 6 T) correspond to data obtained when the cavity is tuned into resonance with LL0 around ν = 2/3; red squares (Bz = 6 T) and green triangles (Bz = 4.9 T) show data obtained when the cavity is tuned into resonance with LL0 around ν = 1 (data analyzed from the same scan as in Fig. 3, E and F).
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Cavity quantum electrodynamics with many-body states of a twodimensional electron gas
incidence (Fig. 2A). We observe two split branches each with an ultrasmall effective polariton mass which is approximately twice as large as that of the bare cavity-photon mass (9). In order to study the density and mobility dependence of the Fermiedge polaritons, we recorded the polariton excitations versus detuning from the Fermi-edge at ne = 1.6 × 1011 cm−2 on the high mobility sample B (Fig. 2C). The data shows a significantly smaller normal mode splitting as compared to sample A (Fig. 2A) which is a consequence of weaker hole confinement due to higher ne and mobility. A detailed study of the normal-mode splitting at resonance as a function of ne is presented in Fig. 2D showing a clear decrease from 2g = (2.0 ± 0.1) meV to 2g = (1.3 ± 0.2) meV as the density is increased from ne = 0.8 × 1011 cm−2 to ne = 1.6 × 1011 cm−2 (9). The scattering of an electron with momentum ∼kF on the Fermi surface is accompanied by a hole momentum kick, leading to a hole recoil energy Er (15). A larger hole recoil partially inhibits the formation of the electron screening cloud as ne(kF) is increased resulting in the observed reduction of the normal-mode splitting. Remarkably, polariton excitations depend strongly on temperature as well: when we compare the ne = 0.9 × 1011 cm−2 dR spectrum at T = 0.2 K with the one at T = 4 K (Fig. 2D, inset), we observe that the normalmode splitting disappears at T = 4 K. This observation is striking since the corresponding splitting at T = 0.2K (2g = 1.3 ± 0.2meV) is larger than the thermal energy ET = 345 μeV at T = 4 K. Our experimental findings point to a scenario where the nonperturbative cavity-QW coupling leads to the formation of Fermi-edge polaritons for ne > 8 × 1010 cm−2. In contrast to the Fermi-edge singularity associated with screening of a localized hole by the Fermi sea (4, 6), these new many-body optical excitations are delocalized as demonstrated by the polariton dispersion (Fig. 2B). A key requirement for the observation of Fermi-edge polaritons is that the cavity-QW coupling strength exceeds Er: in this limit, the enhancement of the 2DEG-cavity coupling by optical excitations with energy ≥ EF + Er ensures strong coupling, albeit with reduced normal-mode splitting. This conclusion is supported by the reduction of normal mode splitting (Fig. 2D) with increasing ne, or equivalently, increasing Er observed in the high-mobility sample B. Conversely, the persistence of polariton splitting implies that the hole population in the QW exhibits Rabi oscillations, leading to a time-dependent scattering potential. In contrast to the Bz = 0 limit where the initial state of the optical transition is a Fermi sea without interaction-induced correlations, the electronic states for Bz ≠ 0 in the absence of photonic excitations are QH states. We analyze polariton excitations out of these many-body states by performing polarization-resolved dR measurements. Since the absorption of right-hand (left-hand) circularly-polarized σ+(σ−) light probes the transition from a heavy-hole ⇑
( ⇓ ) state to a spin-down
↓ (spin-up
↑ ) electronic LL state, the observed spectrum is highly sensitive to spin polarization-dependent phase-space-filling. When the magnetic field is increased, the filling factor ν is reduced, allowing optical absorption into LLs that are no longer (completely) filled. In Fig. 3, A and B, the cavity resonance on sample B is tuned to E = 1523.8 meV. In the low field range (Bz = 0 − 2 T) absorption into higher lying empty LLs (ν ≥ 2), enabled by the weak off-resonant coupling of these resonances to the cavity mode, is visible. The reflection spectrum from the lowest LL (LL0) at ne = 1.4 × 1011 cm−2 (Fig. 3, A and B) shows that a finite polariton splitting emerges at Bz ∼ 3 T (ν ∼ 2). To study the resonant interaction between the cavity mode and the optical excitations of QH states, we fixed Bz = 3 T, tuned the cavity to resonance with the LL0 spin-up (σ−) transition at ν = 2 and varied ne through Vg. Fig. 3, C and D, show that for both light polarizations only the bare cavity resonance is visible for ν ≥ 2, since phasespace-filling blocks all 2DEG optical excitations at Ec. In contrast, a
normal-mode splitting appears for ν < 2; the splitting grows monotonically as ne is reduced, since the number of available final states is thereby increased. We remark that the normal-mode splitting of σ+ polarization starts at higher ν than the one of σ−, suggesting that 2DEG electrons remain spin-polarized for 1.7 ≤ ν < 2.0. Unlike the Bz = 0 case, the normal-mode splitting at Bz ≠ 0 is comparable to that of a heavy-hole exciton polariton in the limit ne → 0, pointing to a reduced screening of electron-hole attraction. The dominant optical excitations into LL0 for 1 < ν < 2 can be described as heavy-hole singlet trions (16), as the strongest binding energy and the highest optical oscillator strength would be obtained for a complex of two electrons of opposite spin in the same electronic orbital m of size am = eBz and a heavy-hole with a strong spatial overlap. Such a trion could have an absorption strength comparable to that obtained for an exciton since the magnetic length am is comparable to the exciton Bohr radius for Bz ∼ 5T. By increasing the magnetic field to Bz = 4.9 T we explore polariton excitations around ν = 1. In Fig. 3, E and F, we see a drastic change in the optical response: while the normal-mode splitting collapses for σ− light, it increases sharply for σ+. This feature shows the emergence of a high degree of spin polarization, which we associate with the ν = 1 integer QH state (17, 18). In case of full spin polarization the oscillator strength (∝ g2) of the σ+ polariton excitations is maximum since no ↓ states are occupied. For ν = 1, the degree of spin polarization,
(
P =− g σ2+ g σ2−
) (g
2 σ+
)
+ g σ2− , is determined from the measured mini-
mum normal-mode splittings and found to be P = 90 ± 10%. The error arises primarily from the measurement uncertainty of the individual splittings and simultaneous coupling to higher lying optical transitions. As we move away to both sides from ν = 1 by changing either Vg or Bz, we observe a symmetric decrease (increase) of the normal-mode splitting for the σ+ (σ−) polariton excitations which we attribute to spindepolarization due to skyrmion excitations (19). In order to investigate fractional QH states we tuned Ec into resonance with the ↓ -transition of LL0 at ν = 2/3. Fig. 4, A and B, present dR spectra at Bz = 6 T as a function of ne: for ν ≅ 2/3, we observe a small decrease (increase) in the normal-mode splitting for the σ− (σ+) polariton modes, which points to enhanced spin polarization associated with the ν = 2/3 fractional QH state (20, 21). The small change in the splitting stems from the sizeable degree of spin polarization that persists for ∀ν < 1 and the fact that even for complete spin polarization at ν = 2/3 one third of the states in the spin-up LL0 are available for excitonic excitations. To highlight the variations in the normal-mode splitting, we plot in Fig. 4C the energy difference ∆ELP between the dR peaks of the σ+ and σ− lower polariton branches as a function of ν at Bz = 6 T (blue circles): a local maximum in ∆ELP around the fractional QH state ν = 2/3 signifies spin polarization (22–25). To confirm that the peak in ∆ELP at ν = 2/3 does not have an auxiliary origin, we spatially tune the cavity mode to Eν = 1 (red squares in Fig. 4C) and find that the local maximum in ∆ELP at ν = 2/3 persists. It has been previously established that a phase transition from an unpolarized ν = 2/3 to a polarized ν = 2/3 QH ground state takes place at a critical magnetic field Bc that depends strongly on the effective width of the electron wave function (26, 27). Strikingly, at a lower magnetic field of Bz = 4.9 T (Figs. 3, E and F), we do not observe a local maximum in ∆ELP at ν = 2/3 (Fig. 4C), suggesting that the excess spin polarization is lost: this observation is consistent with a phase transition from an unpolarized to a polarized ν = 2/3 state. Our Bz = 0 experiments reveal a novel cavity excitation induced strongly correlated state which requires a new theoretical framework describing a mobile 2DEG impurity in a cavity-QED setting. On the
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other hand, Bz ≠ 0 experiments establish cavity-QED as an invaluable spectroscopic tool for investigating and manipulating properties of many-body states in a 2DEG. By increasing the cavity lifetime it should be possible to enhance the measurement sensitivity and to observe the incompressibility induced modification of the polariton splitting. Semiintegrated cavity structures not only allow for a factor of 20 increase in lifetime, but also enable tuning the electric-dipole coupling strength (28). More intriguing possibilities raised by our observations include observation of nonequilibrium dynamics following an optical quench into a fractional QH state. Finally, the system we consider in the Bz = 0 limit could also be viewed as a new nonlinear optical system where ne of the 2DEG can be used to change the effective Bohr radius and consequently the strength of polariton-polariton interactions (29). References and Notes 1. C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314–3317 (1992). Medline doi:10.1103/PhysRevLett.69.3314 2. H. Deng, H. Haug, Y. Yamamoto, Exciton-polariton Bose-Einstein condensation. Rev. Mod. Phys. 82, 1489–1537 (2010). doi:10.1103/RevModPhys.82.1489 3. K. Ohtaka, Y. Tanabe, Theory of the soft-x-ray edge problem in simple metals: Historical survey and recent developments. Rev. Mod. Phys. 62, 929–991 (1990). doi:10.1103/RevModPhys.62.929 4. N. S. Averkiev, M. M. Glazov, Light-matter interaction in doped microcavities. Phys. Rev. B 76, 045320 (2007). doi:10.1103/PhysRevB.76.045320 5. A. Gabbay, Y. Preezant, E. Cohen, B. M. Ashkinadze, L. N. Pfeiffer, Fermi edge polaritons in a microcavity containing a high density two-dimensional electron gas. Phys. Rev. Lett. 99, 157402 (2007). Medline doi:10.1103/PhysRevLett.99.157402 6. M. Baeten, M. Wouters, (available at http://arxiv.org/abs/1404.2048). 7. B. Sbierski, M. Hanl, A. Weichselbaum, H. E. Türeci, M. Goldstein, L. I. Glazman, J. von Delft, A. Imamoğlu, Proposed Rabi-Kondo correlated state in a laser-driven semiconductor quantum dot. Phys. Rev. Lett. 111, 157402 (2013). Medline doi:10.1103/PhysRevLett.111.157402 8. S. Das Sarma, A. Pinczuk, (eds), Perspectives on Quantum Hall Effects (Wiley, New York, 1996). 9. See supplementary materials on Science Online. 10. B. R. Bennett, R. A. Soref, J. A. del Alamo, Carrier-induced change in refractive index of InP, GaAs and InGaAsP. IEEE J. Quantum Electron. 26, 113–122 (1990). doi:10.1109/3.44924 11. P. W. Anderson, Infrared Catastrophe in Fermi Gases with Local Scattering Potentials. Phys. Rev. Lett. 18, 1049–1051 (1967). doi:10.1103/PhysRevLett.18.1049 12. M. S. Skolnick, J. M. Rorison, K. J. Nash, D. J. Mowbray, P. R. Tapster, S. J. Bass, A. D. Pitt, Observation of a many-body edge singularity in quantum well luminescence spectra. Phys. Rev. Lett. 58, 2130–2133 (1987). Medline doi:10.1103/PhysRevLett.58.2130 13. M. Perrin, P. Senellart, A. Lemaître, J. Bloch, Polariton relaxation in semiconductor microcavities: Efficiency of electron-polariton scattering. Phys. Rev. B 72, 075340 (2005). doi:10.1103/PhysRevB.72.075340 14. R. Houdré, C. Weisbuch, R. P. Stanley, U. Oesterle, P. Pellandini, M. Ilegems, Measurement of cavity-polariton dispersion curve from angle resolved photoluminescence experiments. Phys. Rev. Lett. 73, 2043–2046 (1994). Medline doi:10.1103/PhysRevLett.73.2043 15. A. Rosch, T. Kopp, Heavy Particle in a d-Dimensional Fermionic Bath: A Strong Coupling Approach. Phys. Rev. Lett. 75, 1988–1991 (1995). Medline doi:10.1103/PhysRevLett.75.1988 16. I. Bar-Joseph, Trions in GaAs quantum wells. Semicond. Sci. Technol. 20, R29–R39 (2005). doi:10.1088/0268-1242/20/6/R01 17. P. Plochocka, J. M. Schneider, D. K. Maude, M. Potemski, M. Rappaport, V. Umansky, I. Bar-Joseph, J. G. Groshaus, Y. Gallais, A. Pinczuk, Optical absorption to probe the quantum Hall ferromagnet at filling factor nu=1. Phys. Rev. Lett. 102, 126806 (2009). Medline doi:10.1103/PhysRevLett.102.126806 18. L. Tiemann, G. Gamez, N. Kumada, K. Muraki, Unraveling the spin polarization of the ν = 5/2 fractional quantum Hall state. Science 335, 828– 831 (2012). Medline doi:10.1126/science.1216697
19. E. H. Aifer, B. B. Goldberg, D. A. Broido, Evidence of Skyrmion excitations about nu =1 in n-modulation-doped single quantum wells by interband optical transmission. Phys. Rev. Lett. 76, 680–683 (1996). Medline doi:10.1103/PhysRevLett.76.680 20. B. I. Halperin, Hel. Phys. Acta 56, 75–102 (1983). 21. J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, K. W. West, Evidence for a spin transition in the ν=2/3 fractional quantum Hall effect. Phys. Rev. B 41, 7910– 7913 (1990). doi:10.1103/PhysRevB.41.7910 22. I. V. Kukushkin, K. von Klitzing, K. Eberl, Spin Polarization of Composite Fermions: Measurements of the Fermi Energy. Phys. Rev. Lett. 82, 3665– 3668 (1999). doi:10.1103/PhysRevLett.82.3665 23. J. H. Smet, R. A. Deutschmann, F. Ertl, W. Wegscheider, G. Abstreiter, K. von Klitzing, Gate-voltage control of spin interactions between electrons and nuclei in a semiconductor. Nature 415, 281–286 (2002). Medline doi:10.1038/415281a 24. B. Verdene, J. Martin, G. Gamez, J. Smet, K. von Klitzing, D. Mahalu, D. Schuh, G. Abstreiter, A. Yacoby, Microscopic manifestation of the spin phase transition at filling factor 2/3. Nat. Phys. 3, 392–396 (2007). doi:10.1038/nphys588 25. J. Hayakawa, K. Muraki, G. Yusa, Real-space imaging of fractional quantum Hall liquids. Nat. Nanotechnol. 8, 31–35 (2013). Medline doi:10.1038/nnano.2012.209 26. T. Chakraborty, Spin-reversed ground state and energy gap in the fractional quantum Hall effect. Surf. Sci. 229, 16–20 (1990). doi:10.1016/00396028(90)90821-O 27. N. Kumada, D. Terasawa, Y. Shimoda, H. Azuhata, A. Sawada, Z. F. Ezawa, K. Muraki, T. Saku, Y. Hirayama, Phase diagram of interacting composite fermions in the bilayer nu=2/3 quantum hall effect. Phys. Rev. Lett. 89, 116802 (2002). Medline doi:10.1103/PhysRevLett.89.116802 28. B. Besga et al., (available at http://arxiv.org/abs/1312.0819). 29. I. Carusotto, C. Ciuti, Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013). doi:10.1103/RevModPhys.85.299 30. C. Latta, F. Haupt, M. Hanl, A. Weichselbaum, M. Claassen, W. Wuester, P. Fallahi, S. Faelt, L. Glazman, J. von Delft, H. E. Türeci, A. Imamoglu, Quantum quench of Kondo correlations in optical absorption. Nature 474, 627–630 (2011). Medline doi:10.1038/nature10204 31. F. Haupt, S. Smolka, M. Hanl, W. Wüster, J. Miguel-Sanchez, A. Weichselbaum, J. von Delft, A. Imamoglu, Nonequilibrium dynamics in an optical transition from a neutral quantum dot to a correlated many-body state. Phys. Rev. B 88, 161304 (2013). doi:10.1103/PhysRevB.88.161304 32. S. Pau, G. Björk, J. Jacobson, H. Cao, Y. Yamamoto, Microcavity excitonpolariton splitting in the linear regime. Phys. Rev. B 51, 14437–14447 (1995). doi:10.1103/PhysRevB.51.14437 33. D. Snoke, Spontaneous Bose coherence of excitons and polaritons. Science 298, 1368–1372 (2002). Medline doi:10.1126/science.1078082 34. I. V. Kukushkin, K. von Klitzing, K. Ploog, V. E. Kirpichev, B. N. Shepel, Reduction of the electron density in GaAs-AlxGa1-xAs single heterojunctions by continuous photoexcitation. Phys. Rev. B 40, 4179–4182 (1989). doi:10.1103/PhysRevB.40.4179 35. A. J. Shields, J. L. Osborne, M. Y. Simmons, D. A. Ritchie, M. Pepper, Comparison of optical and transport measurements of electron densities in quantum wells. Semicond. Sci. Technol. 11, 890–896 (1996). doi:10.1088/0268-1242/11/6/007 Acknowledgments: The Authors acknowledge insightful discussions with L. Glazman, M. Goldstein, A. Rosch, A. Srivastava and J. von Delft. This work is supported by NCCR Quantum Photonics (NCCR QP), research instrument of the Swiss National Science Foundation (SNSF). Supplementary Materials www.sciencemag.org/content/science.1258595/DC1 Materials and Methods Supplementary Text Figs. S1 to S8 References (30–35) 10 July 2014; accepted 17 September 2014 Published online 3 October 2014 10.1126/science.1258595
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Fig. 1. Cavity QED in a two-dimensional electron system. (A) The sample consists of a modulation doped 20 nm GaAs QW inside a DBR microcavity. (B) left: Dispersion of the conduction and valence bands at Bz = 0 T; EG and EF denote the band gap and Fermi energy. Right: For Bz ≠ 0 the bands split into discrete Landau energy levels; electron-hole pairs
−
, can be excited with σ (σ ) polarized +
light into the corresponding (electron) spin-split Landau levels (LLO). (C) Magnetotransport 11 −2 measurements of the resistivity ρxx and ρxy of wafer B at an electron density of ne = 2.3 × 10 cm . (D) dR (blue) and photoluminescence (orange) spectra of sample A for different electron densities at T = 0.2 K. The spectra are vertically shifted for clarity. The uncertainty in the estimated electron density for the 10 −2 10 −2 plots with ne ≥ 4 × 10 cm is ∼ 2 × 10 cm .
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Fig. 2. Experimental signatures of Fermi-edge polaritons. (A) dR spectra taken from sample A as a function of cavity detuning δ from the Fermi-edge at T = 0.2 K 11 −2 and ne = 1.2 × 10 cm . (B) Fermi-edge polariton energy dispersion obtained at T = 11 −2 1.6 K, ne = 1.2 × 10 cm and δ = 0. The dashed straight line depicts the Fermi energy. The blue (red) color-coded area displays low (high) dR signal. (C) dR spectra measured on sample B versus cavity detuning at T = 0.2 K and ne = 1.6 × 11 −2 10 cm . (D) dR spectra for different densities from sample B at δ = 0 and T = 0.2 11 −2 11 −2 11 K for ne=0.8 × 10 cm (black), ne = 1.2 × 10 cm (green) and ne = 1.6 × 10 −2 cm (red). The spectra shift to higher energies as the density increases due to the increasing EF. (Inset) dR spectra for two different temperatures of T = 0.2 K (red) 11 −2 and T = 4 K (black) at δ = 0. In both cases we have ne = 0.9 × 10 cm .
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Fig. 3. Integer QH polariton excitations. (A and B) Polarization-resolved dR measurements versus magnetic field of sample B at a constant cavity 11 −2 energy of Ec = 1523.8 meV and an electron density of ne = 1.4 × 10 cm (Vg = −4 V). (C and D) Polarization-resolved dR measurements versus Vg around ν = 2 when the cavity energy is on resonance with the lowest Landau level (Bz = 3 T). The blue (red) color-code represents low (high) dR signal. + − σ (σ ) corresponds to right-hand (left-hand) circularly polarized light. (E and F) Polarization-resolved dR measurements as a function of the filling factor + obtained by changing Vg around ν = 1 at a magnetic field of Bz = 4.9 T for σ − in (E) and σ -polarized light in (F). The spectral white light power in panels (A −1 and B) is P ≈ 15 pW nm . In panels (C) to (F) we used a spectral power P ≈ −1 0.2 pW nm ; the plotted data are an average over 150 measurements.
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Fig. 4. Fractional QH polariton excitations. (A and B) Polarization-resolved dR measurements versus Vg performed on sample B when the cavity energy is on resonance with the lowest Landau level (Bz = 6 T). The color-code is the same as in − + − + Fig. 3. (C) The energy difference, ∆ELP = E(LPσ ) − E(LPσ ), of the σ and σ lower polariton branches versus filling factor for different magnetic fields and resonance conditions. Blue circles (Bz = 6 T) correspond to data obtained when the cavity is tuned into resonance with LL0 around ν = 2/3; red squares (Bz = 6 T) and green triangles (Bz = 4.9 T) show data obtained when the cavity is tuned into resonance with LL0 around ν = 1 (data analyzed from the same scan as in Fig. 3, E and F).
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