Polaron-Polaritons in the Integer and Fractional Quantum Hall Regimes
Elementary quasi-particles in a two dimensional electron system can be described as exciton-polarons since electron-exciton interactions ensures dressing of excitons by Fermi-sea electron-hole pair excitations. A relevant open question is the modification of this description when the electrons occupy flat-bands and electron-electron interactions become prominent. Here, we perform cavity spectroscopy of a two dimensional electron system in the strong-coupling regime where polariton resonances carry signatures of strongly correlated quantum Hall phases. By measuring the evolution of the polariton splitting under an external magnetic field, we demonstrate the modification of electron-exciton interactions that we associate with phase space filling at integer filling factors and polaron dressing at fractional filling factors. The observed non-linear behavior shows great promise for enhancing polariton-polariton interactions.
Strong coupling of excitons in a semiconductor quantum well (QW) to a microcavity mode leads to formation of quasiparticles called cavity exciton polaritons Weisbuch et al. (1992). Polaritons have played a central role in the investigation of nonequilibrium condensation and superfluidity of photonic excitations Deng et al. (2010); Carusotto and Ciuti (2013). While polaritons acquire a finite nonlinearity due to their exciton character, interactions between polaritons in undoped QWs are not strong enough for realizing strongly interacting photonic systems Amo and Bloch (2016).
Two-dimensional electron systems (2DES) evolving in large magnetic fields, in contrast, are a fertile ground for many-body physics due to prominence of electron-electron interactions. Formation of skyrmion excitations in the vicinity of filling factor is a consequence of such interactions. More spectacularly, electron correlations lead to the formation of fractional quantum Hall (FQH) states where the ground state exhibits topological order Prange et al. (1989); Das Sarma and Pinczuk (1997); Sondhi et al. (1997). Moreover, it has been proposed that a sub-class of FQH states exhibit non-abelian quasi-particles which can be used to implement topological quantum computation Nayak et al. (2008). The nature of optical excitations of a 2DES have also recently generated lots of interest: experimental and theoretical Sidler et al. (2017); Efimkin and MacDonald (2016) studies in transition metal dichalcogenide (TMD) monolayers have established that these excitations should be described in the framework of the Fermi polaron problem, as a collective excitation resulting from exciton-electron interactions Schirotzek et al. (2009); Koschorreck et al. (2012); Schmidt et al. (2012); Massignan et al. (2014). In this context, optical excitation of an exciton leads to generation of an electron screening cloud that results in formation of a lower energy attractive exciton-polaron Sidler et al. (2017); Efimkin and MacDonald (2016). In this work, we report corresponding signatures in GaAs, where energy scales are known to differ significantly compared to TMD monolayers Koudinov et al. (2014); Suris (2003), due to a particularly small binding energy of the bound-molecular trion state. More importantly, our work presents experimental signatures of polaron formation at nonzero magnetic fields where electrons are confined to the lowest Landau level.
It has recently been demonstrated that embedding a 2DES inside a microcavity realizes an alternate method for probing quantum Hall (QH) states Smolka et al. (2014). In the strong coupling regime, polariton excitations are sensitive to elementary properties of the many-body ground state, such as spin-polarization and incompressibility due to their part-exciton character. In contrast to bare excitons though, polaritons are immune to decoherence processes such as phonon or impurity scattering due to their ultra-light mass, ensuring that they are delocalized. Consequently, the energy resolution achievable in polariton-based spectroscopy is only limited by the polariton decay rate due to mirror losses, which can be on the order of 20 mK in state-of-the-art microcavities Steger et al. (2013). In the present work, we also demonstrate a new feature of cavity-polariton-spectroscopy of FQH states: by adjusting the separation distance between the 2DES and the doping layers we substantially reduce unwanted light-induced variations of the 2DES electron density Sup ().
Our structure consists of a 2DES in a 20 nm modulation doped GaAs QW, embedded at the center of a microcavity Sup (). The front (back) distributed Bragg reflector (DBR) is composed of 19 (25) pairs of layers leading to the measured quality factor for the cavity. The QW features a double-sided silicon -doping with a set-back distance of above and below the center of the cavity. From magneto-transport measurements Koon and Knickerbocker (1992), we estimate the 2DES electron density and the mobility . We deliberately choose a relatively low to access the physics of the lowest Landau level in the range of magnetic fields currently available on our experimental setup (). The relatively high ensures that we can probe FQH physics.
We perform polarization-resolved spectroscopy of the 2DES using an infrared light emitting diode centered around 820 nm. We shine excitation light onto the sample placed in a dilution refrigerator with a 30 mK base temperature. An aspheric lens () collects the light reflected off the sample, which is analyzed using a spectrometer. Earlier studies on the optics of 2DES have shown extreme sensitivity of to optical power Kukushkin et al. (1989); Goldberg et al. (1992); Groshaus et al. (2007); Smolka et al. (2014). Increasing the optical power not only changes , but also causes qualitative changes in the reflectivity spectrum Wüster (2015), which is detrimental to the study of fragile QH states. These unwanted effects are attributed to photoexcitation of centers in Si-doped with Ihn (2010). We minimize light-induced variations of by keeping in the structure, and more importantly by placing the dopants in 10 nm GaAs () doping quantum wells (DQW). Further, we locate the DQWs in nodes of the electric field inside the cavity Sup (), which minimizes the intracavity light intensity at the position of the dopants.
We first carry out cavity spectroscopy of the 2DES (see Fig. 1a) when . Fig. 1(b) shows the reflectivity spectrum as a function of . In contrast to undoped QW structures Weisbuch et al. (1992), we observe coupling to three exciton-like resonances as we scan . For the lowest energy anticrossing, we measure a normal mode splitting of . Since is larger than the bare-cavity linewidth , the system is in the strong coupling regime of cavity-QED and the elementary excitations should be characterized as cavity-polaritons Weisbuch et al. (1992). Since the cavity-exciton coupling in this system is comparable to energy level splittings of the three exciton-like resonances, the polariton modes observed in the reflection spectrum can only be described as a superposition of all underlying resonances (see Fig. 1a). We identify the lowest energy exciton-like resonance observed in Fig. 1(b) as the heavy-hole attractive polaron () – a heavy-hole exciton dressed by Fermi sea electron-hole pair excitations Sidler et al. (2017). Since the attractive polaron resonance is associated with the bound-molecular singlet trion channel, it was previously referred to as “trion mode” Esser et al. (2001); Bar-Joseph (2005). We assign the middle-energy excitonic resonance to the heavy hole repulsive-polaron () Sidler et al. (2017); Efimkin and MacDonald (2017). The magnitude of the splitting of this mode from the attractive polaron ( meV) is a factor of 2 larger than the bare trion binding energy and is fully consistent with its identification as the repulsive polaron branch. Finally, we tentatively identify the highest energy excitonic mode to the light-hole exciton 111The absence of attractive and repulsive polaron branches associated with the light-hole exciton may stem from the absence of a bound light-hole trion, or from the lower oscillator strength and the larger broadening of these resonances.
Next, we analyze the case where the electrons are confined to the lowest Landau Level (LL), with filling factor Prange et al. (1989). To explore the interplay between quantum Hall states and polaritonic excitations, we tune to ensure that the cavity mode dressed by nonperturbative coupling to higher energy excitonic modes is resonant with . Since the lowest energy polariton has predominantly character, the spin state of the optically generated electron is determined by the photon polarization Smolka et al. (2014): left-hand circularly polarized light probes transitions to the lower electron Zeeman spin subband () and right-hand circularly polarized light probes transitions to the upper electron Zeeman spin subband (). Consequently, the observed spectral signatures are strongly dependent on how the electrons are arranged in the LLs i.e. on the spin-polarization of the different ground states of the 2DES Goldberg et al. (1992); Groshaus et al. (2007).
Figure 2(a-b) shows the white light reflection spectrum as a function of , varied by scanning . Here, we tuned close to resonance with the -transition of lowest Landau level LL0 at . The most striking feature is the collapse of around , concurrent with the enhancement of . We associate this feature with the QH state, in excellent agreement with the value of the electron density measured independently. The observed behavior is a direct consequence of the high degree of spin-polarization of the QH ferromagnet at : for a fully polarized state, is expected to collapse due to the fact that all -electron states are occupied. Phase space filling thus prevents optical excitation of an electron to that level, and therefore the oscillator strength for that transition collapses (see Fig. 2(c)). Concurrently, all -electron states are free and increases due to the increased number of available states Aifer et al. (1996); Groshaus et al. (2004). Fig. 2(d) shows and extracted from fits of the reflection spectra. From this, we calculate the spin-polarization at Groshaus et al. (2007); Smolka et al. (2014). We obtain Sup (), suggesting that full polarization is not achieved at , contrary to what is expected for the quantum Hall ferromagnet. In our low sample, incomplete polarization may arise due to disorder and reduced screening of impurity potentials Manfra et al. (1997). Furthermore the cyclotron frequency is comparable to the exciton binding energy, ensuring that exciton formation has a sizable contribution from higher LLs. As a consequence, our measurements only yield a lower-bound on , and we do not expect full cancellation of at . We finally observe a rapid, symmetric depolarization on both sides of which is compatible with formation of many-body spin excitations in the ground state (skyrmions and anti-skyrmions) Aifer et al. (1996); Manfra et al. (1997); Zhitomirsky et al. (2002); Groshaus et al. (2004); Plochocka et al. (2009) as a consequence of the competition between Coulomb and Zeeman energies. Finally, coupling to integer QH states is also visible in Fig. 2(a-b) as variations of the lower polariton energies vs as a consequence of phase space-filling Sup (). We emphasize that these spectral features are robust against increased optical powers, which demonstrates that our sample structure provides, through “cavity protection” of the 2DES, a unique platform for optical studies of QH physics Sup ().
We investigate FQH states by scanning to up to 5 T for an increased value of as shown in Fig. 3(a-b). Increasing reduces , thus leading to absorption in a partially filled lowest LL Goldberg et al. (1990); Yusa et al. (2001); Byszewski et al. (2006). Cavity coupling to several FQH states is observed in Fig. 2 and Fig. 3 as a -dependent normal mode splitting in both polarizations. Such spectral signatures are particularly striking when reaches the fractional values , 2/5, 2/3 and 5/3. We observe that and differ significantly at , 2/5 and 5/3, which shows that these fractional QH states experience sizable spin-polarization Sup (). On the contrary, at shows that this state is not polarized, as expected for samples with in the range of the one studied here. Increasing should allow us to probe the phase transition from an unpolarized to a polarized -state Eisenstein et al. (1990); Smolka et al. (2014).
We now focus on filling factor , see Fig. 3(a-b). In stark contrast with the integer QH states, both and states are available and phase-space filling only plays a marginal role here. Fig. 3(d) shows polariton splittings and extracted from fits of the reflectivity spectra. One striking feature is that the collapse of around is not accompanied with an appreciable increase in , contrary to what was observed for . Because the LLs are partially filled, the mechanism leading to modification of the polariton splitting is indeed modified.
We argue that the decrease for a spin-polarized state is due to the polaron nature of optical excitations that are accessible when promoting an electron into the -state with -polarized light. For a fully polarized state, all electrons are in the same -state and there are no electrons in the -state. Since the oscillator strength of the singlet is proportional to the density of electrons, perfect spin polarization would lead to vanishing cavity coupling. In contrast, promotion of an electron in the -state with -polarized light always leads to formation of a singlet polaron excitation with electrons available in the -state, and the polariton splitting is only marginally modified. Fig. 3(e) plots the evolution of the polariton peak areas around . The decrease in polariton splitting in -polarization is accompanied with a loss (gain) of weight of the lower (upper) polariton. This observation is fully consistent with a reduced cavity-polaron coupling strength and a finite detuning between the bare polaron and cavity resonances, ensuring that the lower (upper) polariton has predominantly polaron (cavity) character at . The absence of a similar oscillator strength transfer in -polarization on resonance further supports the interpretation of our data in terms of inhibition of polaron-dressing by -electrons at .
Finally, we address the question of the modification of the polaron-polariton effective mass in the vicinity of . We use a lens to excite a broad range of in-plane momenta using the same broadband light emitting diode. A low NA lens couples the reflected light into a fiber, which enables angle selective measurements. The dispersion relation in Fig. 4(a) at clearly shows the anticrossings with and as pointed out already in Fig. 1(b). We fit a parabola to the lower polariton dispersion at (dashed orange line) and compare it, in Fig. 4(b), to the dispersions measured at filling factors slightly above (green) and below (blue). Strikingly, we find an increase of the effective mass at (orange) by a factor of compared to (green) and (blue) 222Note that estimation of the lower polariton mass at is rendered difficult due to the low curvature of the parabola.. This observation illustrates further the strong reduction in the oscillator strength of the attractive-polaron resonance which reduces the cavity-character and enhances .
We emphasize that theory of exciton-polarons has been previously developed for excitons interacting with a 2DES in the limit Sidler et al. (2017); Efimkin and MacDonald (2016). A quantitative modeling of our experiment requires extending prior theoretical work to the case of screening of excitons by electrons occupying a single LL: a significant advance in this direction was the recent development of the theory of exciton-polarons in the limit of strong magnetic fields but without taking into account electron-electron interactions leading to FQH states Efimkin and MacDonald (2017). Our work focused on the singlet channel which plays a prominent role in the limit of moderate magnetic fields ( T) used in our experiments. Yet, we expect triplet channels to play a key role in determining the full polariton spectrum, particularly at higher B-fields relevant for samples with higher electron density. A more challenging problem is exciton-electron interactions in the vicinity of FQH states: polaron-polariton formation in this limit may be described using polariton dressing by fractionally charged quasi-particle-hole pairs Grusdt et al. (2016). The latter problem is related to identification of signatures of incompressibility of the many-body ground state in the polariton excitation spectrum.
On the technical side, we demonstrate that cavity electrodynamics is an invaluable platform to probe fragile fractional states. This could potentially enable optical manipulation of anyonic quasi-particles associated with strongly-correlated phases. Furthermore, increasing the quality factor of the cavity could further enhance the sensitivity of our measurements Steger et al. (2013). Finally, the observed filling factor-dependent polariton splitting could be particularly useful to engineer single-photon non-linearities between polaritons and allow for experimental realization of strongly correlated driven-dissipative photonic systems in arrays of semiconductor microcavities Amo and Bloch (2016).
Acknowledgements.The Authors acknowledge many useful discussions with Hadis Abbaspour, Valentin Goblot, Wolf Wuester and Sina Zeytinoglu. This work was supported by NCCR Quantum Photonics (NCCR QP), an ETH Fellowship (S. R.), and an ERC Advanced investigator grant (POLTDES).
- Weisbuch et al. (1992) C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69, 3314 (1992).
- Deng et al. (2010) H. Deng, H. Haug, and Y. Yamamoto, ‘‘Exciton-polariton bose-einstein condensation,” Rev. Mod. Phys. 82, 1489 (2010).
- Carusotto and Ciuti (2013) I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
- Amo and Bloch (2016) A. Amo and J. Bloch, “Exciton-polaritons in lattices: A non-linear photonic simulator,” Comptes Rendus Physique 17, 934 (2016).
- Prange et al. (1989) R. Prange, M. Cage, K. Klitzing, S. Girvin, A. Chang, F. Duncan, M. Haldane, R. Laughlin, A. Pruisken, and D. Thouless, The Quantum Hall Effect, Graduate Texts in Contemporary Physics (Springer New York, 1989).
- Das Sarma and Pinczuk (1997) S. Das Sarma and A. Pinczuk, Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-dimensional Semiconductor Structures, A Wiley-Interscience publication (Wiley, 1997).
- Sondhi et al. (1997) S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, “Continuous quantum phase transitions,” Rev. Mod. Phys. 69, 315 (1997).
- Nayak et al. (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
- Sidler et al. (2017) 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 semiconductors,” Nat Phys 13, 255 (2017).
- Efimkin and MacDonald (2016) D. K. Efimkin and A. H. MacDonald, “Many-Body Theory of Trion Absorption Features in Two-Dimensional Semiconductors,” arXiv:1609.06329 (2016) .
- Schirotzek et al. (2009) A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwierlein, “Observation of fermi polarons in a tunable fermi liquid of ultracold atoms,” Phys. Rev. Lett. 102, 230402 (2009).
- Koschorreck et al. (2012) M. Koschorreck, D. Pertot, E. Vogt, B. Frohlich, M. Feld, and M. Kohl, “Attractive and repulsive fermi polarons in two dimensions,” Nature 485, 619 (2012).
- Schmidt et al. (2012) R. Schmidt, T. Enss, V. Pietilä, and E. Demler, “Fermi polarons in two dimensions,” Phys. Rev. A 85, 021602 (2012).
- Massignan et al. (2014) P. Massignan, M. Zaccanti, and G. M. Bruun, “Polarons, dressed molecules and itinerant ferromagnetism in ultracold fermi gases,” Reports on Progress in Physics 77, 034401 (2014).
- Koudinov et al. (2014) A. V. Koudinov, C. Kehl, A. V. Rodina, J. Geurts, D. Wolverson, and G. Karczewski, “Suris tetrons: Possible spectroscopic evidence for four-particle optical excitations of a two-dimensional electron gas,” Phys. Rev. Lett. 112, 147402 (2014).
- Suris (2003) R. A. Suris, “Correlation between trion and hole in fermi distribution 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.
- Smolka et al. (2014) S. Smolka, W. Wuester, F. Haupt, S. Faelt, W. Wegscheider, and A. Imamoglu, “Cavity quantum electrodynamics with many-body states of a two-dimensional electron gas,” Science 346, 332 (2014) .
- Steger et al. (2013) M. Steger, G. Liu, B. Nelsen, C. Gautham, D. W. Snoke, R. Balili, L. Pfeiffer, and K. West, “Long-range ballistic motion and coherent flow of long-lifetime polaritons,” Phys. Rev. B 88, 235314 (2013).
- (19) See Supplemental Material at [URL will be inserted by publisher] for additional information about the sample structure, the light sensitivity of and the measurements of .
- Koon and Knickerbocker (1992) D. W. Koon and C. J. Knickerbocker, “What do you measure when you measure resistivity?” Review of Scientific Instruments 63, 207 (1992) .
- Kukushkin et al. (1989) I. V. Kukushkin, K. von Klitzing, K. Ploog, V. E. Kirpichev, and B. N. Shepel, “Reduction of the electron density in gaas-as single heterojunctions by continuous photoexcitation,” Phys. Rev. B 40, 4179 (1989).
- Goldberg et al. (1992) B. Goldberg, D. Heiman, A. Pinczuk, L. Pfeiffer, and K. West, “Magneto-optics in the integer and fractional quantum hall and electron solid regimes,” Surface Science 263, 9 (1992).
- Groshaus et al. (2007) J. G. Groshaus, P. Plochocka-Polack, M. Rappaport, V. Umansky, I. Bar-Joseph, B. S. Dennis, L. N. Pfeiffer, K. W. West, Y. Gallais, and A. Pinczuk, “Absorption in the fractional quantum hall regime: Trion dichroism and spin polarization,” Phys. Rev. Lett. 98, 156803 (2007).
- Wüster (2015) W. Wüster, Cavity quantum electrodynamics with many-body states of a two-dimensional electron system, Ph.D. thesis, ETH-Zürich (2015).
- Ihn (2010) T. Ihn, Semiconductor Nanostructures: Quantum States and Electronic Transport (OUP Oxford, 2010).
- Esser et al. (2001) A. Esser, R. Zimmermann, and E. Runge, “Theory of Trion Spectra in Semiconductor Nanostructures,” Physica Status Solidi B Basic Research 227, 317 (2001).
- Bar-Joseph (2005) I. Bar-Joseph, “Trions in gaas quantum wells,” Semiconductor Science and Technology 20, R29 (2005).
- Efimkin and MacDonald (2017) D. K. Efimkin and A. H. MacDonald, “Many-body theory of trion absorption in a strong magnetic field,” arXiv:1707.05845 (2017) .
- (29) The absence of attractive and repulsive polaron branches associated with the light-hole exciton may stem from the absence of a bound light-hole trion, or from the lower oscillator strength and the larger broadening of these resonances.
- Aifer et al. (1996) E. H. Aifer, B. B. Goldberg, and D. A. Broido, “Evidence of skyrmion excitations about in -modulation-doped single quantum wells by interband optical transmission,” Phys. Rev. Lett. 76, 680 (1996).
- Groshaus et al. (2004) J. G. Groshaus, V. Umansky, H. Shtrikman, Y. Levinson, and I. Bar-Joseph, “Absorption spectrum around : Evidence for a small-size skyrmion,” Phys. Rev. Lett. 93, 096802 (2004).
- Manfra et al. (1997) M. Manfra, B. Goldberg, L. Pfeiffer, and K. West, “Optical determination of the spin polarization of a quantum hall ferromagnet,” Physica E: Low-dimensional Systems and Nanostructures 1, 28 (1997).
- Zhitomirsky et al. (2002) V. Zhitomirsky, R. Chughtai, R. Nicholas, and M. Henini, “Spin polarization of 2d electrons in the quantum hall ferromagnet: evidence for a partially polarized state around filling factor one,” Physica E: Low-dimensional Systems and Nanostructures 12, 12 (2002).
- Plochocka et al. (2009) 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 the quantum hall ferromagnet at filling factor ,” Phys. Rev. Lett. 102, 126806 (2009).
- Goldberg et al. (1990) B. B. Goldberg, D. Heiman, A. Pinczuk, L. Pfeiffer, and K. West, “Optical investigations of the integer and fractional quantum hall effects: Energy plateaus, intensity minima, and line splitting in band-gap emission,” Phys. Rev. Lett. 65, 641 (1990).
- Yusa et al. (2001) G. Yusa, H. Shtrikman, and I. Bar-Joseph, “Charged excitons in the fractional quantum hall regime,” Phys. Rev. Lett. 87, 216402 (2001).
- Byszewski et al. (2006) M. Byszewski, B. Chwalisz, D. K. Maude, M. L. Sadowski, M. Potemski, T. Saku, Y. Hirayama, S. Studenikin, D. G. Austing, A. S. Sachrajda, and P. Hawrylak, “Optical probing of composite fermions in a two-dimensional electron gas,” Nat Phys 2, 239 (2006).
- Eisenstein et al. (1990) J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, and K. W. West, ‘‘Evidence for a spin transition in the =2/3 fractional quantum hall effect,” Phys. Rev. B 41, 7910 (1990).
- (39) Note that the estimation of the lower polariton mass at is rendered difficult by the low curvature of the parabola.
- Grusdt et al. (2016) F. Grusdt, N. Y. Yao, D. Abanin, M. Fleischhauer, and E. Demler, “Interferometric measurements of many-body topological invariants using mobile impurities,” Nature Communications 7, 11994 EP (2016).