Letter | Published:

Nonlinear optics in the fractional quantum Hall regime


Engineering strong interactions between optical photons is a challenge for quantum science. Polaritonics, which is based on the strong coupling of photons to atomic or electronic excitations in an optical resonator, has emerged as a promising approach to address this challenge, paving the way for applications such as photonic gates for quantum information processing1 and photonic quantum materials for the investigation of strongly correlated driven–dissipative systems2,3. Recent experiments have demonstrated the onset of quantum correlations in exciton-polariton systems4,5, showing that strong polariton blockade6—the prevention of resonant injection of additional polaritons in a well delimited region by the presence of a single polariton—could be achieved if interactions were an order of magnitude stronger. Here we report time-resolved four-wave-mixing experiments on a two-dimensional electron system embedded in an optical cavity7, demonstrating that polariton–polariton interactions are strongly enhanced when the electrons are initially in the fractional quantum Hall regime. Our experiments indicate that, in addition to strong correlations in the electronic ground state, exciton–electron interactions leading to the formation of polaron-polaritons8,9,10,11 have a key role in enhancing the nonlinear optical response of the system. Our findings could facilitate the realization of strongly interacting photonic systems, and suggest that nonlinear optical measurements could provide information about fractional quantum Hall states that is not accessible through their linear optical response.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Data availability

The data that support the findings of this study are available in the ETH Research Collection (http://hdl.handle.net/20.500.11850/338463).


  1. 1.

    O’Brien, J. L., Furusawa, A. & Vučković, J. Photonic quantum technologies. Nat. Photon. 3, 687–695 (2009).

  2. 2.

    Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).

  3. 3.

    Sanvitto, D. & Kéna-Cohen, S. The road towards polaritonic devices. Nat. Mater. 15, 1061–1073 (2016).

  4. 4.

    Muñoz-Matutano, G. et al. Emergence of quantum correlations from interacting fibre-cavity polaritons. Nat. Mater. 18, 213–218 (2019).

  5. 5.

    Delteil, A. et al. Towards polariton blockade of confined exciton-polaritons. Nat. Mater. 18, 219–222 (2019).

  6. 6.

    Verger, A., Ciuti, C. & Carusotto, I. Polariton quantum blockade in a photonic dot. Phys. Rev. B 73, 193306 (2006).

  7. 7.

    Smolka, S. et al. Cavity quantum electrodynamics with many-body states of a two-dimensional electron gas. Science 346, 332–335 (2014).

  8. 8.

    Schmidt, R., Enss, T., Pietilä, V. & Demler, E. Fermi polarons in two dimensions. Phys. Rev. A 85, 021602 (2012).

  9. 9.

    Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors. Nat. Phys. 13, 255–261 (2017).

  10. 10.

    Efimkin, D. K. & MacDonald, A. H. Exciton-polarons in doped semiconductors in a strong magnetic field. Phys. Rev. B 97, 235432 (2018).

  11. 11.

    Ravets, S. et al. Polaron polaritons in the integer and fractional quantum Hall regimes. Phys. Rev. Lett. 120, 057401 (2018).

  12. 12.

    Jia, N. et al. A strongly interacting polaritonic quantum dot. Nat. Phys. 14, 550–554 (2018).

  13. 13.

    Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose–Einstein condensation. Rev. Mod. Phys. 82, 1489–1537 (2010).

  14. 14.

    Ferrier, L. et al. Interactions in confined polariton condensates. Phys. Rev. Lett. 106, 126401 (2011).

  15. 15.

    Sun, Y. et al. Bose–Einstein condensation of long-lifetime polaritons in thermal equilibrium. Phys. Rev. Lett. 118, 016602 (2017).

  16. 16.

    Takemura, N., Trebaol, S., Wouters, M., Portella-Oberli, M. T. & Deveaud, B. Polaritonic Feshbach resonance. Nat. Phys. 10, 500–504 (2014).

  17. 17.

    Cristofolini, P. et al. Coupling quantum tunneling with cavity photons. Science 336, 704–707 (2012).

  18. 18.

    Rosenberg, I. et al. Strongly interacting dipolar-polaritons. Sci. Adv. 4, eaat8880 (2018).

  19. 19.

    Togan, E., Lim, H.-T., Faelt, S., Wegscheider, W. & Imamoglu, A. Enhanced interactions between dipolar polaritons. Phys. Rev. Lett. 121, 227402 (2018).

  20. 20.

    Kukushkin, I. V., V. Klitzing, K. & Eberl, K. Spin polarization of composite fermions: measurements of the Fermi energy. Phys. Rev. Lett. 82, 3665–3668 (1999).

  21. 21.

    Byszewski, M. et al. Optical probing of composite fermions in a two-dimensional electron gas. Nat. Phys. 2, 239–243 (2006).

  22. 22.

    Groshaus, J. G. et al. Absorption in the fractional quantum Hall regime: trion dichroism and spin polarization. Phys. Rev. Lett. 98, 156803 (2007).

  23. 23.

    Bar-Joseph, I. Trions in GaAs quantum wells. Semicond. Sci. Technol. 20, R29–R39 (2005).

  24. 24.

    Hayakawa, J., Muraki, K. & Yusa, G. Real-space imaging of fractional quantum Hall liquids. Nat. Nanotechnol. 8, 31–35 (2013).

  25. 25.

    Bartolo, N. & Ciuti, C. Vacuum-dressed cavity magnetotransport of a two-dimensional electron gas. Phys. Rev. B 98, 205301 (2018).

  26. 26.

    Paravicini-Bagliani, G. L. et al. Magneto-transport controlled by Landau polariton states. Nat. Phys. 15, 186–190 (2019).

  27. 27.

    Rapaport, R. et al. Negatively charged quantum well polaritons in a GaAs/AlAs microcavity: an analog of atoms in a cavity. Phys. Rev. Lett. 84, 1607–1610 (2000).

  28. 28.

    Rapaport, R., Cohen, E., Ron, A., Linder, E. & Pfeiffer, L. N. Negatively charged polaritons in a semiconductor microcavity. Phys. Rev. B 63, 235310 (2001).

  29. 29.

    Suris, R. A. In Optical Properties of 2D Systems with Interacting Electrons (eds Ossau, W. J. & Suris, R.) 111–124 (Springer Science and Business Media, 2003).

  30. 30.

    Rodriguez, S. R. K. et al. Interaction-induced hopping phase in driven-dissipative coupled photonic microcavities. Nat. Commun. 7, 11887 (2016).

  31. 31.

    Brichkin, A. S. et al. Effect of Coulomb interaction on exciton-polariton condensates in GaAs pillar microcavities. Phys. Rev. B 84, 195301 (2011).

  32. 32.

    Walker, P. et al. Dark solitons in high velocity waveguide polariton fluids. Phys. Rev. Lett. 119, 09703 (2017).

  33. 33.

    Stepanov, P. et al. Dispersion relation of the collective excitations in a resonantly driven polariton fluid. Preprint at https://arxiv.org/abs/1810.12570(2018).

  34. 34.

    Boyd, R. W. Nonlinear Optics (Elsevier, 2008).

  35. 35.

    Hall, K. L., Lenz, G., Ippen, E. P. & Raybon, G. Heterodyne pump–probe technique for time-domain studies of optical nonlinearities in waveguides. Opt. Lett. 17, 874–876 (1992).

  36. 36.

    Mecozzi, A. & Mørk, J. Transient and time-resolved four-wave mixing with collinear pump and probe pulses using the heterodyne technique. J. Eur. Opt. Soc. A 7, 335–344 (1998).

  37. 37.

    Patton, B., Woggon, U. & Langbein, W. Coherent control and polarization readout of individual excitonic states. Phys. Rev. Lett. 95, 266401 (2005).

  38. 38.

    Kohnle, V. et al. Four-wave mixing excitations in a dissipative polariton quantum fluid. Phys. Rev. B 86, 064508 (2012).

  39. 39.

    Nardin, G., Autry, T. M., Silverman, K. L. & Cundiff, S. T. Multidimensional coherent photocurrent spectroscopy of a semiconductor nanostructure. Opt. Express 21, 28617 (2013).

  40. 40.

    Smallwood, C. L. & Cundiff, S. T. Multidimensional coherent spectroscopy of semiconductors. Laser Photonics Rev. 12, 1800171 (2018).

  41. 41.

    Wouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose–Einstein condensate of exciton polaritons. Phys. Rev. Lett. 99, 140402 (2007).

  42. 42.

    Keeling, J. & Berloff, N. G. Spontaneous rotating vortex lattices in a pumped decaying condensate. Phys. Rev. Lett. 100, 250401 (2008).

  43. 43.

    Ciuti, C., Savona, V., Piermarocchi, C., Quattropani, A. & Schwendimann, P. Role of the exchange of carriers in elastic exciton-exciton scattering in quantum wells. Phys. Rev. B 58, 7926–7933 (1998).

  44. 44.

    Tassone, F. & Yamamoto, Y. Exciton–exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons. Phys. Rev. B 59, 10830–10842 (1999).

  45. 45.

    Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities. Nat. Phys. 5, 805–810 (2009).

  46. 46.

    Byrnes, T., Kolmakov, G. V., Kezerashvili, R. Y. & Yamamoto, Y. Effective interaction and condensation of dipolaritons in coupled quantum wells. Phys. Rev. B 90, 125314 (2014).

  47. 47.

    Nalitov, A. V., Solnyshkov, D. D., Gippius, N. A. & Malpuech, G. Voltage control of the spin-dependent interaction constants of dipolaritons and its application to optical parametric oscillators. Phys. Rev. B 90, 235304 (2014).

Download references


We acknowledge discussions with J. Bloch, A. Browaeys, T. Chervy, O. Cotlet, A. Delteil, T. Grass, M. Hafezi, E. Togan, S. Zeytinoglu and O. Zilberberg. We thank M. Lupatini for the neutral quantum well reference sample. This work was supported by the Swiss National Science Foundation (NCCR Quantum Science and Technology) through an ETH Fellowship (S.R.). This project received funding from the European Research Council under grant agreement 671000.

Author information

P.K. and S.R. performed and analysed the measurements. S.F. and W.W. grew the sample. S.R., M.K. and A.I. supervised the work. P.K., S.R., M.K. and A.I. wrote the manuscript.

Correspondence to Sylvain Ravets or Atac Imamoglu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Experimental setup.

Schematic of the interferometer used for measuring the nonlinear response of the system. PBS, polarizing beam splitter; BS, beam splitter.

Extended Data Fig. 2 White-light reflectivity measurements.

a, Evolution of the reflectivity spectra while tuning the cavity energy across the exciton resonance. The red line in a marks the cavity energy for the spectrum shown in b. b, Background-subtracted spectrum (blue dots). The black line shows a Lorentzian fit to the spectrum. From the peak areas we determine the exciton content |X|2 = 0.7. The LP amplitude is ηc = 0.24.

Extended Data Fig. 3 Comparison between data from undoped quantum well sample and GPE.

Top row, comparison between measured (green circles) and calculated (red shaded area) \({\mathscr{I}}({\omega }_{{\rm{m}}},\tau )\) for different input powers, used to calibrate the detection efficiency ϕ. Bottom row, comparison between measured (purple circles) and calculated (red shaded area) \({\mathscr{I}}(3{\omega }_{{\rm{m}}},\tau )\) for different input powers, yielding a value of g = 0.54 μeV for the polariton interaction strength.

Extended Data Fig. 4 Estimation of interaction constant at ν = 2/5.

a, White-light reflectivity spectra as a function of magnetic field. b, Line cut of the data (blue circles) at B = 3.2 T and a fit (black line) consisting of three Lorentzian resonances. c, Comparison of the linear (\({\mathscr{I}}({\omega }_{{\rm{m}}},\tau )\); top row, green circles) and nonlinear (\({\mathscr{I}}(3{\omega }_{{\rm{m}}},\tau )\); bottom row, purple circles) response at ν = 2/5 with the GPE model (red).

Extended Data Fig. 5 Increase in polariton coherence time with input power at fractional quantum Hall states.

a, Extraction of TLP , showing an exemplary linear response (in a logarithmic scale) and the fit to the envelope (black line). The inverse slope corresponds to TLP. bd, Dependence of TLP on the input power for the filling factors considered. Blue circles correspond to the magnetic field at the quantum Hall state and orange circles to magnetic fields tuned to nearby filling factors.

Extended Data Fig. 6 Data from a high-electron-density sample.

a, White-light reflectivity spectrum recorded using σ polarized light. At B = 8.6 T, the optical signature of ν = 2/3 shows as a reduction in the polariton splitting around 1,527 meV (note that the upper polariton is particularly faint). b, Four-wave-mixing experiment around filling factor ν = 2/3. The top row shows \({\mathscr{I}}({\omega }_{{\rm{m}}},\tau )\) and the bottom row shows \({\mathscr{I}}(3{\omega }_{{\rm{m}}},\tau )\). All data have been normalized to the maximal value of \({\mathscr{I}}({\omega }_{{\rm{m}}},\tau )\) at B = 8.65 T (red diamond). The integration time is 10 s and the input power is 35 ± 5 nW.

Extended Data Table 1 Comparison of interaction constants and LP linewidths

Source data

Source Data Fig. 1

Source Data Fig. 2

Source Data Fig. 3

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark
Fig. 1: Quantum Hall polaritons.
Fig. 2: Time-resolved measurement of interactions between polaron-polaritons.
Fig. 3: Enhancing interactions between quantum Hall polaritons at fractional filling factors.
Extended Data Fig. 1: Experimental setup.
Extended Data Fig. 2: White-light reflectivity measurements.
Extended Data Fig. 3: Comparison between data from undoped quantum well sample and GPE.
Extended Data Fig. 4: Estimation of interaction constant at ν = 2/5.
Extended Data Fig. 5: Increase in polariton coherence time with input power at fractional quantum Hall states.
Extended Data Fig. 6: Data from a high-electron-density sample.


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.