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# Nonlinear optics in the fractional quantum Hall regime

## Abstract

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.

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## 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).

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## Acknowledgements

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

Authors

### Contributions

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.

### Corresponding authors

Correspondence to Sylvain Ravets or Atac Imamoglu.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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.

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Knüppel, P., Ravets, S., Kroner, M. et al. Nonlinear optics in the fractional quantum Hall regime. Nature 572, 91–94 (2019). https://doi.org/10.1038/s41586-019-1356-3

• Accepted:

• Published:

• Issue Date:

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