Letter | Published:

# Interacting Floquet polaritons

## Abstract

Ordinarily, photons do not interact with one another. However, atoms can be used to mediate photonic interactions1,2, raising the prospect of forming synthetic materials3 and quantum information systems4,5,6,7 from photons. One promising approach combines highly excited Rydberg atoms8,9,10,11,12 with the enhanced light–matter coupling of an optical cavity to convert photons into strongly interacting polaritons13,14,15. However, quantum materials made of optical photons have not yet been realized, because the experimental challenge of coupling a suitable atomic sample with a degenerate cavity has constrained cavity polaritons to a single spatial mode that is resonant with an atomic transition. Here we use Floquet engineering16,17—the periodic modulation of a quantum system—to enable strongly interacting polaritons to access multiple spatial modes of an optical cavity. First, we show that periodically modulating an excited state of rubidium splits its spectral weight to generate new lines—beyond those that are ordinarily characteristic of the atom—separated by multiples of the modulation frequency. Second, we use this capability to simultaneously generate spectral lines that are resonant with two chosen spatial modes of a non-degenerate optical cavity, enabling what we name ‘Floquet polaritons’ to exist in both modes. Because both spectral lines correspond to the same Floquet-engineered atomic state, adding a single-frequency field is sufficient to couple both modes to a Rydberg excitation. We demonstrate that the resulting polaritons interact strongly in both cavity modes simultaneously. The production of Floquet polaritons provides a promising new route to the realization of ordered states of strongly correlated photons, including crystals and topological fluids, as well as quantum information technologies such as multimode photon-by-photon switching.

## Access optionsAccess options

from\$8.99

All prices are NET prices.

## Data availability

The experimental data presented in this manuscript is available from the corresponding author upon request.

## References

1. 1.

Birnbaum, K. M. et al. Photon blockade in an optical cavity with one trapped atom. Nature 436, 87–90 (2005).

2. 2.

Chang, D. E., Vuletić, V. & Lukin, M. D. Quantum nonlinear optics—photon by photon. Nat. Photon. 8, 685–694 (2014).

3. 3.

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

4. 4.

Raimond, J. M., Brune, M. & Haroche, S. Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys. 73, 565–582 (2001).

5. 5.

Duan, L.-M., Lukin, M., Cirac, J. I. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413–418 (2001).

6. 6.

Kimble, H. J. The quantum internet. Nature 453, 1023–1030 (2008).

7. 7.

Saffman, M., Walker, T. G. & Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 2313–2363 (2010).

8. 8.

Peyronel, T. et al. Quantum nonlinear optics with single photons enabled by strongly interacting atoms. Nature 488, 57–60 (2012).

9. 9.

Dudin, Y., Li, L., Bariani, F. & Kuzmich, A. Observation of coherent many-body Rabi oscillations. Nat. Phys. 8, 790–794 (2012).

10. 10.

Tiarks, D., Baur, S., Schneider, K., Dürr, S. & Rempe, G. Single-photon transistor using a Förster resonance. Phys. Rev. Lett. 113, 053602 (2014).

11. 11.

Gorniaczyk, H., Tresp, C., Schmidt, J., Fedder, H. & Hofferberth, S. Single-photon transistor mediated by interstate Rydberg interactions. Phys. Rev. Lett. 113, 053601 (2014).

12. 12.

Thompson, J. D. et al. Symmetry-protected collisions between strongly interacting photons. Nature 542, 206–209 (2017).

13. 13.

Guerlin, C., Brion, E., Esslinger, T. & Mølmer, K. Cavity quantum electrodynamics with a Rydberg-blocked atomic ensemble. Phys. Rev. A 82, 053832 (2010).

14. 14.

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

15. 15.

Georgakopoulos, A., Sommer, A. & Simon, J. Theory of interacting cavity Rydberg polaritons. Quantum Sci. Technol. 4, 014005 (2018).

16. 16.

Silveri, M. P., Tuorila, J. A., Thuneberg, E. V. & Paraoanu, G. S. Quantum systems under frequency modulation. Rep. Prog. Phys. 80, 056002 (2017).

17. 17.

Eckardt, A. Atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys. 89, 011004 (2017).

18. 18.

Fleischhauer, M., Imamoglu, A. & Marangos, J. Electromagnetically induced transparency: optics in coherent media. Rev. Mod. Phys. 77, 633–673 (2005).

19. 19.

Douglas, J. S. et al. Quantum many-body models with cold atoms coupled to photonic crystals. Nat. Photon. 9, 326–331 (2015).

20. 20.

Schine, N., Ryou, A., Gromov, A., Sommer, A. & Simon, J. Synthetic Landau levels for photons. Nature 534, 671–675 (2016).

21. 21.

Lim, H.-T., Togan, E., Kroner, M., Miguel-Sanchez, J. & Imamoğlu, A. Electrically tunable artificial gauge potential for polaritons. Nat. Commun. 8, 14540 (2017).

22. 22.

Schine, N., Chalupnik, M., Can, T., Gromov, A. & Simon, J. Electromagnetic and gravitational responses of photonic Landau levels. Nature 565, 173–179 (2019).

23. 23.

Strand, J. D. et al. First-order sideband transitions with flux-driven asymmetric transmon qubits. Phys. Rev. B 87, 220505 (2013).

24. 24.

Naik, R. et al. Random access quantum information processors using multimode circuit quantum electrodynamics. Nat. Commun. 8, 1904 (2017).

25. 25.

Beaudoin, F., da Silva, M. P., Dutton, Z. & Blais, A. First-order sidebands in circuit QED using qubit frequency modulation. Phys. Rev. A 86, 022305 (2012).

26. 26.

Beaufils, Q. et al. Radio-frequency association of molecules: an assisted Feshbach resonance. Eur. Phys. J. D 56, 99–104 (2010).

27. 27.

Ningyuan, J. et al. Observation and characterization of cavity Rydberg polaritons. Phys. Rev. A 93, 041802 (2016).

28. 28.

Zeuthen, E., Gullans, M. J., Maghrebi, M. F. & Gorshkov, A. V. Correlated photon dynamics in dissipative Rydberg media. Phys. Rev. Lett. 119, 043602 (2017).

29. 29.

Ivanov, P. A., Letscher, F., Simon, J. & Fleischhauer, M. Adiabatic flux insertion and growing of Laughlin states of cavity Rydberg polaritons. Phys. Rev. A 98, 013847 (2018).

30. 30.

Dutta, S. & Mueller, E. Coherent generation of photonic fractional quantum Hall states in a cavity and the search for anyonic quasiparticles. Phys. Rev. A 97, 033825 (2018).

31. 31.

Sommer, A., Büchler, H. P. & Simon, J. Quantum crystals and Laughlin droplets of cavity Rydberg polaritons. Preprint at https://arxiv.org/abs/1506.00341 (2015).

32. 32.

Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

33. 33.

Norcia, M. A., Cline, J. R. K., Bartolotta, J. P., Holland, M. J. & Thompson, J. K. Narrow-line laser cooling by adiabatic transfer. New J. Phys. 20, 023021 (2018).

34. 34.

Ma, R. et al. Photon-assisted tunneling in a biased strongly correlated Bose gas. Phys. Rev. Lett. 107, 095301 (2011).

35. 35.

Parker, C. V., Ha, L.-C. & Chin, C. Direct observation of effective ferromagnetic domains of cold atoms in a shaken optical lattice. Nat. Phys. 9, 769–774 (2013).

36. 36.

Clark, L. W., Feng, L. & Chin, C. Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition. Science 354, 606–610 (2016).

37. 37.

Daley, A. J. & Simon, J. Effective three-body interactions via photon-assisted tunneling in an optical lattice. Phys. Rev. A 89, 053619 (2014).

38. 38.

Meinert, F., Mark, M. J., Lauber, K., Daley, A. J. & Nägerl, H.-C. Floquet engineering of correlated tunneling in the Bose–Hubbard model with ultracold atoms. Phys. Rev. Lett. 116, 205301 (2016).

39. 39.

Clark, L. W., Gaj, A., Feng, L. & Chin, C. Collective emission of matter-wave jets from driven Bose–Einstein condensates. Nature 551, 356–359 (2017).

40. 40.

Lignier, H. et al. Dynamical control of matter-wave tunneling in periodic potentials. Phys. Rev. Lett. 99, 220403 (2007).

41. 41.

Struck, J. et al. Tunable gauge potential for neutral and spinless particles in driven optical lattices. Phys. Rev. Lett. 108, 225304 (2012).

42. 42.

Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).

43. 43.

Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302 (2013).

44. 44.

Tai, M. E. et al. Microscopy of the interacting Harper–Hofstadter model in the two-body limit. Nature 546, 519–523 (2017).

45. 45.

Clark, L. W. et al. Observation of density-dependent gauge fields in a Bose–Einstein condensate based on micromotion control in a shaken two-dimensional lattice. Phys. Rev. Lett. 121, 030402 (2018).

46. 46.

Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

47. 47.

Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).

48. 48.

Tarnowski, M. et al. Measuring topology from dynamics by obtaining the Chern number from a linking number. Nat. Commun. 10, 1728 (2019).

49. 49.

Fläschner, N. et al. Observation of dynamical vortices after quenches in a system with topology. Nat. Phys. 14, 265–268 (2018).

## Acknowledgements

We thank L. Feng for comments on the manuscript. This work was supported by DOE grant DE-SC0010267 for apparatus construction, AFOSR grant FA9550-18-1-0317 for modelling and MURI grant FA9550-16-1-0323 for data collection and analysis. N.S. acknowledges support from a University of Chicago Grainger graduate fellowship and C.B. acknowledges support from the NSF GRFP.

### Reviewer information

Nature thanks Oliver Morsch, Michael Sentef and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

The experiment was designed and built by all authors. L.W.C., N.J. and N.S. collected the data. L.W.C. and N.J. analysed the data. L.W.C. and J.S. developed the theory. L.W.C. prepared the manuscript, and all authors contributed to the final manuscript.

Correspondence to Logan W. Clark.

## 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 Atomic level diagram.

a, Three key electronic transitions of 87Rb atoms enable the formation of Floquet polaritons. First, cavity photons near 780 nm couple to the 5S1/2 → 5P3/2 atomic transition. Second, a beam near 480 nm drives the 5P3/2 → nS1/2 transition to the Rydberg level with principal quantum number n at coupling strength Ω. Third, a multichromatic field near the 5P3/2 → 5D5/2 transition modulates the energy of the 5P3/2 state. b, The multichromatic field has two components with approximately opposite detunings ±δ: a red-detuned component with constant intensity Ir, and a blue-detuned component with sinusoidally modulated intensity Ib[1 + cos(ωt)], where ω = 2πf.

## Supplementary information

### Supplementary Information

The supplementary information document provides experimental and theoretical details for this work. It contains three experimental sections discussing the analysis of the band strengths, correlations, and making comparisons to other possible modulation schemes. It contains nine theoretical sections discussing our model for the atom-cavity system, the nature of collective atomic excitations, and redistribution of a state using frequency modulation, Floquet polaritons in the high frequency approximation, the quasienergy spectrum, applications for Floquet polaritons, the connection to shaken optical lattices, the cause of asymmetric band strengths, and the multimode non-Hermitian perturbation theory.

## Rights and permissions

Reprints and Permissions