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Observation of Laughlin states made of light

Abstract

Much of the richness in nature emerges because simple constituents form an endless variety of ordered states1. Whereas many such states are fully characterized by symmetries2, interacting quantum systems can exhibit topological order and are instead characterized by intricate patterns of entanglement3,4. A paradigmatic example of topological order is the Laughlin state5, which minimizes the interaction energy of charged particles in a magnetic field and underlies the fractional quantum Hall effect6. Efforts have been made to enhance our understanding of topological order by forming Laughlin states in synthetic systems of ultracold atoms7,8 or photons9,10,11. Nonetheless, electron gases remain the only systems in which such topological states have been definitively observed6,12,13,14. Here we create Laughlin-ordered photon pairs using a gas of strongly interacting, lowest-Landau-level polaritons as a photon collider. Initially uncorrelated photons enter a cavity and hybridize with atomic Rydberg excitations to form polaritons15,16,17, quasiparticles that here behave like electrons in the lowest Landau level owing to a synthetic magnetic field created by Floquet engineering18 a twisted cavity11,19 and by Rydberg-mediated interactions between them16,17,20,21. Polariton pairs collide and self-organize to avoid each other while conserving angular momentum. Our finite-lifetime polaritons only weakly prefer such organization. Therefore, we harness the unique tunability of Floquet polaritons to distil high-fidelity Laughlin states of photons outside the cavity. Particle-resolved measurements show that these photons avoid each other and exhibit angular momentum correlations, the hallmarks of Laughlin physics. This work provides broad prospects for the study of topological quantum light22.

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Fig. 1: Components for Laughlin states of light.
Fig. 2: Collisions between polaritons in the lowest Landau level.
Fig. 3: Laughlin state characterization in angular momentum space.
Fig. 4: Spatial correlations of a photonic Laughlin puddle.

Data availability

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

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Acknowledgements

We thank L. Feng and M. Jaffe for feedback on the manuscript. This work was supported by AFOSR grant FA9550-18-1-0317 and AFOSR MURI grant FA9550-16-1-0323. N.S. acknowledges support from the University of Chicago Grainger graduate fellowship and C.B. acknowledges support from the NSF GRFP.

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Contributions

The experiment was designed and built by all authors. N.S. built the primary cavity. L.W.C., N.S. and C.B. collected the data. L.W.C. and N.S. analysed the data. L.W.C. wrote, and all authors contributed to, the manuscript.

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Correspondence to Jonathan Simon.

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Extended data figures and tables

Extended Data Fig. 1 Single-mode polariton spectrum.

a, Photons at the waist of our twisted optical cavity (red) couple with strength g to the 5S1/2 → 5P3/2 transition of a gas of cold 87Rb atoms, which is subsequently coupled with strength Ω to the highly excited 111D5/2 Rydberg state using an additional laser (blue). b, These couplings cause excitations of the atom-cavity system to propagate as polaritons—quasiparticles combining photons with collective atomic excitations. c, The transmission spectrum of the cavity with atoms present directly reveals the narrow dark polariton flanked by two broad bright polariton peaks. The solid curve shows a fit of the cavity electromagnetically induced transparency spectrum to the measured transmission (see ref. 32, Supplementary equation (7)).

Extended Data Fig. 2 Essential features of the Floquet scheme.

a, Our Floquet scheme utilizes an additional laser beam (green) incident on the atoms with a wavelength of λ = 1,529 nm close to the 5P3/2 → 4D transition. b, This beam induces a sinusoidally modulated a.c. Stark shift Ep = ηsin(2πfmodt) of the 5P3/2 state with amplitude η and frequency fmod. c, As a result of this modulation, the ordinary 5P3/2 state is split into three bands with energies separated by the modulation frequency. The additional bands enable the atoms to couple with cavity photons at frequencies shifted by ±fmod from the ordinary 5S1/2 → 5P3/2 resonance frequency. For more details on the Floquet scheme see ref. 18.

Extended Data Fig. 3 Scheme for forming the Landau level of Floquet polaritons.

a, The bare cavity modes are not degenerate in this work, but instead the length of the cavity is increased so that there is a fcav ≈ 70 MHz splitting between every third angular momentum mode. b, To form polaritons in three modes, even though only the l = 6 mode is resonant with the un-modulated 5S1/2 → 5P3/2 transition, we utilize the Floquet scheme depicted in Extended Data Fig. 218. Modulating the 5P3/2 state at fmod ≈ 70 MHz splits it into three bands (grey), each of which is resonant with one of the three chosen cavity modes. The coupling strengths gl to each mode l are controlled by the modulation amplitude; in this work, g3 = g9 = 0.37(4)g6. Note that each mode couples to a unique collective atomic excitation, as depicted at the top (blue atoms are included in the corresponding collective excitation, while grey atoms are not). c, This scheme produces polaritons in the l = 3, l = 6 and l = 9 modes. The dark polaritons can be made effectively degenerate (see Extended Data Fig. 4) without making the corresponding cavity modes degenerate, which protects the polaritons from intracavity aberrations (see Supplementary Information section B2).

Extended Data Fig. 4 Understanding and controlling polariton spectra with the Floquet scheme.

a, Cavity transmission spectrum in the presence of the modulated atoms (see Extended Data Fig. 3), reproducing Fig. 2b. The spectrum was collected in three parts, corresponding to injection of photons into l = 3 (left, green), l = 6 (middle, black) and l = 9 (right, violet). Dark polaritons in the l = 6 mode are less photon-like than those in the other two modes, reducing their relative transmission (Supplementary Information section A1); we multiply the l = 6 transmission by four to improve visibility. The lower x axis indicates the frequency f of the probe laser relative to the l = 6 dark polariton resonance at f6. The top x axis indicates the quasifrequency \(\tilde{f}\), proportional to the quasienergy of the polaritons from a treatment using Floquet theory; \(\tilde{f}\) is equal to f modulo the modulation frequency fmod. The solid curves show three independent fits of the cavity electromagnetically induced transparency spectrum to the measured transmission in each angular momentum mode (see ref. 32, Supplementary equation (7)). b, c, Illustration of the theoretical dependence of the quasifrequencies of the three dark polariton features on the Rydberg beam detuning (b) and the modulation frequency fmod (c). d, Example transmission spectra for the l = 6 (black, lower), l = 3 (green, middle) and l = 9 (violet, upper) dark polaritons as a function of quasifrequency. The scans are scaled to make their heights equal and have additional vertical offsets for clarity. Shortly before performing each of the experiments reported in the main text, we collect a sequence of plots similar to those displayed here and adjust the Rydberg detuning and modulation frequency to make all three dark polaritons have the same quasifrequency (right-most plot). The only experiments reported in the main text that did not use this sequence are those shown in Fig. 2c, where instead we varied δr to intentionally vary the energy mismatch between the polaritons. Throughout this figure, quasifrequencies \(\tilde{f}\) are reported relative to the l = 6 dark polariton resonance \({\tilde{f}}_{6}\). Solid curves provide a guide to the eye.

Supplementary information

Supplementary Information

The supplementary information document provides experimental and theoretical details for this work. It contains eight experimental sections describing the relationship between photonic and polaritonic states in the Floquet scheme, the sorting of photons by angular momentum, the details of our two-photon correlation analysis, the reconstruction of the density matrix, the experimental setup and a typical operation sequence, the details of our optical cavity design, the modes in the lower resonator waist, and electric field management. It contains six theoretical sections describing collective atomic excitations, the protection of Floquet polaritons from intracavity aberrations, the many-body spectrum, the varieties of two particle Laughlin states, how we analyze the collision data, and how to understand the width of the energy-conservation feature.

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Clark, L.W., Schine, N., Baum, C. et al. Observation of Laughlin states made of light. Nature 582, 41–45 (2020). https://doi.org/10.1038/s41586-020-2318-5

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