Synthetic Landau levels for photons

Journal name:
Nature
Volume:
534,
Pages:
671–675
Date published:
DOI:
doi:10.1038/nature17943
Received
Accepted
Published online

Synthetic photonic materials are an emerging platform for exploring the interface between microscopic quantum dynamics and macroscopic material properties1, 2, 3, 4, 5. Photons experiencing a Lorentz force develop handedness, providing opportunities to study quantum Hall physics and topological quantum science6, 7, 8. Here we present an experimental realization of a magnetic field for continuum photons. We trap optical photons in a multimode ring resonator to make a two-dimensional gas of massive bosons, and then employ a non-planar geometry to induce an image rotation on each round-trip9. This results in photonic Coriolis/Lorentz and centrifugal forces and so realizes the Fock–Darwin Hamiltonian for photons in a magnetic field and harmonic trap10. Using spatial- and energy-resolved spectroscopy, we track the resulting photonic eigenstates as radial trapping is reduced, finally observing a photonic Landau level at degeneracy. To circumvent the challenge of trap instability at the centrifugal limit10, 11, we constrain the photons to move on a cone. Spectroscopic probes demonstrate flat space (zero curvature) away from the cone tip. At the cone tip, we observe that spatial curvature increases the local density of states, and we measure fractional state number excess consistent with the Wen–Zee theory, providing an experimental test of this theory of electrons in both a magnetic field and curved space12, 13, 14, 15. This work opens the door to exploration of the interplay of geometry and topology, and in conjunction with Rydberg electromagnetically induced transparency, enables studies of photonic fractional quantum Hall fluids16, 17 and direct detection of anyons18, 19.

At a glance

Figures

  1. Resonator structure and transverse manifold geometry.
    Figure 1: Resonator structure and transverse manifold geometry.

    a, Top, ray trajectories (black lines) in a curved mirror resonator oscillate transversely (green arrows). In a particular transverse plane, the stroboscopic time evolution of the ray positions samples a harmonic oscillator trajectory (blue points). In paraxial optics, the solutions for the transverse modes are Hermite–Gauss profiles (red curve). The transverse degrees of freedom of a resonator are precisely those of a 2D quantum harmonic oscillator (bottom). b, Top, as a four mirror resonator is made non-planar (purple arrows), the light rays are induced to rotate (blue arrow) about the optic axis. In the transverse plane (represented below), this corresponds to flattening the 2D harmonic potential (centrifugal force) and the introduction of an effective magnetic field (Coriolis force). c, Our non-planar resonator consists of four mirrors (blue and purple) in a stretched tetrahedral configuration of on-axis length La and opening half-angle θ. The image rotates about the optic axis (red) on every round trip. d, Left, we depict the transverse plane at the resonator waist pierced by a uniform perpendicular (along ) magnetic field B of magnitude B, and show a generic profile (red curve) with three-fold symmetry. When the plane is cut arbitrarily into three equal sections, the entire profile is fully determined within any one-third section of the plane: when a trajectory leaves one side of a section, it reappears on the other side. Each section may be wrapped into a cone on which the original profile appears once (right; this would be true for any discrete rotational symmetry). The effective magnetic field is everywhere perpendicular to the cone’s surface.

  2. Building a Landau level.
    Figure 2: Building a Landau level.

    The modes of our resonator follow the Fock–Darwin Hamiltonian of a massive, harmonically trapped particle in magnetic field: the magnetic field creates a ladder of Landau levels uniformly spaced by the cyclotron frequency, ωc, while the harmonic trap of frequency ωtrap uniformly splits levels within each Landau level by (see Supplementary Information). We probe this spectrum versus resonator length Lrt, and demonstrate that, for each Lrt, the spectrum is determined by two energies ν(1,0) and ν(0,1) according to ν(α,β)= αν(1,0) + βν(0,1) mod νFSR, where ωc= 2π × ν(1,1) gives the cyclotron frequency and provides the harmonic trapping frequency. Furthermore, fine-tuning Lrt drives ωtrap to zero, bringing specific sets of angular momentum eigenmodes into degeneracy, thereby forming Landau levels. a, The frequency separations between several modes and a reference l = 0 mode are plotted as the harmonic confinement is coarsely tuned relative to an approximately degenerate reference length Lrt= 78.460 mm (corresponding free spectral range νFSR= 3.8209 GHz). Solid lines are obtained as integer linear combinations of fits to the modes labelled (1,0) and (0,1) and the free spectral range. For details on mode indexing, see Supplementary Information. b, Main panel, we plot the transmission spectrum of the first ~10 modes in the lowest Landau level against small deviations from nominal degeneracy. Top inset, low order modes become degenerate to within a resonator linewidth, κ ≈ 200 kHz, while in the main panel, we observe weak level repulsion (approximately equal to the resonator linewidth) in the higher order modes consistent with mode mixing due to mirror imperfections of ~λ/5,000. ωtrap is presented on the upper horizontal axis. Bottom insets, as the resonator is tuned through degeneracy, the harmonic potential (orange surface) changes sign, while the magnetic field (blue arrows) remains nearly unchanged. c, The lifetimes (and corresponding finesses) of representative modes decrease for higher mode numbers both away from degeneracy (blue circles) and near degeneracy (green squares). Here ΔL is the offset of the round-trip resonator length from nominal degeneracy. d, With significant residual harmonic trapping (ΔL = 124 μm), angular momentum modes are simple rings. As the trapping is reduced (ΔL = 32 μm), high angular momentum modes begin to mix owing to local disorder. When the trapping is precisely cancelled (ΔL = −3 μm), mirror imperfection consistent with a single nanoscopic scratch dramatically alters the modes’ shape away from the predicted near-Laguerre–Gauss profiles. Even the first resonator mode is noticeably triangular, indicating at least a mixing of Laguerre–Gauss l = 0 and l = 3 modes. Overcoming this disorder necessitates only ~MHz photon–photon interactions to explore strongly correlated physics.

  3. Photonic lowest Landau levels on a cone.
    Figure 3: Photonic lowest Landau levels on a cone.

    a, At degeneracy, all resonator modes display three-fold symmetry. We present a very large displaced angular momentum mode with radial extent up to 8 times the mode waist, w0, implying that ~20 modes must be degenerate. The rapid phase winding for large l modes causes the strong fringing pattern when the mode self-interferes. Inset, an l = 0 mode at the same scale. b, We project another large-angular-momentum mode onto a cone and view it from above the apex. We observe a general property that circular orbits must encircle the cone apex either zero or three times. Inset, the original image of the mode. The pair of rays overlaying the inset image corresponds to the cut in the main image. c, The twisted resonator corresponds to Landau levels on three cones with differing quantities of magnetic flux threaded through the tip. The cone built out of l = 0, 3, 6, … has no flux threading; the cone built out of l = 1, 4, 7, … is threaded by Φ0/3; and the cone built out of l = 2, 5, 8, … is threaded by 2Φ0/3, where Φ0 is the magnetic flux quantum. d, With the resonator tuned to degeneracy, we identify the energies of the l = c modes for c = 0, 1, or 2 by the transmission peaks (blue, orange, and green curves, respectively) that correspond the correct observed transmitted modes’ profiles (single images, labelled). The degenerate sets starting with these modes each form a lowest Landau level on different cones. Except at the apex, each cone is flat, so away from the tip each lowest Landau level supports modes of—and therefore the dynamics of—a planar lowest Landau level with l = 0, 1, 2, … defined about a displaced point. On each cone, we show displaced l = 0 (bottom two) and l = 1 (top two) modes. For large displacements (right two), these modes are undistorted; however, for small displacements (left two), where there is significant mode amplitude at the tip, we observe distortions due to self-interference, similar to a. e, f, Displaced l = 0 and l = 1 modes from d are projected onto a cone to show how observed mode images may be interpreted on a conical surface. gi, We explore the effects of curvature and flux threading near the tip by measuring the local density of photonic states. For the c = 0 cone (i), we find an approximately threefold increase in local state density near the cone apex above a constant background plateau of density. This corresponds to an additional one-third of a state localized near the apex. For the cones with c = 1 and 2 (h and g, respectively), we find a vanishing local density of states near the apex, reflecting the negative magnetic flux threading through the cone apex. Each unit of flux removes one-third of a state local to the apex so that the c = 1 cone has no additional states, and the c = 2 cone is missing one-third of one state. The data to the right display a slice through the middle of each image; the grey curves are fits to the expected analytic form (see Supplementary Information).

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Author information

Affiliations

  1. Department of Physics and James Franck Institute, University of Chicago, Chicago, Illinois, USA

    • Nathan Schine,
    • Albert Ryou,
    • Ariel Sommer &
    • Jonathan Simon
  2. Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois, USA

    • Andrey Gromov

Contributions

The experiment was designed and built by N.S., J.S., A.R. and A.S. Measurement and analysis of the data was performed by N.S. Theoretical development and interpretation of results were performed by J.S., A.S., N.S. and A.G. All authors contributed to the manuscript.

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The authors declare no competing financial interests.

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Supplementary information

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  1. Supplementary Information (1.7 MB)

    This file contains Supplementary Text and Data, Supplementary Figures 1-3 and additional references.

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