Photons carry no charge, so they do not bend in a magnetic field. They can be made to behave as if they did. Arrange ring resonators in a square array, couple neighbours through link rings, and shift each link by a little: a photon hopping once around a plaquette comes back with a phase it did not start with, which is exactly what a charged particle does in a field. The array is then an integer quantum Hall system for light, with a bulk band and two chiral edge bands, one going each way round.
This is a device that exists: a hundred silicon nitride rings in a ten-by-ten array on a chip, made in a commercial foundry. What this paper is about is that such a chip does not have one gauge field. It has hundreds.
The lattice, and the phase
the Harper–Hofstadter Hamiltonian in the Landau gauge , with the coupling between neighbouring site rings and the phase around one plaquette. The devices here are designed for — a quarter of a flux quantum — and between 10 and 20 GHz, set by a coupling gap between 300 and 600 nm. A ten-by-ten array gives a hundred super-modes: two edge bands of about ten each, separated by a bulk band.
Eq. (1) · equation numbers throughout are those of arXiv:2606.23960v3
Five scales
A ring resonator is not one mode. It has a comb of longitudinal modes, spaced by its free spectral range, and each of them sees the array. The trouble is arithmetic: the structure the gauge field makes is narrow, the modes are far apart, and the light spans further still.
- spacing of the edge modes ≈ 5 GHz the round-trip time of one lap around the edge of the array
- coupling between rings, J 10–20 GHz set by the gap between neighbours, 300 to 600 nm
- width of a topological band ≈ 100 GHz narrow enough that J and φ look constant across it
- free spectral range of one ring 773.92 GHz the step from one gauge field to the next
- the span of the comb ≈ 100 THz nearly an octave; more than a hundred free spectral ranges across
Read that ladder from the middle outwards. Within one band, treating and as numbers is fine — the band is a hundred times narrower than the spacing to the next mode. Across the whole comb it is not fine at all: by the far end of an octave the waveguide has a different index, the coupling has a different strength, and the phase around a plaquette is not what it was. The usual fix keeps one Hamiltonian per mode and lets only the on-site frequency drift with dispersion. That still assumes the field itself is fixed.
Letting the field drift
with , the free spectral range and the integrated dispersion — the deviation of the resonances from an evenly spaced grid. Every parameter now carries the mode index: , , , all three taken from a finite-difference simulation of a single unit cell of waveguide rather than of the whole chip, which is about a hundred times cheaper and gives the same answer.
Eq. (2)
The test is a long sweep. Tuning a low-power laser across nineteen terahertz — twenty-six modes of the ring — and reading the drop port gives the resonances of every one of those twenty-six lattices. The edge–bulk–edge structure survives the whole way, and the edge resonances drift within their band as the mode number changes: the two chiral bands drift in opposite directions, which is what a Landau fan does when you change the field. The dispersion-corrected model follows the measurement; the uncorrected one does not.
All of them at once
Sweeping a laser is slow, and it can only reach as far as the laser tunes. The alternative is to stop being linear. Pump one mode hard enough and the Kerr effect in the silicon nitride generates the rest by cascaded four-wave mixing — a frequency comb, from one pulse. Here the threshold is about 200 mW and the comb grows with power until it spans nearly an octave.
What comes out is a comb of combs. Each tooth is one mode , and each tooth is itself resolved into about ten lines spaced by 5 GHz — the edge modes of the lattice at that colour. More than a hundred gauge fields, ten thousand photonic modes, and up to a thousand comb lines, from a single shot.
The nonlinear model is the dispersive Hamiltonian above inside a Lugiato–Lefever equation, one field amplitude per ring per mode:
Eq. (3)
Two chips differing only in the gap between rings behave differently: at 400 nm the comb follows the lattice’s own dispersion, at 300 nm it locks to an evenly spaced grid instead. The dispersion is being engineered at the scale of the lattice, not of the waveguide.
What the topology survives
The sharpest result is a thing that does not happen. Pump one chiral edge band and the other one stays dark, and so does the bulk, by up to 30 dB. Nothing backscatters and nothing mixes into the bulk — under a nonlinearity strong enough to generate an octave of new light. That is topological protection holding up in a regime where there was no particular reason to expect it to.
And the separation between the two edge combs is itself a measurement of the gauge field’s drift: 140 GHz at mode −26, 75 GHz at mode 2, 50 GHz at mode 13. One number per mode, read off an optical spectrum analyser, showing how far the field has moved between one end of the comb and the other.
Three nominally identical chips give the same behaviour with about 100 GHz of chip-to-chip shift from fabrication spread — small compared with the free spectral range, which is why a single shot is enough and why this is something a foundry could make in quantity.
The same ring-array platform is what Quantum Metamorphosis proposes to nest, one lattice inside another, using the link rings to set a ratio of scales rather than a single field.