Can topological corner states survive inside the bulk energy band?
Light injected at a corner of a topological photonic lattice stays trapped there even though the corner states share their energy with bulk states, until the bandgap is made too small.
Source
Quantum superposition demonstrated higher-order topological bound states in the continuum
Study at a glance
- Design
- Other — Laser-written 2D waveguide lattices (C4-symmetric 2D SSH model) probed with heralded single photons injected at corners, at several propagation lengths
- N
- No participant N; physical samples are lattices of 8 x 8 waveguides with propagation lengths from 10 to 30 mm, in topological, trivial and near-transition parameter sets
- Population
- Femtosecond-laser-written photonic waveguide lattices in glass
- Outcome
- Photon intensity distribution at the output and a localization index measuring how much light stays at the injected corner
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What they did
The team wrote two-dimensional arrays of optical waveguides into glass so that alternating weak and strong couplings mimic a 2D Su-Schrieffer-Heeger model with fourfold rotation symmetry. They sent single photons into one corner, or a four-way equal superposition into all four corners, and imaged where the light ended up after different propagation lengths. They compared a topological lattice, a trivial lattice with the couplings swapped, and a lattice close to the phase transition.
What they found
In the topological lattice the light stayed on the injected corner and the localization index remained close to one at every propagation length, whereas in the trivial lattice light spread through the whole array and the index fell towards zero. Injecting a prepared superposition excited just one zero-energy corner eigenstate, which kept its shape as it propagated. When the coupling ratio was raised to 0.68, shrinking the bandgap, light leaked from the corner to the other corners and edges, showing the bound states break down.
The limits
What it doesn't show
The lattices are small (8 by 8 sites), so finite-size effects near the transition are significant and the results are demonstrations rather than scaling studies. Only a few coupling ratios and propagation lengths were tested, so the exact point where protection fails is not mapped. The single-photon source behaves like classical light in a linear lattice, so the 'quantum superposition' framing does not show any effect that needs entanglement. Much of the theoretical argument (orthogonality to bulk states, the topological index) is in supplementary material not included here.
Key terms
- Higher-order topological insulator
- A topological phase whose protected boundary states live two or more dimensions below the bulk, such as corner states in a 2D lattice.
- Bound state in the continuum (BIC)
- A localized state whose energy lies inside a band of extended states but which does not leak into them, usually because of symmetry.
- SSH model
- A tight-binding lattice with alternating strong and weak couplings; which coupling is stronger inside the unit cell decides whether it is topological.
- Localization index
- The fraction of output light found at (or near) the site where it was injected.
- Chiral symmetry
- A symmetry that forces the energy spectrum to be mirror-symmetric about zero, pinning corner states at zero energy.
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Quiz yourself
What is a bound state in the continuum?
Common questions
If the corner states have the same energy as bulk states, why doesn't the light just spread out?
Symmetry makes the corner states orthogonal to the bulk states, so light that starts in a corner state has no route into the bulk even though energies match.
Why do the waveguide lengths matter?
In a waveguide array, distance along the guide plays the role of time, so imaging at several lengths is like taking snapshots of the evolution.
What happens near the topological transition?
As the two couplings become similar the bandgap shrinks, corner states move away from zero energy and mix with edge and bulk states, so light escapes the corner.
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