Diffusion of waves is a natural result in propagation, like sound, water, and light, while the localization of waves is an unconventional phenomenon in periodic media. It has been an important scientific problem for a long time to control the propagation of waves and localize it in a certain range.
There are several routes to realize the localization, such as Anderson localization, topological boundary state, Moire lattice, and bound-state in the continuum (BIC). Among these, the topological lattice and BIC lattice allow the media to be periodic, which is an interesting phenomenon.
The localized states in topological lattice appear in the bandgap, meanwhile, the localization states can also be embedded into the bulk spectrum as BIC. It seems impossible for a topological lattice to own the localized states embed into the bulk spectrum. Such that, the combination of topological phase based on band theory and the BIC is the topic attracting a wide range of research interests. The combination of higher-order topological phase and BIC sheds light on new materials design. Specifically, it will be promising for practical applications, such as designing lower-dimensional higher-Q-factor topological cavities if the corner states in higher-order topological BIC can be experimentally independently triggered
Recently, we experimentally demonstrate a kind higher-order topological bound states in the continuum. Specifically, we demonstrate two ways of identifying the bound states in the continuum photonic lattice, with single-site and superposition-state injection. We show that the corner states, lying inside the continuum and coexists with extended waves, can be well excited and perfectly confined.
In the tight-bounding 2D Su-Schrieffer-Heeger (SSH) lattice, the second-order topological corner states are pinned on the zero-energy level and embedded into the bulk states. Intuitively, such corner states may not be observed since they have the same energy as bulk states, however, we find that the corner states are orthogonal to bulk states in Hilbert space, forming the bound states in the continuum, which makes it possible to be independently excited by eigenmodes injections from degenerate bulk states.
In this post, I would especially like to show you the way we identify the single bound states in the continuum photonic lattice with superposition-state injection. As the Figure is shown, we inject the photons into a 3D 1×4 photonic coupler and obtain the photon superposition state. Subsequently, the prepared photon superposition is injected into four corners of the lattice. The distribution probability of photon in the lattice is identical to the zero-energy corner state with the same phase. The system now is excited into the zero-energy corner state and the photon superposition state in the lattice will be maintained due to the orthogonality among eigenstates, which means that the single corner state is identified. The measured photon distribution probability in the experiment shows that the output probability distribution of photons follows the distribution of corner states and is maintained under the change of the evolution distance. In this way, only one zero-energy corner mode has been excited by precisely preparing the system into its eigen-wave-functions.
Interestingly, if we make little change to such a lattice, the corner modes would separate from the bulk mode. Such as we can pull the topological corner modes into the gap by increasing the on-site energy of corner sites. Moreover, with the increase of off-diagonal lattice disorder, which breaks the lattice symmetry, the zero-energy bulk modes will also gradually deviate from the zero energy. The BIC is easily broken while the localized topological corner states are robust, in experiment, we introduce the disorder into the pure lattice and demonstrate this point.
Moreover, I want to share with you the view of explaining the photon evolution in topological lattice form quantum dynamics. According to the principle of quantum mechanics, if we inject the photons into the lattice from the corners, due to the overlap of spatial distribution, the initial injected states can be expressed in the form of a superposition of almost only corner states. The probability amplitude proportion and the relative phase of the corner states are maintained with evolution. In this case, the photons will be confined in the excited corner of lattice, and the evolution is stable.
In theory, we point out that the corner state can be embedded in the bulk states while being decoupled and can be excited individually from them, regarded as the higher-order topological bound states in the continuum. In experiment, we propose and experimentally demonstrate a new way to identify the topological corner states by exciting them separately from the bulk states with a single-photon superposition state.
Our results extend the conventional topological BICs into higher-order cases, providing an unprecedented mechanism to achieve robust and localized states in a bulk spectrum. In terms of physical explanation, we combine topological photonics and quantum dynamics and provide a method to identify the single corner state with the help of a photonic quantum superposition state. In terms of applications, our demonstration may support high-quality factor modes and lower threshold lasers. Moreover, higher-order BICs can be used to enhance light-matter interaction and lead to non-linearity enhancement, nanophotonic circuits and quantum information processing.