Tying knots in energy bands

The generation of knots in optical systems has fascinated me since my PhD days. Influential early works focused on optical vortex knots. Optical vortices trace out curves in 3D space; in certain circumstances the curves trace out knotted trajectories. Influential early works include the generation of knotted vortices in light beams using a spatial light modulator, large-scale statistical analysis of the probability of knots forming in random optical beams, and the spontaneous formation of vortex knots in the tails of self-trapped nonlinear optical beams. 

More recently, with the surge in interest in topological photonics, interest has turned to the study of knots in other objects, including the energy bands of periodic systems.

An article just published in Nature reports the observation of braided energy bands using a pair of coupled modulated ring resonators. Braids are closely related to knots (knots can be obtained by connecting the ends of non-trivial braids). Braids are more natural to study in the context of band structures, owing to the periodicity of the Brillouin zone.

In order to generate knots or braids, one needs to consider curves in a 3D space. In this work, one of the dimensions is provided by the momentum (k) space, while the other two dimensions are played by the real and imaginary parts of the complex energy eigenvalues E. Using phase and amplitude modulators it is now possible to implement near-arbitrary non-Hermitian Hamiltonians using coupled ring resonators, allowing exquisite of the trajectories of the energy eigenvalues in the complex plane.

To observe the energy band braiding, the authors probed their ring resonator system using a continuous wave laser beam, measuring the time-dependent transmission, with time playing the role of k. Scanning the frequency of the probe beam, a peak in transition occurs at frequencies corresponding to resonances (real parts of the eigenvalues) of the system, with the linewidth of the resonance corresponding to the imaginary part of the eigenvalue. This allows the reconstruction of various braids, including those corresponding the simplest Hopf link (two threaded rings) and more complex objects such as trefoil knots.