Higher-order topological insulators in synthetic dimensions

Conventional topological insulators support boundary states with dimension one lower than that of the bulk system that hosts them, and these states are topologically protected due to quantized bulk dipole moments. Recently, higher-order topological insulators have been proposed as a way of realizing topological states with dimensions two or more lower than that of the bulk due to the quantization of bulk quadrupole or octupole moments. However, all these proposals as well as experimental realizations have been restricted to real-space dimensions. Here, we construct photonic higher-order topological insulators (PHOTIs) in synthetic dimensions. We show the emergence of a quadrupole PHOTI supporting topologically protected corner modes in an array of modulated photonic molecules with a synthetic frequency dimension, where each photonic molecule comprises two coupled rings. By changing the phase difference of the modulation between adjacent coupled photonic molecules, we predict a dynamical topological phase transition in the PHOTI. Furthermore, we show that the concept of synthetic dimensions can be exploited to realize even higher-order multipole moments such as a fourth-order hexadecapole (16-pole) insulator supporting 0D corner modes in a 4D hypercubic synthetic lattice that cannot be realized in real-space lattices. Opening new dimensions for topological insulatorsA theoretical structure that provides extra dimensions to topological insulators could enable novel quantum states to be manipulated. The usual definition of topological insulators is that they act as insulators on the three-dimensional inner bulk, but are highly conductive on their two-dimensional surfaces. However, when photonic topological insulators are constructed, other non-spatial dimensions can be harnessed related to frequency, orbital angular momentum or spin. Shanhui Fan and co-workers at Stanford University, USA, designed photonic higher-dimensional topological insulators made from arrays of ring-shaped resonators. The team showed that each pair of rings acts like a 'photonic molecule', generating complex isolated quantum states that could be switched on and off. Their work suggests that these synthetic dimensions could be used to explore exotic new phases of matter, with applications such as quantum computing.