Direct extraction of topological Zak phase with the synthetic dimension

Measuring topological invariants is an essential task in characterizing topological phases of matter.They are usually obtained from the number of edge states due to the bulk-edge correspondence or from interference since they are integrals of the geometric phases in the energy band.It is commonly believed that the bulk band structures could not be directly used to obtain the topological invariants.Here,we implement the experimental extraction of Zak phase from the bulk band structures of a Su-Schrieffer-Heeger(SSH)model in the synthetic frequency dimension.Such synthetic SSH lattices are constructed in the frequency axis of light,by controlling the coupling strengths between the symmetric and antisymmetric supermodes of two bichromatically driven rings.We measure the transmission spectra and obtain the projection of the time-resolved band structure on lattice sites,where a strong contrast between the non-trivial and trivial topological phases is observed.The topological Zak phase is naturally encoded in the bulk band structures of the synthetic SSH lattices,which can hence be experimentally extracted from the transmission spectra in a fiber-based modulated ring platform using a laser with telecom wavelength.Our method of extracting topological phases from the bulk band structure can be further extended to characterize topological invariants in higher dimensions,while the exhibited trivial and non-trivial transmission spectra from the topological transition may find future applications in optical communications.

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,higherorder 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.

Creating locally interacting Hamiltonians in the synthetic frequency dimension for photons

The recent emerging field of synthetic dimension in photonics offers a variety of opportunities for manipulating different internal degrees of freedom of photons such as the spectrum of light.While nonlinear optical effects can be incorporated into these photonic systems with synthetic dimensions,these nonlinear effects typically result in long-range interactions along the frequency axis.Thus,it has been difficult to use the synthetic dimension concept to study a large class of Hamiltonians that involves local interactions.Here we show that a Hamiltonian that is locally interacting along the synthetic dimension can be achieved in a dynamically modulated ring resonator incorporatingχ3nonlinearity,provided that the group velocity dispersion of the waveguide forming the ring is specifically designed.As a demonstration we numerically implement a Bose–Hubbard model and explore photon blockade effect in the synthetic frequency space.Our work opens new possibilities for studying fundamental many-body physics in the synthetic space in photonics,with potential applications in optical quantum communication and quantum computation.