The nontrivial topological features in the energy band of non-Hermitian systems provide promising pathways to achieve robust physical behaviors in classical or quantum open systems. A key topological feature of non-Hermitian systems is the nontrivial winding of the energy band in the complex energy plane. We provide experimental demonstrations of such nontrivial winding by implementing non-Hermitian lattice Hamiltonians along a frequency synthetic dimension formed in a ring resonator undergoing simultaneous phase and amplitude modulations, and by directly characterizing the complex band structures. Moreover, we show that the topological winding can be controlled by changing the modulation waveform. Our results allow for the synthesis and characterization of topologically nontrivial phases in nonconservative systems.

https://science.sciencemag.org/content/sci/371/6535/1240.full.pdf

Generating arbitrary topological windings of a non-Hermitian band.

Effects connected with the mathematical theory of knots1 emerge in many areas of science, from physics2,3 to biology4. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems5, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹2. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems. Experiments using two coupled optical ring resonators and based on the concept of synthetic dimension reveal non-Hermitian energy band structures exhibiting topologically non-trivial knots and links.

Topological complex-energy braiding of non-Hermitian bands

The physics of a photonic structure is commonly described in terms of its apparent geometric dimensionality. On the other hand, with the concept of synthetic dimension, it is in fact possible to explore physics in a space with a dimensionality that is higher as compared to the apparent geometrical dimensionality of the structures. In this review, we discuss the basic concepts of synthetic dimension in photonics, and highlighting the various approaches towards demonstrating such synthetic dimension for fundamental physics and potential applications.

Synthetic Dimension in Photonics

Topological photonics has emerged as a novel paradigm for the design of electromagnetic systems from microwaves to nanophotonics. Studies to date have largely focused on the demonstration of fundamental concepts, such as non-reciprocity and waveguiding protected against fabrication disorder. Moving forward, there is a pressing need to identify applications where topological designs can lead to useful improvements in device performance. Here we review applications of topological photonics to ring resonator-based systems, including one- and two-dimensional resonator arrays, and dynamically-modulated resonators. We evaluate potential applications such as quantum light generation, disorder-robust delay lines, and optical isolation, as well as future research directions and open problems that need to be addressed.

/pdf/topological-phases-in-ring-resonators-recent-progress-and-19h6cw6htq.pdf

Topological phases in ring resonators: recent progress and future prospects

The concept of synthetic dimensions in photonics has attracted rapidly growing interest in the past few years. Among a variety of photonic systems, the ring resonator system under dynamic modulation has been investigated in depth both in theory and experiment and has proven to be a powerful way to build synthetic frequency dimensions. In this Tutorial, we start with a pedagogical introduction to the theoretical approaches in describing the dynamically modulated ring resonator system and then review experimental methods in building such a system. Moreover, we discuss important physical phenomena in synthetic dimensions, including nontrivial topological physics. This Tutorial provides a pathway toward studying the dynamically modulated ring resonator system and understanding synthetic dimensions in photonics and discusses future prospects for both fundamental research and practical applications using synthetic dimensions.

Synthetic frequency dimensions in dynamically modulated ring resonators

Band structure theory plays an essential role in exploring physics in both solid-state systems and photonics. Here, we demonstrate a direct experimental measurement of the dynamic band structure in a synthetic space including the frequency axis of light, realized in a ring resonator under near-resonant dynamic modulation. This synthetic lattice exhibits the physical picture of the evolution of the wave vector reciprocal to the frequency axis in the band structure, analogous to a one-dimensional lattice under an external force. We experimentally measure the trajectories of the dynamic band structure by selectively exciting the band with a continuous wave source with its frequency scanning across the entire energy regime of the band. Our results not only provide a new perspective for exploring the dynamics in fundamental physics of solid-state and photonic systems with the concept of the synthetic dimension but also enable great capability in band structure engineering in photonics.

https://advances.sciencemag.org/content/advances/7/2/eabe4335.full.pdf

Dynamic band structure measurement in the synthetic space

The exploration of flatband in photonics is fundamentally important, aiming to control the localization of light for potential applications in optical communications. We study the flatband physics in a synthetic space including the frequency axis of light. A ring-resonator array is used to construct a synthetic Lieb-type lattice, where the modulation phase distribution supports a locally non-zero effective magnetic flux pointing into the synthetic space. We find that the flatband is isolated from other dispersive bands and the light can still be localized even with the perturbation from the group velocity dispersion of the waveguide. Our work points out a route towards manipulating the localization performance of a wavepacket of light along both the spatial dimension and the frequency dimension, which holds the potential implication for controlling the storage of optical information in dispersive materials.

Isolated Photonic Flatband with the Effective Magnetic Flux in a Synthetic Space Including the Frequency Dimension

We recently proposed a two-dimensional synthetic space including one spatial axis and one synthetic frequency dimension in a one-dimensional ring resonator array [Opt. Lett., Vol. 41, No. 4, 741–744, 2016]. Nevertheless, the group velocity dispersion (GVD) of the waveguides that compose rings was ignored for simplicity. In this paper, we extend the previous work and study the topological one-way edge states in such a synthetic space involving GVD. We show that the GVD brings a natural vague boundary in the frequency dimension, so the topological edge state still propagates at several frequency modes unidirectionally along the spatial axis. Positions of such vague boundary can be controlled by changing the magnitude of the GVD. In particular, a relatively strong GVD can degrade this two-dimensional synthetic space to one-dimensional spatial lattice, but yet the one-way state is still preserved in simulations. Our work therefore exhibits the impact of the GVD on topological photonics in the synthetic space, which will be important for future practical experimental implementations.

/pdf/one-way-topological-states-along-vague-boundaries-in-2k2jovu02f.pdf

One-way topological states along vague boundaries in synthetic frequency dimensions including group velocity dispersion (invited)

The notion of topological phases extended to dynamical systems stimulates extensive studies, of which the characterization of nonequilibrium topological invariants is a central issue and usually necessitates the information of quantum dynamics in both the time and momentum dimensions. Here, we propose the topological holographic quench dynamics in synthetic dimension, and also show it provides a highly efficient scheme to characterize photonic topological phases. A pseudospin model is constructed with ring resonators in a synthetic lattice formed by frequencies of light, and the quench dynamics is induced by initializing a trivial state, which evolves under a topological Hamiltonian. Our key prediction is that the complete topological information of the Hamiltonian is encoded in quench dynamics solely in the time dimension, and is further mapped to lower-dimensional space, manifesting the holographic features of the dynamics. In particular, two fundamental time scales emerge in the dynamical evolution, with one mimicking the topological band on the momentum dimension and the other characterizing the residue time evolution of the state after the quench. For this, a universal duality between the quench dynamics and the equilibrium topological phase of the spin model is obtained in the time dimension by extracting information from the field evolution dynamics in modulated ring systems in simulations. This work also shows that the photonic synthetic frequency dimension provides an efficient and powerful way to explore the topological nonequilibrium dynamics. Topological holographic quench dynamics in a synthetic frequency dimension studied in modulated ring system shows the universal bulk-surface duality with complete topological information being encoded in the single time variable.

/pdf/topological-holographic-quench-dynamics-in-a-synthetic-2p8zd0ypvo.pdf

Danying Yu

Papers

Dynamic band structure measurement in the synthetic space

Isolated Photonic Flatband with the Effective Magnetic Flux in a Synthetic Space Including the Frequency Dimension

One-way topological states along vague boundaries in synthetic frequency dimensions including group velocity dispersion (invited)

Topological holographic quench dynamics in a synthetic frequency dimension.

Isolated Photonic Flatband with the Effective Magnetic Flux in A Synthetic Space including the Frequency Dimension.