Multidimensional synthetic chiral-tube lattices via nonlinear frequency conversion

NOON-state interference in the frequency domain

The examination of entanglement across various degrees of freedom has been pivotal in augmenting our understanding of fundamental physics, extending to high dimensional quantum states, and promising the scalability of quantum technologies. In this paper, we demonstrate the photon number path entanglement in the frequency domain by implementing a frequency beam splitter that converts the single-photon frequency to another with 50% probability using Bragg scattering four-wave mixing. The two-photon NOON state in a single-mode fiber is generated in the frequency domain, manifesting the two-photon interference with two-fold enhanced resolution compared to that of single-photon interference, showing the outstanding stability of the interferometer. This successful translation of quantum states in the frequency domain will pave the way toward the discovery of fascinating quantum phenomena and scalable quantum information processing.

Spectral Engineering of Optical Microresonators in Anisotropic Lithium Niobate Crystal

On-chip optical microresonators are essential building blocks in integrated optics. The ability to arbitrarily engineer their resonant frequencies is crucial for exploring novel physics in synthetic frequency dimensions and practical applications like nonlinear optical parametric processes and dispersion-engineered frequency comb generation. Photonic crystal ring (PhCR) resonators are a versatile tool for such arbitrary frequency engineering, by controllably creating mode splitting at selected resonances. To date, these PhCRs have mostly been demonstrated in isotropic photonic materials, while such engineering can be significantly more complicated in anisotropic platforms that often offer more fruitful optical properties. Here, the spectral engineering of chip-scale optical microresonators is realized in the anisotropic lithium niobate (LN) crystal by a gradient design that precisely compensates for variations in both refractive index and perturbation strength. Controllable frequency splitting is experimentally demonstrated at single and multiple selected resonances in LN PhCR resonators with different sizes, while maintaining high quality-factors up to 1 × 106. Moreover, a sharp boundary is experimentally constructed in the synthetic frequency dimension based on an actively modulated x-cut LN gradient-PhCR, opening up new paths toward the arbitrary control of electro-optic comb spectral shapes and exploration of novel physics in the frequency degree of freedom.

Artificial gauge fields in the t-z mapping for optical pulses: Spatiotemporal wave packet control and quantum Hall physics

We extend the t-z mapping of time-dependent paraxial optics by engineering a synthetic magnetic vector potential, leading to a nontrivial band topology. We consider an inhomogeneous 1D array of coupled optical waveguides and show that the wave equation describing paraxial propagation of optical pulses can be recast as a Schrödinger equation, including a synthetic magnetic field whose strength can be controlled via the spatial gradient of the waveguide properties across the array. We use an experimentally motivated model of a laser-written array to demonstrate that this synthetic magnetic field can be engineered in realistic setups and can produce interesting physics such as cyclotron motion, a controllable Hall drift of the pulse in space or time, and propagation in chiral edge states. These results substantially extend the physics that can be explored within propagating geometries and pave the way for higher-dimensional topological physics and strongly correlated fluids of light.

Multi-dimensional band structure spectroscopy in the synthetic frequency dimension

The concept of synthetic dimensions in photonics provides a versatile platform in exploring multi-dimensional physics. Many of these physics are characterized by band structures in more than one dimensions. Existing efforts on band structure measurements in the photonic synthetic frequency dimension however are limited to either one-dimensional Brillouin zones or one-dimensional subsets of multi-dimensional Brillouin zones. Here we theoretically propose and experimentally demonstrate a method to fully measure multi-dimensional band structures in the synthetic frequency dimension. We use a single photonic resonator under dynamical modulation to create a multi-dimensional synthetic frequency lattice. We show that the band structure of such a lattice over the entire multi-dimensional Brillouin zone can be measured by introducing a gauge potential into the lattice Hamiltonian. Using this method, we perform experimental measurements of two-dimensional band structures of a Hermitian and a non-Hermitian Hamiltonian. The measurements reveal some of the general properties of point-gap topology of the non-Hermitian Hamiltonian in more than one dimensions. Our results demonstrate experimental capabilities to fully characterize high-dimensional physical phenomena in the photonic synthetic frequency dimension.

Artificial Non-Abelian Lattice Gauge Fields for Photons in the Synthetic Frequency Dimension

Non-Abelian gauge fields give rise to nontrivial topological physics. Here we develop a scheme to create an arbitrary SU(2) lattice gauge field for photons in the synthetic frequency dimension using an array of dynamically modulated ring resonators. The photon polarization is taken as the spin basis to implement the matrix-valued gauge fields. Using a non-Abelian generalization of the Harper-Hofstadter Hamiltonian as a specific example, we show that the measurement of the steady-state photon amplitudes inside the resonators can reveal the band structures of the Hamiltonian, which show signatures of the underlying non-Abelian gauge field. These results provide opportunities to explore novel topological phenomena associated with non-Abelian lattice gauge fields in photonic systems.

Mirror-induced reflection in the frequency domain

Mirrors are ubiquitous in optics and are used to control the propagation of optical signals in space. Here we propose and demonstrate frequency domain mirrors that provide reflections of the optical energy in a frequency synthetic dimension, using electro-optic modulation. First, we theoretically explore the concept of frequency mirrors with the investigation of propagation loss, and reflectivity in the frequency domain. Next, we explore the mirror formed through polarization mode-splitting in a thin-film lithium niobate micro-resonator. By exciting the Bloch waves of the synthetic frequency crystal with different wave vectors, we show various states formed by the interference between forward propagating and reflected waves. Finally, we expand on this idea, and generate tunable frequency mirrors as well as demonstrate trapped states formed by these mirrors using coupled lithium niobate micro-resonators. The ability to control the flow of light in the frequency domain could enable a wide range of applications, including the study of random walks, boson sampling, frequency comb sources, optical computation, and topological photonics. Furthermore, demonstration of optical elements such as cavities, lasers, and photonic crystals in the frequency domain, may be possible.

Creating boundaries along a synthetic frequency dimension

Synthetic dimensions have garnered widespread interest for implementing high dimensional classical and quantum dynamics on low-dimensional geometries. Synthetic frequency dimensions, in particular, have been used to experimentally realize a plethora of bulk physics effects. However, in synthetic frequency dimension there has not been a demonstration of a boundary which is of paramount importance in topological physics due to the bulk-edge correspondence. Here we construct boundaries in the frequency dimension of dynamically modulated ring resonators by strongly coupling an auxiliary ring. We explore various effects associated with such boundaries, including confinement of the spectrum of light, discretization of the band structure, and the interaction of boundaries with one-way chiral modes in a quantum Hall ladder, which exhibits topologically robust spectral transport. Our demonstration of sharp boundaries fundamentally expands the capability of exploring topological physics, and has applications in classical and quantum information processing in synthetic frequency dimensions.

Topological complex-energy braiding of non-Hermitian bands

Effects connected with the mathematical theory of knots¹ emerge in many areas of science, from physics^2,3 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 dimension^6,7. 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.

Arbitrary linear transformations for photons in the frequency synthetic dimension

Arbitrary linear transformations are of crucial importance in a plethora of photonic applications spanning classical signal processing, communication systems, quantum information processing and machine learning. Here, we present a photonic architecture to achieve arbitrary linear transformations by harnessing the synthetic frequency dimension of photons. Our structure consists of dynamically modulated micro-ring resonators that implement tunable couplings between multiple frequency modes carried by a single waveguide. By inverse design of these short- and long-range couplings using automatic differentiation, we realize arbitrary scattering matrices in synthetic space between the input and output frequency modes with near-unity fidelity and favorable scaling. We show that the same physical structure can be reconfigured to implement a wide variety of manipulations including single-frequency conversion, nonreciprocal frequency translations, and unitary as well as non-unitary transformations. Our approach enables compact, scalable and reconfigurable integrated photonic architectures to achieve arbitrary linear transformations in both the classical and quantum domains using current state-of-the-art technology.

Generating arbitrary topological windings of a non-Hermitian band

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.

Real-time observation of frequency Bloch oscillations with fibre loop modulation

Bloch oscillations (BOs) were initially predicted for electrons in a solid lattice to which a static electric field is applied. The observation of BOs in solids remains challenging due to the collision scattering and barrier tunnelling of electrons. Nevertheless, analogies of electron BOs for photons, acoustic phonons and cold atoms have been experimentally demonstrated in various lattice systems. Recently, BOs in the frequency dimension have been proposed and studied by using an optical micro-resonator, which provides a unique approach to controlling the light frequency. However, the finite resonator lifetime and intrinsic loss hinder the effect from being observed practically. Here, we experimentally demonstrate BOs in a synthetic frequency lattice by employing a fibre-loop circuit with detuned phase modulation. We show that a detuning between the modulation period and the fibre-loop roundtrip time acts as an effective vector potential and hence a constant effective force that can yield BOs in the modulation-induced frequency lattices. With a dispersive Fourier transformation, the pulse spectrum can be mapped into the time dimension, and its transient evolution can be precisely measured. This study offers a promising approach to realising BOs in synthetic dimensions and may find applications in frequency manipulations in optical fibre communication systems.

Dynamic band structure measurement in the synthetic space

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.