Dynamic band structure measurement in the synthetic space

Nonlinear Landau-Zener-Stückelberg-Majorana tunneling and interferometry of extended Bose-Hubbard flux ladders

The nonlinear Landau-Zener-Stückelberg-Majorana (LZSM) tunneling dynamics and interferometry of an extended Bose-Hubbard flux ladder are studied. Based on the mean-field theory, the dispersion relation of the system is given, and it is found that loop structures periodically appear in the band structure and the nonlinear LZSM interference occurs naturally without Floquet engineering, which can be effectively modulated by atomic interactions. The nonlinear energy bands and the unique chirality feature of the flux ladder system can be identified through the dynamics of nonlinear Landau-Zener tunneling. Remarkably, the critical position of the noise in the interference pattern can be employed to identify the loop structure in the energy band, establishing an effective link between the nonlinear loop structure and LZSM interferometry. The position, intensity, symmetry, and width of interference patterns strongly depend on the magnetic field, atomic interactions, rung-to-leg coupling ratio, and energy bias, which provides an effective way to measure these parameters using the nonlinear LZSM interferometry. This paper further expands the dynamics of flux ladder systems to complex interaction regions and has potential applications in the precise measurement of related nonlinear systems.

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.

Moiré Lattice in One-Dimensional Synthetic Frequency Dimension

The moiré lattice has recently attracted broad interest in both solid-state physics and photonics where exotic phenomena in manipulating the quantum states are explored. In this work, we study the one-dimensional (1D) analogs of "moiré" lattices in a synthetic frequency dimension constructed by coupling two resonantly modulated ring resonators with different lengths. Unique features associated with the flatband manipulation as well as the flexible control of localization position inside each unit cell in the frequency dimension have been found, which can be controlled via the choice of flatband. Our work therefore provides insight into simulating moiré physics in 1D synthetic frequency space, which holds important promise for potential applications toward optical information processing.

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.

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.

Technologically feasible quasi-edge states and topological Bloch oscillation in the synthetic space

The dimensionality of a physical system is one of the major parameters defining its physical properties. The recently introduced concept of synthetic dimension has made it possible to arbitrarily manipulate the system of interest and harness light propagation in different ways. It also facilitates the transformative architecture of system-on-a-chip devices enabling far reaching applications such as optical isolation. In this report, a novel architecture based on dynamically-modulated waveguide arrays with the Su-Schrieffer-Heeger configuration in the spatial dimension is proposed and investigated with an eye on a practical implementation. The propagation of light through the one-dimensional waveguide arrays mimics time evolution of the field in a synthetic two-dimensional lattice. The addition of the effective gauge potential leads to an exotic topologically protected one-way transmission along adjacent boundary. A cosine-shape isolated band, which supports the topological Bloch oscillation in the frequency dimension under the effective constant force, appears and is localized at the spatial boundary being robust against small perturbations. This work paves the way to improved light transmission capabilities under topological protections in both spatial and spectral regimes and provides a novel platform based on a technologically feasible lithium niobate platform for optical computing and communication.

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 band structure via twisted photons in a degenerate cavity

Synthetic dimensions based on particles' internal degrees of freedom, such as frequency, spatial modes and arrival time, have attracted significant attention. They offer ideal large-scale lattices to simulate nontrivial topological phenomena. Exploring more synthetic dimensions is one of the paths toward higher dimensional physics. In this work, we design and experimentally control the coupling among synthetic dimensions consisting of the intrinsic photonic orbital angular momentum and spin angular momentum degrees of freedom in a degenerate optical resonant cavity, which generates a periodically driven spin-orbital coupling system. We directly characterize the system's properties, including the density of states, energy band structures and topological windings, through the transmission intensity measurements. Our work demonstrates a mechanism for exploring the spatial modes of twisted photons as the synthetic dimension, which paves the way to design rich topological physics in a highly compact platform.

Synthetic dimension band structures on a Si CMOS photonic platform

Synthetic dimensions, which simulate spatial coordinates using nonspatial degrees of freedom, are drawing interest in topological science and other fields for modeling higher-dimensional phenomena on simple structures. We present the first realization of a synthetic frequency dimension on a silicon ring resonator integrated photonic device fabricated using a CMOS process. We confirm that its coupled modes correspond to a one-dimensional tight-binding model through acquisition of up to 280-GHz bandwidth optical frequency comb-like spectra and by measuring synthetic band structures. Furthermore, we realized two types of gauge potentials along the frequency dimension and probed their effects through the associated band structures. An electric field analog was produced via modulation detuning, whereas effective magnetic fields were induced using synchronized nearest- and second nearest-neighbor couplings. Creation of coupled mode lattices and two effective forces on a monolithic Si CMOS device represents a key step toward wider adoption of topological principles.

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.

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.