Spatio-temporal modulation has shown great promise as a strong time-reversal symmetry breaking mechanism that enables integrated nonreciprocal devices and topological materials at optical frequencies. However, optical modulation has its own constraints in terms of modulation index and frequency, which limit the bandwidth and miniaturization of circulators and isolators, not unlike the magneto-optical schemes that it promises to replace. Here we propose and numerically demonstrate a Floquet circulator that leverages the untapped degrees of freedom unique to time-modulated resonators. Excited by sideband-selective waveguides, the system supports broadband nonreciprocal transmission without relying on the mirror or rotational symmetries required in conventional circulators. Cascading two resonators, we create a linear three-port circulator that exhibits complete and frequency-independent forward transmission between two of the ports. This approach enables wavelength-scale circulators that can rely on a variety of modulation mechanisms.
It has been demonstrated that dynamic refractive index modulation, which breaks time-reversal symmetry, can be used to create on-chip non-reciprocal photonic devices. In order to achieve amplitude non-reciprocity, all such devices moreover require modulations that break spatial symmetries, which adds complexity in implementations. Here we introduce a modal circulator, which achieves amplitude non-reciprocity through a circulation motion among three modes. We show that such a circulator can be achieved in a dynamically-modulated structure that preserves mirror symmetry, and as a result can be i…
A rigorous Floquet mode analysis is proposed for a zero thickness space-time modulated Huygens' metasurface to model and determine the strengths of the new harmonic components of the scattered fields. The proposed method is based on Generalized Sheet Transition Conditions (GSTCs) treating a metasurface as a spatial discontinuity. The metasurface is described in terms of Lorentzian electric and magnetic surface susceptibilities, $χ_\text{ee}$ and $χ_\text{mm}$, respectively, and its resonant frequencies are periodically modulated in both space and time. The unknown scattered fields are then exp…
A rigorous semi-analytical Floquet analysis is proposed for a zero-thickness space-time modulated Huygens' metasurface to model and determine the strengths of the new harmonic components of the scattered fields. The proposed method is based on Generalized Sheet Transition Conditions (GSTCs) treating a metasurface as a spatial discontinuity. The metasurface is described in terms of Lorentzian electric and magnetic surface susceptibilities, $χ_\text{e}$ and $χ_\text{m}$, respectively, with parameters (e.g. resonant frequency) that are periodically modulated in both space and time. The unknown sc…
In this paper, the aperture field estimation technique and generalized adiabatic Floquet-wave expansion method is developed for two-dimensional temporally modulated anisotropic holograms. The determinant equations for propagation eigenstates in the presence of space-time modulation are obtained, and the behaviors of the corresponding dispersion curves are investigated. Injecting temporal modulation causes asymmetrical displacement of the dispersion curve, which is studied in detail in this paper. The consequence of temporal modulation is the Doppler-shift effect and nonreciprocal response obse…
Using the Floquet Hamiltonian derived based on the time-dependent perturbation theory, we investigated the quasienergy bands of a one-dimensional time-Floquet photonic crystal. The time-Floquet photonic crystal contains two alternating layers labeled as A and B, and the permittivity of A layer is modulated periodically in time. We showed that the quasienergy bands are reciprocal when the modulation function is a function of time only, while the quasienergy bands could be nonreciprocal when the permittivity is modulated in both time and space through an unique combination. In the former case, t…
We theoretically report an analog of photon-assisted tunneling (PAT) in a periodically driven lattice array without a static biased potential by studying a three-channel waveguide system. This analog of PAT can be achieved by only periodically modulating the top waveguide and adjusting the distance between the bottom and its adjacent waveguide. It is numerically shown that the PAT resonances also exist in the five-channel waveguide system and probably exist in the waveguide arrays with other odd numbers of waveguides, but they will become weak as the number of waveguides increases. With origin…
Parity-Time (PT) symmetric wave devices, which exploit balanced interactions between material gain and loss, exhibit extraordinary properties, including lasing and flux-conserving scattering processes. In a seemingly different research field, periodically driven systems, also known as time-Floquet systems, have been widely studied as a relevant platform for reconfigurable active wave control and manipulation. In this article, we explore the connection between PT-symmetry and parametric time-Floquet systems. Instead of relying on material gain, we use parametric amplification by considering a t…
Controlling the balanced gain and loss in a PT-symmetric system is a rather challenging task. Utilizing Floquet theory, we explore the constructive role of periodic modulation in controlling the gain and loss of a PT-symmetric optical coupler. It is found that the gain and loss of the system can be manipulated by applying a periodic modulation. Further, such an original non-Hermitian system can even be modulated into an effective Hermitian system derived by the high-frequency Floquet method. Therefore, compared with other PT symmetry control schemes, our protocol can modulate the unbroken PT-s…
We consider periodically modulated Su-Schrieffer-Heeger (SSH) model with gain and loss. This model, which can be realized with current technology in photonics using waveguides, allows us to study Floquet topological insulating phase. By using Floquet theory, we find the quasi-energy spectrum of this one dimensional PT symmetric topological insulator. We show that stable Floquet topological phase exists in our model provided that oscillation frequency is large and the non-Hermitian degree is below than a critical value.
Non-Hermitian Hamiltonians provide a simple picture for analyzing systems with natural or induced gain and loss; however, in general, such Hamiltonians feature complex energies and a corresponding non-orthonormal eigenbasis. Provided that the Hamiltonian has $\mathcal{PT}$ symmetry, it is possible to find a regime in which the eigenspectrum is completely real. In the case of static $\mathcal{PT}$-symmetric extensions of the simple Su-Schrieffer-Heeger model, it has been shown that the energies associated with any edge states are guaranteed to be complex. Moving to a time-dependent system means…
We investigate wave transport properties of Parity-Time (PT) symmetric lattices that are periodically modulated along the direction of propagation. We demonstrate that in the regime of unbroken PT-symmetry the system Floquet-Bloch modes may interfere constructively leading to either controlled oscillations or linear power absorption and amplification occurring exactly at the phase transition point. The differential power response is effected by the overlap of the gain and loss system distribution with wave intensity pattern that is formed through Rabi oscillations engaging the coupled Floquet-…
Temporal Faraday effect enabled by Floquet-induced chirality
State Key Laboratory of Radio Frequency Heterogeneous Integration, College of Physics and Optoelectronic Engineering, Shenzhen University（深圳大学物理与光电工程学院射频异质集成国家重点实验室）
Non-Hermitian Interaction between Light and Photonic Time Crystal Beyond the Floquet Quasinormal Mode Approximation
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所凝聚态物理国家实验室）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；State Key Laboratory for Mesoscopic Physics and Frontiers Science Center for Nano-optoelectronics, School of Physics, Peking University（北京大学物理学院介观物理国家重点实验室及纳米光电子前沿科学中心）
Exceptional Points, Lasing, and Coherent Perfect Absorption in Floquet Scattering Systems
Vienna University of Technology (TU Wien)（维也纳工业大学）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；University of Graz（格拉茨大学）；NAWI Graz（NAWI格拉茨）；Riga Technical University（里加技术大学）
The design of time-independent effective Hamiltonians that describe periodically modulated systems, provides a promising approach to realize new forms of matter. This, so-called, Floquet engineering approach is currently limited to the description of wavepacket dynamics. Here, we utilize the notion of effective Hamiltonians and develop a Floquet engineering scattering formalism that relies on a systematic high-frequency expansion of the scattering matrix. The method unveils the critical role of micromotion. An application to the case of non-reciprocal transport is presented.
Cavity optomechanical systems enable fine manipulation of nanomechanical degrees of freedom with light, adding operational functionality and impacting their appeal in photonic technologies. We show that distinct mechanical modes can be exploited with a temporally modulated laser drive to steer between bistable steady states induced by changes of cavity radiation pressure. We investigate the influence of thermo-optic nonlinearity on these Floquet dynamics and find that it can inhibit or enhance the performance of the coupling mechanism in contrast to their often performance limiting character. …
We investigate symmetry-selective surface-mode excitation in a general periodically time-modulated double-layer system, where the modulation of the two layers has a constant phase difference. By deriving a semi-analytic trans-fer matrix formalism of a Drude-dispersive double-layer structure with periodic time-modulation, we calculate the scattering amplitudes and the corresponding transmission coefficient. Our results show that the phase-difference between the modulation of the two sheets plays an essential role in significantly enhancing and selectively exciting either the even or odd surface…
In recent years a significant progress has been made in the development of magnet-less nonreciprocity using space-time modulation, both in electromagnetics and acoustics. This approach has so far resulted in a plethora of non-reciprocal devices, such as isolators and circulators, over different parts of the spectrum, for guided waves. On the other hand, very little work has been performed on non-reciprocal devices for waves propagating in free space, which can also have many practical applications. For example, it was shown theoretically that non-reciprocal scattering by a metasurface can be o…
This work theoretically and experimentally studies metasurfaces with spatially-discrete, traveling-wave modulation (SD-TWM). A representative metasurface is considered consisting of columns of time-modulated subwavelength unit cells, referred to as stixels. SD-TWM is achieved by enforcing a time delay between temporal waveforms applied to adjacent columns. In contrast to the continuous traveling-wave modulation commonly assumed in studies of space-time metasurfaces, here the modulation is spatially discretized. In order to account for the discretized spatial modulation, a modified Floquet anal…
Perturbative Born theory for light scattering by time-modulated scatterers
Institute of Nanoscience and Nanotechnology, NCSR “Demokritos”（德谟克利特国家科学研究中心纳米科学与纳米技术研究所）；Section of Condensed Matter Physics, National and Kapodistrian University of Athens（雅典国立卡波季斯特里安大学凝聚态物理系）
Floquet modulation has been widely used in optical lattices for coherent control of quantum gases, in particular for synthesizing artificial gauge fields and simulating topological matters. However, such modulation induces heating which can overwhelm the signal of quantum dynamics in ultracold atoms. Here we report that the thermal motion, instead of being a noise source, provides a new control knob in Floquet-modulated superradiance lattices, which are momentum-space tight-binding lattices of collectively excited states of atoms. The Doppler shifts combined with Floquet modulation provide eff…
Eigenmode coalescence imparts remarkable properties to non-hermitian time evolution, culminating in a purely non-hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution at the EP and classify two-level non-hermitian Hamiltonians in terms of the Möbius group. We then leverage that classification to study dynamical EP encircling, by applying it to periodically-modulated (Floquet) Hamiltonians. This reveals that Floquet non-hermitian systems exhibit rich physics whose complexity is not captured by an EP-encircling rule. For example, Floquet EPs can occur …
Spatio-temporally modulated impedance surfaces can be good candidates for generation of radiating waves with arbitrary eigenstates by breaking momentum and energy conservations. Here, we present a theoretical framework based on the holographic technique and generalized Floquet-wave expansion to analyze spatio-temporally modulated impedance surfaces. The holographic technique estimates the required impedance distribution to achieve the desired momentum. Injecting temporal modulation deviates the eigenvalues and changes the radiation frequency. Using the proposed analytical model, the eigenvalue…
We study the Floquet edge states in arrays of periodically curved optical waveguides described by the modulated Su-Schrieffer-Heeger model. Beyond the bulk-edge correspondence, our study explores the interplay between band topology and periodic modulations. By analysing the quasi-energy spectra and Zak phase, we reveal that, although topological and non-topological edge states can exist for the same parameters, \emph{they can not appear in the same spectral gap}. In the high-frequency limit, we find analytically all boundaries between the different phases and study the coexistence of topologic…
Effective magnetic fields have enabled unprecedented manipulation of neutral particles including photons. In most studied cases, the effective gauge fields are defined through the phase of mode coupling between spatially discrete elements, such as optical resonators and waveguides in the case for photons. Here, in the paradigm of Bloch-wave modulated photonic crystals, we show creation of effective magnetic fields for photons in conventional dielectric continua for the first time, via Floquet band engineering. By controlling the phase and wavevector of Bloch waves, we demonstrated anomalous qu…
Generalized Conductivity Modeling and Selective Harmonic Amplification in Time-Modulated Graphene Cavities
Aristotle University of Thessaloniki（萨洛尼卡亚里士多德大学）；Sorbonne Université, CNRS（索邦大学，法国国家科学研究中心）；Université Paris-Saclay, CentraleSupélec, CNRS（巴黎萨克雷大学，中央理工-高等电力学院，法国国家科学研究中心）
In this paper, we study the scattering and diffraction phenomena in time-modulated metamaterials of metallic nature by means of Floquet equivalent circuits. Concretely, we focus on a time-periodic screen that alternates between "metal" and "air" states. We generalize our previous approaches by introducing the concepts of "macroperiod" and "duty cycle" to the time modulation. This allows to analyze time-periodic metallic metamaterials whose modulation ratios are, in general, rational numbers. Furthermore, with the introduction of the duty cycle, perfect temporal symmetry is broken within the ti…
The non-Hermitian models, which are symmetric under parity (P) and time-reversal (T) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise contorol of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge t…
We show that the time-dependence of electromagnetic field in a parametrically modulated cavity can be effectively analyzed using a $Floquet$ $map$. The map relates the field states separated by one period of the drive; iterative application of the map allows to determine field configuration after arbitrary number of drive periods. For resonant and near-resonant drives, the map has stable and unstable fixed points, which are the loci of infinite energy concentration in the long time limit. The Floquet map method can be applied both to classical and quantum massless field problems, including the…
Swiss Federal Institute of Technology in Lausanne (EPFL)（洛桑联邦理工学院）
Wave-based analog signal processing holds the promise of extremely fast, on-the-fly, power-efficient data processing, occurring as a wave propagates through an artificially engineered medium. Yet, due to the fundamentally weak non-linearities of traditional wave materials, such analog processors have been so far largely confined to simple linear projections such as image edge detection or matrix multiplications. Complex neuromorphic computing tasks, which inherently require strong non-linearities, have so far remained out-of-reach of wave-based solutions, with a few attempts that implemented n…
