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We study plasmonic time crystals, which constitute an extension of the dielectric-based photonic time crystals to plasmonic media. A salient feature of plasmonic time crystals is their ability to support amplification of both longitudinal and transverse modes. In particular, we show that such systems support collective resonances of longitudinal modes, independently of the wave vector $k$. These resonances originate from the interaction between the positive and negative frequency branches of the plasmonic dispersion relation of the unmodulated system, and from the divergence of the density of states near the plasma frequency $\omega_p$. The strongest resonance occurs at a modulation frequency of $\Omega = 2 \omega_p$, associated with a direct inter-band transition. Higher-order resonances are associated with related mechanisms but at lower modulation frequencies. We demonstrate these resonances for various periodic modulation profiles, and also provide a generic perturbative formula for resonance widths in the limit of weak modulation amplitude, in the absence of losses. Our findings offer insights into using time-modulated plasmonic media to enhance optical gain., Comment: main text (7pp, 2 figures) + supplementary material (7pp + 4 figures)

Here, we investigate the effective response of three-dimensional spacetime crystals formed by spherical scatterers under a travelling-wave modulation. We develop an analytical formalism to homogenize the spacetime crystals that extends the renowned Clausius-Mossotti formula to time-varying platforms. Our formalism shows that travelling-wave spacetime crystals can be used to engineer a wide range of classes of nonreciprocal bianisotropic couplings in the long wavelength limit. In particular, our theory reveals the possibility of realizing a purely isotropic Tellegen response in crystals formed by interlaced sub-lattices of scatterers subjected to different modulation velocities. Furthermore, we introduce a class of generalized Minkowskian crystals that displays invariance under arbitrary Lorentz boosts aligned with a fixed spatial direction. We prove that such systems are formed by pseudo-uniaxial materials with the principal axis aligned parallel to the modulation velocity. The electromagnetic response of such generalized Minkowskian crystals is indistinguishable from that of moving photonic crystals.

Here, we present an exploratory study of the potential applications of electrically biased materials that possess a nonreciprocal and non-Hermitian electromagnetic response analogous to the electronic response of field-effect transistors. The most distinctive feature of such materials is their chiral-gain, meaning that their response can be active or dissipative depending on the handedness of the wave polarization. Here, we show how the chiral-gain can be harnessed to develop novel electromagnetic devices with unique properties such as chiral lasers, polarization-dependent mirrors, and coherent-perfect-absorber lasers. Furthermore, it is demonstrated that materials with chiral-gain can bypass a reciprocity constraint that typically limits the external coupling strength, thus facilitating the excitation of cavities with extremely large quality factors., Comment: 39 pages, 10 figures

We investigate the frictional force arising from quantum fluctuations when two dissipative metallic plates are set in a shear motion. While early studies showed that the electromagnetic fields in the quantum friction setup reach nonequilibrium steady states, yielding a time-independent force, other works have demonstrated the failure to attain steady states, leading to instability and time-varying friction under sufficiently low-loss conditions. Here, we develop a fully quantum-mechanical theory without perturbative approximations and unveil the transition from stable to unstable regimes of the quantum friction setup. Due to the relative motion of the plates, their electromagnetic response may be active in some conditions, resulting in optical gain. We prove that the standard fluctuation-dissipation leads to inconsistent results when applied to our system, and, in particular, it predicts a vanishing frictional force. Using a modified fluctuation-dissipation relation tailored for gain media, we calculate the frictional force in terms of the system Green's function, thereby recovering early works on quantum friction. Remarkably, we also find that the frictional force diverges to infinity as the relative velocity of the plates approaches a threshold. This threshold is determined by the damping strength and the distance between the metal surfaces. Beyond this critical velocity, the system exhibits instability, akin to the behaviour of a laser cavity, where no steady state exists. In such a scenario, the frictional force escalates exponentially. Our findings pave the way for experimental exploration of the frictional force in proximity to this critical regime.

Photonic systems with time-varying modulations have attracted considerable attention as they allow for the design of non-reciprocal devices without the need for an external magnetic bias. Unlike time-invariant systems, such modulations couple modes with different frequencies. Here, we discuss how this coupling and particle-hole symmetry may lead to the resonant interaction of positive and negative frequency oscillators. To illustrate this idea, we analyze a dispersive spacetime crystal described by a Drude-Lorentz model with a travelingwave modulation. Our findings demonstrate that the interaction between positive and negative frequency bands can induce parametric instabilities under certain conditions, stemming from the interplay between dispersion and spacetime modulations. In particular, we find that material dispersion creates the conditions for the formation of instabilities for arbitrarily small modulations speeds in the absence of dissipative channels., Comment: 15 pages, 4 figures

Here, we present an in-depth theoretical analysis of the linear electrooptic effect in low-symmetry three-dimensional (3D) conductive materials with large Berry curvature dipoles. Our study identifies two distinct kinetic contributions to the linear electrooptic effect: a gyrotropic Hermitian (conservative) piece and a non-Hermitian term that can originate optical gain. We concentrate on the study of 3D materials belonging to the 32 ($D_3$) point group subject to a static electric bias along the trigonal axis. Our investigation shows that doped trigonal tellurium has promising properties, with its gyrotropic electrooptic response offering the potential for realizing electrically-biased electromagnetic isolators and inducing significant optical dichroism. Most notably, it is demonstrated that under sufficiently large static electric bias, tellurium's non-Hermitian electrooptic response may lead to optical gain. Using first-principles calculations, it is shown that n-doped tellurium is particularly promising, as it can host significantly larger Berry curvature dipoles than the more common p-doped tellurium., Comment: 38 pages, 9 figures

Materials with Berry curvature dipoles (BDs) support a non-Hermitian electro-optic (EO) effect that is investigated here for lasing at terahertz (THz) frequencies. Such a system is here conceived as a stack of low-symmetry 2D materials. We show that a cavity made of such a material supports a nonreciprocal growing mode with elliptical polarization that generates an unstable resonance leading to self-sustained oscillations. Notably, we demonstrate that the chiral nature of the gain derived from the Berry dipole allows for the manipulation of the laser light's handedness by a simple reversal of the electric field bias.

Time-variant systems have recently garnered considerable attention due to their unique potentials in manipulating electromagnetic waves. Here, we introduce a novel class of topological systems that rely on spacetime crystals with a traveling-wave modulation that emulates certain aspects of physical motion. Challenging intuition, our findings reveal that, even though such systems rely on a linear momentum bias, it is feasible to engineer an internal angular momentum and non-trivial topological phases by leveraging the symmetry of its structural elements. Furthermore, we establish that the proposed platforms exhibit a gauge degree of freedom associated with the arbitrariness in the choice of the coordinate transformation that eliminates the time dependence of the Hamiltonian that describes the system. The topology is intricately governed by a synthetic magnetic potential whose field lines can be controlled by manipulating material anisotropy. Remarkably, we demonstrate that the proposed spacetime crystals host an unconventional class of scattering-immune edge states. The oscillation frequency of the edge states adapts continuously along the propagation path, shaped by the geometric attributes of the path itself., Comment: 37 pages, 9 figures

We present a theoretical study of the interaction between an atom characterized by a degenerate ground state and a reciprocal environment, such as a semiconductor nanoparticle, without the presence of external bias. Our analysis reveals that the combined influence of the electron's intrinsic spin magnetic moment on the environment and the chiral atomic dipolar transitions may lead to either the spontaneous breaking of time-reversal symmetry or the emergence of time-crystal-like states with remarkably long relaxation times. The different behavior is ruled by the handedness of the precession motion of the atom's spin vector, which is induced by virtual chiral-dipolar transitions. Specifically, when the relative orientation of the precession angular velocity and the electron spin vector is as in a spinning top, the system manifests time-crystal-like states. Conversely, with the opposite relative orientation, the system experiences spontaneous symmetry breaking of time-reversal symmetry. Our findings introduce a novel mechanism for the spontaneous breaking of time-reversal symmetry in atomic systems, and unveil an exciting opportunity to engineer a nonreciprocal response at the nanoscale, exclusively driven by the quantum vacuum fluctuations., Comment: 62 pages

We introduce a homogenization approach to characterize the dynamical response of a generic dispersive spacetime crystal in the long-wavelength limit. The theory is applied to dispersive spacetime platforms with a travelling-wave modulation. It is shown that for long wavelengths the effective response may be described by a frequency dependent permittivity. Due to the active nature of spacetime systems, the permittivity is not bound by the same constraints as in standard time-invariant metamaterials. In particular, we find that dispersive spacetime crystals can exhibit rather peculiar physics, such as an anomalous (non-Foster) permittivity dispersion with a negative stored energy density, alternate between gain and loss regimes, and present multiple resonances in the quasistatic regime. Furthermore, it is verified with numerical simulations that the effective theory captures faithfully the exact dispersion of the first few photonic bands., Comment: "in press" at Physical Review B, 43 pages, 8 figures

We theoretically investigate the role of chiral-transitions in the quantum friction force that acts on a two-level atom that moves with relative velocity v parallel to a planar metallic surface. We find that the friction force has a component that is sensitive to the handedness of the atomic transition dipole moment. In the particular, we show that the friction force due to a particular atomic transition can be enhanced by a dipole moment with a certain handedness, and almost suppressed by the dipole moment with the opposite handedness. Curiously, the handedness of the transition dipole moment that boosts the ground-state friction force is the opposite of what is classically expected from the spin-momentum locking. We explain this discrepancy in terms of the interaction between positive and negative frequency oscillators.

In this work, we develop a homogenization model to determine the effective response of a metallic nanowire array embedded in an electric gyrotropic material. We study the interaction of electromagnetic waves with the metamaterial and demonstrate that the nanowire array can greatly enhance the nonreciprocal response of the gyrotropic substrate. In particular, the metamaterial can either absorb the incoming energy almost entirely or reflect it with little loss, depending on the sign of the incidence angle. We explore the implications of our findings in the context of Kirchhoff's law of thermal radiation. Our results demonstrate that the wire array can boost the difference between the emissivity and absorptivity in a broad spectrum of frequencies and incidence angles as compared to an unstructured gyrotropic substrate. These findings suggest potential applications for the nonreciprocal wire medium in thermal management, radiative cooling, and others., Comment: 34 pages, 11 figures

Over the past decade, there has been a great interest in topological effects, with concepts originally developed in the context of electron transport in condensed matter platforms now being extended to optical systems. While topological properties in electronic systems are often linked to the quantization of electric conductivity observed in the integer quantum Hall effect, a direct analogue in optics remains elusive. In this study, we bridge this gap by demonstrating that the response of the Poynting vector (which may be regarded as a "photon current") to the mechanical acceleration of a medium provides a precise photonic analogue of the electric conductivity. In particular, it is shown that the photonic conductivity determines the energy irreversibly transferred from a periodic mechanical driving of the medium to the electromagnetic field. Furthermore, it is demonstrated that for nonreciprocal systems enclosed in a cavity, the constant acceleration of the system induces a flow of photons along a direction perpendicular to the acceleration, analogous to the Hall effect but for light. The spectral density of the photonic conductivity is quantized in the band gaps of the bulk region with the conductivity quantum determined by the gap Chern number., Comment: 46 pages

This paper presents an analytical framework for the study of scattering and diffraction phenomena in spacetime-modulated metallic gratings. Using a Lorentz transformation, it is shown that a particular class of spacetime-modulated gratings behave effectively as moving media. We take advantage of this property to derive a closed analytical solution for the wave scattering problem. In particular, using our formalism it is possible to avoid spacetime Floquet-Bloch expansions, as the solution of the problem in the original laboratory frame (grating parameters are periodic in space and time) is directly linked to a co-moving frame where the metallic grating is time-invariant (grating parameters are periodic only in space). In this way, we identify a fundamental connection between moving metallic gratings and spacetime-modulated metamaterials, and exploit this link to study the nonreciprocal response of the structure. Some limitations and difficulties of the alternative nonrelativistic Galilean approach are discussed and the benefits of the Lorentz approach are highlighted. Finally, some analytical results are presented in order to validate the formalism. The results include scenarios involving TM(p) and TE(s) normal and oblique incidence, even beyond the onset of the diffraction regime. Furthermore, we show how the synthetic Fresnel drag can tailor the Goos-H\"anchen effect and create a specular point shifted towards the direction of the synthetic motion, independent of the sign of the incidence angle., Comment: Published in Physical Review Applied, 20, 014063, 2023

Here, we show that isorefractive spacetime crystals with a travelling-wave modulation may mimic rigorously the response of moving material systems. While generic spacetime crystals are characterized by a bi-anisotropic coupling in the co-moving frame, isorefractive crystals have a response that is observer independent, which leads to isotropic constitutive relations free of bianisotropy. We show how to take advantage of this property in the calculation of the band diagrams of isorefractive spacetime crystals in the laboratory frame and in the study of the synthetic Fresnel drag. Furthermore, we discuss the impact of considering either a Galilean or a Lorentz transformation in the homogenization of spacetime crystals, showing that the effective response is independent of the considered transformation.

Photonic platforms invariant under parity ($\mathcal{P}$), time-reversal ($\mathcal{T}$), and duality ($\mathcal{D}$) can support topological phases analogous to those found in time-reversal invariant ${\mathbb{Z}_2}$ electronic systems with conserved spin. Here, we demonstrate the resilience of the underlying spin Chern phases against non-Hermitian effects, notably material dissipation. We identify that non-Hermitian, $\mathcal{P}\mathcal{D}$-symmetric, and reciprocal photonic insulators fall into two topologically distinct classes. Our analysis focuses on the topology of a $\mathcal{P}\mathcal{D}$-symmetric and reciprocal parallel-plate waveguide (PPW). We discover a critical loss level in the plates that marks a topological phase transition. The Hamiltonian of the $\mathcal{P}\mathcal{T}\mathcal{D}$-symmetric system is found to consist of an infinite direct sum of Kane-Mele type Hamiltonians with a common band gap. This structure leads to the topological charge of the waveguide being an ill-defined sum of integers due to the particle-hole symmetry. Each component of this series corresponds to a spin-polarized edge state. Our findings present a unique instance of a topological photonic system that can host an infinite number of edge states in its band gap.

Here, we investigate the nonreciprocal propagation and amplification of surface plasmons in drift-current biased graphene, using both Galilean and relativistic-type Doppler shift transformations of the graphene's conductivity. Consistent with previous studies, both conductivity models predict strongly nonreciprocal propagation of surface plasmons due to the drag effect caused by the drifting electrons. In particular, the Galilean Doppler shift model leads to stronger spectral asymmetries in the plasmon dispersion with regimes of unidirectional propagation. Remarkably, it is shown that both conductivity models predict regimes of nonreciprocal plasmon amplification in a wide angular sector of in-plane directions when the drift-current biased graphene sheet is coupled to a plasmonic substrate (namely, SiC), with the plasmon amplification rate being substantially higher for the relativistic Doppler shift model., Comment: 29 pages, 10 figures

Here, we show that spacetime modulations provide an exciting route to realize complex nonreciprocal couplings, and in particular the elusive axion response. We develop an analytical formalism to homogenize anisotropic spacetime crystals in the long wavelength limit. It is found that spacetime crystals with suitable glide-rotation symmetry can have a giant axion-type response, several orders of magnitude larger than in natural materials. The nonreciprocal axion response may have interesting applications in optics, for example in electromagnetic isolation, and in addition may enable exciting new forms of light-wave interactions.

Topological photonics has recently emerged as a very general framework for the design of unidirectional edge waveguides immune to back-scattering and deformations, as well as other platforms that feature extreme nonreciprocal wave phenomena. While the topological classification of time invariant crystals has been widely discussed in the literature, the study of spacetime crystals formed by time-variant materials remains largely unexplored. Here, we extend the methods of topological band theory to photonic crystals formed by inclusions that are subject to a spacetime rotating-wave modulation that imitates a physical rotating motion. By resorting to an approximate nonhomogeneous effective description of the electromagnetic response of the inclusions, it is shown that they possess a bianisotropic response that breaks the time-reversal symmetry and may give rise to non-trivial topologies. In particular, we propose an implementation of the Haldane model in a spacetime modulated photonic crystal., Comment: 29 pages, 6 figures

Topological photonics provides a powerful framework to describe and understand many nontrivial wave phenomena in complex electromagnetic platforms. The topological index of a physical system is an abstract global property that depends on the family of operators that describes the propagation of Bloch waves. Here, we highlight that there is a profound geometrical connection between topological physics and the topological theory of mathematical surfaces. We show that topological band theory can be understood as a generalization of the topological theory of surfaces and that the genus of a surface can be regarded as a Chern number of a suitable operator defined over the surface. We point out some nontrivial implications of topology in the context of radiation problems and discuss why for physical problems the topological index is often associated with a bulk-edge correspondence., Comment: 24 pages

Semiconductor transistors are essential elements of electronic circuits as they enable, for example, the isolation or amplification of voltage signals. While conventional transistors are point-type (lumped-element) devices, it may be highly interesting to realize a distributed transistor-type optical response in a bulk material. Here, we show that low-symmetry two-dimensional metallic systems may be the ideal solution to implement such a distributed-transistor response. To this end, using the semiclassical Boltzmann equation approach, we characterize the optical conductivity of a two-dimensional material under a static electric bias. It is found that similar to the nonlinear Hall effect, the electron transport depends on the Berry curvature dipole. Our analysis reveals that the electro-optic effect modifies the optical conductivity of the material, breaking the electromagnetic reciprocity and yielding a dynamical response that imitates that of a transistor but in a distributed volume. Furthermore, the effective conductivity tensor can be non-Hermitian, opening the possibility of optical gain. To maximize the non-Hermitian response, we explore the specific case of strained twisted bilayer graphene. Our analysis reveals that the optical gain for incident light transmitted through the biased system depends on the light polarization, and can be quite large, especially for multilayer configurations.

Here, inspired by the operation of conventional semiconductor transistors, we introduce a novel class of bulk materials with nonreciprocal and non-Hermitian electromagnetic response. Our analysis shows that material nonlinearities combined with a static electric bias may lead to a linearized permittivity tensor that lacks the Hermitian and transpose symmetries. Remarkably, the material can either dissipate or generate energy, depending on the relative phase of the electric field components. We introduce a simple design for an electromagnetic isolator based on an idealized "MOSFET-metamaterial" and show that its performance can in principle surpass conventional Faraday isolators due to the material gain. Furthermore, it is suggested that analogous material responses may be engineered in natural media in nonequilibrium situations. Our solution determines an entirely novel paradigm to break the electromagnetic reciprocity in a bulk nonlinear material using a static electric bias.

In this article, it is experimentally verified that nonreciprocal photonic systems with a continuous translation symmetry may have an ill-defined topology. The topological classification of such systems is only feasible when the material response is regularized with a spatial-frequency cutoff. Here, we experimentally demonstrate that inserting a small air gap in between two materials may effectively imitate an idealized spatial cutoff that suppresses the nonreciprocal response for short wavelengths and regularizes the topology. Furthermore, it is experimentally verified that nonreciprocal systems with an ill-defined topology may be used to abruptly halt the energy flow in a unidirectional waveguide due to the violation of the bulk-edge correspondence. In particular, we report the formation of an energy sink that absorbs the incoming electromagnetic waves with a large field enhancement at the singularity., Comment: 19 pages, 6 figures

In recent years there has been a great interest in topological materials and in their fascinating properties. Topological band theory was initially developed for condensed matter systems, but it can be readily applied to arbitrary wave platforms with little modifications. Thus, the topological classification of optical systems is usually regarded as being mathematically equivalent to that of condensed matter systems. Surprisingly, here we find that both the particle-hole symmetry and the dispersive nature of nonreciprocal photonic materials may lead to situations where the usual topological methods break-down and the Chern topology becomes ill-defined. It is shown that due to the divergence of the density of photonic states in plasmonic systems the gap Chern numbers can be non-integer notwithstanding that the relevant parametric space is compact. In order that the topology of a dispersive photonic crystal is well defined, it is essential to take into account the nonlocal effects in the bulk-materials. We propose two different regularization methods to fix the encountered problems. Our results highlight that the regularized topologies may depend critically on the response of the bulk materials for large k.

In this paper we propose a new paradigm to create superlenses inspired by n-p-n junctions of graphene. We show that by adjoining a n-type region and a p-type region with a crystal dislocation, it is possible to mimic the interaction of complementary Hamiltonians and achieve subwavelength imaging. We introduce an effective model of the system, and show that it predicts perfect lensing for both propagating and evanescent waves due to the excitation of a resonant mode at the interface between each region. This phenomenon is the consequence of a nontrivial boundary condition at the n-p interfaces due to a dislocation of the graphene "atoms". We discuss practical realizations of such superlenses in electronic and photonic platforms. Using full wave simulations, we study in detail the performance of a photonic realization of the lens based on a honeycomb array of dielectric cylinders embedded in a metal., Comment: 6 figures

Time-varying media have recently emerged as a new paradigm for wave manipulation, thanks to thesynergy between the discovery of novel, highly nonlinear materials, such as epsilon-near-zero materials, and the questfor novel wave applications, such as magnet-free nonreciprocity, multi-mode light shaping, and ultrafast switching. Inthis review we provide a comprehensive discussion of the recent progress achieved with photonic metamaterials whoseproperties stem from their modulation in time. We review the basic concepts underpinning temporal switching and itsrelation with spatial scattering, and deploy the resulting insight to review photonic time-crystals and their emergentresearch avenues such as topological and non-Hermitian physics. We then extend our discussion to account for spa-tiotemporal modulation and its applications to nonreciprocity, synthetic motion, giant anisotropy, amplification andother effects. Finally, we conclude with a review of the most attractive experimental avenues recently demonstrated,and provide a few perspectives on emerging trends for future implementations of time-modulation in photonics.

We study the role of time-reversal symmetry on the dynamical response of nonlinear optical systems that behave as unidirectional ("one-way") devices. It is shown that lossless nonlinear materials, despite being nonreciprocal, are typically time-reversal invariant. This property raises an apparent paradox because time-reversal invariant systems are forcibly bi-directional. Here, we present a solution for this conundrum, and theoretically explain why the "one-way" behavior can indeed be compatible with the time-reversal invariance. It is found that in the time-reversed problem the incident waves have a variation in time that is generally incompatible with the adiabatic approximation. Due to this reason the adiabatic approximation fails to predict the bi-directional nature of nonlinear system. We discuss the implications of this finding in the performance of practical nonlinear "one-way" devices., Comment: 24 pages, 7 figures

In recent years, the confinement of light in open systems with no radiation leakage has raised great interest in the scientific community, both due to its peculiar and intriguing physics and due to its important technological applications. In particular, materials with near-zero permittivity offer a unique opportunity for light localization, as they enable the formation of embedded eigenstates in core-shell systems with suppressed radiation loss. For all the solutions presented thus far in the literature, the exact suppression of the radiation leakage can occur only when the size of the resonator is delicately tuned. Surprisingly, here it is shown that the tuning of the resonator radius may be unnecessary, and that nonlocal metal spherical nanostructures of any size may support multiple embedded eigenstates with a monopole-type symmetry.

Photonic topological materials with a broken time reversal symmetry are characterized by nontrivial topological phases, such that they do not support propagation in the bulk region but forcibly support a nontrivial net number of unidirectional edge states when enclosed by an opaque type boundary, e.g., an electric wall. The Haldane model played a central role in the development of topological methods in condensed matter systems, as it unveiled that a broken time reversal is the essential ingredient to have a quantized electronic Hall phase. Recently, it was proved that the magnetic field of the Haldane model can be imitated in photonics with a spatially varying pseudo Tellegen coupling. Here, we use a Greens function method to determine from first principles the band diagram and the topological invariants of the photonic Haldane model, implemented as a Tellegen photonic crystal. Furthermore, the topological phase diagram of the system is found, and it is shown with first principles calculations that the granular structure of the photonic crystal can create nontrivial phase transitions controlled by the amplitude of the pseudo Tellegen parameter.