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We investigate entanglement mediated by DC current induced nonreciprocal graphene plasmon polaritons. Nonreciprocal systems are ideal for the enhancement, control, and preservation of entanglement due to the potential for unidirectional beam-like wave propagation, i.e., efficiently transporting photons from one emitter to another. Using a quantum master equation and three-dimensional Green's function analysis, we investigate a system consisting of two two-level emitters dominantly interacting via electric current induced nonreciprocal plasmonic modes of a graphene waveguide. We use concurrence as a measure of entanglement. We show that nonreciprocal graphene plasmon polaritons are a promising candidate to generate and mediate concurrence, where it is shown that there is good enhancement and control of entanglement over vacuum, which is beneficial for the broad applications of entanglement as a quantum resource. We believe our findings contribute to the development of quantum devices, enabling efficient and tunable entanglement between two-level systems, which is a central goal in quantum technologies.

This article aims at developing a simple, automated test bench procedure to measure polarization rotation of an electromagnetic wave upon propagation through a nonreciprocal medium. A comprehensive measurement approach is developed from the scattering matrix. The proposed measurement procedure is demonstrated using an electronically tunable nonreciprocal metamaterial, and the accuracy of the proposed method is compared with the well-accepted conventional measurement technique., Comment: 5 pages, 6 figures

In this work, we study the in-plane optical phonon modes of current-carrying single-layer graphene whose coupling to the $\pi$ electron gas is strong. Such modes are expected to undergo a frequency shift compared to the non-current-carrying state due to the non-equilibrium occupation of the Dirac cone electronic eigen-states with the flowing $\pi$ electron gas. Large electron-phonon coupling (EPC) can be identified by an abrupt change in the slope of the phonon mode dispersion known as the Kohn anomaly, which mainly occurs for (i) the in-plane longitudinal/transverse optical (LO/TO) modes at the Brillouin zone (BZ) center ($\Gamma$ point), and (ii) the TO modes at the BZ corners ($K$ points). We show that the breaking of the rotational symmetry by the DC current results in different frequency shifts to the $\Gamma$-TO and $\Gamma$-LO modes. More specifically, the DC current breaks the TO-LO mode degeneracy at the $\Gamma$ point which ideally would be manifested as the splitting of the Raman G peak.

Dynamic manipulation of the surface plasmon polariton (SPP) and wave steering are important in plasmonic applications. In this work, we excite a curved SPP in topological continua by applying a radial magnetic bias. We believe that it is a new technique to create a unidirectional SPP traveling along a curved trajectory. We also derive a Greens function model for radially-biased plasma, applicable to curved SPPs. We compare the properties of unidirectional curved SPPs with the usual case when an axial bias is applied.

The bulk-edge correspondence is a fundamental principle of topological wave physics, which states that the difference in gap Chern numbers between the interfaced materials is equal to the net number of topological edge modes. Although this principle works well for periodic photonic topological insulators, difficulties arise in the case of continuous systems with no intrinsic periodicity, due to the absence of a finite Brillouin zone, which may lead to an ill-behaved response for diverging wavenumbers. This problem has been solved previously by introducing an ad hoc material model including a spatial cutoff wavenumber. However, this method introduces other difficulties, such as an unphysical response at large wavenumbers, and the need to interpolate the interfaced materials permittivity functions. In this work, we show that the inclusion of realistic spatial dispersion (e.g., hydrodynamic nonlocality) formally restores the bulk-edge correspondence by avoiding an unphysical response at large wavenumbers, forming a complete, self-consistent model. This model clarifies the subtle and rich topological wave physics of continuous media.

In this work, a plasmonic right-angled waveguide bend and divider are proposed. Using the Transformation Optics (TO) approach the transformation media of a bend and a T-shaped divider are obtained. Such media with continuous refractive index are realized with the help of graphene in the terahertz frequency range, key to effectively guiding the surface plasmon polariton (SPP) propagation on the 90 degree curves. Components with such capability are promising for THz device applications., Comment: 10 pages, 5 figures, 2 Tables

Magnetic-field-biased indium antimonide (InSb) is one of the most widely-discussed materials for supporting nonreciprocal surface plasmon polaritons (SPPs), which have recently been shown to be topological. In this work, we provide a critical assessment of InSb as a magneto-optical SPP platform, and show that it is only viable under a narrow set of conditions., Comment: 6 pages, 4 figures and 2 tables

The Langevin noise approach for quantization of macroscopic electromagnetics for three-dimensional, inhomogeneous environments is compared with normal mode quantization. Recent works on the applicability of the method are discussed, and several examples are provided showing that for closed systems the Langevin noise approach reduces to the usual cavity mode expansion method when loss is eliminated.

The exchange splitting and resulting electromagnetic response of graphene in a monolayer chromium triiodide(CrI3) graphene van der Waals(vdW) heterostructure are considered using a combination of density functional theory and electromagnetic calculations. Although the effective exchange fields are in the hundreds of Tesla, for the equilibrium separation non reciprocal effects are found to be weak compared to those for a comparable external magnetic bias. For non equilibrium separations, nonreciprocal effects can be substantial., Comment: 19 pages included Supplement Information, 11 main-text figures and 9 SI figures

We report the magneto-optical response of Gadolinium Gallium Garnet (GGG) and Gadolinium Scandium Gallium Garnet (SGGG) at frequencies ranging from $300 \, \mathrm{GHz}$ to $1 \, \mathrm{THz}$, and determine the material response tensor. Within this frequency window, the materials exhibit nondispersive and low-loss optical responses. At low temperatures, significant THz Faraday rotations are found in the (S)GGG samples. Such strong gyroelectric response is likely associated with the high-spin paramagnetic state of the Gd$^{3+}$ ions. A model of the material response tensor is determined, together with the Verdet and magneto-optic constants., Comment: 9 pages, 5 figures and 3 tables

A simple analytical model based on the transmission-matrix approach is proposed for equivalent wire-medium (WM) interfaces. The obtained ABCD matrices for equivalent interfaces capture the non-local effects due to the evanescent transverse magnetic (TM) WM mode and in part due to the propagating transverse electromagnetic (TEM) WM mode. This enables one to characterize the overall response of bounded WM structures by cascading the ABCD matrices of equivalent WM interfaces and WM slabs as transmission lines supporting only the propagating TEM WM mode, resulting in a simple circuit-model formalism for bounded WM structures with arbitrary terminations, including the open-end, patch/slot arrays, and thin metal/2D material, among others. The individual ABCD matrices for equivalent WM interfaces apparently violate the conservation of energy and reciprocity, and therefore, the equivalent interfaces apparently behave as non-reciprocal lossy or active systems. However, the overall response of a bounded WM structure is consistent with the lossless property maintaining the conservation of energy and reciprocity. These unusual features are explained by the fact that in the non-local WM the Poynting vector has an additional correction term which takes into account a "hidden power" due to non-local effects. Results are obtained for various numerical examples demonstrating a rapid and efficient solution for bounded WM structures, including the case of geometrically complex multilayer configurations with arbitrary terminations, subject to the condition that WM interfaces are decoupled by the evanescent TM WM mode below the plasma frequency.

A local permittivity model is proposed to accurately characterize spatial dispersion in non-local wire-medium (WM) structures with arbitrary terminations. A closed-form expression for the local thickness-dependent permittivity is derived for a general case of a bounded WM with lumped impedance insertions and terminated with impedance surfaces, which takes into account the effects of spatial dispersion and loads/terminations in the averaged sense per length of the wire medium. The proposed approach results in a local model formalism and accurately predicts the response of WM structures for near-field and far-field excitation. It is also shown that a traditional transmission network and circuit model can be effectively used to quantify the interaction of propagating and evanescent waves with WM structures. In addition, the derived analytical expression for the local thickness-dependent permittivity has been used in the full-wave numerical solver (CST Microwave Studio) demonstrating a drastic reduction in the computation time and memory in the solution of near-field and far-field problems involving wire media.

Unidirectional surface plasmon polaritons (SPPs) at the interface between a gyrotropic medium and a simple medium are studied in a newly-recognized frequency regime wherein the SPPs form narrow, beam-like patterns due to hyperbolic dispersion. The SPP beams are steerable by controlling parameters such as the cyclotron frequency (external bias) or the frequency of operation. The bulk band structure along different propagation directions is examined to ascertain a common bandgap, valid for all propagation directions, which the SPPs cross. The case of a finite-thickness gyrotropic slab is also considered, for which we present the Green function and examine the thickness and loss level required to maintain a unidirectional SPP., Comment: 11 pages, 10 figures

The transient Casimir-Polder force on a two-level atom introduced into a three-dimensional, inhomogeneous, generally non-reciprocal environment is evaluated using non-Markovian Weisskopf-Wigner theory in the strong and weak coupling regimes. Ground-state and excited atoms are considered as two separate initial-value problems, and both the short-time and long-time atomic population and force are evaluated. The results are compared with various Markov approximation of the Weisskopf-Wigner theory, and with previous Markov results from the Heisenberg picture.

Light-matter interactions in conventional nanophotonic structures typically lack directionality. Furthermore, surface waves supported by conventional material substrates do not usually have a preferential direction of propagation, and their wavefront tends to spread as it propagates along the surface, unless the surface or the excitation are properly engineered and structured. In this article, we theoretically demonstrate the possibility of realizing \emph{unidirectional and diffractionless surface-plasmon-polariton modes} on a nonreciprocal platform, namely, a gyrotropic magnetized plasma. Based on a rigorous Green function approach, we provide a comprehensive and systematic analysis of all the available physical mechanisms that may bestow the system with directionality, both in the sense of one-way excitation of surface waves, and in the sense of directive diffractionless propagation along the surface. The considered mechanisms include (i) the effect of strong and weak forms of nonreciprocity, (ii) the elliptic-like or hyperbolic-like topology of the modal dispersion surfaces, and (iii) the source polarization state, with the associated possibility of chiral surface-wave excitation governed by angular-momentum matching. We find that three-dimensional gyrotropic plasmonic platforms support a previously-unnoticed wave-propagation regime that exhibit several of these physical mechanisms simultaneously, allowing us to theoretically demonstrate, for the first time, unidirectional surface-plasmon-polariton modes that propagate as a single ultra-narrow diffractionless beam. We also assess the impact of dissipation and nonlocal effects. Our theoretical findings may enable a new generation of plasmonic structures and devices with highly directional response.

We investigate the quantum optical torque on an atom interacting with an inhomogeneous electromagnetic environment described by the most general linear constitutive relations. The atom is modeled as a two-level system prepared in an arbitrary initial energy state. Using the Heisenberg equation of motion (HEM) and under the Markov approximation, we show that the optical torque has a resonant and non-resonant part, associated respectively with a spontaneous-emission process and Casimir-type interactions with the quantum vacuum, which can both be written explicitly in terms of the system Green function. Our formulation is valid for any inhomogeneous, dissipative, dispersive, nonreciprocal, and bianisotropic structure. We apply this general theory to a scenario in which the atom interacts with a material characterized by strong nonreciprocity and modal unidirectionality. In this case, the main decay channel of the atom energy is represented by the unidirectional surface waves launched at the nonreciprocal material-vacuum interface. To provide relevant physical insight into the role of these unidirectional surface waves in the emergence of non-trivial optical torque, we derive closed-form expressions for the induced torque under the quasi-static approximation. Finally, we investigate the equilibrium states of the atom polarization, along which the atom spontaneously tends to align due to the action of the torque. Our theoretical predictions may be experimentally tested with cold Rydberg atoms and superconducting qubits near a nonreciprocal material. We believe that our general theory may find broad application in the context of nano-mechanical and bio-mechanical systems.

The connection between topology and nonreciprocity in photonic systems is reviewed. Topological properties such as Chern number, and momentum-space properties such as Berry phase and Berry connection, are used to explain back-scattering immune edge states and their topological protection. We consider several examples to illustrate the role of momentum-space topology on wave propagation, and discus recent magnet-less approaches.

We demonstrate several new aspects of exceptional points of degeneracy (EPD) pertaining to propagation in two uniform coupled transmission-line structures. We describe an EPD using two different approaches - by solving an eigenvalue problem based on the system matrix, and as a singular point from bifurcation theory, and the link between these two disparate viewpoints. Cast as an eigenvalue problem, we show that eigenvalue degeneracies are always coincident with eigenvector degeneracies, so that all eigenvalue degeneracies are implicitly EPDs in two uniform coupled transmission lines. Furthermore, we discuss in some detail the fact that EPDs define branch points (BPs) in the complex-frequency plane; we provide simple formulas for these points, and show that parity-time (PT) symmetry leads to real-valued EPDs occurring on the real-frequency axis. We discuss the connection of the linear algebra approach to previous waveguide analysis based on singular points from bifurcation theory, which provides a complementary viewpoint of EPD phenomena, showing that EPDs are singular points of the dispersion function associated with the fold bifurcation. This provides an important connection of various modal interaction phenomena known in guided-wave structures with recent interesting effects observed in quantum mechanics, photonics, and metamaterials systems described in terms of the EPD formalism.

We investigate the quantum recoil force acting on an excited atom close to the surface of a nonreciprocal photonic topological insulator (PTI). The main atomic emission channel is the unidirectional surface-plasmon propagating at the PTI-vacuum interface, and we show that it enables a spontaneous lateral recoil force that scales at short distance as $1/d^4$, where $d$ is the atom-PTI separation. Remarkably, the sign of the recoil force is polarization and orientation-independent, and it occurs in a translation-invariant homogeneous system in thermal equilibrium. Surprisingly, the recoil force persists for very small values of the gyration pseudovector, which, for a biased plasma, corresponds to very low cyclotron frequencies. The ultra-strong recoil force is rooted on the quasi-hyperbolic dispersion of the surface-plasmons. We consider both an initially excited atom and a continuous pump scenario, the latter giving rise to a continuous lateral force whose direction can be changed at will by simply varying the orientation of the biasing magnetic field. Our predictions may be tested in experiments with cold Rydberg atoms and superconducting qubits.

We theoretically study the Casimir-Polder force on an atom in a arbitrary initial state in a rather general electromagnetic environment wherein the materials may have a nonreciprocal bianisotropic dispersive response. It is shown that under the Markov approximation the force has resonant and nonresonant contributions. We obtain explicit expressions for the optical force both in terms of the system Green function and of the electromagnetic modes. We apply the theory to the particular case wherein a two-level system interacts with a topological gyrotropic material, showing that the nonreciprocity enables exotic light-matter interactions and the opportunity to sculpt and tune the Casimir-Polder forces on the nanoscale. With a quasi-static approximation, we obtain a simple analytical expression for the optical force and unveil the crucial role of surface plasmons in fluctuation induced forces. Finally, we derive the Green function for a gyrotropic material half-space in terms of a Sommerfeld integral.

Based on the theory of macroscopic quantum electrodynamics, we generalize the expression of the Casimir force for nonreciprocal media. The essential ingredient of this result is the Green's tensor between two nonreciprocal semi-infinite slabs including a reflexion matrix with four coefficients that mixes optical polarizations. This Green's tensor does not obey Lorentz's reciprocity and thus violates time-reversal symmetry. The general result for the Casimir force is analyzed in the retarded and nonretarded limits, concentrating on the influences arising from reflections with or without change of polarization. In a second step we apply our general result to a photonic topological insulator whose nonreciprocity stems from an anisotropic permittivity tensor, namely InSb. We show that there is a regime for the distance between the slabs where the magnitude of the Casimir force is tunable by an external magnetic field. Furthermore the strength of this tuning depends on the orientation of the magnetic field with respect to the slab surfaces., Comment: 14 pages, 9 figures

We show that the energy-transport efficiency in a chain of two-level emitters can be drastically enhanced by the presence of a photonic topological insulator (PTI). This is obtained by exploiting the peculiar properties of its nonreciprocal surface plasmon polariton (SPP), which is unidirectional, and immune to backscattering, and propagates in the bulk band gap. This amplification of transport efficiency can be as much as 2 orders of magnitude with respect to reciprocal SPPs. Moreover, we demonstrate that despite the presence of considerable imperfections at the interface of the PTI, the efficiency of the SPP-assisted energy transport is almost unaffected by discontinuities. We also show that the SPP properties allow energy transport over considerably much larger distances than in the reciprocal case, and we point out a particularly simple way to tune the transport. Finally, we analyze the specific case of a two-emitter chain and unveil the origin of the efficiency amplification. The efficiency amplification and the practical advantages highlighted in this work might be particularly useful in the development of new devices intended to manage energy at the atomic scale.

We investigate spontaneous and pumped entanglement of two level systems in the vicinity of a photonic topological insulator interface, which supports a nonreciprocal (unidirectional), scattering-immune and topologically-protected surface plasmon polariton in the bandgap of the bulk material. To this end, we derive a master equation for qubit interactions in a general three-dimensional, nonreciprocal, inhomogeneous and lossy environment. The environment is represented exactly, via the photonic Green function. The resulting entanglement is shown to be extremely robust to defects occurring in the material system, such that strong entanglement is maintained even if the interface exhibits electrically-large and geometrically sharp discontinuities. Alternatively, depending on the initial excitation state, using a non-reciprocal environment allows two qubits to remain unentangled even for very close spacing. The topological nature of the material is manifest in the insensitivity of the entanglement to variations in the material parameters that preserve the gap Chern number. Our formulation and results should be useful for both fundamental investigations of quantum dynamics in nonreciprocal environments, and technological applications related to entanglement in two-level systems.

The properties that quantify photonic topological insulators (PTIs), Berry phase, Berry connection, and Chern number, are typically obtained by making analogies between classical Maxwell's equations and the quantum mechanical Schr\"{o}dinger equation, writing both in Hamiltonian form. However, the aforementioned quantities are not necessarily quantum in nature, and for photonic systems they can be explained using only classical concepts. Here we provide a derivation and description of PTI quantities using classical Maxwell's equations, we demonstrate how an electromagnetic mode can acquire Berry phase, and we discuss the ramifications of this effect. We consider several examples, including wave propagation in a biased plasma, and radiation by a rotating isotropic emitter. These concepts are discussed without invoking quantum mechanics, and can be easily understood from an engineering electromagnetics perspective.

We have investigated the one-way surface plasmon-polariton (SPP) at the interface of a continuous magneto-plasma material and metal. We demonstrated that TM modes inside a continuous material can be assigned non-trivial Chern numbers analogous to those of topological photonic crystals; moreover these Chern numbers can be calculated analytically. This leads to the appearance of topologically protected surface modes propagating at frequencies inside the bandgap of the magneto-plasma. Previous works considered 2D structures; here we consider the effects of 3D defects, and show that, although backward propagation/reflection cannot occur, side scattering does take place and it has significant effect on the propagation of the surface mode. Several different waveguiding geometries are considered for reducing the effects of side-scattering, and we also consider the effects of metal loss.

The topic of photonic topological insulators and scattering-protected edge states bridges concepts from condensed matter physics and electromagnetics, and necessitates understanding the Berry potential and related concepts. These notes are an attempt at a moderately self-contained introduction to the topic, including two detailed photonic examples drawn from the literature. We made these notes in the process of trying to understand this topic ourselves, and we are posting this material in the spirit of helping other researchers start to understand this material. We claim no novelty in the material or its presentation.

We study the electromagnetic response of two- and quasi-two-dimensional hyperbolic materials, on which a simple dipole source can excite a well-confined and tunable surface plasmon polariton (SPP). The analysis is based on the Green's function for an anisotropic two-dimensional surface, which nominally requires the evaluation of a two-dimensional Sommerfeld integral. We show that for the SPP contribution this integral can be evaluated efficiently in a mixed continuous-discrete form as a continuous spectrum contribution (branch cut integral) of a residue term, in distinction to the isotropic case, where the SPP is simply given as a discrete residue term. The regime of strong SPP excitation is discussed, and complex-plane singularities are identified, leading to physical insight into the excited SPP. We also present a stationary phase solution valid for large radial distances. Examples are presented using graphene strips to form a hyperbolic metasurface, and thin-film black phosphorus. The Green's function and complex-plane analysis developed allows for the exploration of hyperbolic plasmons in general 2D materials.

Entanglement between two qubits (two level atoms) mediated by surface plasmons in three-dimensional plasmonic waveguides is studied using a quantum master equation formalism. Two types of waveguides, a nanowire and a V-shaped channel cut in a flat metal plane, are considered. The Green functions for the waveguides, which rigorously describes the dissipative qubit environment, are calculated numerically using a direct finite-difference time-domain (FDTD) solution of Maxwell's equations. Finite-length effects are shown to play a crucial role in enhancing entanglement, and resonant-length plasmonic waveguides can provide higher entanglement between qubits than infinite-length waveguides. It is also shown that coupling slots can improve entanglement via stronger qubit-waveguide coupling, for both the infinite- and finite-waveguide cases.

We investigate the excitation of quantum plasmonic states of light in graphene using end-fire and prism coupling. In order to model the excitation process quantum mechanically we quantize the transverse-electric and transverse-magnetic surface plasmon polariton (SPP) modes in graphene. A selection of regimes are then studied that enable the excitation of SPPs by photons and we show that efficient coupling of photons to graphene SPPs is possible at the quantum level. Futhermore, we study the excitation of quantum states and their propagation under the effects of loss induced from the electronic degrees of freedom in the graphene. Here, we investigate whether it is possible to protect quantum information using quantum error correction techniques. We find that these techniques provide a robust-to-loss method for transferring quantum states of light in graphene over large distances., Comment: 14 pages, 8 figures

Using a realistic quantum master equation we show that the resonance fluorescence spectra of a two-level artificial atom (quantum dot) can be tuned by adjusting its photonic local density of states via biasing of one or more graphene monolayers. The structured photon reservoir is included using a photon Green function theory which fully accounts for the loss and dispersion. The field-driven Mollow triplet spectrum can be actively controlled by the graphene bias in the THz frequency regime. We also consider the effect of a dielectric support environment, and multiple graphene layers, on the emitted fluorescence. Finally, thermal bath effects are considered and shown to be important for low THz frequencies.

The canalization of terahertz surface plasmon polaritons using a modulated graphene monolayer is investigated for subwavelength imaging. An anisotropic surface conductivity formed by a set of parallel nanoribbons with alternating positive and negative imaginary conductivities is used to realize the canalization regime required for hyperlensing. The ribbons are narrow compared to the wavelength, and are created electronically by gating a graphene layer over a corrugated ground plane. Good quality canalization of surface plasmon polaritons is shown in the terahertz even in the presence of realistic loss in graphene, with relevant implications for subwavelength imaging applications.

A graphene sheet gated with a ridged ground plane, creating a soft-boundary (SB) graphene nanoribbon, is considered. By adjusting the ridge parameters and bias voltage a channel can be created on the graphene which can guide TM surface plasmon polaritons (SPP). Two types of modes are found; fundemental and higher-order modes with no apparent cutoff frequency and with energy distributed over the created channel, and edge modes with energy concen-trated at the soft-boundary edge. Dispersion curves, electric near-field patterns, and current distributions of these modes are determined. Since the location where energy is concentrated in the edge modes can be easily controlled electronically by the bias voltage and frequency, the edge-mode phenomena is used to propose a novel voltage controlled plasmonic switch and a plasmonic frequency demultiplexer.

An infinite sheet of graphene lying above a perturbed ground plane is studied. The perturbation is a two dimensional ridge, and a bias voltage is applied between the graphene and the ground plane, resulting in a graphene nanoribbonlike structure with a soft-boundary (SB) The spatial distribution of the graphene conductivity forming the soft-boundary is studied as a function of the ridge parameters and the bias voltage. The current distribution of the fundamental TM surface plasmon polariton (SPP) is considered. The effect of the ridge parameters and shape of the soft boundary on the current distributions are investigated, and the conditions are studied under which the mode remains confined to the vicinity of the ridge region.

An exact solution is obtained for the electromagnetic field due to an electric current in the presence of a surface conductivity model of graphene. The graphene is represented by an infinitesimally-thin, local and isotropic two-sided conductivity surface. The field is obtained in terms of dyadic Green's functions represented as Sommerfeld integrals. The solution of plane-wave reflection and transmission is presented, and surface wave propagation along graphene is studied via the poles of the Sommerfeld integrals. For isolated graphene characterized by complex surface conductivity, a proper transverse-electric (TE) surface wave exists if and only if the imaginary part of conductivity is positive (associated with interband conductivity), and a proper transverse-magnetic (TM) surface wave exists when the imaginary part of conductivity is negative (associated with intraband conductivity). By tuning the chemical potential at infrared frequencies, the sign of the imaginary part of conductivity can be varied, allowing for some control over surface wave properties., Comment: 9 figures