arXiv subjects

Sebastian Franke

Publications and source records attributed to Sebastian Franke.

18 recordsLinked to original sources

arXiv2026

Correlation functions for quantum dynamics of coupled quasinormal modes and quantum emitters interacting via finite-delay propagating photons

A time-dependent theory for the interactions between spatially separated lossy cavities in a homogeneous background medium using quantized quasinormal modes (QNMs) is presented. The cavities interact via a bath of traveling photons, described by non-bosonic operators that are orthogonal to the open-cavity QNMs. The retarded (i.e., time-delayed) inter-cavity dynamics are fully described by system-bath correlation functions, in which the emission from one cavity appears as the input field for another. Coupling between quantum emitters (described as two-level systems), placed inside a cavity or embedded in an external medium, and the electromagnetic field (cavity modes and bath photons) is included in the theory, which gives rise to both bath-mediated and QNM-mediated interactions between the emitters.

cond-mat.mes-hall↗

arXiv2025

Dissipation in the Broadband and Ultrastrong Coupling Regimes of Cavity Quantum Electrodynamics: An Ab Initio Quantized Quasinormal Mode Approach

Phenomenological approaches to photon loss have long been the workhorse of cavity-QED, but prove inadequate in the presence of sufficiently broadband light-matter interactions. We present a rigorous and ab initio derivation of a quantum master equation for a quantized optical cavity mode coupled to a dipole, using a quasinormal mode (QNM) quantization procedure for plasmonic and dielectric open-system cavity-QED, which is valid in broadband light-matter interaction regimes, including ultrastrong coupling (USC). The theory supports general three-dimensional resonators with arbitrary dispersion and loss, and thus can be applied to a wide range of open cavities. Our ab initio and gauge-invariant approach fully recovers the recent result of Phys. Rev. Lett. 134, 123601 (2025) for the spectral density of a quantized cavity with a single dipole, exhibits a dissipative classical-quantum correspondence for bosonic Hopfield model systems, and reveals important departures from previous heuristic assumptions about system-bath coupling. We identify a new criterion for what we term the "broadband" dissipative regime of cavity-QED, where phenomenological models require corrections in accordance with the intrinsic and spatially-dependent complex phase of the QNM, and also shed light on fundamental limits to single-mode models in extreme coupling regimes. Using plasmonic and dielectric cavity examples, we show validity ranges of our QNM master equation and spectral USC calculations, and discuss prospects for near-term experimental observation of broadband dissipative effects.

quant-ph↗

arXiv2025

Gain-modified emission dynamics between two quantum emitters in a plasmonic gain cavity system

We present a general theory of gain-modified emission dynamics between two quantum emitters (two level systems) in a linear gain medium, demonstrating how gain modifies the usual radiative decay rates and inter-emitter coupling and decay rates, that are well known from purely lossy systems. We derive a Born-Markov master equation that shows explicitly how gain modifies the usual rates appearing for two emitters in a loss only medium, and introduces new gain terms. We then present Bloch equations in the bare state basis as well as a dressed-state basis (using superradiant and subradiant states). As an application of the theory, we show calculations for a fully three-dimensional metal dimer in a gain-compensated medium, containing two quantum emitters, and show explicit solutions in terms of the quasinormal modes, which easily allows one to obtain all the relevant decay rates. We study the modified decay dynamics, gain excited pumping, as well as the entanglement entropy and spectral emission. We also demonstrate how a breakdown of the weak excitation approximation in the presence of gain requires a careful treatment beyond a linear response, which yields non-trivial incoherent coupling terms that typically are not included in dipole-dipole coupling models or heuristic gain pumping models.

quant-ph↗

arXiv2024

Quantization of optical quasinormal modes for spatially separated cavity systems with finite retardation

A multi-cavity quantization scheme is developed using quasinormal modes (QNMs) of optical cavities embedded in a homogeneous background medium for cases where retardation is significant in the inter-cavity coupling. Using quantities that can be calculated in computational optics with numerical Maxwell solvers, we extend previous QNM quantization schemes and define a quantitative measure to determine if a separate quantization of QNM cavities is justified or if a joint quantization of the system is necessary. We test this measure for the examples of two coupled one-dimensional dielectric slabs and a dimer of metal nanorods acting as QNM cavities. For sufficiently large separations, the new scheme allows for an efficient treatment of multi-cavity phenomena using parameters defined for the individual cavities. Formulating the Hamiltonian in a familiar system-bath form, the scheme connects the rigorous QNM theory and widespread phenomenological models of open cavities coupled to a shared photonic bath with parameters obtained directly from Maxwell calculations.

cond-mat.mes-hall↗

arXiv2023

Gain-compensated metal cavity modes and a million-fold improvement of Purcell factors

Using a rigorous mode theory for gain-compensated plasmonic dimers, we demonstrate how quality factors and Purcell factors can be dramatically increased, improving the quality factors from 10 to over 26,000 and the peak Purcell factors from around 3000 to over 10 billion. Full three-dimensional calculations are presented for gold dimers in a finite-size gain medium, which allows one to easily surpass fundamental Purcell factor limits of lossy media. Within a regime of linear system response, we show how the Purcell factors are modified from the contributions from the projected local density of states as well as a non-local gain. Further, we show that the effective mode volume and radiative beta factors remain relatively constant, despite the significant enhancement of the Purcell factors.

physics.optics↗

arXiv2023

Classical Purcell factors and spontaneous emission decay rates in a linear gain medium

Recently the photonic golden rule, which predicts that the spontaneous emission rate of an atom depends on the projected local density of states (LDOS), was shown to fail in an optical medium with a linear gain amplifier. We present a classical light-matter theory to fix this widely used spontaneous emission rate, fully recovering the quantum mechanical rate reported in Franke et al., Phys. Rev. Lett. 127, 013602 (2021). The corrected classical Purcell factor, for media containing linear amplifiers, is obtained in two different forms, both of which can easily be calculated in any standard classical Maxwell solver. We also derive explicit analytical results in terms of quasinormal modes, which are useful for studying practical cavity structures in an efficient way, including the presence of local field effects for finite-size dipole emitters embedded inside lossy or gain materials (using a real cavity model). Finally, we derive a full classical correspondence from the viewpoint of quantized quasinormal modes in the bad cavity limit. Example numerical calculations are shown for coupled loss-gain microdisk resonators, showing excellent agreement between few mode expansions and full numerical dipole simulations.

physics.optics↗

arXiv2022

Gauge-invariant theory of truncated quantum light-matter interactions in arbitrary media

The loss of gauge invariance in models of light-matter interaction which arises from material and photonic space truncation can pose significant challenges to conventional quantum optical models when matter and light strongly hybridize. In structured photonic environments, necessary in practice to achieve strong light-matter coupling, a rigorous model of field quantization within the medium is also needed. Here, we use the framework of macroscopic QED by quantizing the fields in an arbitrary material system, with a spatially-dependent dispersive and absorptive dielectric, starting from a fundamental light-matter action. We truncate the material and mode degrees of freedom while respecting the gauge principle by imposing a partial gauge fixing constraint during canonical quantization, which admits a large number of gauges including the Coulomb and multipolar gauges. We also consider gauge conditions with explicit time-dependence, enabling us to unambiguously introduce phenomenologically time-dependent light-matter interactions in any gauge. Our results allow one to derive rigorous models of ultrastrong light-matter interactions in structured photonic environments with no gauge ambiguity. Results for two-level systems and the dipole approximation are discussed, as well as how to go beyond the dipole approximation for effective single-particle models. By comparing with the limiting case of an inhomogeneous dielectric, where dispersion and absorption can be neglected and the fields can be expanded in terms of the generalized transverse eigenfunctions of the dielectric, we show how lossy systems can introduce an additional gauge ambiguity, which we resolve and predict to have fundamental implications for open quantum system models. Finally, we show how observables in mode-truncated systems can be calculated without ambiguity by using a simple gauge-invariant photodetection model.

quant-ph↗

arXiv2022

Impact of mode regularization for quasinormal mode perturbation theories

We give insight into the critical problem of an open resonator that is subject to a perturbation outside of its cavity region. We utilize the framework of quasinormal modes (QNMs), which are the natural mode solutions to the open boundary problem with complex eigenfrequencies. We first highlight some fundamental problems with currently adopted formulas using QNM perturbation theory, when perturbations are added outside the resonator structure and present a first potential step for solving this problem connected to a regularization of the QNMs. We then show an example for a full three-dimensional plasmonic resonator of arbitrary shape and complex dispersion and loss, clearly displaying the divergent nature of the first-order mode change predicted from QNM perturbation theory. Subsequently, we concentrate on the illustrative case of a one-dimensional dielectric barrier, where analytical QNM solutions are possible. We inspect the change of the mode frequency as function of distance between the cavity and another smaller barrier structure. The results obtained from a few QNM expansion are compared with exact analytical solutions from a transfer matrix approach. We show explicitly how regularization prevents a problematic spatial divergence for QNM perturbations in the far field, though eventually higher-order effects and multimodes can also play a role in the full scattering solution, and retaining a pure discrete QNM picture becomes questionable in such situations, since the input-output coupling ultimately involves reservoir modes.

physics.optics↗

arXiv2021

Quasinormal mode analysis of chiral power flow from linearly polarized dipole emitters coupled to index-modulated microring resonators close to an exceptional point

Chiral emission can be achieved from a circularly polarized dipole emitter in a nanophotonic structure that possess special polarization properties such as a polarization singularity, namely with right or left circularly polarization (C-points). Recently, Chen et al. [Nature Physics 16, 571 (2020)] demonstrated the surprising result of chiral radiation from a linearly-polarized (LP) dipole emitter, and argued that this effect is caused by a decoupling with the underlying eigenmodes of a non-Hermitian system, working at an exceptional point (EP). Here we present a quasinormal mode (QNM) approach to model a similar index-modulated ring resonator working near an EP and show the same unusual chiral power flow properties from LP emitters, in direct agreement with the experimental results. We explain these results quantitatively without invoking the interpretation of a missing dimension (the Jordan vector) and a decoupling from the cavity eigenmodes, since the correct eigenmodes are the QNMs which explain the chiral emission using only two cavity modes. By coupling a LP emitter with the dominant two QNMs of the ring resonator, we show how the chiral emission depend on the position and orientation of the emitter, which is also verified by the excellent agreement with respect to the power flow between the QNM theory and full numerical dipole solutions. We also show how a normal mode solution will fail to capture the correct chirality since it does not take into account the essential QNM phase. Moreover, we demonstrate how one can achieve frequency-dependent chiral emission, and replace lossy materials with gain materials in the index modulation to reverse the chirality.

physics.optics↗

arXiv2021

Quantized quasinormal mode theory of coupled lossy and amplifying resonators

In the presence of arbitrary three-dimensional linear media with material loss and amplification, we present an electromagnetic field quantization scheme for quasinormal modes (QNMs), extending previous work for lossy media [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)]. Applying a symmetrization transformation, we show two fundamentally different ways for constructing a QNM photon Fock space, including (i) where there is a separate operator basis for both gain and loss, and (ii) where the loss and gain degrees of freedom are combined into a single basis. These QNM operator bases are subsequently used to derive the associated QNM master equations, including the interaction with a quantum emitter, modelled as a quantized two-level system (TLS). We then compare the two different quantization approaches, and also show how commonly used phenomenological methods to quantize light in gain-loss resonators are corrected by several important aspects, such as a loss-induced and gain-induced intermode coupling, which appears through the rigorous treatment of loss and amplification on a dissipative mode level. For specific resonator designs, modelled in a fully consistent way with the classical Maxwell equations with open boundary conditions, we then present numerical results for the quantum parameters and observables of a TLS weakly interacting with the medium-assisted field in a gain-loss microdisk resonator system, and discuss the validity of the different quantization approaches for several gain-loss parameter regimes.

quant-ph↗

arXiv2021

Connecting classical and quantum mode theories for coupled lossy cavity resonators using quasinormal modes

We present a quantized quasinormal approach to rigorously describe coupled lossy resonators, and quantify the quantum coupling parameters as a function of distance between the resonators. We also make a direct connection between classical and quantum quasinormal modes parameters and theories, offering new and unique insights into coupled open cavity resonators. We present detailed calculations for coupled microdisk resonators and show striking interference effects that depend on the phase of the quasinormal modes, an effect that is also significant for high quality factor modes. Our results demonstrate that commonly adopted master equations for such systems are generally not applicable and we discuss the new physics that is captured using the quantized quasinormal mode coupling parameters and show how these relate to the classical mode parameters. Using these new insights, we also present several models to fix the failures of the dissipative Jaynes-Cummings type models for coupled cavity resonators. Additionally, we show how to improve the classical and quantum lossless mode models (i.e., using normal modes) by employing a non-diagonal mode expansion based on the knowledge of the quasinormal mode eigenfrequencies, and analytical coupled mode theory, to accurately capture the mode interference effects for high quality factors.

quant-ph↗

arXiv2021

Fermi's golden rule for spontaneous emission in absorptive and amplifying media

We demonstrate a fundamental breakdown of the photonic spontaneous emission (SE) formula derived from Fermi's golden rule, in absorptive and amplifying media, where one assumes the SE rate scales with the local photon density of states, an approach often used in more complex, semiclassical nanophotonics simulations. Using a rigorous quantization of the macroscopic Maxwell equations in the presence of arbitrary linear media, we derive a corrected Fermi's golden rule and master equation for a quantum two-level system (TLS) that yields a quantum pumping term and a modified decay rate that is net positive. We show rigorous numerical results of the temporal dynamics of the TLS for an example of two coupled microdisk resonators, forming a gain-loss medium, and demonstrate the clear failure of the commonly adopted formulas based solely on the local density of states.

physics.optics↗

arXiv2021

Quasinormal modes and Purcell factors of coupled loss-gain resonators and index-modulated ring resonators near exceptional points

We first present a quasinormal mode (QNM) theory for coupled loss-gain resonators working near an exceptional point. Assuming linear media, which can be fully quantified using the complex pole properties of the QNMs, we show how the QNMs yield a quantitatively good model to a full dipole spontaneous emission response in Maxwell's equations at various spatial positions and frequencies (linear response). We also develop a highly accurate and intuitive QNM coupled-mode theory, which can be used to rigorously model such systems using only the QNMs of the bare resonators, where the hybrid QNMs of the complete system are automatically obtained. Near a lossy exceptional point, we analytically show how the QNMs yield a Lorentzian-like and a Lorentzian-squared-like response for the spontaneous emission lineshape, consistent with other works. However, using rigorous analytical and numerical solutions for microdisk resonators, we demonstrate that the general lineshapes are far richer than what has been previously predicted. Indeed, the classical picture of spontaneous emission can take on a wide range of positive and negative Purcell factors from the hybrid modes of the coupled loss-gain system. These negative Purcell factors are unphysical and signal a clear breakdown of the classical dipole picture of spontaneous emission in such media, though the negative local density of states is correct. We also show the rich spectral features of the Green function propagators, which can be used to model various physical observables. Second, we present a QNM approach to model index modulated ring resonators working near an exceptional point and show unusual chiral power flow from linearly polarized emitters, in agreement with recent experiments, which is quantitatively explained without invoking the interpretation of a missing dimension (the Jordan vector) and a decoupling from the cavity eigenmodes.

physics.optics↗

arXiv2020

Fluctuation-dissipation theorem and fundamental photon commutation relations in lossy nanostructures using quasinormal modes

We provide theory and formal insight on the Green function quantization method for absorptive and dispersive spatial-inhomogeneous media in the context of dielectric media. We show that a fundamental Green function identity, which appears, e.g., in the fundamental commutation relation of the electromagnetic fields, is also valid in the limit of non-absorbing media. We also demonstrate how the zero-point field fluctuations yields a non-vanishing surface term in configurations without absorption, when using a more formal procedure of the Green function quantization method. We then apply the presented method to a recently developed theory of photon quantization using quasinormal modes [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)] for finite nanostructures embedded in a lossless background medium. We discuss the strict dielectric limit of the commutation relations of the quasinormal mode operators and present different methods to obtain them, connected to the radiative loss for non-absorptive but open resonators. We show exemplary calculations of a fully three-dimensional photonic crystal beam cavity, including the lossless limit, which supports a single quasinormal mode and discuss the limits of the commutation relation for vanishing damping (no material loss and no radiative loss).

cond-mat.mes-hall↗

arXiv2020

Quantized quasinormal mode description of non-linear cavity QED effects from coupled resonators with a Fano-like resonance

We employ a recently developed quantization scheme for quasinormal modes (QNMs) to study a nonperturbative open cavity-QED system consisting of a hybrid metal-dielectric resonator coupled to a quantum emitter. This hybrid cavity system allows one to explore the complex coupling between a low $Q$ (quality factor) resonance and a high $Q$ resonance, manifesting in a striking Fano resonance, an effect that is not captured by traditional quantization schemes using normal modes or a Jaynes-Cummings (JC) type model. The QNM quantization approach rigorously includes dissipative coupling between the QNMs, and is supplemented with generalized input-output relations for the output electric field operator for multiple modes in the system, and correlation functions outside the system. The role of the dissipation-induced mode coupling is explored in the strong coupling regime between the photons and emitter beyond the first rung of the JC dressed-state ladder. Important differences in the quantum master equation and input-output relations between the QNM quantum model and phenomenological dissipative JC models are found. In a second step, numerical results for the Fock distributions and system as well as output correlation functions obtained from the quantized QNM model for the hybrid structure are compared with results from a phenomenological approach. We demonstrate explicitly how the quantized QNM model manifests in multiphoton quantum correlations beyond what is predicted by the usual JC models.

cond-mat.mes-hall↗

arXiv2019

Efficient near-field to far-field transformations for quasinormal modes of optical cavities and plasmonic resonators

We describe an efficient near-field to far-field transformation for optical quasinormal modes, which are the dissipative modes of open cavities and plasmonic resonators with complex eigenfrequencies. As an application of the theory, we show how one can compute the reservoir modes (or regularized quasinormal modes) outside the resonator, which are essential to use in both classical and quantum optics. We subsequently demonstrate how to efficiently compute the quantum optical parameters necessary in the theory of quantized quasinormal modes [Franke et al., Phys. Rev. Lett. 122, 213901 (2019)]. To confirm the accuracy of our technique, we directly compare with a Dyson equation approach currently used in the literature (in regimes where this is possible), and demonstrate several order of magnitude improvement for the calculation run times. We also introduce an efficient pole approximation for computing the quantized quasinormal mode parameters, since they require an integration over a range of frequencies. Using this approach, we show how to compute regularized quasinormal modes and quantum optical parameters for a full 3D metal dimer in under one minute on a standard desktop computer. Our technique is exemplified by studying the quasinormal modes of metal dimers and a hybrid structure consisting of a gold dimer on top of a photonic crystal beam. In the latter example, we show how to compute the quantum optical parameters that describe a pronounced Fano resonance, using structural geometries that cannot practically be solved using a Dyson equation approach. All calculations for the spontaneous emission rates are confirmed with full-dipole calculations in Maxwell's equations and are shown to be in excellent agreement.

physics.optics↗

arXiv2019

Theory and limits of on-demand single photon sources using plasmonic resonators: a quantized quasinormal mode approach

Quantum emitters coupled to plasmonic resonators are known to allow enhanced broadband Purcell factors, and such systems have been recently suggested as possible candidates for on-demand single photon sources, with fast operation speeds. However, a true single photon source has strict requirements of high efficiency (brightness) and quantum indistinguishability of the emitted photons, which can be quantified through two-photon interference experiments. To help address this problem, we employ and extend a recently developed quantized quasinormal mode approach, which rigorously quantizes arbitrarily lossy open system modes, to compute the key parameters that accurately quantify the figures of merit for plasmon-based single photon sources. We also present a quantized input-output theory to quantify the radiative and nonradiative quantum efficiencies. We exemplify the theory using a nanoplasmonic dimer resonator made up of two gold nanorods, which yields large Purcell factors and good radiative output beta factors. Considering an optically pulsed excitation scheme, we explore the key roles of pulse duration and pure dephasing on the single photon properties, and show that ultrashort pulses (sub-ps) are generally required for such structures, even for low temperature operation. We also quantify the role of the nonradiative beta factor both for single photon and two-photon emission processes. Our general approach can be applied to a wide variety of plasmon systems, including metal-dielectrics, and cavity-waveguide systems, without recourse to phenomenological quantization schemes.

quant-ph↗

arXiv2018

Quantization of quasinormal modes for open cavities and plasmonic cavity-QED

We introduce a second quantization scheme based on quasinormal modes, which are the dissipative modes of leaky optical cavities and plasmonic resonators with complex eigenfrequencies. The theory enables the construction of multi-plasmon/photon Fock states for arbitrary three-dimensional dissipative resonators and gives a solid understanding to the limits of phenomenological dissipative Jaynes-Cummings models. In the general case, we show how different quasinormal modes interfere through an off-diagonal mode coupling and demonstrate how these results affect cavity-modified spontaneous emission. To illustrate the practical application of the theory, we show examples using a gold nanorod dimer and a hybrid dielectric-metal cavity structure.

cond-mat.mes-hall↗