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.
We demonstrate a fundamental breakdown of the photonic spontaneous emission (SE) formula derived from Fermis golden rule, in absorptive and amplifying media, where one assumes the SE rate scales with the local photon density of states, an approach of
ten 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 Fermis 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.
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 quan
titatively good model to a full dipole spontaneous emission response in Maxwells 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.
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 t
he 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).
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.
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 c
an 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 Maxwells equations and are shown to be in excellent agreement.
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 t
rue 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.
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 s
tates 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.
