We develop a numerical framework to analyze the dynamics of a dissipative cavity quantum electrodynamics without the Markovian approximation. For the correct quantization of electromagnetic fields in the presence of lossy dielectric objects in the free space, we employ the so-called BAMA formulation [1] which takes into account fluctuations to both radiation and medium losses. With the coarse-graining strategy, the BAMA fields (or modes) can be evaluated by using conventional computational electromagnetics numerical modes, e.g., FEM, FDTD, or MoM. We then incorporate the BAMA modes into the multimode Jaynes-Cummings model and solve the time-dependent Schrödinger equation to find probability amplitudes.
We develop a numerical framework to analyze the dynamics of a dissipative cavity quantum electrodynamics without the Markovian approximation. For the correct quantization of electromagnetic fields in the presence of lossy dielectric objects in the free space, we employ the so-called BAMA formulation [1] which takes into account fluctuations to both radiation and medium losses. With the coarse-graining strategy, the BAMA fields (or modes) can be evaluated by using conventional computational electromagnetics numerical modes, e.g., FEM, FDTD, or MoM. We then incorporate the BAMA modes into the multimode Jaynes-Cummings model and solve the time-dependent Schrödinger equation to find probability amplitudes.
Quantization of lossy electromagnetic (EM) systems is a challenging task since the standard (eigen)mode decomposition approach [1], [2] is no longer available. The use of eigenmodes from non-Hermitian systems does not preserve the equal-time commutators; consequently, the Heisenberg uncertainty principle breaks down. Recently, we have proposed the so-called BAMA formulation [3] to account for the radiation and medium loss effects in the quantization of electromagnetic fields based on the fluctuation-dissipation theorem and Green's function approach [4]. In this presentation, we show how to incorporate the BAMA formulation into the multimode Jaynes-Cummings model to describe the dynamics of two-level systems placed inside a lossy cavity from the first principle, i.e., without the master equation based on the Markovian approximation. s
We present a novel numerical framework that integrates the modified Langevin noise formalism into the multimode Jaynes- and Tavis-Cummings models, enabling a first-principles, non-Markovian analysis of atom-field interactions in dissipative electromagnetic (EM) environments that account for both radiative losses and absorptive dissipation in lossy dielectric media (or satisfying general inhomogeneous causal media exhibiting both dispersion and absorption effects). In the modified Langevin noise formalism, the boundary- and medium-assisted (BA and MA) fields, which constitute a continuum set of EM modes in dissipative EM environments, are numerically obtained using the finite-element method (FEM). Specifically, BA field modes are extracted by solving plane-wave scattering problems, while MA field modes are determined through point-source radiation problems. These numerically obtained BA and MA field modes are then incorporated into the multimode Jaynes- and Tavis-Cummings models such that the coupling strength between atoms and BA-MA field modes can be calculated for the study of atom-field interactions in dissipative EM environments. The proposed methodology captures non-Markovian atomic dynamics that cannot be described by traditional quantum master equations under the Markovian approximation. To validate the accuracy of the proposed numerical framework, we present four numerical examples: (i) a two-level system (TLS) in a perfect electric conductor (PEC) half-space; (ii) dissipative cavity electrodynamics with two limiting cases approaching spontaneous emission in free space and ideal Rabi oscillations; (iii) super-radiance in TLS arrays; and (iv) entanglement sudden death of two TLSs inside dissipative cavities. The proposed methodology can serve as a ground-truth numerical simulator for studying atom-field interactions in general dissipative EM environments.
We present a numerical framework that integrates a modified Langevin noise formalism into multimode Jaynes- and Tavis-Cummings models for first-principles, non-Markovian analysis of atom-field interactions in dissipative electromagnetic environments under weak coupling. This formalism enables the quantization of electric field operators in terms of boundary- and medium-assisted modes with real-valued eigenfrequencies, accounting for both radiative and absorptive losses. To efficiently handle the continuum nature of the boundary- and medium-assisted mode spectrum, we apply adaptive frequency refinement and mode-space coarse-graining based on finite-element simulations to obtain a tractable discrete set of field modes. These modes are then used to construct matrix representations of the Schr\"odinger equation, enabling direct computation of atomic observables and field quantities---such as single-photon amplitudes and higher-order correlation functions---beyond the Markovian approximation. Importantly, this framework does not rely on any phenomenologically designed memory kernel. Instead, the spectrum of boundary- and medium-assisted modes interacting with atoms inherently and rigorously captures memory effects arising from the open and dissipative electromagnetic environment. We present four numerical examples to validate the proposed framework and demonstrate its ability to model non-Markovian atomic dynamics in open and dissipative electromagnetic environments: (i) a two-level system near the surface of a perfect electric conductor, (ii) dissipative cavity quantum electrodynamics for a single two-level system, (iii) super- and subradiance in two-level system arrays, and (iv) entanglement sudden death between two two-level systems. This framework provides a versatile simulation tool for the design of quantum photonic devices, including nitrogen-vacancy center single-photon sources coupled to plasmonic structures, entangled photon-pair generation near lossy media, and decoherence modeling of solid-state quantum memories based on rare-earth ions and color centers. It is particularly well suited to dissipative quantum electrodynamics in photonic integrated circuits, where accurate modeling of scattering and diffraction is essential.
In this work, we present a numerical framework for modeling single photon emission from a two level system in open and dissipative systems beyond the Markovian approximation. The method can be readily integrated into standard computational electromagnetic (CEM) solvers such as finite difference time domain (FDTD) and finite element method (FEM). We numerically verify the completeness of boundary and medium assisted modes in the modified Langevin noise formalism by reconstructing the imaginary part of the dyadic Greens function through modal expansion in three dimensions. This reconstruction enables a first principles description of atom field interaction via the multi mode Jaynes Cummings model in open and dissipative environments. Within the single excitation manifold, we show that the memory kernel of a two level system is determined by the imaginary part of the Greens function, implying that radiative modes alone govern the relevant dynamics. The proposed framework thus provides a Greens function based approach for describing atomic population and single photon dynamics, directly compatible with Maxwell solvers. We then present concrete strategies for implementing our method in both FDTD and FEM frameworks, demonstrating its practical applicability. We further verify numerical results for a lossy Lorentz Drude type mirror, including both the case of a TLS near a finite sized metallic mirror and that of a TLS centered in a Fabry Perot cavity. This work establishes a rigorous foundation for incorporating quantum emitter dynamics into computational electromagnetics, thereby extending classical solvers toward quantum light matter interactions.
The modified Langevin noise formalism [A. Drezet, Phys. Rev. A 95, 023831 (2017); O. D. Stefano, S. Savasta, and R. Girlanda, J. Mod. Opt. 48, 67 (2001)] has been proposed for the correct characterization of quantum electromagnetic fields in the presence of finite-size lossy dielectric objects in free space. The main modification to the original one [T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818 (1996); H. T. Dung, L. Kn\"oll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998)] (also known as the Green's function approach for a bulk inhomogeneous lossy dielectric medium) was to introduce another fluctuating source in reaction to the radiation loss. Consequently, the resulting electric field operator is now determined by (i) boundary-assisted and (ii) medium-assisted fields on an equal footing, which originate from radiation and medium losses, respectively. However, due to the lengthy mathematical manipulation and complicated concepts, the validity of the modified Langevin noise formalism has not been clearly confirmed yet. In this work, we propose and develop a numerical framework for the modified Langevin noise formalism by exploiting computational electromagnetic methods. Specifically, we present utilization of the finite-element method to numerically solve plane-wave-scattering and point-source-radiation problems whose solutions are boundary-assisted and medium-assisted fields, respectively. We numerically validated the modified Langevin noise model calculating the Purcell factor of a two-level atom inside or outside a lossy dielectric slab. The proposed numerical framework is particularly useful for analyzing the dynamics of multilevel atoms near plasmonic structures or metasurfaces in the open space.
The unprecedented pace of evolution in nanoscale architectures for cavity quantum electrodynamics (cQED) has posed crucial challenges for theory, where the quantum dynamics arising from the non-perturbative dressing of matter by cavity electric and magnetic fields as well as the fundamentally non-Hermitian character of the system are to be treated without significant approximation. The lossy electromagnetic resonances of photonic, plasmonic, or magnonic nanostructures are described as quasinormal modes (QNMs), whose properties and interactions with quantum emitters and spin qubits are central to the understanding of dissipative nano-optics and magnetodielectric cQED. Despite recent advancements toward a fully quantum framework for QNMs, a general and universally accepted approach to QNM quantization for arbitrary linear media remains elusive. In this work, we introduce a unified theoretical framework, based on macroscopic QED and complex coordinate transformations, which achieves QNM quantization for a wide class of spatially inhomogeneous, dissipative and dispersive, linear, magnetodielectric resonators. The complex coordinate transformations equivalently convert the radiative losses into non-radiative material dissipation, and via a suitable transformation that reflects all the losses of the resonator, we define creation and annihilation operators that allow the construction of modal Fock states for the joint excitations of field-dressed matter. By directly addressing the intricacies of modal loss in a fully quantum theory of magnetodielectric cQED, our approach enables the exploration of modern, quantum nano-optical experiments utilizing dielectric, plasmonic, magnetic, or hybrid cQED architectures and paves the way toward a rigorous assessment of room-temperature, quantum nanophotonic technologies without recourse to ad hoc quantization schemes.