Bridging quantum noise and classical electrodynamics with stochastic methods
Felix Hitzelhammer, Johannes Stowasser, Lukas Hanschke, Katarina Boos, Tobias C. Sutter, Michael Haider, Christian Jirauschek, Kai Müller, Gabriela M. Slavcheva, Ulrich Hohenester
The development of emerging technologies in quantum optics demands accurate models that faithfully capture genuine quantum effects. Mature semiclassical approaches reach their limits when confronted with quantized electromagnetic fields, while full Hilbert space treatments are often computationally prohibitive. To address these challenges, we develop a framework based on coupled stochastic processes with a common cross-covariance structure that can be easily coupled to various types of Maxwell solvers. Our approach accounts for the non-commutativity in the quantum-to-classical transition in a natural way, and has the ability to capture quantum optical signatures while retaining compatibility with classical electromagnetics. For benchmarking, we compare our simulation results with experimental emission spectra of a strongly driven InGaAs quantum dot, finding excellent agreement. Our results highlight the potential of tailored stochastic processes for simulating non-classical light in complex photonic environments.
The development of emerging technologies in quantum optics demands accurate models that faithfully capture genuine quantum effects. Mature semiclassical approaches reach their limits when confronted with quantized electromagnetic fields, while full Hilbert space treatments are often computationally prohibitive. To address these challenges, we develop a framework based on coupled stochastic processes with a common cross-covariance structure that can be easily coupled to various types of Maxwell solvers. Our approach accounts for the non-commutativity in the quantum-to-classical transition in a natural way, and has the ability to capture quantum optical signatures while retaining compatibility with classical electromagnetics. For benchmarking, we compare our simulation results with experimental emission spectra of a strongly driven InGaAs quantum dot, finding excellent agreement. Our results highlight the potential of tailored stochastic processes for simulating non-classical light in complex photonic environments.
We present a full-wave Maxwell-density matrix simulation tool including c-number stochastic noise terms for the modeling of the spatiotemporal dynamics in active photonic devices, such as quantum cascade lasers and quantum dot structures. The coherent light–matter interaction in such devices plays an important role in the generation of frequency combs and other nonlinear and non-classical optical phenomena. Since the emergence of nonlinear and non-classical features is directly linked to the noise properties, detailed simulations of the noise characteristics are required for the development of low-noise quantum optoelectronic sources. Our semiclassical simulation framework is based on the Lindblad equation for the electron dynamics, coupled with Maxwell’s equations for optical propagation in the laser waveguide. Fluctuations arising from the interactions of the optical field and quantum system with their reservoirs are treated within the quantum Langevin theory. Here, the fluctuations are included by adding stochastic c-number terms to the Maxwell-density matrix equations. The implementation in the mbsolve dynamic simulation framework is publicly available.
Spontaneous light emission is central to a vast range of physical systems and is a founding pillar for the theory of light–matter interactions. In the presence of complex photonic media, the description of spontaneous light emission usually requires advanced theoretical quantum optics tools such as macroscopic quantum electrodynamics, involving quantized electromagnetic fields. Although rigorous and comprehensive, the complexity of such models can obscure the intuitive understanding of many quantum-optical phenomena. Here, we review a method for calculating spontaneous emission and other quantum-optical processes without making explicit use of quantized electromagnetic fields. Instead, we introduce the concept of transition currents, comprising charges in matter that undergo transitions between initial and final quantum states. We show how predictions that usually demand advanced methods in quantum electrodynamics or quantum optics can be reproduced by feeding these transition currents as sources to the classical Maxwell equations. One then obtains the relevant quantum observables from the resulting classical field amplitudes, without washing out quantum optical effects. We show that this procedure allows for a straightforward description of quantum phenomena, even when going beyond the dipole approximation and single emitters. As illustrative examples, we calculate emission patterns and Purcell-enhanced emission rates in both bound-electron and free-electron systems. For the latter, we derive cathodoluminescence emission and energy-loss probabilities of free electrons interacting with nanostructured samples. In addition, we calculate quantum-beat phenomena in bound-electron systems and wave function-dependent optical coherence in free-electron systems. Remarkably, the transition-current formalism captures more complex phenomena, such as many-body interference effects and super-radiance of both bound- and free-electron systems, second-order processes such as two-photon emission, and quantum recoil corrections to free-electron radiation. We review a variety of light–matter interactions in fields ranging from electron microscopy to nanophotonics and quantum optics, for which the transition-current theoretical formalism facilitates practical simulations and a deeper understanding of novel applications.
A light-matter interaction modified by the material environment is one of the central topics in quantum electrodynamics. While a strong coupling between a single emitter and a cavity and the Markovian (exponential) relaxation regime are most straightforwardly covered by theory, real physical systems that include also various line broadening effects may possess a much more complicated behavior. Here we propose a theoretical framework to account for nonradiative interaction effects in emission in photonic and plasmonic cavities. The quantum electrodynamics model formulated via a stochastic Hamiltonian approach has been developed with nonradiative line broadening introduced via the Kubo oscillator model. The impact of competing radiative and nonradiative processes on the emitter dynamics has been studied, showing that nonradiative relaxations, having significant impact on processes in high-Q photonic cavities, are much less influential in the plasmonic regime. The developed theoretical framework is not restricted to the emitter in a cavity example, but represents a general tool for multiple stochastic Hamiltonian evolution, important for various types of interactions where either classical or quantum noise is present.
The last two decades have witnessed increasing experimental interest in the study and control of many-electron systems strongly interacting with quantum electromagnetic fields. This includes notable experiments in the areas of cavity and circuit quantum electrodynamics, quantum computing via photon mediated atom entanglement, electromagnetically induced transparency, quantum plasmonics, quantum simulators and chemistry. The description of realistic coupled matter-photon systems requires combining electronic structure methods from material science with quantum-optical methods. In this work, we propose a formally exact and numerically feasible approach, termed quantum electrodynamical time-dependent density functional theory (QED-TDDFT), that generalizes successful TDDFT for electronic structure calculations, to the electron-photon coupling. We develop a general framework for the description of coupled matter-photon systems in all possible realizations of interest, ranging from the fully relativistic case, to well-known quantum-optical model (e.g., the Rabi model). Of particular interest for condensed-matter applications is the density functionalisation of non-relativistic QED. The underlying theory in fact corresponds to standard non-relativistic quantum mechanics for the electrons, but with additional terms, which correct for relativity and electron-photon interactions. However, any application of density-functional theory requires approximations to the exchange-correlation (xc) functional. In this work we construct such an approximation for the description of electron-photon interaction effects in an optical cavity (the theory of reference is QED-TDDFT for many-electron systems coupled to cavity photons). The derived electron-photon optimized effective potential (OEP) reduces in the static limit to the Lamb shift of the ground state. The OEP is then tested on the Rabi model from weak to strong coupling regimes. The results are promising and open a path for describing complex strongly coupled matter-photon systems. Further, we consider the inclusion of relativistic spin-spin effects into the xc functional of standard spin density functional theory. In this way we provide quantum corrections that are essential for the description of magnetic inhomogeneities at the nanoscale. Such an approach improves over the current (classical) micromagnetic treatment of domain walls and skyrmions for spintronic applications.
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.
One of the most intriguing quantum phenomena of light is the wave–particle duality which is related to the inability of the classical wave and particle concepts to fully describe the behavior of quantum objects. Theoretically, the wave features of the electromagnetic field, such as interference, follow from the solution of Maxwell's equations while particle features are a consequence of the field quantization, in which the electromagnetic field is described to consist of discrete energy packets, photons. Due to the wave–particle duality, for example, simultaneous description of optical absorption, emission, and interference of propagating fields has conventionally not been considered feasible using local optical models but has required using Maxwell's equations with stochastic source terms accounting for the wave–particle duality. Also the description of the momentum of light in a medium has been problematic.
In the research described in this thesis, the quantized fluctuational electrodynamics (QFED) model is derived based on combining the electromagnetic Green's functions to the field quantization to describe the position dependence of the photon number and local thermal balance in general non-equilibrium conditions, and to separate the electromagnetic field into parts propagating in different directions in resonant structures, which has conventionally been problematic due to interference effects. The QFED method is shown to resolve the previously found anomalies in the canonical commutation relations of photon ladder operators. The QFED method is also used to derive quantum optical field–matter interaction parameters fully capturing the interference, emission, damping, and scattering of propagating photons in stratified media leading to an interference-exact radiative transfer equation (RTE) model. Therefore, the derived interaction parameters solve the problem of simultaneously describing interference and losses in the widely used RTE framework.
The particle aspects of light are also studied by directly applying the conservation laws of nature and the special theory of relativity to show that light propagating in a medium must be described as mass-polariton (MP) quasiparticles, covariant coupled states of the field and matter. These quasiparticles are shown to have momentum of the Minkowski form and a nonzero mass that is carried by a mass density wave associated with light. The field–matter interaction related to the MP quasiparticles also leads to the dissipation of photon energy. These particle model results are also derived using a wave-based optoelastic continuum dynamics (OCD) approach following from the electrodynamics of continuous media and the continuum mechanics. The obtained results verify the full agreement between the wave and particle models when they are correctly applied. The key finding that a light pulse propagating in a medium is inevitably associated with an experimentally measurable mass implies a fundamental change in the prevailing perceptions of light.; Eras kiehtovimmista valon…