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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Semi-analytical Approach for Modeling Strong Coupling of Quantum Emitters in Electromagnetic Resonators

Mohammad Abutoama, George Kountouris, Jesper Mørk, Philip Trøst Kristensen

Published Jun 26, 2023• 0 citations
Publisher

Abstract

We theoretically examine the interaction between a single quantum emitter (QE) and a general electromagnetic resonator and provide a semi-analytical approach to calculate the coupling strength in the combined system. The model relies on the Lippmann-Schwinger equation to calculate the quasi-normal modes of the coupled system with no fitting parameters. This is in stark contrast to standard approaches in the literature, in which the properties of the coupled system are typically fitted from scattering spectra.

DOI
10.1109/cleo/europe-eqec57999.2023.10232532

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•Jul 7, 2021•1 citation
We theoretically examine the interaction between a single quantum emitter (QE) and a general electromagnetic resonator and provide a semi-analytical approach to calculate the coupling strength in the combined system. The model relies on the Lippmann-Schwinger equation to calculate the quasi-normal modes of the coupled system with no fitting parameters. This is in stark contrast to standard approaches in the literature, in which the properties of the coupled system are typically fitted from scattering spectra.
We present a semi-analytical framework for studying interactions between quantum emitters and general electromagnetic resonators. The method relies on the Lippmann-Schwinger equation to calculate the complex resonance frequencies of the coupled system based only on a single calculation for the electromagnetic resonator without the quantum emitter and with no fitting parameters. This is in stark contrast to standard approaches in the literature, in which the properties of the coupled system are fitted from calculated spectra. As an application example, we consider a recent dielectric cavity design featuring deep subwavelength confinement of light. We find the expected anti-crossing of the emitter and cavity resonance frequencies, and comparing to independent reference calculations, we find an extraordinary quantitative agreement with a relative error below one part in ten thousand. In order to unambiguously connect with the Jaynes-Cummings model, we derive an explicit expression relating the classical description of the emitter, as modeled by a spherical inclusion with a Lorentzian material response, to the dipole moment of the corresponding quantum optical model. The combined framework therefore enables classical calculations to be used for evaluating the coupling strength entering quantum optical theories in a transparent way.
We present a modal analysis of a popular modeling framework for studying light-matter interaction between quantum emitters and general electromagnetic resonators, in which the active materials are modeled via Lorentzian material responses. The method relies on the Lippmann-Schwinger equation to calculate the complex resonance frequencies of the coupled system based only on a single calculation for the electromagnetic resonator without the active material and without fitting to calculated spectra. As an application example, we consider a recent dielectric cavity design featuring deep subwavelength confinement of light. We find the expected anticrossing of the emitter and cavity resonance frequencies, and comparing with independent reference calculations, we find an extraordinary quantitative agreement with a relative error below one part in ten thousand. To connect with the Jaynes-Cummings model, we relate the classical description of the emitter to the dipole moment in the corresponding quantum optical model. The combined framework therefore enables classical calculations to be used for evaluating the coupling strength entering quantum optical theories in a transparent way. The methodology is expected to be useful in the design and interpretation of ongoing and future experiments and devices involving single-photon sources for quantum information and quantum communication technology.
We present a systematic study of the properties of systems composed of NNN two-level quantum emitters coupled to a single cavity mode, for light-matter interaction strengths ranging from the weak to the ultrastrong and deep-strong coupling regimes. Beginning with an analysis of the energy spectrum as a function of the light-matter coupling strength, we examine systems with varying numbers of emitters, from a pair to large collections, approaching the thermodynamic limit $(N\ensuremath{\rightarrow}\ensuremath{\infty})$. Additionally, we explore the emission properties of these systems under incoherent excitation of the emitters, employing a general theoretical framework for open cavity-QED systems, which is valid across all light-matter interaction regimes and preserves gauge invariance within truncated Hilbert spaces. Furthermore, we study the influence of the emitter-environment interaction on the spectral properties of the system. Specifically, when each emitter interacts independently with its own reservoir, we observe the emergence of an emission peak at the cavity's resonant frequency for even values of NNN. Our analysis also clarifies the evolution of the system as the number of emitters increases, ultimately converging toward an equivalent system composed of two interacting single-mode bosonic fields.
The interaction of an ensemble of NNN two-level quantum emitters with a single-mode electromagnetic field is described by the Tavis-Cummings model. There, the collectively enhanced light-matter coupling strength is given by gN=Ng‾1{g}_{N}=\sqrt{N}{\overline{g}}_{1}gN​=N​g​1​, where g‾1{\overline{g}}_{1}g​1​ is the ensemble-averaged single-emitter coupling strength. This model has been employed to describe and to analyze numerous cavity-based experiments. Here, we show that this is only justified if the effective scattering rate into noncavity modes is negligible compared to the cavity's free-spectral range. In terms of experimental parameters, this requires that the optical depth of the ensemble is low, a condition that is violated in several state-of-the-art experiments. We give quantitative conditions for the validity of the Tavis-Cummings model and derive a more general Hamiltonian description that takes into account the cascaded interaction of the photons with all consecutive emitters. We show that the predictions of our cascaded model can differ quantitatively and even qualitatively from those obtained with the Tavis-Cummings model. Finally, we present experimental data, for which the deviation from the predictions of the Tavis-Cummings model is apparent. Our findings are relevant for all experiments in which optically dense ensembles of quantum emitters are coupled to an optical resonator.
We provide a simple semi-classical formalism to describe the coupling between one or several quantum emitters and a structured environment. Describing the emitter by an electric polarizability, and the surrounding medium by a Green function, we show that an intuitive scattering picture allows one to derive a coupling equation from which the eigenfrequencies of the coupled system can be extracted. The model covers a variety of regimes observed in light-matter interaction, including weak and strong coupling, coherent collective interactions, and incoherent energy transfer. It provides a unified description of many processes, showing that different interaction regimes are actually rooted on the same ground. It can also serve as a basis for the development of more refined models in a full quantum electrodynamics framework.
Quantum cooperativity is evident in light-matter platforms where quantum emitter ensembles are interfaced with confined optical modes and are coupled via the ubiquitous electromagnetic quantum vacuum. Cooperative aspects such as dipole-dipole interactions and subradiance find applications in the design of nanoscale coherent light sources and highly-reflective quantum metasurfaces made up of hundreds of optically trapped atoms, in the implementation of topological quantum optics on subwavelength arrays of emitters, in quantum metrology and quantum information. The quick bursts of radiation from a collection of quasi-indistiguishable emitters provides an alternative approach to standard lasers by introducing superradiant lasers operating at extremely low intracavity power. This tutorial provides a set of theoretical tools to tackle the behavior responsible for the onset of cooperativity in light-matter systems by extending open quantum system dynamics methods, such as the master equation and quantum Langevin equations, to electron-photon interactions in strongly coupled and correlated quantum emitters ensembles. These analytical approaches are then also extended to frequency disordered or vibronically coupled quantum emitter ensembles with wide relevance ranging from atoms in optical lattices, quantum dots in solid state environments or molecular quantum systems.