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.
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.
Devices involving bound or propagating photons are among the basic building blocks for modern quantum technologies. For example, the bound modes of open optical resonators, coupled to quantum emitters such as atoms, molecules, or quantum dots, are used for an enhanced emission in lasing applications, for increased sensitivity in quantum sensing, and as qubits in quantum computing. Meanwhile, propagating photons are crucial for the transmission of quantum states between spatially separated systems. In this thesis, we develop a rigorous theory of multiple, spatially separated, and individually addressable cavities (or plasmonic resonators) for applications in modern quantum devices. Losses are modeled via the use of quasinormal modes (QNMs), which are solutions to the source-free Helmholtz equation under open boundary conditions, making them the natural modes of open resonators. Quantum emitters are modeled as two-level systems (TLSs) which couple to the electric field via dipole coupling. The multi-cavity theory is developed in three steps: 1. We present a classical scattering theory for a system of multiple QNM cavities, where the multi-cavity Green's function is formulated in terms of the single-cavity modes. The inter-cavity scattering is frequency-dependent to account for retardation delays, and is efficiently calculated numerically, using common (commercial) solvers. 2. We formulate a quantization scheme for QNMs of spatially separated open resonators. We introduce the \textit{cavity separation parameter} as a simple measure for when QNM cavities can be quantized separately and when they have to be treated as a single system with a joint set of operators. We derive a general system-bath Hamiltonian that contains coupling between bound modes, quantum emitters, and a bath of propagating photons, expressed via rigorous, numerically calculable coupling parameters. 3. We discuss the time-dependent theory of QNMs and TLSs coupled to propagating photons. The system-bath interaction is formulated in terms of correlation functions that are again rigorously defined in terms of numerically calculable QNM parameters. The correlation functions are immediately applicable to common methods for calculating the dynamics of open quantum systems. Steps 1-3 are all formulated for general cavity shapes and materials, as well as (in principle) arbitrary background structures. Hence, the theory is applicable to a wide range of devices, including cavities or plasmonic resonators in free space, on a photonic crystal, or on a substrate. For each step, we show example calculations for one-dimensional or three-dimensional cavities.
Open optical or plasmonic resonators are placed on and connected through surfaces or via waveguides, forming complex lightguiding nanostructures, e.g. for integrated photonic quantum devices. We derive general boundary conditions for quasinormal modes that account for the structure's specific geometry. We then present a general quantization scheme for multiple, interacting quasinormal-mode cavities coupled to quantum emitters and to a non-bosonic bath of propagating photons on waveguides or a surface. We derive a system-bath Hamiltonian with rigorously defined coupling elements that can be computed using Maxwell solvers, including light-matter coupling between the electromagnetic field and quantum emitters. We define system-bath correlation functions for an effective, bath-mediated, and time-delayed interaction between the quasinormal modes and quantum emitters, which is a main ingredient commonly used to simulate open quantum system dynamics.
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 [S. 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, modeled 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, modeled 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.
Cavity quantum magnonics has ushered in an unprecedented era of hybrid light-matter interfaces in which microwave photons, collective spin excitations, and superconducting or spin qubits can be coupled within a single device architecture, facilitating a versatile platform for quantum information processing. Realizing their full potential, however, requires a theory that can treat non-reciprocal magnetic media, material dissipation and open-cavity radiation losses, as well as finite-temperature fluctuations rigorously and without significant approximation. The dissipative magnonic resonances of such systems are described as quasinormal modes (QNMs), yet a consistent quantization procedure for lossy gyrotropic resonators has remained elusive. In this work, we present a finite-temperature quantum QNM theory, based on macroscopic quantum electrodynamics for non-reciprocal media and complex coordinate transformations, that achieves QNM quantization for a wide class of spatially inhomogeneous, dissipative and dispersive, cavity magnonic systems. We demonstrate the predictive capabilities of our theory via applications to hybrid dielectric-magnonic resonators interacting with both spin and superconducting qubits, evidencing modal interference effects, a continuous transition from coherent to dissipative coupling, spatiotemporal quantum field dynamics, and magnon-mediated qubit synchronization, at both zero and finite temperatures. By addressing the intricacies of modal loss and non-reciprocity in a fully quantum theory of cavity magnonics, our approach enables the quantitative design of hybrid quantum devices that operate under realistic open-system and thermal conditions, without recourse to ad hoc quantization schemes.
Abstract This work introduces a theoretical framework to model the collective dynamics of quantum emitters in highly non-Markovian environments, interacting through the exchange of photons with significant retardations. The formalism consists on a set of coupled delay differential equations for the emitter’s raising/lowering operators σ i ± , supplemented by input–output relations that describe the field mediating the interactions. These equations capture the dynamics of both linear (bosonic) and nonlinear (two-level) emitter arrays. It is exact in some limits—e.g. bosonic emitters or generic systems with up to one collective excitation—and can be integrated to provide accurate results for larger numbers of photons. These equations support a study of collective spontaneous emission of emitter arrays in open waveguide-QED environments. This study uncovers an effect we term cascaded super- and sub-radiance, characterized by light-cone-limited propagation and increasingly correlated photon emission across distant emitters. The collective nature of this dynamics for two-level systems is evident both in the enhancement of collective emission rates, as well as in a superradiant burst with a faster than linear growth. While these effects should be observable in existing circuit QED devices or slight generalizations thereof, the formalism put forward in this work can be extended to model other systems such as network of quantum emitters or the generation of correlated photon states.