Quantum Quasinormal Mode Theory for Dissipative Nano-Optics and Magnetodielectric Cavity Quantum Electrodynamics

TL;DR

The work develops a unified quantum theory for quasinormal modes in dissipative magnetodielectric resonators by integrating macroscopic QED with exterior complex coordinate transformations (PMLs) to regularize and quantize modes. It constructs a complete, orthonormal modal basis, defines bosonic operators, and derives a Lindblad master equation describing mode–emitter dynamics, including gain/loss effects. Numerical demonstrations in 1D and 3D geometries validate the approach against exact macroscopic QED, capturing Purcell enhancements and strong-coupling phenomena with high fidelity. This framework provides a rigorous, general tool for predicting quantum light–matter interactions in contemporary nano-optical devices, with potential impact on room-temperature quantum technologies.

Abstract

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, that achieves QNM quantization for a wide class of spatially inhomogeneous, dissipative (with possible gain components) 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 towards a rigorous assessment of room-temperature quantum nanophotonic technologies without recourse to ad hoc quantization schemes.

Figures (5)

• Figure 1: Concept of the background-field formulation and the PML-based truncation of the domain used for QNM calculation, regularization and quantization. (a) Schematic of an example background, described by material parameters $(\varepsilon_\mathrm{b}, \mu_\mathrm{b})$, in which Maxwell's equations have to be satisfied by the background/incident field $\bm{E}_\mathrm{b}$ in the infinite space. The background can be non-trivial and contain substrates or waveguides. (b) The scattered-field problem in which Maxwell's equations have to be satisfied by $\bm{E}_\mathrm{sca}$ in the full geometry, including the resonator with compact domain $\Omega_\mathrm{res}$ and containing the effective current density arising from $\bm{E}_\mathrm{b}$. Suitable outgoing-wave conditions in the infinite space are applied. (c) The corresponding problem for $\bm{E}_\mathrm{sca}$ with PMLs imposed, in which the infinite space is truncated to a compact domain $\Omega$. Using a complex coordinate transformation, which leaves the inner domain $\Omega \setminus \Omega_\mathrm{PML}$ invariant and causes outgoing waves to be damped in $\Omega_\mathrm{PML}$, is equivalent to the introduction of an effective material $(\tilde{\varepsilon}, \tilde{\mu})$. The damping of the outgoing waves effectively converts the outgoing-wave condition to a Dirichlet condition (perfect electrical conductor) at infinity, and due to the exponential nature of the damping, this boundary condition can very well be approximated using a finite truncated domain $\Omega$ and imposing the condition at $\partial\Omega$.
• Figure 2: Schematic representation of the quantum description of an electric/magnetic dipolar QE, modeled as a TLS, interacting with an electromagnetic resonator using PMLs. (a) Most general scenario, in which the resonator could have electric and magnetic gain components in distinct spatial or spectral regions (e.g., a pair of coupled, spherical resonators composed of different materials). In this case, every eigenmode of the resonator (both QNMs and PML modes) is described by four bosonic operators $\hat{a}_{n,\lambda,X}$ with $\lambda= \mathrm{e, m}$ and $X = \mathrm{L, G}$, corresponding to differently symmetrized modes according to Eqs. \ref{['eqn:symmetrized_modes_general']} and \ref{['eqn:symmetrized_modes_general_gain']}. The presence of gain/loss in the system necessitates a symmetrization and the consequent coupling between the symmetrized modes described by $\chi_{n n'}^{\lambda,X,+}$ in the effective Hamiltonian of Eq. \ref{['eqn:H_eff']}. The losses of the system are described by Lindblad dissipators proportional to $\chi_{n n'}^{\lambda,\mathrm{L},-}$, as given in Eqs. \ref{['eqn:dissipator_general']}, which also involve coupling between the symmetrized modes. The gain components of the resonator are described by negative-energy contributions in the effective Hamiltonian. All the different bosonic modes couple to the QE with the coupling constants $g_{n,\lambda,X}$, as given in Eq. \ref{['eqn:QE_couplings_general']}. Additional non-radiative decay or dephasing processes of the QE, here labeled by $\Gamma_\mathrm{nr}$, can be treated in the usual way by adding corresponding Lindblad terms to the master equation. (b) Scenario in which the electric/magnetic and gain/loss modal operators can be combined intos a single operator per mode $\hat{a}_n$ (e.g., two spheres composed of passive, magnetodielectric media). In this case, the gain of the resonator is described by incoherent pumping terms proportional to $\tilde{\chi}_{n n'}^{\mathrm{G},-}$.
• Figure 3: Numerical exploration of a 1D, half-open, diamond cavity in air, with an electric-dipole QE embedded therein. (a) Schematic of the cavity-QE system with characteristic parameters indicated. The dashed line shows the electric field intensity profile of the second-order QNM. (b) Complex frequencies $\tilde{\omega}_n$ of the three, lowest-order QNMs (upper panel) and the frequency-dependent Purcell factor of the QE, calculated in accordance with the quantum QNM and semi-classical theories (lower panel).
• Figure 4: Spontaneous decay of a QE with $\omega_\mathrm{QE} = 1.95\,\mathrm{eV}$ and $|d|/\sqrt{\hbar \varepsilon_0 c A} = 0.3$ in the 1D, half-open cavity of Fig. \ref{['fig:1D_Cavity']}(a). (a) QE and symmetrized mode dynamics obtained by solving the QNM master equation, compared to the QE dynamics obtained from the integral equation Eq. \ref{['eqn:mQED_Integral_Eq']} in mQED. (b) Normalized output electric field intensity at $x = L$ obtained from the QNM master equation, compared to the exact mQED result from Eq. \ref{['eqn:I_out_mQED']}. Also shown is the electric field intensity calculated from the quantum QNM model retaining only the second-order mode $\hat{a}_2$.
• Figure 5: Numerical exploration of a 3D, spherical, Si cavity in air, with an electric-dipole QE embedded therein. (a) Schematic of the cavity-QE system with characteristic parameters indicated and two distinct coupling configurations, featuring a QE located at a radial distance of $r_{\bm{d}_\parallel} = 380\,\mathrm{nm}$ or $r_{\bm{d}_\perp} = 545\,\mathrm{nm}$ and with orthogonal dipole orientations. (b) Complex frequencies $\tilde{\omega}_{l m n}$ of the three considered $\mathrm{TM}_{l = 1}$ QNMs (upper panel) and the frequency-dependent contributions of these modes to the Purcell factor of the QE in each configuration, calculated in accordance with the quantum QNM and semi-classical theories (lower panel).