Position-Resolved Resonance Quantization for Lossy Cavities

TL;DR

The paper addresses the challenge of modeling highly open, lossy cavities where traditional few-mode approaches fail to capture non-Markovian dynamics and spatially extended emitters.It introduces generalized pseudomodes (gPM) – a position-resolved, discrete-mode expansion of the cavity field – and ties them to a Lindblad-type dynamics for the quantum system, constrained by a frequency-domain matching condition to reproduce the continuum environment.Key developments include the Hermitization condition, a meromorphic pole expansion linked to quasi-normal modes, and a practical procedure to construct gPM parameters from QNMs, supported by an extended-index-domain solution and a positive-definite reconstruction via V.The method is demonstrated on a one-dimensional slab cavity, where the position-dependent spectral density is accurately reproduced (with M=30 terms), validating the approach and highlighting its potential for extended geometries and dispersive media.

Abstract

Modern experiments in resonators are moving to ever more extreme quantum regimes, posing major challenges to established theoretical approaches, such as so-called few-mode models. While these models have driven major insights for traditional regimes, they are now hitting their limitations for highly open cavities and extended systems, as encountered in cavity experiments with molecules and solid-state systems. Here, we present a novel method that significantly extends the conceptual underpinning of these discrete-mode models, promoting them to a systematic treatment. We develop an ansatz which allows to quantize the resonator's resonances with position-resolved discrete modes, thus naturally incorporating losses in the formalism. Such a construction effectively unifies key ideas from pseudomodes and quantized quasi-normal modes theory. We further present a criterion for construction of the ansatz parameters at every point in space, and semi-analytically benchmark the resulting solution for a paradigmatic one-dimensional example resonator.

Figures (6)

• Figure 1: Setup of the one-dimensional example cavity WeitzelThesis2025. The horizontal axis indicates the position coordinate $x$, while $n_{\mathrm{R}}$ and $n_{\mathrm{B}}$ are the refractive indices of the slab and its surroundings, respectively.
• Figure 2: Spectral density for the one-dimensional slab cavity depicted in Fig. \ref{['fig:fig_1-1D_example']}, for different frequencies and positions, considering $n_{\mathrm{R}}=4$ and $n_{\mathrm{B}}=1$WeitzelThesis2025. Only positions between the center ($x/L=0$) and the edge ($x/L=0.5$) of the cavity are shown, due to the symmetry of the latter. In (a), we show the exact spectral density obtained from the QNMs Green's function, via the correspondence in Eq. \ref{['eq:correlator_to_green']}. (b) depicts the spectral density obtained from a pole expansion constructed via Eq. \ref{['eq:hermitized_pole_correlator_1d']}, using $M=30$ terms in the summation. In (c), the difference between the spectral densities in (a) and (b) is shown. The colorbar is in units of $\hbar/(\epsilon_0 L)$.
• Figure 3: Spectral density for the one-dimensional example cavity depicted in Fig. \ref{['fig:fig_1-1D_example']}, as a function of frequency, choosing $n_{\mathrm{R}}=4$ and $n_{\mathrm{B}}=1$WeitzelThesis2025. The plot corresponds to a horizontal cross section of the spectral densities in Fig. \ref{['fig:fig_3-nR_4_2d_xw_comparison_exact_hermitized']} at position $x/L=0.15$, zooming into the region around $\omega L/c=0$. We used $M=30$ terms the expansion given by Eq. \ref{['eq:hermitized_pole_correlator_1d']}.
• Figure 4: Block corresponding to $\mu,\nu>0$ of the matrix solution $\tilde{T}^{\rm (slab)}_{\mu\nu}$, given by Eq. \ref{['eq:solution_1D']}, and of its Hermitian part $\tilde{T}^{\rm (slab),\,H}_{\mu\nu}$, given by Eq. \ref{['eq:solution_1d_hermitian']}WeitzelThesis2025. (a) and (b), respectively, show the real and imaginary parts of the former, while (c) and (d) show the real and imaginary parts of the latter. The parameters $n_{\mathrm{R}}=4$ and $n_{\mathrm{B}}=1$ were chosen. The colorbar indicates the numerical value of each matrix element, in units of $c/L$.
• Figure 5: Imaginary part of the one-sided Fourier transform of the advanced correlator for the one-dimensional cavity slab depicted in Fig. \ref{['fig:fig_1-1D_example']}, for different position arguments $x$, $x'$, and a fixed frequency WeitzelThesis2025. We consider $n_{\mathrm{R}}=4$, $n_{\mathrm{B}}=1$ and set $\omega L/c = 15$. In (a), we depict the exact computation from the Green's function, via Eq. \ref{['eq:correlator_to_green']}; in (b), the approximation via Eq. \ref{['eq:hermitized_pole_correlator_1d']}, using $M=30$ terms, and, in (c), the difference between the two previous plots. The colorbar is in units of $\hbar/(\epsilon_0 L)$.