Spectroscopic Signatures of a Liouvillian Exceptional Spectral Phase in a Collective Spin

Abstract

Non-Hermitian degeneracies of Lindblad generators (Liouvillian exceptional points) can induce non-exponential relaxation and higher-order poles in dynamical response functions. A collective spin coupled to a polarized Markovian bath exhibits an \emph{exceptional spectral phase} in which defective Liouvillian modes imprint super-Lorentzian features in frequency-resolved spectra. We compute the emission spectrum via the Liouvillian resolvent, identify symmetry-sector selection rules, and demonstrate that exceptional-point signatures are strongly state-dependent: they are suppressed in steady-state fluorescence yet become unambiguous for generic (infinite-temperature or random) initial states. Our results provide an experimentally accessible spectroscopic diagnostic of many-body Liouvillian exceptional phases and clarify when steady-state emission can (and cannot) reveal them.

Figures (3)

• Figure 1: Emission spectra $S(\omega)$ computed from the Liouvillian resolvent for two representative values of the incoherent pumping parameter $p$. In each panel, spectra are shown for the steady state $\rho_{\mathrm{ss}}$, the infinite-temperature state $\rho_{\infty}=\mathbb{1}/d$, and a random reference state, together with fits to a single Lorentzian and to a Lorentzian-plus-superlorentzian model. For weak pumping ($p=0.2$, upper panel), all spectra are well described by a single Lorentzian, indicating isolated Liouvillian resonances. At stronger pumping ($p=0.9$, lower panel), the steady-state spectrum develops clear deviations from a single-Lorentzian lineshape, while the infinite-temperature and random-state spectra remain comparatively featureless. This behavior reflects the increasing role of collective and near-degenerate Liouvillian modes at large $p$.
• Figure 2: Emission spectra and Liouvillian exceptional-point diagnostics. (a) Steady-state emission spectrum $S(\omega)$ and Lorentzian fit. (b) Spectrum $S(\omega;\rho_0)$ for $\rho_0=\mathbb{I}/d$ with Lorentzian and Lorentzian+super-Lorentzian fits. (lower panel) Extracted EP weight $r$ and information-criterion difference $\Delta\mathrm{BIC}$.
• Figure 3: Liouvillian spectrum for a collective spin system with $j=20$, obtained by explicit vectorization of the master equation and direct diagonalization of the Liouvillian superoperator. The real and imaginary parts of the eigenvalues $\lambda$ are shown (rescaled by $j$) for three representative values of the incoherent pumping parameter: $p=0$, $p=0.5$, and $p=0.99$. All eigenvalues are plotted as black points, while eigenvalues that have at least one neighbor in the complex plane closer than $|\Delta\lambda|<10^{-6}$ are highlighted in red. The latter provide a numerical diagnostic for near-degeneracies of the Liouvillian spectrum, consistent with the emergence of exceptional-point--like behavior as $p$ is increased. Parameters are $h=1$, $\Gamma=0.1$, and $\Gamma_0=0$.