Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: I. Inherited and genuine singularities

TL;DR

The paper analyzes inherited and genuine singularities—quantum exceptional points (QEPs), diabolical points (QDPs), and hybrid points (QHPs)—in multimode bosonic systems described by quadratic non-Hermitian Hamiltonians. By exploiting a block structure in the Heisenberg-Langevin dynamics, the authors reduce the effective dimension and derive analytic expressions for eigenvalues and eigenvectors of the dynamical matrices up to five modes. They identify conditions under which second- and third-order QEPs occur as inherited features, with second-order diabolic degeneracies producing QHPs, and show how these degeneracies manifest in the dynamics of field-operator moments (FOMs) across first- and second-order moments, including ellipsoidal loci in parameter space. The study highlights rich, configuration-dependent degeneracy patterns (linear, circular, pyramid) and provides tables of QEP/QHP degeneracies, setting the stage for Part II to explore higher-order effects, partial PT-symmetry dynamics, and unidirectional coupling. Overall, the work reveals a systematic pathway to engineer and observe quantum degeneracies in pumped-dissipative bosonic networks with potential sensing and nonlinear enhancement implications.

Abstract

The existence and degeneracies of quantum exceptional, diabolical, and hybrid (i.e., diabolically degenerated exceptional) singularities of simple bosonic systems composed of up to five modes with damping and/or amplification are analyzed. Their dynamics governed by quadratic non-Hermitian Hamiltonians is followed using the Heisenberg-Langevin equations. Their dynamical matrices generally exhibit specific structures that allow for an effective reduction of their dimension by half. This facilitates analytical treatment and enables efficient spectral analysis based on characteristic second-order diabolical degeneracies. Conditions for the observation of inherited quantum hybrid points, observed directly in the dynamics of field operators, having up to third-order exceptional and second-order diabolical degeneracies are revealed. Surprisingly, exceptional degeneracies of only second and third orders are revealed, even though the systems with up to five modes are considered. Exceptional and diabolical genuine points and their degeneracies observed in the dynamics of second-order field-operator moments are also analyzed. Each analyzed bosonic system exhibits its own unique and complex dynamical behavior.

Figures (3)

• Figure 1: Schematic diagrams of the bosonic systems composed of (a) two, (b) three, (c---e) four, and (f,g) five modes with typical linear, circular, and pyramid configurations that exhibit quantum exceptional points (QEPs) and quantum hybrid points (QHPs) with various exceptional degeneracies (EDs) and diabolical degeneracies (DDs) observed in the dynamics of field-operator moments (FOMs) of different orders. Strengths $\epsilon$ and $\kappa$ characterize, respectively, the linear and nonlinear coupling between the modes, $\gamma$, with subscripts indicating the mode number(s), are the damping or amplification rates, and annihilation operators $\hat{a}$ identify the mode number via their subscripts, and $\gamma_{jk}$ indicates that $\gamma_j = \gamma_k$.
• Figure 2: Schematic diagram of the structure of eigenvalues $\Lambda_{j}^{M^{(n)}}$ and the corresponding eigenvectors $\bm{Y_{j}^{M^{(n)}}}$, $j=1,\ldots,2n$, of a general dynamical $2n\times 2n$ matrix $\bm{M^{(n)}}$ built from the eigenvalues $\lambda_{k}^{M^{(n)}}$ and the corresponding eigenvectors $\bm{y_{k}^{M^{(n)}}}$, $k=1,\ldots,n$, of the matrix $\bm{M^{(n)}}$ considered as an $n\times n$ matrix composed of $2\times 2$ submatrices at the positions of its elements; $n=2,3,\ldots$. The eigenvalues $\lambda^{\xi}_{1,2}$ and the corresponding eigenvectors $\bm{y^{\xi}_{1,2}}$ belong to the submatrix $\bm{\xi}$.
• Figure 3: Real parts $\lambda^{\rm r}$ of the eigenvalues (a) $\lambda_{1,2}^{M^{(2)}}$ of the matrix $\bm{M^{(2)}}$, given in Eq. (\ref{['5']}), for the two-mode bosonic system, (b) $\lambda_{1,2,3}^{M^{(3)}}$ of the matrix $\bm{M^{(3)}}$, given in Eq. (\ref{['23']}), for the three-mode linear bosonic system, (c) [(d)] $\lambda_{1,\ldots,4}^{M^{(4)}_{\rm l1}}$ [$\lambda_{1,\ldots,4}^{M^{(4)}_{\rm l2}}$] of the matrix $\bm{M^{(4)}_{\rm l1}}$ [$\bm{M^{(4)}_{\rm l2}}$], given in Eq. (\ref{['30']}) [(\ref{['34']})] for the four-mode linear bosonic system with neighbor modes having equal [different] damping and/or amplification rates, (e) $\lambda_{1,\ldots,4}^{M^{(4)}_{\rm c1}}$ of the matrix $\bm{M^{(4)}_{\rm c1}}$, given in Eq. (\ref{['40']}), for the four-mode circular bosonic system with neighbor modes having equal damping and/or amplification rates, (f) $\lambda_{2,\ldots, 5}^{M^{(5)}_{\rm l}}$ of the matrix $\bm{M^{(5)}_{\rm l}}$, given by Eq. (\ref{['47']}), for the five-mode linear bosonic system, and (g,h) $\lambda_{2,\ldots, 5}^{M^{(5)}_{\rm p}}$ of the matrix $\bm{M^{(5)}_{\rm p}}$, given in Eq. (\ref{['54']}), for the five-mode pyramid bosonic system assuming (g) $\beta_1=0$ and (h) $\beta_2=0$. The eigenvalues are drawn in the parameter space $(\kappa/\epsilon,\gamma_-/\epsilon)$, where $\epsilon$ ($\kappa$) is the linear (nonlinear) coupling strength and $\gamma_-$ the difference of the damping/amplification rates in individual models. Dashed red curves indicate the positions of the QHPs given by (a,e,g) Eq. (\ref{['13']}), (b) Eq. (\ref{['25']}), (c) Eq. (\ref{['32']}), (d) Eq. (\ref{['36']}), (f) Eq. (\ref{['49']}), and (h) Eq. (\ref{['56']}).