Multiple quantum exceptional, diabolical, and hybrid points in multimode bosonic systems: II. Nonconventional PT-symmetric dynamics and unidirectional coupling

TL;DR

The work probes QEPs, QHPs, and diabolical points in multimode bosonic systems under nonconventional PT-symmetric dynamics and unidirectional coupling. It combines analytical diagonalization of Heisenberg-Langevin dynamics with numerical Jordan-form techniques to locate degeneracies, showing that higher-order EDs can be engineered by concatenating simple subsystems, though such constructions are valid mainly in short-time regimes due to reservoir limitations. A general framework links the order of field-operator moments to the underlying spectral degeneracies, revealing that a second-order ED in first-order moments can yield a fourth-order ED in second-order moments, and so on. These results extend the understanding of PT-symmetric bosonic systems, offering routes to control spectral singularities while highlighting practical limitations for long-time, physically consistent realizations.

Abstract

We analyze the existence and degeneracies of quantum exceptional, diabolical, and hybrid points in simple bosonic systems - comprising up to six modes with damping and/or amplification - under two complementary scenarios to those described in Quantum 9, 1932 (2025): (i) nonconventional PT-symmetric dynamics confined to a subspace of the full Liouville space, and (ii) systems featuring unidirectional coupling.} The system dynamics described by quadratic non-Hermitian Hamiltonians is governed by the Heisenberg-Langevin equations. Conditions for the observation of inherited quantum hybrid points with up to sixth-order exceptional and second-order diabolical degeneracies are revealed, though relevant only for short-time dynamics. This raises the question of whether higher-order inherited singularities exist in bosonic systems under general conditions. Nevertheless, for short times, unidirectional coupling of various types enables the concatenation of simple bosonic systems with second- and third-order exceptional degeneracies such that arbitrarily high exceptional degeneracies are reached. Methods for numerical identifying the quantum exceptional and hybrid points together with their degeneracies are addressed. Following Quantum 9, 1932 (2025) rich dynamics of second-order field-operator moments is analyzed from the point of view of the presence of exceptional and diabolical points and their degeneracies.

Figures (6)

• Figure 1: Schematic diagrams of the four-mode bosonic systems in (a) circular and (b) tetrahedral configurations that exhibit quantum exceptional points (QEPs) and quantum hybrid points (QHPs) with various exceptional degeneracies (EDs) and diabolical degeneracies (DDs) observed in their nonconventional $\mathcal{PT}$-symmetric dynamics of field-operator moments (FOMs) of different orders. Strengths $\epsilon$ and $\kappa$ characterize, respectively, the linear and nonlinear coupling between the modes, while $\gamma$, with subscripts indicating the mode number(s), are the damping or amplification rates, and the annihilation operators $\hat{a}$ identify the mode number via their subscripts.
• Figure 2: Real parts $\lambda^{\rm r}$ of the eigenvalues (a) $\lambda_{3,4}^{ M^{(4)}_{\rm c}}$ of the matrix $\bm{M^{(4)}_{\rm c}}$, given in Eq. (\ref{['6']}), for the four-mode bosonic system in the circular configuration with different damping and/or amplification rates of neighbor modes and (b) $\lambda_{3,4}^{ M^{(4)}_{\rm t}}$ of the matrix $\bm{M^{(4)}_{\rm t}}$, given in Eq. (\ref{['17']}), for $\xi=\pm\zeta$ for the four-mode bosonic system in the tetrahedral configuration with the same damping and/or amplification rates of neighbor modes are drawn in the parameter space $(\kappa/\epsilon,\gamma_-/\epsilon)$. The dashed red curves indicate the values at the positions of the QHPs of the four-mode bosonic systems as given by Eq. (\ref{['8']}).
• Figure 3: Schematic diagrams of bosonic systems composed of two subsystems mutually coupled by unidirectional coupling and having: (a) two, (b) three, (c) four, (d) five, and (e) six modes in typical linear 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. The coupling strengths $\epsilon$ and $\kappa$ characterize both unidirectional (red dashed arrows) and bidirectional (back full double arrows) coupling between modes, $\gamma$, with subscripts indicating the mode number(s), give the damping or amplification rates, and the annihilation operators $\hat{a}$ identify the mode number via their subscripts.
• Figure 4: Scheme for constructing general $k$th-order FOMs for an $n$-mode bosonic system with $\Sigma_\Lambda$ eigenvalues $\Lambda_1,\ldots,\Lambda_{\Sigma_\Lambda}$ having $n_1, \ldots, n_{\Sigma_\Lambda}$ coalescing eigenvectors, i.e. $\sum_{j=1}^{\Sigma_\Lambda} n_j = 2n$. The field operators $\hat{b}_l$ and $\hat{b}_l^\dagger$ for $l=1,\ldots, n$ in the diagonalized basis form elements of the field-operator vectors $\bm{\hat{B}_1}, \ldots, \bm{\hat{B}_{\Sigma_\Lambda}}$ composed of in turn $n_1, \ldots, n_{\Sigma_\Lambda}$ elements. The $k$th-order FOMs are divided into the groups identified with the vectors $\bm{k} \equiv [k_1,\ldots,k_{\Sigma_\Lambda}]$, $\sum_{j=1}^{\Sigma_\Lambda} k_j = k$, containing FOMs of the form $\langle \bm{\hat{B}}_{1}^{k_1} \cdots \bm{\hat{B}}_{\Sigma_\Lambda}^{k_{\Sigma_\Lambda}} \rangle$.
• Figure 5: (a) Real $\lambda^{\rm r}$ and (b) imaginary $\lambda^{\rm i}$ parts of eigenvalues $\lambda_{1,2}$ of the dynamical matrices $\bm{M^{(2)}_{\delta,1} }$ ($\gamma_{+}=0$, $\gamma_{-}=0.5$, blue solid curves), $\bm{M^{(2)}_{\delta,2}}$ ($\gamma_{+}=0$, $\gamma_{-}=0.5$, red dashed curves), and $\bm{M^{(1+1)}_{{\rm u},\delta} }$ ($\gamma_{2}=0$, green dot-dashed curves) given in turn in Eqs. (\ref{['B5']}), (\ref{['B8']}), and (\ref{['B11']}) as they depend on perturbation parameter $\delta$. In (c) the overlap $F$ of the corresponding eigenvectors is plotted. In (a) and (b), the numbers denote the curves of the corresponding eigenvalues.