Exceptional-Point-Induced Nonequilibrium Entanglement Dynamics in Bosonic Networks

TL;DR

The work investigates how exceptional points in a non-Hermitian bosonic network govern nonequilibrium entanglement dynamics. By mapping the Heisenberg equations to a non-Hermitian BdG dynamical matrix $M(g,\eta,J)$, it uncovers second-order and higher-order EPs that partition the spectrum into purely imaginary, purely real, and mixed regimes, each driving distinct entanglement behavior—exponential, oscillatory, or hybrid. Higher-order EPs induced by noninteger-$\pi$ hopping phases or nonuniform couplings significantly enhance multimode entanglement, achieving up to roughly a 2.5-fold increase over EP$_2$ under optimal timing. The results offer a route to engineer quantum correlations in bosonic networks and suggest practical implementations in optical, optomechanical, and superconducting platforms, with potential metrological and noise-resilience advantages enabled by the non-Hermitian spectral structure.

Abstract

Exceptional points (EPs), arising in non-Hermitian systems, have garnered significant attention in recent years, enabling advancements in sensing, wave manipulation, and mode selectivity. However, their role in quantum systems, particularly in influencing quantum correlations, remains underexplored. In this work, we investigate how EPs control multimode entanglement in bosonic chains. Using a Bogoliubov-de Gennes (BdG) framework to describe the Heisenberg equations, we identify EPs of varying orders and uncover spectral transitions between purely real, purely imaginary, and mixed eigenvalue spectra. These spectral regions, divided by EPs, correspond to three distinct entanglement dynamics: oscillatory, exponential, and hybrid. Remarkably, we demonstrate that higher-order EPs, realized by non-integer-pi hopping phases or nonuniform interaction strengths, significantly enhance the degree of multimode entanglement compared to second-order EPs. Our findings provide a pathway to leveraging EPs for entanglement control and exhibit the potential of non-Hermitian physics in advancing quantum technologies.

Figures (4)

• Figure 1: Schematic of $N$-mode bosonic chain and the dynamical spectrum divided by EPs.a Schematic of an $N$-mode bosonic chain with nearest-neighbor mode-hopping and squeezing terms. The system is represented graphically using the BdG formalism, which includes BS ($g$), TMS ($J$), and SMS ($\eta$) interactions (Supplementary Note 1). b-c Illustration of the dynamical spectrum ($M$) change affected by the EPs. For a system with only BS and TMS terms, the spectrum is divided into two regions by $X$-fold EP$_2$s appearing at $g=g_{0}$. In contrast, the introduction of the SMS term splits the $X$-fold EP$_2$s into $X$-unfold EP$_2$s located in the region of $g_0^- \leq g \leq g_0^+$, corresponding to three different regions of the spectrum. Here $X=N$ for even $N$ and $X=N-1$ for odd $N$.
• Figure 2: Exceptional points and entanglement dynamics for the two-mode system. The real part a and imaginary part b of four eigenvalues of $M$ versus $g/J$ with $\eta/J = 0.2$. The spectrum is divided into three regions by two EP$_2$s at $g_0^-/J = 0.8$ and $g_0^+/J = 1.2$. c The time evolution of $\nu_-$ for $g/J = 0.79$ (blue), $g/J = 1.19$ (orange) and $g/J = 1.59$ (red), showing three distinct types of the entanglement dynamics, i.e. exponential behavior in region I (blue), oscillatory behavior in region II (red) and mixed behavior in region III (orange). d The long time evolution of $\nu_-$, to show more difference between region II (red) and III (orange). Other parameters are the same as c. e The values of $\nu_-$ versus $g/J$ and $t/J$ with $\eta/J = 0.2$. The representative parameters ($g/J = 0.79$ and $g/J = 1.19$) near the EP$_2$s in c are chosen to illustrate the contrasting entanglement dynamics across these regions, while $g/J = 1.59$ is just chosen to ensure a consistent parametric spacing.
• Figure 3: Entanglement enhancement by higher-order EPs.a Logarithmic negativity $-\log\nu_-$ of $N$-mode BKC model as a function of hopping phase $\phi$ for $N = 2$ (blue), $N = 3$ (light blue), $N = 4$ (yellow), $N = 5$ (orange), and $N = 6$ (red). Here, we set $Jt=3.5$ and any point on the solid line corresponds to 2-fold EP$_N$s except for $\phi=0,\pi$ (X-fold EP$_2$s instead). b The time evolution of optimized entanglement enhancement ratio $R(t)=\log[\nu_-(\pi/2,t)]/\log[\nu_-(0,t)]$ with different lengths of $N=3$ (light blue), $N=4$ (yellow), $N=5$ (orange) and $N=6$ (red). The inset shows the ratio R as a function of $N$ ($2\sim30$) with fixed time, i.e., $Jt=3.5$, which can be fitted as $R\approx-4.11 e^{-0.4633 N}+2.493$, revealing a trend of nearly 2.5 times enhancement by highest-order EPs in thermodynamic limits. The other parameters are fixed at $g/J=1$ and $\eta=0$.
• Figure 4: Exceptional points and entanglement dynamics in the nonuniform BKC.a Schematic of a three-mode system and the corresponding conditions for EPs in parameter space. The light red surface represents the 2-fold third-order ESs, where each point corresponds to 2-fold EP$_3$s, except for the EAs (red line) consisting of 2-fold EP$_2$s. The parameter space is divided into two regions: eigenvalues are purely imaginary above the ESs, while they are purely real below the ESs, with the exception of two eigenvalues always being 0. The light blue cut at $J_1/J_2 =1$ is shown as a visual guide for the data presented in panel b. b The entanglement witness $\nu_-^{13|2}$ at $Jt = 5$ versus interaction strengths, along the blue cut in panel a. In region I, entanglement evolves exponentially, while in region II, it oscillates. The white dashed line, marking the intersection of the light red surface and the light blue plane, indicates the positions of EPs, with $\varphi$ representing the angle between EP$_3$s and EP$_2$s. c Entanglement witness $-\log(\nu_-^{13|2})$ as a function of $\varphi$ along the white dashed line in b. The degree of entanglement increases as $\varphi$ moves further from EP$_2$s. d Time evolution of $-\log(\nu_-^{13|2})$ at $\varphi = 0$ (blue line), $\varphi = \pi/8$ (orange line), and $\varphi = \pi/4$ (red line), with other parameters being the same as in c.