Higher order coherence as witness of exceptional point in Hermitian bosonic Kitaev dimer

TL;DR

EPs, traditionally non-Hermitian degeneracies, can induce non-analytic effects in Hermitian systems when a non-Hermitian core is embedded. Here, a minimal Hermitian bosonic Kitaev dimer reveals EPs at $\mu=0$ and $\mu=\pm2$ that partition the parameter space into four regions with HO or IHO dynamics, and a nonequilibrium quantum phase transition is probed via second-order coherence after a quench. The two-site Bose-Kitaev dimer is diagonalized via a Bogoliubov-like transform into decoupled $H^{+}$ and $H^{-}$ blocks with EPs; exact solutions show HO vs IHO regimes and EP boundaries. The second-order coherence function $g^{(2)}(t_1,t_2)$ and its time-averaged derivative $D_{\mu}$ serve as dynamical witnesses, with $D_{\mu}$ displaying minima near $\mu=\pm2$ and diverging in the infinite-time limit, signaling a nonequilibrium quantum phase transition. This minimal Hermitian platform provides a practical route to observe EP-induced nonanalytic dynamics via time-domain correlations, with potential experimental realization.

Abstract

The non-analyticity induced by exceptional points (EPs) has manifestations not only in non-Hermitian but also in Hermitian systems. In this work, we focus on a minimal Hermitian bosonic Kitaev model to reveal the dynamical demonstration of EPs in a Hermitian system. It is shown that the EPs separate the parameter space into four regions, in which the systems are characterized by different equivalent Hamiltonians, including the harmonic oscillator, the inverted harmonic oscillator, and their respective counterparts. We employ the second-order intensity correlation to characterize a nonequilibrium quantum phase transition by calculating the time evolution of a trivial initial state. The results indicate that the concept of the EP can be detected in a small Hermitian bosonic system.

Figures (3)

• Figure 1: Phase diagram of the Hamiltonian in Eq. (\ref{['H dimer']}) on the parameter $\mu$ line. Different colors distinguish different phases, and the distinction between phases lies in the different forms of Eqs. (\ref{['H mu>2']}) and (\ref{['H mu<2']}). Overall, green indicates that both $H^{+}$ and $H^{-}$ are HO, yellow indicates that one of $H^{+}$ and $H^{-}$ is a HO and the other is an IHO, and phase boundaries are marked by red solid (or dashed) lines.
• Figure 2: Plots of the second-order coherence function $g^{(2)}(0,t)$ and the quantity $D_{\mu }$, given by Eq. (\ref{['second order']}) and Eq.(\ref{['Dm']}). The numerical simulation is performed by tracking the time evolution of the initial state Eq. (\ref{['is']}). The quantity $g^{(2)}(0,t)$ for a given $\mu$ is obtained by exact diagonalization of the finite-dimensional matrix representation of Hamiltonian. (a) A three-dimensional (3D) plot of $g^{(2)}(0,t)$ as a function of time $t$ and $\mu$. (b) Plots of $D_{\mu }$, the derivative second-order coherence function $g^{(2)}(0,t)$ over a period of time $T$ with respect to $\mu$, as defined by Eq. (\ref{['Dm']}). The time interval $T$ are indicated in the legend. The results show that each $D_{\mu }$ has a minimum near the EP at $\mu =2$. The valleys become deeper as $T$ increases, implying that $D_{\mu }$ is divergent at the EP in the case of infinite $T$. Inset: The minimum value of $D_{\mu }$ and the corresponding value of $\mu$ for different truncation times $T$. The blue circles represent the scatter plots of minimum $D_{\mu }$ and the corresponding value of $\mu$ for Eq. (39), respectively, while the blue dashed lines are their respective fitting curves.
• Figure 3: Plots of the second-order coherence function $g^{(2)}(t,0)$ and the quantity $D_{\mu }$, given by Eq. (\ref{['second order']}) and Eq.(\ref{['Dm']}). The numerical simulation is performed by tracking the time evolution of the initial state Eq. (\ref{['is']}). The quantity $g^{(2)}(0,t)$ for a given $\mu$ is obtained by exact diagonalization of the finite-dimensional matrix representation of Hamiltonian. (a) A three-dimensional (3D) plot of $g^{(2)}(t,0)$ as a function of time $t$ and $\mu$. (b) Plots of $D_{\mu }$, the derivative second-order coherence function $g^{(2)}(0,t)$ over a period of time $T$ with respect to $\mu$, as defined by Eq. (\ref{['Dm']}). The time interval $T$ are indicated in the legend. The results show that each $D_{\mu }$ has a minimum near the EP at $\mu =2$. The valleys become deeper as $T$ increases, implying that $D_{\mu }$ is divergent at the EP in the case of infinite $T$. Inset: The minimum value of $D_{\mu }$ and the corresponding value of $\mu$ for different truncation times $T$. The blue circles represent the scatter plots of minimum $D_{\mu }$ and the corresponding value of $\mu$ for Eq. (39), respectively, while the blue dashed lines are their respective fitting curves.