Higher-order Exceptional Points Induced by Non-Markovian Environments

Abstract

Exceptional points (EPs) are central to non-Hermitian physics because of their unique properties and broad application prospects. While extensively studied in parity-time ($\mathcal{P}\mathcal{T}$)-symmetric systems and under Markovian dynamics, their exploration in broader pseudo-Hermitian settings particularly those involving non-Markovian environments remains largely unexplored. In this study, we investigate a pseudo-Hermitian system consisting of three coupled optical cavities interacting with non-Markovian environments. Compared to the Markovian baseline, we demonstrate that the emergence of non-Markovian memory effects enlarges the dimensionality of the parameter space of the system, thereby giving rise to higher-order EPs. Moreover, we observe that the pseudo-Hermitian system with an effective gain induced by coherent perfect absorption enables the higher-order EPs to be directly read out from the output spectrum. Additionally, we find that breaking the symmetry of the parameter space of the system reduces the order of the EPs. Possible experimental implementations based on a superconducting circuit are also discussed. Our findings reveal how non-Markovianity enhances the sensitivity of the system and provide a theoretical insight into the experimental observation of higher-order EPs.

Figures (6)

• Figure 1: Schematic of a three-mode optical cavity coupling setup in an open quantum system Qin20005692021Li0335052021Jiang2011062023Zhong0132202021Soleymani5992022Tang66602023Jia2932023Stalhammar2011042021Chen0315012022Zhong0140702020. Cavity mode $a$ couples to cavity modes $b_{1}$ and $b_{2}$ with strengths $g_{1}$ and $g_{2}$, respectively. On the other hand, cavity modes $b_{1}$ and $b_{2}$ are coupled with strength $J$. In addition, the input fields $a_{1}^{(\mathrm{in})}$ and $a_{2}^{(\mathrm{in})}$ generate non-Markovian baths, which couples to cavity mode $a$ through two ports with strengths $C_{k}$ and $D_s$, respectively. Furthermore, cavity modes $b_{1}$ and $b_{2}$ are each coupled to Markovian environments with dissipation rates $\gamma_{1}$ and $\gamma_{2}$.
• Figure 2: The total output spectrum (\ref{['eq:output-total']}) varies with the coupling strength $g_1$ and the frequency detuning $(\omega-\Delta_a)$ between the input field and the cavity $a$, for (a) symmetric and (b) asymmetric cases. The parameter settings for (a) are $\gamma_1/2\pi=\gamma_2/2\pi=1.885$ MHz, $\lambda_1/2\pi=\lambda_2/2\pi=3.77{\rm{MHz}}$, $\Gamma_1/2\pi=\Gamma_2/2\pi=1.637$ MHz, $g_1=g_2$, $J=0$, ${\Delta _{{b_2}}} = - {\Delta _{{b_1}}}$, and ${\Delta _a} = 0$. Parameters for (b) are chosen as ${{{\gamma _1}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {2\pi }} = {{{\gamma _2}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} \pi } = 3$MHz, $\lambda_1/2\pi=\lambda_2/2\pi=4.5{\rm{MHz}}$, $\Gamma_1/2\pi=\Gamma_2/2\pi=1.942$ MHz, $g_1=g_2$, $J=0$, and $\;{\Delta _{{b_2}}} = \; - 2{\Delta _{{b_1}}} = 2{\Delta _a}$.
• Figure 3: The real and imaginary parts of the eigenvalues obtained from the characteristic equation (\ref{['eq:characteristic-f']}) as functions of the coupling strength $g_1$. Panels (a) and (b) correspond to the symmetric case with parameters as in Fig. \ref{['fig:Fig2']}(a), whereas (c) and (d) show the asymmetric case with parameters as in Fig. \ref{['fig:Fig2']}(b).
• Figure 4: The Markovian counterparts of Fig. \ref{['fig:Fig3']}. Panels (a) and (b) present symmetric cases, where the parameters are set to ${{{\lambda _1}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {2\pi }} = {{{\lambda _2}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {2\pi }} = 200$MHz and ${\Gamma _1} = {\Gamma _2} = 2{\gamma _1}=2{\gamma _2}$. The asymmetric case is illustrated in panels (c) and (d), with the parameter chosen as ${{{\lambda _1}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {2\pi }} = {{{\lambda _2}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {2\pi }} = 200$MHz, ${{{\Gamma _1}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} {2\pi }} = {{{\Gamma _2}} \mathord{\left/ {\newline} \right. \nulldelimiterspace} \pi } = 4.5$MHz, and ${\gamma _1} = 2{\gamma _2}$. The remaining parameters are chosen to be the same as those in Fig. \ref{['fig:Fig2']}.
• Figure 5: Based on Eq. (\ref{['5*5NM']}), the conversion of eigenvalues from the non-Markovian to the Markovian regime at the EPs is analyzed as a function of the coupling strength $g_1$. (a) and (b) respectively plot the real and imaginary parts of the eigenvalues of the effective Hamiltonian ${H}_{N5}$ in the non-Markovian case with $\lambda_1/2\pi = 0.079$ MHz and $\lambda_2 /2\pi= 2.921$ MHz. (c) and (d) display the real and imaginary parts of the eigenvalues during the conversion to the Markovian regime, corresponding to $\lambda_1/2\pi = 200$ MHz and $\lambda_2 /2\pi= 220$ MHz. (e) and (f) present the real and imaginary parts under the Markovian approximation, with parameters $\gamma_1/2\pi = 2$ MHz and $\gamma_2 /2\pi= 1$ MHz.