High-order exceptional points and stochastic resonance in pseudo-Hermitian systems

Abstract

Exceptional points, a remarkable phenomenon in physical systems, have been exploited for sensing applications. It has been demonstrated recently that it can also utilize as sensory threshold in which the interplay between exceptional-point dynamics and noise can lead to enhanced performance. Most existing works focused on second-order exceptional points. We investigate the stochastic dynamics associated with high-order exceptional points with a particular eye towards optimizing sensing performance by developing a theoretical framework based on pseudo-Hermiticity. Our analysis reveals three distinct types of frequency responses to external perturbations. A broad type of stochastic resonance is uncovered where, as the noise amplitude increases, the signal-to-noise ratio reaches a global maximum rapidly but with a slow decaying process afterwards, indicating achievable high performance in a wide range of the noise level. These results suggest that stochastic high-order exceptional-point dynamics can be exploited for applications in signal processing and sensor technologies.

Figures (9)

• Figure 1: Schematic illustration of the real (top) and imaginary (bottom) components of the eigenvalues as a function of the perturbation $\varepsilon$. The critical value of the perturbation at which the eigenvalues and corresponding eigenvectors of the system coalesce is $\varepsilon_{EP}$.
• Figure 2: Emergence of an EP through a branch structure. (a) Real part of frequency splitting ${\rm Re}\{\delta \omega \}$ versus the magnitude $\varepsilon$ of the perturbation, where an EP arises at $\varepsilon_{EP}$. (b) Consequence of the EP: for $\varepsilon \agt\varepsilon_{EP}$, a pair of dips in the reflection coefficient are created at $f_{EP} \pm f_1$, where $f_1$ is the absolute value of ${\rm Re}\{\delta\omega\}$ at $\varepsilon \agt\varepsilon_{EP}$ in (a).
• Figure 3: Monotonic scenario. (a) Real frequency shift ${\rm Re}\{\delta \omega \}$ versus the perturbation about the critical value $\varepsilon_{EP}$. Depending on whether $\varepsilon$ is below or above $\varepsilon_{EP}$, the amount of the shift is negative or positive, respectively. (b) The resulting dips in the reflection coefficient, one for $\varepsilon < \varepsilon_{EP}$ and another for $\varepsilon > \varepsilon_{EP}$. Frequency filtering can be employed to remove a dip.
• Figure 4: A non-injective structure. (a) Real frequency splitting ${\rm Re}\{\delta\omega\}$ for different perturbations. (b) A schematic illustration of the response of a system in terms of the reflection coefficient under a perturbation. Any perturbation leads to a frequency shift to the right. An example for $\varepsilon < \varepsilon_{EP}$ ($\varepsilon > \varepsilon_{EP}$) is shown in yellow (green).
• Figure 5: System of three inductively coupled resonators as a wireless sensor. (a) A schematic illustration of the circuit system. (b,c) Real and imaginary parts of the eigenfrequency of the system \ref{['Eq: schrodinger']} as a function of mutual coupling parameter $\kappa_{12}$, respectively. Other parameter values are $\alpha=50$, $\gamma = 0.1$, $\kappa_{13} = 0$, and $\kappa_{23} = (1+\alpha)^{3/2}\kappa_{12}$. The response of the three coupled RLC resonators exhibits a branch structure: for $\kappa_{12} < \kappa_{EP}$, the perturbed system has one real eigenfrequency ($\omega_3 = 0$) and two complex conjugate eigenfrequencies, whereas for $\kappa_{12} > \kappa_{EP}$, it has three distinct real eigenfrequencies.