Hybrid exceptional point and its dynamical encircling in a two-state system

Hybrid exceptional points (HEPs) are non-Hermitian degeneracies that exhibit different dispersion relations (e.g., linear or square-root) along different directions in the parameter space. Here, we show that a two-state system consisting of coupled ferromagnetic waveguides applied with a bias magnetic field can support the presence of a HEP. A configuration is designed to dynamically encircle the HEP, showing a nonchiral behavior due to the unique topological structure of energy surfaces around the HEP. This is in contrast to the chiral dynamics found when an exceptional point is encircled dynamically. We performed numerical simulations and microwave experiments to demonstrate the topological transition enabled by the HEP in the proposed system.

Figure 1

(a) Hamiltonian parameter $\kappa$ as a function of $\sigma$ (see text). The horizontal lines mark different values of $\gamma$ of the Hamiltonian. The crossing points between $\left|\kappa \right|$ and $\gamma$ are EPs, whereas their point of tangency is a HEP. (b) Eigenvalues of the model Hamiltonian with $\gamma =0$ as a function of $\sigma$ and $\delta$.

Figure 2

Real (left panels) and imaginary (right panels) parts of the eigenvalues of the model Hamiltonian as a function of $\sigma$ and $\delta$ with (a), (b) $\gamma =0.5$, (c), (d) $\gamma =1$, and (e), (f) $\gamma =1.5$.

Figure 3

(a), (b) Imaginary parts of the eigenvalues as a function of (a) $\sigma$ with $\delta =0$ and (b) $\delta$ with $\sigma =0$. (c), (d) Phase rigidities (see text) of the two eigenstates as a function of (c) $\sigma$ with $\delta =0$ and (d) $\delta$ with $\sigma =0$. We choose $\gamma =1$ for all the calculations.

Figure 4

(a) Cross-section view of coupled YIG waveguides applied with a transverse bias magnetic field (red arrow). A microwave absorber (yellow region) is attached to the side of YIG waveguide 2 to introduce asymmetric loss. (b) Effective mode index in a lossless system (i.e., without the absorber) as a function of the bias field and the scale factor. (c)–(e) Imaginary part of the effective mode index in a lossy system with (c) $w=0.4\mathrm{mm}$, (d) $w=1.335\mathrm{mm}$, and (e) $w=2\mathrm{mm}$. The two insets in panel (d) are side views of the Riemann sheets. In these simulations, the structure parameters are $W=8\mathrm{mm}$, $H=4\mathrm{mm}$, $g=0.5\mathrm{mm}$, and $h=2\mathrm{mm}$. The frequency is 10 GHz.

Figure 5

(a) Parameter space of the system with $w=1.335\mathrm{mm}$. (b), (c) Variation of (b) the bias field $B_0$ and (c) the scale factor $\alpha$ along the $z$ axis, following the formula $B_0\left(z\right)=-B_{m}cos\left(2\pi z/L\right)$ and $\alpha \left(z\right)=1-{\alpha }_{m}sin\left(2\pi z/L\right)$, respectively, where $B_m=0.07\mathrm{T}$, ${\alpha }_m=0.125$, and $L=600\mathrm{mm}$. (d) Schematic diagram of coupled YIG waveguides in which the bias magnetic field and the width of YIG waveguide 2 vary continuously along the waveguiding direction and follow the designed parameters in panels (b) and (c). Wave transmissions through the system can be described by a trajectory in the parameter space as shown by the curve in panel (a), with the starting and end point located at $B_0=0$ and $\alpha =1$. Injections from the left-hand side of the system correspond to counterclockwise loops while those from the right-hand side of the system correspond to clockwise loops.

Figure 6

(a)–(d) Simulated $H_y$ field distributions for (a) a counterclockwise loop with a gain injection, (b) a counterclockwise loop with a loss injection, (c) a clockwise loop with a gain injection, and (d) a clockwise loop with a loss injection. (e) Amplitudes of the two eigenmodes along the $z$ axis for the four cases. (f), (g) Trajectories of the state evolution on the Riemann sheets for (f) a counterclockwise loop with a gain injection and (g) a counterclockwise loop with a loss injection.

Figure 7

(a) Schematic diagram of coupled YIG waveguides in which the width of YIG waveguide 2 varies linearly from 7 to 9 mm along the waveguiding direction. Other structure parameters are $H=4\mathrm{mm}$, $g=0.5\mathrm{mm}$, and $h=2\mathrm{mm}$. Wave scatterings through the system correspond to a line trajectory in the parameter space with a starting point at $B_0=0$, $\alpha =0.875$ and an end point at $B_0=0$, $\alpha =1.125$. (b) Calculated transmission intensities of the system as a function of $w$, with a gain mode propagating in waveguide 1 as injections. (c), (d) Simulated power flow distributions of the system with (c) $w=0.1\mathrm{mm}$ and (d) $w=2\mathrm{mm}$. The frequency is 10 GHz in these simulations.

Figure 8

(a), (b) Experimentally measured transmission intensities at the output side of the two waveguides for the system (a) without and (b) with the absorber. (c), (d) Experimentally measured electric-field intensities on the surface of the two waveguides along the waveguiding direction for the system (c) without and (d) with the absorber. The resonances in these figures are due to the Fabry-Pérot effect. The frequency is 10 GHz in panels (c) and (d).

Figure 9

(a) Real and (b) imaginary parts of the eigenvalues of the model Hamiltonian as a function of $\kappa$ and $\delta$ with $\gamma =1$. (c) Parameter space formed by $\delta$ and $\sigma$, where the loop trajectory is generated by $\delta =\rho sin\theta$ and $\sigma =-\rho cos\theta +\chi$. (d) Parameter space formed by $\delta$ and $\kappa$, where the loop trajectory is generated by $\delta =\rho sin\theta$ and $\kappa ={\sigma }^2-1={(-\rho cos\theta +\chi )}^2-1$. The parameters of the loop are $\rho =0.5$, $\chi =0.1$, and $\theta$, which ranges from zero to $2\pi$. The dashed lines mark the $PT$-broken phase. The shaded region in panel (d) corresponds to the part of the parameter space that is not physically realizable. As the ($\delta$, $\kappa$) space does not have a $PT$-symmetric phase, we label the $(\delta ,\kappa )=(0,-1)$ point as an “EP-like” degeneracy.