Hamiltonian Hopping for Efficient Chiral Mode Switching in Encircling Exceptional Points

Dynamically encircling exceptional points (EPs) can lead to chiral mode switching as the system parameters are varied along a path that encircles EP. However, conventional encircling protocols result in low transmittance due to path-dependent losses. Here, we present a paradigm to encircle EPs that includes fast Hamiltonian variations on the parameter boundaries, termed Hamiltonian hopping, enabling ultrahigh-efficiency chiral mode switching. This protocol avoids path-dependent loss and allows us to experimentally demonstrate nearly 90% efficiency at 1550 nm in the clockwise direction, overcoming a long-standing challenge of non-Hermitian optical systems and powering up new opportunities for EP physics.

Figure 1

(a),(b) Amplitudes of $X_1=[X_1(1),X_1(2){]}^T$ and $X_2=[X_2(1),X_2(2){]}^T$ of $H_2$ in the parameter space described by $\beta /\kappa$ and $\gamma /\kappa$. The surfaces covered by solid and dashed meshes represent the amplitudes of the first and second terms of $X_1$ (a) [$X_2$ (b)], respectively. Red regions represent the parameter space boundaries that share the same convergent eigenstates as ${[0,1]}^T$ and ${[1,0]}^T$. (c),(d) CW and (e),(f) ACW Hamiltonian-hopping-assisted loop around EP in the Riemann surfaces formed by the real part $\mathrm{Re}(E)$ and imaginary part $\mathrm{Im}(E)$ of the energy spectra of $H_2$. The red and blue surfaces denote $\mathrm{Im}(E)\ge 0$ and $\mathrm{Im}(E)\le 0$, respectively. The surfaces formed by the dotted lines symbolically show $\beta /\kappa$ or $\gamma /\kappa$ extending to infinity. The two eigenvalues, associated with ${[1,0]}^T$ and ${[0,1]}^T$, are denoted by magenta and cyan lines, respectively. The dashed lines indicate the Hamiltonian hopping among $A^{+}$, $A^{-}$, and $\{B\}$.

Figure 2

(a) Double-coupled silicon waveguides for state evolution in a Hamiltonian-hopping-assisted loop around EP. The width of the first (second) waveguide is $W_1$ ($W_2$) and $d$ is the separation distance between the two waveguides. (b),(c) $\beta$ and $\kappa$ as a function of $\mathrm{\Delta }W$ and $d$ at 1550 nm. (d),(e) Retrieved $\beta$ and $\kappa$ along the $z$ direction in the region of the coupled waveguides. (f) Hamiltonian-hopping-assisted loop around EP for different wavelengths.

Figure 3

(a),(b) Simulated field distributions of the transverse components of electric field ($E_x$) at 1550 nm. The ${\mathrm{TE}}_0$ mode inputs from the left port (a) and right port (b), respectively. (c) SEM image of the device. (d) Simulated transmittance spectra for ${\mathrm{TE}}_0$ and ${\mathrm{TE}}_1$ mode at the output port over 1200–1700 nm wavelength range. Solid (dashed) lines represent input from the left (right) port. (e) Measured transmittance spectra over 1525–1575 nm wavelength range. Numerical simulations are conducted by finite-difference time domain with commercial software Lumerical FDTD Solutions.

Figure 4

(a),(b) Simulated field distributions of the transverse components of electric field ($E_x$) at 1550 nm. (c) SEM image of the device. (d) Simulated transmittance spectra for ${\mathrm{TE}}_0$, ${\mathrm{TE}}_1$, and ${\mathrm{TE}}_2$ mode at the output port over 1200–1700 nm wavelength range. (e) Measured transmittance spectra over 1525–1575 nm wavelength range.