Energy Band Attraction Effect in Non-Hermitian Systems

The energy band attraction (EBA) caused by the nonorthogonal eigenvectors is a unique phenomenon in the non-Hermitian (NH) system. However, restricted by the required tight-binding approximation and meticulously engineered complex potentials, such an effect has never been experimentally demonstrated before. Here by a suitable design of all-dielectric Mie resonators in a parallel-plate transmission line, we for the first time verify the photonic analog of the EBA effects both theoretically and experimentally. The evolution of the EBA effect in a two-level NH system from gapped bands to gapless bands to flat bands is observed by precisely tuning the loss of the Mie resonators. The transmission spectra can be theoretically connected to the eigenvalues and eigenvectors of the NH Hamiltonian. Furthermore we extend our methods to a graphenelike two-dimensional NH system. Our works show a metamaterial approach toward NH topological photonics and foster a deeper understanding of band theory in open systems.

Figure 1

Non-Hermitian SSH model and EBA effect in quantum systems. (a) Lattice representation of the non-Hermitian model in Eq. (1). All sites in blue (red) have particle loss (gain) with a rate of $\gamma$. The dotted box indicates the unit cell and the single line (double lines) indicates the weak (strong) coupling between the nearest neighbors. (b) Band structure of NH SSH model as functions of $\gamma$ and $k_a$, $\gamma$ varies in [0,5]; $t_b=\frac{1}{3}{t}_a=1$. The transition point ($\gamma =2$) between the PT-symmetry phase and the PT-symmetry spontaneously broken phase is indicated by an arrow. (c) The transmission spectra $|S_{21}|$ as a function of NH parameter $\gamma$ calculated from Eq. (5). The color scale describes the transmission amplitude, while the white dashed lines indicate the real part of the eigenvalues of system Hamiltonian $H$. The EBA effect can be divided into three regions according to the eigenvalues’ property in the active PT-symmetry model: in the PT-symmetry region $\gamma \in [0,0.053)$, all the eigenvalues are real; in the partially PT-symmetry-broken region $\gamma \in [0.053,0.109)$, some eigenvalues are real, and the others are complex; in the fully PT-symmetry-broken region $\gamma \in [0.053,0.109)$, all the eigenvalues are complex and have identical real parts.

Figure 2

Schematic of experimental analog and the passive PT-symmetry Mie resonator lattices. (a) The Mie resonator lattice in the parallel-plate transmission line. The cutoff frequency of the parallel-plate transmission line, determined by the separation $h$ of the parallel plates, is lower than the Mie resonance frequency to ensure resonator coupling. (b) Absorption implementation. Absorption is introduced on selected sites by gold sputter deposition on top of the resonators and can be controlled by varying the deposition area [44]. The losses of the particles were precharacterized to make sure the resonance frequency deviation was less than 5 MHz. (c) Schematic of complex SSH chain with staggered resonator coupling amplitudes $t_a$ and $t_b$. The coupling amplitudes can be controlled by varying the resonator spacings [44].

Figure 3

Observation of EBA as NH term $\gamma$ varying through $S$ parameter measurements. The blue lines (orange lines) indicate theoretical prediction (experimentally measured transmission). The lattice is composed of 16 identical coupled dielectric cuboid resonators, whose dimensions are $5.00\times 7.35\times 7.35{\mathrm{mm}}^3$ and refractive index is 6.50 (the dielectric loss is negligible). Accordingly, the first-order Mie magnetic dipole resonance frequency of the isolated resonator is $f_0=6.493\mathrm{GHz}$. The resonators are placed at alternating distances: $d_1=11.50\mathrm{mm}$ for $t_a=41.0\mathrm{MHz}$ and $d_2=14.50\mathrm{mm}$ for $t_b=15.0\mathrm{MHz}$. Two conducting plates are separated by $h=17.00\mathrm{mm}$ corresponding to a cutoff frequency of 8.75 GHz. (a) $\gamma =0$, Hermitian case. In this case, all the eigenvalues are real, and each resonance peak corresponds to the eigenvalue of the Mie resonance lattice. (b) $\gamma \approx 15\mathrm{MHz}$ belongs to the PT-symmetry region. The resonance peaks in this region are indistinguishable because of the additional background damping $-i\gamma$. (c) $\gamma \approx 60\mathrm{MHz}$ belongs to the partially PT-symmetry-broken region. The gapped bands coalesce, and the gap is closed. (d) $\gamma \approx 121\mathrm{MHz}$ belongs to fully PT-symmetry-broken region. All the eigenvalues are complex with identical real parts, which corresponds to a single peak.

Figure 4

Measured magnetic field distribution associated with edge states at bare frequency $f_0=6.493\mathrm{GHz}$. (a),(c) Resonator configuration that supports zero-energy edge modes. (b),(d) Normalized experimental field distribution corresponds to (a) and (c), and the white dashed lines indicate resonators. Loss strength ${\gamma }_1\approx 121\mathrm{MHz}$, ${\gamma }_2\approx 15\mathrm{MHz}$, and coupling strength $t_a=41.0\mathrm{MHz}$, $t_b=15.0\mathrm{MHz}$. $Q$ stands for the global Berry phase. (b) The magnetic field of the edge state is localized around the topological interface and has nonvanishing components only on sublattice A. (c) The configuration does not support the topological edge state since it is trivial of $Q=0$. However, it supports the zero-energy edge state (not a topological one) caused by the PT phase transition. (d) The edge state also exists around the PT interface even if the configuration is topologically trivial.

Figure 5

Theoretical and numerically calculated band structure in Hermitian (left panel) and non-Hermitian (right panel) honeycomb lattice. Only the real parts of the eigenvalues are plotted. (a)–(d) Theoretical results of NH graphene, with $t_1=2.7\mathrm{eV}$, $t_2=-0.54\mathrm{eV}$, $t_3=0.1\mathrm{eV}$, and $\gamma =0.5\mathrm{eV}$. The analysis predicts that the real parts of the eigenvalues stay as a constant inside the exceptional points ring, indicating flat bands in dispersion. (e),(f) Simulation results of Hermitian and NH resonances’ lattice, with $t_1=16\mathrm{MHz}$, $t_2=-1.45\mathrm{MHz}$, $t_3=1\mathrm{MHz}$, $\gamma =21\mathrm{MHz}$, and lattice constant $a=15\mathrm{mm}$ [44]. The arrows indicate Dirac point (DP) and the ring of exceptional points.