Characteristic dynamics near exceptional points in non-Hermitian systems
We review some recent work on coalescing eigenstates at exceptional points (EPs) in non-Hermitian systems and their dynamical influence. We first consider an example involving the Su-Schrieffer-Heeger (SSH) model, which, in its original form, exhibits topologically-protected edge states that are either right or left-handed. Our model, consisting of a PT-symmetric trimer connected to two SSH reservoirs, exhibits two zero-energy states, one of which is localized on the central PT potential and one of which is anti-localized, as well as four other solutions. For certain parameter values, pairs of eigenstates from the remaining solutions can coalesce with either the localized or anti-localized zero-energy state to form an exceptional point of order N ? 3 with N odd. The localized zero-energy EPs imprint on the survival probability dynamics a simple power law growth of the form t^{2N-2}[1].
Next we turn to the dynamical properties occurring near the edge of a photonic band gap. Relying on a simplified model of a quantum emitter coupled to a one-dimensional continuum, we show that the divergence in the density of states leads to a triple level splitting at the band edge with the appearance of a resonance, anti-resonance and a bound state. The decay width of the resonance is proportional to g^{4/3}, which suggests the influence of a third-order EP. However, in the limit in which the coupling g vanishes, only two states actually coalesce as the third instead merges with the continuum. The combined influence of this anomalous-order EP and the continuum threshold results in quantum emitter decay of the non-Markovian form 1 – C t^{3/2} [2].
[1] S. Garmon and K. Noba, arXiv:2108.01930 [ https://arxiv.org/abs/2108.01930 ]
[2] S. Garmon, G. Ordonez, and N. Hatano, Phys. Rev. Research 3, 033029 (2021) [ https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.033029 ]
[3] S. Garmon, T. Sawada, K. Noba, and G. Ordonez, J. Phys.: Conf. Ser. 2038, 012011 (2021) [ https://iopscience.iop.org/article/10.1088/1742-6596/2038/1/012011 ]