Anomalous Fermi's golden rule and transition dynamics at non-Hermitian exceptional points

Fermi’s golden rule (FGR), whose origins date back to the early days of quantum mechanics, has recently been extended to non-Hermitian systems. However, generalizations of FGR to non-Hermitian systems have largely been built upon the biorthogonal basis formalism, which presupposes the diagonalizability of the Hamiltonian. The problem becomes much more complicated when the Hamiltonian possesses exceptional points (EPs), where the eigenvectors coalesce and the Hamiltonian becomes defective (non-diagonalizable). In this work, by computing the evolution of the associated Jordan block in the interaction picture, we derived FGR for systems containing an EP of order 2. We found that in the short-time limit, the transition probability P exhibits an anomalous t⁴ scaling, rather than P ∝ t² as in non-Hermitian systems without EPs. On the other hand, in the long-time limit, the system exhibits a Squared-Lorentzian resonance lineshape that is sharper than the conventional Lorentzian profile observed in a non-Hermitian system without EPs. Numerical simulations verify the analytical results.