Generic temporal quantization criterion in dynamic resonant tunneling for current suppression

It is well known that resonant tunneling can significantly increase the tunneling current when the incident particle's energy (Ω) is equal to the resonant energy (Ω_R). However, recent studies have shown that complex current structures can emerge when the resonant energy varies in time. While these temporal variations typically increase the current, there is a specific condition under which the current is significantly suppressed. This condition, which has previously only been presented for specific cases, is characterized by the equation [Formula presented] for =1,2,…, where t_1,2 are the solutions of Ω_R(t_1,2)=Ω. In this paper, we present for the first time a full derivation of this expression using the Quasi-Bound Super State method and demonstrate it for various perturbation scenarios with the following temporal profiles: Gaussian, hyperbolic secant, harmonic, and smooth rectangular. The derivation of this quantization rule has practical implications beyond the intellectual realm. The specific spectral structure of these systems has implications for nanotechnology (e.g., frequency-controlled transistors) and the understanding of biological and biochemical processes (e.g., odor detection).