Smooth double barriers in quantum mechanics

Quantum mechanical tunneling across smooth double barrier potentials modeled using Gaussian functions, is analyzed numerically and by using the WKB approximation. The transmission probability, resonances as a function of incident particle energy, and their dependence on the barrier parameters are obtained for various cases. We also discuss the tunneling time, for which we obtain generalizations of the known results for rectangular barriers.Comment: 23 pages, 8 figures, a slightly reduced version to appear in American Journal of Physics, references correcte

Time reflection and refraction in synthetic frequency dimension

The duality of space and time in Maxwell's equations has prompted interest in time boundaries and the accompanying temporal analog of spatial reflection and refraction. However, achieving observable time boundary effects at optical frequencies in real materials is challenging. In this work, we demonstrate that time reflection and refraction can be observed in a two-band model centered around a non-zero reference energy. Our model can be physically implemented in the synthetic frequency dimension as a system of two coupled dynamically-modulated ring resonators. We find that modulation at microwave frequencies is sufficient to observe time boundary effects for optical waves in synthetic frequency dimension. Our work shows that implementing multi-band models in synthetic dimensions opens a new avenue for further exploration of time boundaries.Comment: 9 pages, 4 figures; fixed typo in equation labe