Where semiclassical physics breaks down for quantum barrier transmission

Displaced Fock states transmitted through an inverted-oscillator barrier serve as a clean probe of where semiclassical methods fail, because their Wigner functions carry negative regions that have no classical counterpart. Numerical simulations show that semiclassics captures the broad transmission envelope but misses short-time plateaus that appear exactly when Wigner-negative regions cross the barrier. Adding a Kerr nonlinearity pushes interference into classically forbidden phase-space regions, a feature entirely inaccessible to classical approaches. Despite these discrepancies, the maximum transmission probability remains bounded by the initial positive-energy fraction already encoded in the state's phase-space structure.