Shortcuts to Adiabaticity across a Separatrix

Shortcuts to adiabaticity are strategies for conserving adiabatic invariants under nonadiabatic (i.e. fast-driving) conditions. Here, we show how to extend classical, Hamiltonian shortcuts to adiabaticity to allow the crossing of a phase-space separatrix - a situation in which a corresponding adiabatic protocol does not exist. Specifically, we show how to construct a time-dependent Hamiltonian that evolves one energy shell to another energy shell across a separatrix. Leveraging this method, we design an erasure procedure whose energy cost bound and fidelity do not depend on the protocol's duration.