We perform a phase-space analysis of strong-field enhanced ionisation in molecules, with emphasis on quantum-interference effects. Using Wigner quasi-probability distributions and the quantum Liouville equation, we show that the momentum gates reported in a previous publication [N. Takemoto and A. Becker, Phys. Rev. A textbf{84}, 023401 (2011)] may occur for static driving fields, and even for no external field at all. Their primary cause is an interference-induced bridging mechanism that occurs if both wells in the molecule are populated. In the phase-space regions for which quantum bridges occur, the Wigner functions perform a clockwise rotation whose period is intrinsic to the molecule. This evolution is essentially non-classical and non-adiabatic, as it does not follow equienergy curves or field gradients. Quasi-probability transfer via quantum bridges is favoured if the electrons initial state is either spatially delocalised, or situated at the upfield molecular well. Enhanced ionisation results from the interplay of this cyclic motion, adiabatic tunnel ionisation and population trapping. Optimal conditions require minimising population trapping and using the bridging mechanism to feed into ionisation pathways along the field gradient.
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