Sub-cycle attosecond control in frustrated double ionization of molecules

We demonstrate sub-cycle control of frustrated double ionization (FDI) of the two-electron triatomic molecule $D_3^{+}$ when driven by two orthogonally polarized two-color laser fields, where a weak mid-infrared laser field is employed to probe the FDI process triggered by a strong near-infrared laser field. We use a three-dimensional semi-classical model that fully accounts for the electron and nuclear motion in strong fields. We analyze the FDI probability and the distribution of the momentum of the escaping electron along the polarization direction of the mid-infrared laser field. These observables when considered in conjunction bear clear signatures of sub-cycle control of FDI. We find that the momentum distribution of the escaping electron has a striking hive-shape with features that can accurately be mapped to the time that one of the two electrons tunnel-ionizes at the start of the break-up process. This mapping distinguishes consecutive tunnel-ionization times within a cycle of the mid-infrared laser field but not tunnel-ionization times differing by an integer number of cycles. (paper)