It is shown that electron injection into a laser-driven plasma bubble can be manipulated by applying an external magnetic field in the presence of a plasma density gradient. The down-ramp of the density-tailored plasma locally reduces the plasma wave phase velocity, which triggers injection. The longitudinal magnetic field dynamically induces an expanding hole in the electron density distribution at the rear of the wake bubble, which reduces the peak electron velocity in its vicinity. Electron injection is suppressed when the electron velocity drops below the phase velocity, which depends on the size of the density hole. This enables the start and end of electron injection to be independently controlled, which allows generation of sub-femtosecond electron bunches with peak currents of a few kilo-Ampere, for an applied magnetic field of $sim 10$ Tesla.
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