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A monostable multivibrator ("one-shot") is the usual way,
but finding one that will respond to a 1ps pulse is not too
likely (for that matter, where are you getting such a pulse
in an electrical system)?
Figure that pulse probably can't charge anything much,
so a simple diode, cap, trsnsistor one-shot probably is
not viable. Maybe, if Zsource was super low. But any
inductance at all (like nH bond wires) and forget it.
There's the timer circuit where the incoming pulse supplies power to build a charge on a capacitor.
There's also the type where the incoming pulse triggers a timer circuit which proceeds independently. No power is taken from the incoming pulse. This method seems suitable for an extremely weak or brief pulse.
1 pico-Sec is so slight that light travels 1/100 of an inch in that time. To begin with you need a sensitive detector. Something akin to the principles of a photo-multiplier tube, or cosmic particle detector.
As long as you don't specify electrical parameters, it's a purely hypothetical problem. Pulse current or voltage? Source impedance? 1 ps corresponds to 0.3 mm wave length in air. The pulse can't be reproduced over any technical coaxial cable and connector. If we hear about 1 ps in a measurement application, it's an optical pulse.