What is the correct instrumentation to measure a pulse height of 10's of microvolts with a pulse width of 10's of picosenconds?
We'll assume a bandwidth of 10 GHz. And a system noise resistance of 62 ohms.

10Ghz requires a (single pole) timeconstant of 16 picoseconds. For each 16 picoseconds of settling time, you will improve the amplitude measurement by 9dB (1 neper).

We will assume your pulse flat-top has long enough duration to provide the amplitude resolution.

The total integrated noise of 10GHz bandwidth, using 62 ohms and its exactly 1nanoVolt RMS noise density, will be

Vnoise_total = sqrt(10GHz) * 1nanoVolt

Vnoise_total = 100,000 * 1 nanovolt = 1E+5 * 1E-9 = 1E-4 volts RMS

Vnoise_total = 100 microVolts RMS

Can you live with that?