This was a tall order, so I went to hit the books. It turned out that there was a special tube designed for just such a purpose, known as a krytron. This was an obscure beast, which actually used radioactive nickel to keep the gas in the tube partially ionized, ready to switch at any moment. It had originally been designed for firing explosives in nuclear weapons, and had been classified. But this was an advanced lab and had accumulated a great assortment of oddball parts. Some time spent asking questions and rummaging around actually managed to produce a krytron.
This was a little thing, about the size of a peanut. I built a charged transmission line setup, with the tube switching it into a 50 ohm load, and ran the thing at a few thousand pulses a second with a sampling plugin to an oscilloscope to fine-tune it and measure the actual risetime (which turned out to be an astonishing 370 picoseconds or so).
So I went to show the finished apparatus to the researcher, only to find out he had basically given me the assignment as a prank, figuring a young college student who wasn't even in his research program wouldn't be able to solve such a difficult and arcane problem. He'd also given it to one of his EE grad students, who'd assembled this huge board with a chain of "avalanche" transistors in series to do the switching. It took about ten minutes between pulses, and would fry the transistors every dozen cycles or so. My board had run for hours at thousands of pulses per second, and was still on the original tube.