Once you're through the door, take note of the 'Emergency Exit' on the right. Then head left down the stairs and into the office not far beyond.

This Terminal is the key to getting the Tesla Coil. And it's not even locked. You can just use it. Activate the Fail Safe and then go through the door to the left of the Terminal. It's very well hidden in dark shadows. Head through the next door beyond it as well.

And there it is. The Tesla Coil. Grab it.

Now head back to that 'Emergency Exit' we saw earlier. It will take you back outside to the wasteland. A warning: there may be several Death Claws in the immediate vicinity when you exit.

Head back to the Citadel and talk to Paladin Tristan. He'll be happy to have the coil. And he'll talk to you about your next quest.

You get the quest complete notification after your conversation with Paladin Tristan.