Are there rules on how the unchained action economy interacts with the staggered condition?
There are rules on how haste works, but I don't remember there being something for slow/staggered.
I assume that your acts/turn are reduced. But how much? 1 act for one attack or move? That would mean no spellcasting which is possible for staggered casters under the standard action economy. But that would mean move and 1 attack for martials.
Standard action economy:
In a normal round, you can perform a standard action and a move action, or you can perform a full-round action. You can also perform one swift action and one or more free actions. You can always take a move action in place of a standard action.
Staggered: A staggered creature may take a single move action or standard action each round (but not both, nor can he take full-round actions). A staggered creature can still take free, swift and immediate actions. A creature with nonlethal damage exactly equal to its current hit points gains the staggered condition
Under the revised action economy you have 3 acts and different kinds of actions cost different amounts of acts (attack 1 act, cast a spell 2 acts, move 1 act)
One possibility to give someone the staggered condition is the spell slow.
From the PC I'm on I can't access the d20pfsrd so I can't link to the revised (unchained) action economy.