Regardless of the situation you have provided, so you can use it in more situations, it depends on the players and how the hand played. There is not one answer, every situation will be different.
Simple answer is whatever amount you think will make your opponent fold. Some opponents will see a small bet as scary and that you want to be paid off, some will be the opposite.
It depends hugely on the level of your opponent. Against weaker players betting big will often work as they'll likely (depending if they're tight) fold thinking you'll only bet like that with the nuts as that's what they would do. They'll just see the bet and won't recall how the hand played. Likewise against a thinking player, they might think along the lines of they bet big to look strong, I know that they're capable of doing this, they're loose player, so he could very well be weak, I'll call. They'll also look at what happened on every street of the hand, and remember previous hands and use that information to make their decisions.
The amount depends and differs based on your opponents level of poker and how they're thinking of the game.
With regards to your situation, you got very lucky here as anyone who understands the basics of poker and even the basics of the maths would know this situation is extremely unlikely. If you have the quad aces oh well, cooler. I worked out the maths here, take a look to see how unlikely it is/the mistake the other player made.