You own them as long as they don't fall under NAGPRA and there are no human remains. Even then, most states have laws restricting what you're able to do with a site.
In many European countries finding a pot of gold buried in your back yard means you have to hand it over to the authorities and maybe get a "finder's fee". You're not allowed to keep or sell such historical artifacts regardless of their monetary value.
I don't know what happens when human remains are involved. Common sense tells me that they would never become the "property" of the land owner.
And of course there is a chance that some artifacts are treated differently than others. A T-Rex fossil is 100% naturally occurring while minted coins or a sarcophagus are not. They may have even been at some point property of the state and as such could be treated as national patrimony.
I have been told that this is also true in my home state- Minnesota. Land owners do not automatically retain mineral rights when they purchase property.
