Edit to add: Warning - Might be boring.
We're potato farmers in Idaho. I don't know what your climate is like but the plants will just naturally start to take a dive at the end of their growing cycle. Yellow, wither, and start to die. You can keep watering them lightly though. It's fine to leave them in the ground after the foliage looks dead and you probably should. They sort of "cure" in the ground and the skins toughen up. You really don't need to worry about digging them until a really hard frost is forecasted. They don't like that. They are about 80% water so a hard freeze turns them to mush, eventually. Stinky mush. So, so stinky.
Maybe you're not interested, but here's a little info. We grow around 500 acres of potatoes. We start getting ready to harvest our crop around the last week or so in September. If we've had a frost it usually kills the vines. If not, we have to spray on a chemical that kills the vines. Then we use a piece of equipment that beats the vines off (grinds them into pieces) so the long vines don't get tangled in the chains of the harvester or bind up on the drive lines of the trucks. Then we let them sit and "cure" for about 10 days in the ground so the skins toughen up. Potatoes get bounced along on a lot of chains and belts before they make it to storage so the skins need to be tough or they bruise and then ultimately rot. That's where the stink comes in. So, so gross. We start digging them out of the ground around the first week in October.
They are stored in a huge cellar and that big of a pile can generate heat so they have to go through a "sweat" period, where the field heat leaves the tubers. It puts the potatoes in dormancy and prevents sprouting during early months in storage. They need to be stored at about 40 degrees. Ideal conditions are ventilated, cool temperatures, high humidity, and no light. Our storage facilities are set up to create all these conditions. We harvest in October, store them, and sell them the next year in August. Usually. Hope I didn't bore you.