Place the 4 raw artichoke hearts and the acidulated water into a large saute pan. Add the white wine, garlic cloves, and mint. Bring to a boil, then reduce the heat and simmer until the hearts are tender enough to be pierced with a skewer, about 20-25 minutes. If your hearts are quite different in size, they may not all be done at the same time. As they finish cooking, remove the hearts and set aside. When all have been removed, use a slotted spoon to extract the garlic cloves and herbs from the cooking liquid. Increase the heat, and reduce the liquid to a glaze, 1 to 2 tablespoons.
Thinly slice the artichoke hearts.
Bring a large pot of water for the pasta to boil over high heat.
Place the tablespoon of olive oil in a large saute pan and heat over medium high heat. Add the shrimp and cook until just cooked through – the time will depend upon the size of the shrimp, but be careful not to overcook.
Add the reserved artichoke cooking liquid, the arugula, and the sliced artichokes. Cook until the arugula has wilted and the sauce is warm. Season with salt and pepper and drizzle in some more olive oil, and keep just warm while the pasta cooks.
When the pasta water is boiling, add salt, return to a boil, and then add the pasta. Cook until al dente. Remove and drain, reserving 1 cup of pasta cooking water. Add the drained pasta to the sauce in the saute pan, stirring to combine. If the sauce seems a bit thick, you can use a little of the reserved pasta cooking water to loosen it up a bit, but this may not be necessary.
Serve in four bowls, distributing the shrimp on top of each.