Instead, why not put CO2 dry-ice cubes in your drink? Or a spritz of liquid nitrogen? Or drop in a shot glass of liquid helium? Or super-cooled metal balls?
One of the other problems is that if you cool too quickly, you get an ice layer around the cooling, which then inhibits you from cooling any further. (Note I say "inhibits", not "prevents".)
but it'll make water pretty carbonated (though not to soda's level). or at least have the same bitter, bubbly flavor (never really looked up if that's actually what happens)