How to train cat to stop going after birds?

My girlfriend recently bought a cat that is now 1 year old, and lives in our house. The problem is that I own lots of birds including three cages of zebra , gouldian, and star finches, lorikeets, and parrotlets.For the past two months, every single day, the cat goes and sits next to the bird cages, and tries to terrorize them. I moved my birds into our bedroom, but this hasn't helped either. I lock the door when I go out, but obviously when I'm doing things around the house and can't keep the door locked, the cat runs in and sits on top of the cages to har the birds. He constantly does this and wont stop. I've tried being nice by telling him no and picking him up and carrying him out of the room. That didn't work so I tried chasing him out with a broom. I even bought him little toy birds that he could play with, but it still wont work. A few days ago, I found him running out of the room with one of my finches in his mouth. Apparently, he opened up the cage and caught one. I immediately secured the bird, picked the cat up by the skin of his neck, and shook him a bit until he was dazed. I've tried all methods of trying to get him to stop, but he continues to get worse and worse. It seems now that the punishments need to be harsher to get him to stop, so now when he tries to go after the birds, I wring him out by the neck for a few seconds then carry him outside. (It's not too forceful because I don't want to hurt him) Does anyone have any advice to get him to stop?