How to create temperatures below absolute zero: Negative-temperature systems have the opposite behaviour. Adding energy reduces their disorder. But they are not cold in the conventional sense that heat will flow into them from systems at positive temperatures. In fact, systems with negative absolute temperatures contain more atoms in high-energy states than is possible even at the hottest positive temperatures, so heat should always flow from them to systems above zero kelvin.
Creating negative-temperature systems to see what other "bizarro world" properties they might have is tricky. It is certainly not done by cooling an object down to absolute zero. It is, however, possible to leap straight from positive to negative absolute temperatures.