For example when they cool down atoms to fractions of a degree above 0K, how do they achieve this?
Of course, usually before they even start to bring the lasers into play, they use magnetic traps and standard evaporative cooling methods. Basically, in a magnetic trap, the hotter atoms are allowed to escape, cooling the sample. Since the temperatures of any sample are arranged in a bell curve like distribution, there will always be atoms that are warmer than average. These warmer atoms get their extra thermal energy from random collisions, some of which give atoms energy, some of which take it away. By consistantly removing the atoms that gained energy in collisions, the energy of the population falls. This is the same way that evaporating water ( some of the atoms at the end of the bell curve actually get hot enough to boil, even if the average temperature of the water is relatively cold)causes things to cool, though researchers bringing things to near absolute zero take it to an extreme.
Ah…I see friedo’s cite gets into all that a few pages in, the funny heads made me fear to click the next button, but I found my courage. 
We usually use the adiabatic (no heat flow into) expansion of liquid helium. Liquid helium is at 4.2 degrees absolute (about -269 deg C or -450 deg F). This is close to absolute zero. If we allow the liquid helium to boil off into a partial vaccum, it gets ccoler. We get down to about 1.5 degrees absolute. This is done with a 200K$ superconducting magnet system. Liquid helium is about $4.50/liter.
The evaporative cooling in a magnetic trap is after the lasers have been used. magnetic traps are really shallow, so the atoms have to be cooled before you can load them. Laser cooling alone can get you to a microK or so.
It’s also not really correct to say the warmest atoms escape the trap, since they’re all part of the same thermal distribution. The most energetic atoms escape, which is why the average energy, and thus temperature, decreases.
In comparison to laser cooling and especially Bose-Einstein condensates, 4.2K is positively balmy.
Using refrigration liquids like liquid helium gets you so far. After that, I think, the next technology you bring to bear is magnetic.
This is pretty hazy, but maybe close enough to get you searching for the right terms.
There are certain salts (ionic compounds) that change their temperature as the strength of the magnetic field around them is changed. So, they cool these down with liquid helium while the magnet is on, then turn the magnet off and they get even colder. Or maybe they leave the magnet off and then turn it on - don’t remember which way. Think the salts are called paramagnetic or diamagnetic, something like that.
I think this step takes things from a few Kelvins to a fraction of a Kelvin.
