Cooling of atomic clocks I have read of cooling atoms to one-quarter of one-thousandth of a degree above absolutely zero. According to Physicist William Phillips, this improves the measurement of the ticking frequency. I have three basic questions relative to this subject;


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*How does it improve the measurement? Does it alter the ticking frequency?  What purpose does the cooling serve?

*If we could cool the atomic clock to absolute zero would it, in theory, stop completely?

*In cases where atomic clocks are used to demonstrate time dilation do the clocks ticking frequencies alter with respect to each other?
 A: No. Cooling down the atom will not alter it's frequency. What the scientist meant by saying that the clocks will be better is that they will be much more accurate.
This comes from the principle of Quantum mechanics, in fact one of its most beautiful consequence, Heisenberg's uncertainty principle. Which says (in one of it's variety) that the uncertainty in the energy is inversely proportional to the uncertainty in time. Now if you cool down a atomic clock to its ground state it's uncertainty in energy decreases. Now the frequency is inverse of time so it's uncertainty also decreases and you measure accurately.
But if you reach to 0 degree kelvin. All the atoms will be in ground state, thus there will be no transition from one state to other, and all atomic clocks will stop( so will everything else). So decreasing the temperature close to zero has different effects than taking the temp to absolute zero(which is anyway unattainable). 
A: Addressing the 3rd question, the clocks' frequencies are the same when each is viewed in its own rest frame.  When a clock is viewed in a moving frame, that's when its frequency is changed.  You record the ticks and their locations (which are different since the clock is moving) and you discover that the time between ticks is now longer.
Oh, and addressing the second question, no.  This has to do with the meaning of temperature. A temperature is function telling what the occupation-number is for each energy-level of the system.  What exactly are the thingies that occupy each energy level can be very un-intuitive; it isn't always just whole atoms or whole particles. So, we'll skip that.  But the main point is that for each energy, there is a number, and for each temperature there is a function.
OK, so what happens when the actual occupations don't match any temperature function? The distribution is said to be non-thermal.
In a cesium clock, this is the case.  There is a temperature associated with the atoms before you start measuring, but once you shine this wave on them, the one associated with the hyper-fine transition, then the number for that one energy level goes 'way up.  So we have a mostly thermal shape with one wrong energy-level.  So, it's easy to see why someone would call that thermal and name a temperature.  You just omit the one odd energy-level.
If you do that for absolute zero, you would still be able to have a clock; the clock would still run.  You just have to understand what is meant.
A: The way atomic clocks work is to produce a microwave signal with exactly the frequency of the atomic transition being used. So for caesium this would be 9,192,631,770Hz. Then we can count the oscillations of our microwave generator to measure the time.
Practically you do this by tuning your microwave signal to maximise its absorption by the caesium atoms. The absorption is a maximum when the microwave frequency exactly matches the frequency of the caesium transition.
The trouble is that the absorption line of the caesium atoms has a non-zero width so the absorption of your microwave is high over a range of frequencies, and that makes it hard to tune the microwave frequency exactly. To get the best accuracy you want the width of the absorption line to be as small as possible.
As Ari says, the line has a natural width associated with the lifetime of the hyperfine excited state. We can do little about this, but in practice the line width is greater than the natural width for reasons we can address.
Collisions between atoms will cause the line to become broadened in a process called (unsurprisingly) collision broadening. Obviously cooling will reduce the collision rate by decreasing the thermal velocities of the caesium atoms, but collision broadening isn't a big problem in caesium clocks.
The big problem is Doppler broadening. Even if the random thermal velocities of the caesium atoms is only a few metres per second this creates a measureable Doppler shift in the light being received by the atoms and as a result the width of the absorption line increases. The solution is to cool the atoms to reduce the thermal velocities and hence reduce the line width to something like its natural width.
