When you hear about atomic clocks, it's accuracy is described by saying something like, " it neither gains or loses $x/y$th of a second in $z$ years." How is this error calculated? Does an error imply that we have a more regular physical phenomena with which to compare the atomic clock? Or does this error simply mean there are quantum phenomena that occur extremely rarely in the clock that might throw off its regularity every once and a while?
In an atomic clock, the reference is the D2 line of Cesium, i.e the energy difference between a ground state and an excited state. This is an absolute value up to the Heisenberg limit. So, if you had perfect conditions, you are only limited by Physics. What I mean is that there is a minimum uncertainty in being able to measure the energy difference between the ground state and the excited state.
However, to get to the Heisenberg limit, several technical errors have to be overcome. "Line broadening" (i.e energy measurement uncertainty) occurs due to many factors such as:
Laser issues: Phase noise, intensity noise, frequency drifts. For the most part, these can be controlled very precisely i.e to about ~10Hz/1THz. "Light shifts" can be controlled by lowering the intensity. Frequency shifts are compensated by "locking" the laser to the resonance line using active feedback. Optical feedback lowers the laser linewidth itself.
Magnetic Fields: This is one of the more nasty problems as you have to actively compensate for AC and DC fields. Since the setup involves a MOT (Magneto-Optical Trap), measurements are made after the trapping magnetic fields are turned off, so hysteresis or residual fields can perturb the system by the Zeeman effect. This problem has also been taken care of by the pioneers of the MOT system.
Apart from this, there are a ton of other things that have to be monitored. So yes, in essence, these errors are of a technical nature as opposed to some random occurrence.
The above is wrong; in a Cesium clock the transition is a microwave transition between the two hyperfine levels of the ground state.
The accuracy stems from the frequency stability. By comparing with other good clock, one can discern how stable the frequency stability is. The elapsed time is just the integral of the frequency, so the timing error can be calculated by multiplying by the fractional frequency stability. i.e. if the frequency stability was 1 part in 3.15 x 10^7, you would expect a timing error of up to 1 second in a year.
The limits to better microwave clocks stem from how well we can control perturbations that affect the atoms, such as magnetic and electric fields, collisions between the atoms, problems with the microwave fields not being uniform, etc.