Is the accuracy of atomic clocks about engineering limits? I read that a microwave with specific wavelength is shone on a bunch of atom to make them change energy state from ground to excited state, then they measure this difference in energy to determine its accuracy.
Does it means ideally the microwave laser must only use a specific wavelength and the bunch of atoms must absorb and emit the microwave, and nothing else, to achieve a perfect atomic clock? So the only issues that prevent the clock from being perfect are the environment such as no perfect laser, no perfect Faraday cage, etc.; all of these are engineering issues, is that right?
 A: 
Does it means ideally the microwave laser must only use a specific wavelength and the bunch of atoms must absorb and emit the microwave and nothing else to achieve perfect atomic clock? 

You can't build a perfect atomic clock. But there are various things you can do to try to minimize the imperfections.
Ideally, the microwave beam is perfectly monochromatic, and the atoms are at absolute zero, isolated from all other influences. But of course that's impossible in practice.
As I said here,

The modern caesium fountain uses a small "cloud" of laser cooled atoms in freefall. Using the optical molasses technique, the atom temperature is reduced to ~40 microkelvin. However, even using a small population of such cold atoms, some thermal noise and atomic collisions are inevitable when the cloud is energised by microwaves. But even if we could use a single atom at picokelvin temperatures there would still be some width of the transition frequency, it cannot be a single number, due to time-energy uncertainty.

A simple way to improve precision is to use a higher frequency. If there are more ticks per second, there are more opportunities for minor random variations to cancel each other. OTOH, there are also more opportunities for systematic errors to accumulate. Higher frequency means more energy per photon, and that makes it easier to reduce the relative uncertainty in the energy of each photon. OTOH, it means that a photon has a greater heating effect if it acts thermally. Most modern experimental atomic clocks use optical frequencies.
Here's a table comparing the uncertainties of several types of atomic clock.




Atom
Type
Uncertainty




Cs-133
Beam
1e-13


Rb-87
Beam
1e-12


H-1
Beam
1e-15


Cs-133
Fountain
1e-16


Sr-87
Lattice
1e-17


Mg+Al
Lattice
8.6e-18


Yb-177
Lattice
1.6e-18


Al+
Lattice
9.4e-19


Sr-87
Fermi gas
2.5e-19




"Beam" refers to a standard off-the-shelf beam maser. "Fountain" is an atomic fountain, that value is for NIST-F2. "Lattice" is an optical lattice. "Fermi gas" is a 3D quantum gas optical lattice.
There is a theoretical possibility to use nuclear transitions (rather than electron transitions) to make a nuclear clock. Such a clock would operate at much higher frequencies (ultraviolet, or even gamma ray), and be far less sensitive to environmental influences. But nobody has built one yet.
A: My answer here may be helpful to you.
"So only issues that prevent the clock from being perfect are the environment such as no perfect laser, no perfect Faraday cage etc all these are engineering issues is that right?"
It's a little bit of a funny question. The short answer is that there will ALWAYS be something limiting the accuracy/stability of an atomic clock. In many cases that something will be technical, such as, magnetic fields fluctuation in the vicinity of the atom, or phase noise on the local oscillator (the microwave source in your example), or finite velocity of the atoms. In some cases that something will be more fundamental. For example, the interaction between the electrons you are interrogating are interacting in a deleterious way with other electrons in the same atom that causes frequency fluctuations or zero point motion of the atom. But, scientists will likely be able to come up with tricks to get around ever more and more fundamental limitations to make better and better clocks.
So if your question is: are current clocks limited by technical limitation, then the answer is yes.
If your question is: will future clocks be limited by technical limitations, then the answer is also yes.
If your question is: will we ever solve all technical limitations and achieve a perfect clock? then the answer is no, this will never happen.
I'll emphasize one point that seems a little wrapped up in your question though: Your question seems to presume that atoms are perfect, and it is only because the systems we use to interrogate them are not perfect. This is a myth which has been propagated about atomic clocks. Atoms are not perfect. They are very good, and they can be used to make clocks that are more stable and accurate than other physical system humans have tried to make clocks out of, or physical system that humans have observed in the universe. But they are not perfect, they have flaws. When you excite an atom it has a finite chance to decay (fundamentally this is due to coupling of your atom to the electromagnetic field, the same field you use to control the atom), if this happens during your measurement then your measurement is junk, this contributes to instability. Various internal forces can lead to decoherence of the electronic superposition state. At some level there is fundamental shot noise on your interrogating laser due to the quantum nature of light. So I would hesitate to say that there are ONLY technical limitations to atomic clock performance. There are some limitations which come from the fact that atoms have lots of junk in them when you zoom in on them at the $10^{-20}$ level.
There are proposals to use atomic nuclei as clocks instead of atoms. You might argue that nuclei have less junk in them than atoms. They don't interact as readily with electromagnetic fields. But, at some level of accuracy/stability, we will find nuclei to be full of junk as well such as quarks and other subatomic effects that will eventually limit precision.
Maybe after nuclei we'll find something else, but there will always be error bars. The only thing that is certain is the presence of uncertainty.
