quantum mechanics - Uncertainty principle in atomic clocks?

How does the uncertainty principle limit the accuracy of atomic clocks. I know line width and measurement time are important but not exactly why?

Answer

A caesium clock generates a microwave signal that it tunes to match the absorption peak in the Caesium spectrum. A counter counts every oscillation of the generated microwave, and every 9,192,631,770 counts = 1 second. That's how the clock counts the seconds and keeps time.

But if the peak in the caesium spectrum is broad it's difficult to tune the microwave generator to exactly the peak maximum. That means there is a potential error in the microwave frequency and hence in the measured time. This will make the clock run slow or fast.

So the narrowness of the peak is very important to keeping accurate time. When I last looked the limitation on the peak width was Doppler broadening. This happens because some of the caesium atoms are moving towards the microwave generator and some caesium atoms are moving away. This motion shifts the position of the absorption peak, and the end result is that the peak broadens.

You can design the clock to reduce the speed the caesium atoms move, and in principle you can eliminate Doppler broadening. However the uncertainty principle sets a lower bound for the peak width. For an electronic transition like the one in Caesium the uncertainty principle tells us that:

$$ \Delta E \Delta \tau >= \frac{\hbar}{2} $$

where $\Delta \tau$ is the lifetime of the excited state i.e. the average time taken for the excited state to emit a photon and relax to the lower energy state. This $\Delta \tau$ is a characteristic of the caesium atom and is effectively a constant beyond our control. This matters because the energy $E$ is related to frequency by $E = h\nu$, so we end up with an uncertainty in the frequency given by:

$$ h\Delta \nu >= \frac{\hbar}{2} \frac{1}{\Delta \tau} $$

This tells us we will never be able to tune the microwave frequency to better than $\Delta \nu$, so there is a fundamental source of error that we cannot eliminate.