Quantum-logic clocks for fundamental physics and geodesy

We have compared the rates of two quantum-logic clocks based on the optical $^1$S$_0$-$^3$P$_0$ transition in Al$^+$. The performance of the newer clock is unmatched, and despite many differences, their rates agree to $1.8 \pm 0.7 \times 10^{-17}$, within the accuracy limit of the older clock. The newer clock has an accuracy of $8.6 \times 10^{-18}$ and stability near $10^{-15} (\tau/s)^{-1/2}$. Quantum-correlation spectroscopy yields an improved measurement stability of $3.7\times10^{-16} (\tau/s)^{-1/2}$. This technique also allows Q-factors beyond $6\times10^{15}$ to be seen. This is the highest observed Q-factor in physics. The talk will discuss the basic operation of quantum-logic clocks based on Al$^+$, together with recent results that include a first geo-potential difference measurement, and constraints on the temporal variation of the fine-structure constant. Potential uses of entangled states in such clocks are also explored.