What is meant by quantum coherence in the context of a two-level atomic system?

What is meant by quantum coherence in the context of a two-level atomic system?

Answer

When you prepare a pure quantum state of a two-level system, $| \psi \rangle = a |0 \rangle + b |1 \rangle$, the associated density matrix will be

$$\rho = \begin{pmatrix} |a|^2 & a b^* \\ a^* b & |b|^2 \end{pmatrix}.$$ The non-diagonal terms are usually called the 'coherent' terms, which come from having a pure state instead of a statistical distribution of $|0\rangle$ with probability $|a|^2$ and $|1\rangle$ with probability $|b|^2$.

Interactions with the environment wash away these terms and make you lose coherence (they make the non-diagonal terms $\to 0$). The time scale associated is what is called the coherence time.

Personally, I think a better characterization of this idea is purity, which is given by $\text{Tr}(\rho^2)$ and is basis indepdendent.