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.
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