# What makes a superposition of states a coherent superposition?

Hi everyone
I am investigating spontaneously generated coherence(SGC), I found that it happens when an excited atomic state decays to one or more closed atomic levels so that atom goes to a coherent superposition of states , Effect of State Superpositions Created by Spontaneous Emission on Laser-Driven Transitions.
J. JAVANAINEN
Europhys. Lett., 17 (5), pp. 407-412 (1992)

initial state may give rise to a coherent superposition of two (or more) receiving states
"..
Now I have a question,
I am wondering when can we call a superposition of states a coherent one?

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rubi
In QM, you can usually specify a set of physically relevant observables ##\mathcal A##. A superposition ##\left|\psi\right> = \alpha \left|\psi_1\right> + \beta \left|\psi_2\right>## is said to be a coherent superposition of ##\left|\psi_1\right>## and ##\left|\psi_2\right>## if there is an ##A\in\mathcal A## such that ##\left<\psi_1\right|A\left|\psi_2\right> \neq 0##.

The reason for this definition is that if there is no such ##A##, the state can't be physically distinguished from the statistical mixture ##\rho = |\alpha|^2 \left|\psi_1\right>\left<\psi_1\right|+|\beta|^2 \left|\psi_2\right>\left<\psi_2\right|##.

Thank you dear Rubi
I understand your first statement , it is related to the coherence condition,which is,having non zero off-diagonal elements of density matrix operator, right?