First think about a simple harmonic oscillator. If it is in the ground state (n=0), we say there are zero phonons present. If it is in the first excited state (n=1) we say there is one phonon present. If it is in the (n=2) state, we say there are two phonons present, and so forth.
It is similar for the monatomic vibrational chain. If you consider a particular value of k, if this mode is in its ground state, we say that there are no k-phonons present. If it is in the first excited state, we say there is one k-phonon present, and so forth.
Does that help?
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