Are coherent states destroyed by measurements?

Question

For a quantum harmonic oscillator in a coherent superposition, what happens if the energy is measured? Would it collapse to an energy eigenstate (a single excitation) corresponding to the result of the measurement, thus destroying the coherence? Similarly, for a quantum field in a coherent state (classical EM field wave), does the entire field configuration collapse to a single eigenstate after the field is measured? Wouldn't this destroy the coherence?
EDIT: I do realize that it should collapse, but I'm not sure how to interpret pictures of coherent state measurements, such as this one from Wikipedia ("Coherent State" article):

Shouldn't the coherent state collapse after the very first measurement (at t=0), thus destroying the "classical wave" pattern for the rest of the measurements? Yet it looks classical in this picture...

Nikodem · Accepted Answer

It depends on the type of the interaction. More precisely, on the interaction (part of the) Hamiltonian $H_I$.
Usually, the interaction brings the system to the eigenbasis of $H_I$, the so called pointer states. These can be but not need to be the energy eigenstates of the system Hamiltonian. Depending on that, the coherence is fully lost, partially lost or preserved.

ZeroTheHero · Answer

yes it collapses to an energy eigenstate.

Tim Crosby · Answer

If the wavefunction can collapse for position measurements then why can't coherence be destroyed by energy measurements? It's approximately the same reason.

Are coherent states destroyed by measurements?

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