Do distinct number states of the same wavelength/momentum and same particle type have differing frequencies?

Question

In quantum field theory there are a number of quantum fields, one for each type of particle. I consider just a single field, for example the electromagnetic field, and as a free field. As I understand we can choose 'momentum (or wavevector) and occupancy' as a basis for this field (or any field). Using this basis, let's consider only a single momentum (or wavevector).

This is commonly introduced when explaining quantum field theory as analogous to a quantum harmonic oscillator (as considered in this question). The harmonic oscillator system as a whole has a fixed frequency, corresponding to the frequency of coherent states. However, individual energy states have distinct frequencies. Energy states of the oscillator correspond to occupancy states of the quantum field of a given momentum, so I wanted to check that each of these also has a distinct frequency?

Answer 1

After a bit of research I believe the answer is yes! The number states have uniformly increasing frequencies just like the oscillator ladder states, and the empty / lowest state has frequency equal to half the increase between states. The field mode also has coherent states, and these are combinations of the number states. All the coherent states of a single mode have the same net frequency, although each can be broken down into all the contributions of number states with different frequencies.