I changed again the mind experiment to avoid any explanation involving doppler effect or nonlinear interaction
I expressed the same problem under a different angle but it will probably make the situation more clear in: Creating light pulse via: turning on and off coherent state VS putting photons in many mode, different in the quantum regime? I hope it helps.
There is something that always confused me to link the laws of physics and signal processing.
Consider you have a monochromatic laser of frequency $\omega_0$, for example, made with hydrogen transition. Consider the beam has a given width of $w$.
At $t=-\infty$ you start your laser. You never turn it off.
You have at your disposal a device that measures the intensity of the electric field. I call A the point representing this device.
For $t<0$ the device is outside of the beam. I start to move it so that it crosses in an orthogonal direction the light beam. At $t=0$ it is in the beam and it stays like this until $t=T$, then it is outside of the beam.
If you analyse the electric field in A you will have something like (I work with complex signal for simplicity but replace my $exp$ by $cos$ or $sin$ if you wish):
$$ E(t<t_0)=0 $$ $$ E(t_0<t<t_0+T)=e^{j \omega_0 t} $$ $$ E(t_0+T<t)=0 $$
If I do the spectral analysis of such signal I won't have a single mode $\omega_0$.
But from a physical point of view the only frequency of light I used in all this is $\omega_0$, I didn't excite any other mode than this one.
What does that physically mean?
In the classical regime there is no paradox for me, it is just that I can represent my wave from two points of view: I turn on a monochromatic source for a finite amount of time, or I send many modes on different frequencies with Fourier amplitude corresponding to the Fourier transform of $E(t)$. Both approaches are mathematically and physically totally equivalent here.
The paradox comes in the quantum regime.
I used a device that produces photons at a single frequency of $\omega_0$. But if I study the classical signal on A, I find many modes occupied. It is a kind of paradox because I never created other photons than the ones in $\omega_0$ initially.
Then two possible answers: either photons have been created in the other modes afterward. Either there is indeed the only photon in $\omega_0$ in all the experiments.
I am not convinced by the first possibility, indeed if I reason in the frame associated with A, it will see the signal I described having many frequencies. And it is just a change of frame I cannot create photons in another mode because of this.
It cannot be explained by the Doppler effect as well as in my mind experiment I move at a constant speed in an orthogonal direction to the light beam. Furthermore, the doppler effect is a shift. Here I don't have a shift in frequency, but I have a "creation" of many other frequencies. It is different.
In addition, the laser is based on hydrogen transition which emits photons of a very specific frequency: it couldn't know that a device will cross the beam later on. The emission frequency is independent of what I will do in the lab.
As I proposed in an answer, I think it means that there are different way to describe the signal, but only one of them corresponds to how it was physically produced ?
In practice, in this example, the interpretation is that photons only occupy the mode $\omega_0$, no other photons exist in all this experiment.
The thing is that in A I will see those photons only for $0<t<T$ and not after. By doing a Fourier transform I can describe the light I see as if other light mode were excited. Which is not the case.
Would you agree with this and my proposed answer? People proposed another answer but as I explained in the comment I am not convinced by those answers. I justified my point. If I made a mistake in my answer where is it? I think I am convinced by it but I would like some external point of view.
A different way to ask the question:
The laser being monomode and always turned on, the quantum state inside of its cavity or of the light it radiates can be described using coherent states of frequency $\omega_0$. If photons exist at other frequency could you write down the interaction that created them.
(I don't think those other frequencies photons exist but if they do I would like to see in the answer precisely written the mechanism that created them).
Note: A kinda similar question has been asked here Fourier transform paradox(?) of a wave packet but I am not very convinced by the answers. Also, to avoid explanation like nonlinearity induced by some shutter I would put on the laser path, I choose to take an example with a laser rotation.