Does amplitude modulation change a photon's frequency or the number of photons?

In the following, we assume that the polarization is aligned such that the scalar treatment of the electric field is justified.
Furthermore, we limit the discussion to a fixed coordinate $x=0$ to drop the wave vector dependency.

Classical description

Let’s consider a classical electric field

$$
E(t)=A(t)cosomega_c t
tag{1}
$$

with angular carrier frequency $omega_c$ where we modulate the field amplitude with

$$
A(t)=A_0cosomega_m t
tag{2}
$$

where $omega_m$ is the angular modulation frequency.
For illustration, we assume $omega_mllomega_c$, for example, $omega_c$ could be in the optical whereas $omega_m$ could be in the low-frequency domain.

In this picture, we can think of the electric wave propagating with frequency $omega_c$ while it’s amplitude slowly oscillates with $omega_m$.

Nevertheless, we can also combine eq. (1) with eq. (2) and write

$$
E(t)
=
A_0cosomega_m tcosomega_c t
=
frac{1}{2}A_0bigl(cosomega_+t+cosomega_-tbigr)
tag{3}
$$

where we define $omega_pm=omega_cpmomega_m$.

In this picture, we actually have two waves, one oscillating rapidly with $omega_+$ and one oscillating slowly with $omega_-$.

So far, so good. Although, eq. (1) and eq. (3) suggest a different point of view both are equivalent and should give the same (classical) predictions.

Quantum description

Now, we turn to the quantum description where we define the electrical field operator to be

$$
hat{E}
=
isum_iomega_ileft{hat{a}_ie^{-iomega_it}-text{c.c}right}
tag{4}
$$

with $text{c.c.}$ referring to the complex conjugate term,
and where $hat{a}_i$ is the annihilation operator of mode $i$ that satisfies the canonical commutation relation

$$
left[hat{a}_i,hat{a}_j^daggerright]=delta_{ij}
tag{5}.
$$

We assume a two-mode coherent state $vertalpha_1,alpha_2rangle$ with $alpha_iinmathbb{C}$ and calculate the expectation value of the electric field operator for that state

$$
begin{aligned}
langlealpha_1,alpha_2verthat{E}vertalpha_1,alpha_2rangle
&=
iomega_1left(alpha_1e^{-iomega_1t}-text{c.c.}right)+
iomega_2left(alpha_2e^{-iomega_2t}-text{c.c.}right)
&=
-2omega_1operatorname{Im}left{alpha_1e^{-iomega_1t}right}+
-2omega_2operatorname{Im}left{alpha_2e^{-iomega_2t}right}
end{aligned}tag{6}.
$$

With the choice $omega_1=omega_+$ and $omega_2=omega_-$ as well as $alpha_i=-iA_0/(4omega_i)$ (upto some unit-preserving factors) we can recover our classical result given in eq. (4).

On the other hand, we should also be able to reproduce eq. (1) by asserting a single-mode coherent state $vertalpha(t)rangle$ with $alpha(t)propto A_0cos(omega_mt)/omega$.

However, this time, we could think of an experiment that distinguishes between these two states!

Recap the photoelectric effect describes electron emission by a photon hitting a metal. The photon energy $hbaromega$ needs to be greater than the work potential $W$ which binds the electron to the metal. Suprisingly, the photoelectric effect is (neglecting multi-photon absorption) independent of the intensity.

Let’s assume that we are in possession of a metamaterial where the work potential is tailored to $hbaromega_c$. In this case, we could distinguish between the single- and two-mode coherent state because the amplitude $vertalphavert$ of the coherent state $vertalpharangle$ fixes the mean of the (Poissonian) photon statistics but the energy, which determines if electron emission occurs, is given by the mode frequency.

If we remind ourselves on how we derived eq. (3) from modes in a confined cavity this also makes sense.

Is it correct to conclude that amplitude modulation of an optical laser signal, shifts the energy of the participating photons?