Classical single-mode electromagnetic waves have a phase: the electric and magnetic fields are oscillating in space and time and the phase dictates the current value of the amplitude as it is oscillating. The quantum state representing classical electromagnetic waves, the coherent state, inherits this same phase. But there is no quantum mechanical (Hermitian) operator whose eigenvalue is this phase, this phase is not an observable, and that at the very least means that not every state will have a "phase" that can be measured or defined or discussed. For two modes, a relative phase can be defined both classically and quantum mechanically, because the relative-phase operator is well defined.
The main two "types" of phases people talk about in quantum optics are
- The "single-mode phase" that is the one that classical electromagnetic waves have, is a relative phase between different photon number states, will change with propagation, and is not defined for Fock states.
- A relative phase between two different modes, such as two polarization modes where that relative phase can change with a waveplate.
Let's write the math! For a single-mode state, we can use the photon-number basis with Fock states $|n\rangle=a^{\dagger n}|0\rangle/\sqrt{n!}$, defining the coherent state as
$$|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}|n\rangle.$$ That this has a "phase" is that $\alpha$ is any complex number. If you look at any single Fock state $|n\rangle$, there is no such phase, but in a coherent state it looks like $|0\rangle+\alpha|1\rangle+\cdots$ times a normalization constant, so the relative phase between the photon numbers like $|0\rangle$ and $|1\rangle$ is directly given by the phase of the complex number $\alpha$. The electric field operator is proportional to $a+a^\dagger$, whose expectation value is proportional to the real part of $\alpha$, so the phase matters. The free evolution of an electromagnetic field is that of a harmonic oscillator Hamiltonian $H=\hbar \omega a^\dagger a$, so the field evolves as
$$|\alpha\rangle\to e^{-i H t/\hbar}|\alpha\rangle=|\alpha e^{-i \omega t}\rangle$$ due to the eigenvalue equation $a^\dagger a|n\rangle=|n\rangle$. The amplitude $|\alpha e^{-i \omega t}|$ is constant with time but the phase is not, so the expectation value of the electric field operator oscillates as $|\alpha|\cos(-\omega t+\arg\alpha)$, same as you see in classical electromagnetism. When we include position in the computation, the wave looks like $|\alpha|\cos(kx-\omega t+\arg\alpha)$. When a Fock state is subject to the same evolution, it merely acquires a global phase $e^{-i H t/\hbar}|n\rangle=e^{-i n\omega t}|n\rangle$, which has no effect on any observable.
How is single-mode phase measured, if it's not an observable? It must be through interference experiments, such as by "projecting" the state onto another state of the form $(|0\rangle+\exp(i\theta)|1\rangle)/\sqrt{2}$ for some value of $\theta$. Each would tell you some value $|1+\exp(-i \theta)\alpha|^2$, such as $|1+\alpha|^2=1+|\alpha|^2+2\mathrm{Re}(\alpha)$. Or, you can project onto another coherent state with a known complex parameter $\beta$, because that also displays relative phases between different photon numbers, or you interfere the state with a known coherent state with large $|\beta|$ on a beam splitter to do homodyne measurement, etc.
Now let's get into relative phases between different modes. We will use the two-mode case of polarization, assuming the two modes house horizontally and vertically polarized light as in the original question, but they can be two spatial modes or anything else. The coherent state version is $|\alpha\rangle_H\otimes|\beta\rangle_V$ and we have already discussed how measuring a relative phase between two different coherent states is meaningful (in the context of using one coherent state to measure the other coherent state's phase). When this passes through something birefringent like a waveplate, the effective propagation distance for horizontal versus vertical light is different (due to different refractive indices here), so the phases they accrue are different:
$$|\alpha\rangle_H\otimes|\beta\rangle_V \to |\alpha e^{-i(k_Hx-\omega t)}\rangle_H\otimes|\beta e^{-i(k_Vx-\omega t)}\rangle_V.$$ With different refractive indices, $k_H\neq k_V$, and so the relative phase changes between the two modes. Or, if each mode describes a path in an interferometer and one is longer than the other, then we require a different $x$ for each path, and a relative phase will accrue.
Can we talk about relative phases with Fock states? Well in this picture, if each of the polarization modes has a Fock state, then no: $|n\rangle_H\otimes |m\rangle_V$ acquires a global phase $\exp(-i (n(k_H x-\omega t)+m(k_V x-\omega t)))$ that still cannot be measured. But if you first interfere the Fock states on a beam splitter, then what exits is not $|n\rangle_H\otimes |m\rangle_V$ but some state that exhibits entanglement and superpositions between photon numbers, and then relative phases will matter.
The last point is important for talking about relative phase for a single photon. You see, a single photon can exist in a superposition of two or more modes, and then the relative phase between those modes matters. If you send a single photon $|1\rangle_H\otimes|0\rangle_V$ through a polarizing beam splitter, what gets output could be a state like $(|1\rangle_H\otimes|0\rangle_V+|0\rangle_H\otimes|1\rangle_V)/\sqrt{2}$. Now all of a sudden we can talk about the relative phases between the two terms in the superposition! And, if we go to a "first-quantized" picture in which we simply talk about the state of the photon, this is exactly equivalent to $(|H\rangle+|V\rangle)/\sqrt{2}$, with a specific phase relationship between the horizontal and vertical component. As in the question, you can define another mode to be this superposition mode, like saying "I have a single photon that is diagonally polarized $|D\rangle=(|H\rangle+|V\rangle)/\sqrt{2}$" and that specifies the relative phase between the horizontal and vertical components, so this type of phase is "in the wavefunction" or "in the mode description" or "between modes" depending on how you look at things. Again, this can acquire a relative phase, eg via birefrigence, so then the state would change from $|D\rangle$ and could become $|R\rangle$ or something.
So was it all about single photons vs coherent states? Nope. We could have done all of that two-mode single-photon stuff with coherent states. You see, we could take something like $|\alpha\rangle_H\otimes|\alpha\rangle_V$ and realize that it is equal to $|\sqrt{2}\alpha\rangle_D\otimes|0\rangle_A$ if we use a diagonal/antidiagonal mode decomposition. The diagonal mode automatically defines itself as having its constituent photons have a specific relative phase between $H$ and $V$, and this is true even classically when you don't need to mention photons, where it is just the phase of the polarization state.
In fact, you can always choose a mode decomposition in which your current modes look complex, just like how if you rotated the axes defining your complex plane then the basis vectors from the old basis would change their phases in the new basis.
To summarize, there are at least two concepts of a phase. Something like $|\alpha\rangle_R\otimes|0\rangle_L$ has a phase in $\alpha$ itself, corresponding to possible interference measurements between different photon numbers, corresponding to something that changes with propagation and time, corresponding to something that is the classical electromagnetic phase that tells you how the electric and magnetic fields are oscillating, corresponding to the thing that Fock states lack, corresponding to the number-phase "uncertainty relations" that are approximate, corresponding to the old controversy of defining a single-mode phase operator; and, it has a phase in the definitions of the modes like $R$, which is a complex superposition of horizontal and vertical polarization, corresponding to the phase of a polarization vector, corresponding to the single-photon phase, corresponding to the phases present when writing a wavefunction, corresponding to relative phases between modes, corresponding to the things that get changed with waveplates, corresponding to the things that change between different modes when they each propagate differently. In optics, it's all about knowing what the modes are, and in quantum optics it's all about adding to that the knowledge of the photon numbers and what they are doing.
