Does orbital angular mometum has no meaning for single photons?

Question

1. In the quantization of free electromagnetic field, it is found that the left-circularly polarised photons corrsponds to helicity $\vec{S}\cdot\hat p=+\hbar$ and right-circularly polarised photons corrsponds to $\vec{S}\cdot\hat p=-\hbar$. They respectively corresponds to the states $$a^{\dagger}_{\vec k,+}|0\rangle, \hspace{0.2cm}\text{and}\hspace{0.2cm} a^{\dagger}_{\vec k,-}|0\rangle$$ where $$a^{\dagger}_{\vec k,\pm}=\frac{1}{\sqrt{2}}(a^{\dagger}_{\vec k,1}\pm a^{\dagger}_{\vec k,2}).$$ This little calculation is performed in the QFT book by Maggiore by looking at the action of the spin operator $S^{ij}$ on these states. But nothing is mentioned about the orbital angular momentum of individual photons. My question is whether individual photons also carry orbital angular momentum? If yes, what are the values of orbital angular momentum in one-particle states? Can the superposition of two photons have orbital angular momentum? If yes, how to determine its possible values?

2. In Classical Electrodynamics (Ref. J. D. Jackson, 3rd edition, page 350) or Classical field theory, the angular momentum of the electromagnetic field is defined as $$\vec J=\epsilon_0\int d^3x \vec x\times (\vec E\times \vec B)$$which can be reduced to the form $$\vec J=\epsilon_0\int d^3x [\vec E\times \vec A+\sum\limits_{i=1}^{3}E_j(\vec x\times \vec\nabla)A_j).$$ The fist term can be identified with the spin contribution of the angular momentum of the field which has its origin in the spin angular momentum of individual photons. The second term is identified with orbital angular momentum of the field? Is there a quantum mechanical origin to this orbital angular momentum?

3. If there is no meaning to orbital angular momentum of individual photons? Is it only a property of that emerges only when collection of photons builds up a classical field?

Answer 1

My question is whether individual photons also carry orbital angular momentum?

Yes. See https://en.m.wikipedia.org/wiki/Orbital_angular_momentum_of_light

If yes, what are the values of orbital angular momentum in one-particle states?

To quote the wikipedia page

In particular, in a quantum theory, individual photons may have the following values of the OAM: $\mathbf L_z=m\hbar.$

So it's not just at a classical level.

Can the superposition of two photons have orbital angular momentum?

Sure, they are bosons so they can even have the same orbital angular momentum quantum number.

Answer 2

Yes, single photons can have orbital angular momentum. However, unlike spin, they are not required to have any. Just like in the classical case, the orbital momentum of single photons is determined by the shape of their EM mode- roughly speaking, the wavefront must have a helical aspect to it. In particular, this means that the eigenmodes of light in a 3D box (at least, one with sides of different lengths) won't have any orbital angular momentum, which is one reason that it tends to be ignored in many QFT treatments.

Orbital angular momentum of single photons is a popular research topic right now because of quantum informational applications. For example, here is a recent paper (open-access) in which the authors generate entanglement between pairs of photons in the orbital angular momentum degree of freedom.

Answer 3

For some experimental evidence of photons with orbital angular momentum, consider the gammas emitted in $2^+\rightarrow 0^+$ transitions in even-even nuclei.

I'm not sure of the superposition question, but experimentally the coulomb excitation of a $4^+$ state from a $0^+$ might be explained as via two virtual $E2$ photons. (Comments from other nuclear peeps welcomed. ;) )