# Does orbital angular mometum has no meaning for single photons?

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?

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?

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.

• Short but sweet. The wikipedia page actually does a good job in this case. – Rococo Oct 3 '15 at 20:23
• @Rococo Only as short as it is because I ignored the last question, e.g. 'how to determine its possible values' – Timaeus Oct 3 '15 at 20:24

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.

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. ;) )

• I'm not sure this is right --- I've always thought a high-multipole electromagnetic transition stored the angular momentum in the system consisting of the emitted photon and the emitting nucleus. There are many decades of literature on multipolar electromagnetic transistions, but discussions about orbital angular momentum in free photons seem to be more recent. – rob Jul 20 '18 at 16:10

## protected by Qmechanic♦Oct 3 '15 at 21:46

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