# Evading the $J=0$ to $J'=0$ forbidden selection rule?

One of the first results covered in atomic physics textbooks is the selection rule that it is forbidden to go from $$J=0$$ to $$J'=0$$ via light-matter coupling to any multipolar order. This selection rule arises essentially from conservation of angular momentum, because the photon has helicity 1.

Nonetheless, the lifetime of the Hydrogen (or similarly Helium) $$2s (F=0)$$ state is not infinite - eventually it decays via a two-photon process into the $$1s (F=0)$$ state. However, since two photon events are a quadrupole-like event (which should vanish identically for $$J=0$$ to $$J'=0$$) how is this decay possible?

And, more generally, what are the mechanisms that can actually drive $$J=0$$ to $$J'=0$$ transitions? It seems these events do happen naturally via multi-photon processes, but we also know matrix elements should vanish for any electric or magnetic multipolar coupling.

In a nutshell, you can evade the $$J=0$$ to $$J'=0$$ selection rule via higher photon-number (i.e. $$n\geq 2$$) processes. However, you cannot evade this rule via $$n=1$$ (single-photon) processes, regardless of the multipole order.

In more detail, the matrix element from first-order perturbation theory for light absorption/emission takes the general form:

$$M_{if}\sim \langle f \vert \mathbf{\hat{p}}\cdot \mathbf{\hat{A}}_0 e^{i\mathbf{k}\cdot \mathbf{r}}\vert i\rangle$$

Where $$\mathbf{\hat{A}} = \mathbf{\hat{A}}_0 e^{i\mathbf{k}\cdot \mathbf{r}}\sim \mathbf{\epsilon}_0 (\hat{a}^{\dagger}+\hat{a})e^{i\mathbf{k}\cdot \mathbf{r}}$$ is the vector potential for the photon and $$\mathbf{p}$$ is the momentum operator acting on the atom/material.

If you expand the term $$e^{i\mathbf{k}\cdot \mathbf{r}}$$ above as a Taylor series in powers of $$\mathbf{k}\cdot \mathbf{r}$$ (or, more correctly, in terms of spherical harmonics), then what you get is a multipole expansion of the matrix element where the multipole order is related to the corresponding power of $$\mathbf{k}\cdot \mathbf{r}$$.

However, it is clear that the number of photon creation/annihilation operators does not change as you go to higher powers of $$\mathbf{k}\cdot \mathbf{r}$$ (multipole moment). Thus, a multipole (quadrupole, octupole, etc.) transition described this way only requires a single photon absorption/emission. What is happening is that the photon compensates larger angular momentum changes through its spatial structure (linear/orbital momentum) rather than through its spin as it does in the dipole case.

On the other hand, if you go to higher order in perturbation theory (i.e., higher powers of $$\mathbf{p}\cdot \mathbf{A}$$), then you are considering multi-photon processes, which themselves could be via any type of multipole intermediate transitions. In about other words, the photon number and multipolar order are distinct concepts.

Returning to the $$J=0$$ to $$J'=0$$ forbidden transition - this transition is forbidden to all orders in $$\mathbf{k}\cdot \mathbf{r}$$ for single-photon events (first-order in $$\mathbf{p}\cdot \mathbf{A}$$), but it is not forbidden for higher photon-order processes and can occur even as soon as $$(\mathbf{p}\cdot \mathbf{A})^2$$.

A useful theory discussion behind the specific case of Hydrogen's $$2s$$ to $$1s$$ decay is given in "Two-photon transitions in hydrogen and cosmological recombination" by J. Chluba and R. A. Sunyaev (http://dx.doi.org/10.1051/0004-6361:20077921)