Arbitrary entangled matter-photon emitter

Question

I am recently approaching to photonics and integration between matter qubits together with photons. I have an interest in understanding the assumption I can do when abstracting such technologies.

I read in this paper about the generation of an EPR state between a matter qubit (ion) and a photon $$\frac{1}{\sqrt{2}}(|\uparrow\rangle|H\rangle + |\downarrow\rangle|V\rangle)$$ Where the arrows represent an ion, while H/V labels represent the photon -- see equation (1) from the paper.

This can be therefore be encoded in: $$\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$

I would like to know whether there is any (theoretical or experimental) reference about the creation of an arbitrary state:

$$\alpha|00\rangle + \beta|11\rangle$$

EDIT 1: I consider a valid answer anyone of the following kinds:

1. Nobody ever tried to implement this.

2. Unfortunately, what you ask for is fundamentally impossible to achieve, because...

EDIT 2: This paper seems like a valid reference to me.

Answer 1

The scheme you linked to already allows for significant flexibility by just tweaking the geometry.

As the paper makes clear in its (unnumbered) first displayed equation, the decay scheme they use does not naturally produce a balanced weighting of the two states in question, but rather the state $$ |\psi⟩ = \sqrt{\frac23}|{↓}⟩|\sigma^+⟩ + \sqrt{\frac13}|{↑}⟩|\pi⟩, $$ where $|\sigma^+⟩$ and $|\pi⟩$ represent two separate photonic modes with different polarizations and different angular emission profiles with respect to the axis defined by the external magnetic field. Figure 1(a) of the paper shows how their protocol uses the geometry of the problem to even out the superposition, by observing the emission from a direction in which the $|\sigma^+⟩$ mode is weaker than the $|\pi⟩$ mode:

Thus, by simply changing the angle between the observation axis and the applied magnetic field, the scheme should be able to produce mixtures at all sorts of possible relative weights between the even 50:50 superposition in the paper and the fully-$|\sigma^+⟩$ limit.

And, because the two states are eigenstates at different energies, you are able to set their relative phase arbitrarily by just waiting during an appropriate interval.

Of course, this only gets you half of the Bloch sphere, and not a full control, but this is only a minor change to the protocol. It is definitely possible to produce arbitrary entangled states between an atomic emitter and the polarization of the photon it has produced. I can't point you to a reference that exhibits this (and I won't undertake a search, because your question is phrased in a way that indicates that you have multiple requirements about what you're looking for that will lead you to discount alternatives, but you are not stating those requirements upfront), but I can offer you a guarantee that there is no fundamental roadblock to producing this.

What can offer roadblocks is that not everything is equally publishable. So, even if the specific task that you're interested in is possible and has been achieved, it might not have been deemed interesting enough to publish. If you want to look for it in literature, ask yourself what the task would be useful for, and then use those prospective applications as a guideline when doing your literature search.