Types of photon qubit encoding How many types of qubit encoding on photons exist nowadays? I know only two:


*

*Encoding on polarization:
$$ \lvert \Psi \rangle = \alpha \lvert H \rangle + \beta \lvert V \rangle $$
$$ \lvert H \rangle = \int_{-\infty}^{\infty} d\mathbf{k}\ f(\mathbf{k}) e^{-iw_k t} \hat{a}^\dagger_{H}(\mathbf{k}) \lvert 0 \rangle_\text{Vacuum} $$
$$ \lvert V \rangle = \int_{-\infty}^{\infty} d\mathbf{k}\ f(\mathbf{k}) e^{-iw_k t} \hat{a}^\dagger_{V}(\mathbf{k}) \lvert 0 \rangle_\text{Vacuum} $$

*Time-bin:
$$ \lvert \Psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle $$
$$ \lvert 0 \rangle = \int_{-\infty}^{\infty} dz\ f\left(\frac{t -z/c}{\delta t_{ph}}\right) e^{-i w_0 (t-z/c)} \hat{a}^\dagger(z) \lvert 0 \rangle_\text{Vacuum} $$
$$ \lvert 1 \rangle = \int_{-\infty}^{\infty} dz\ f\left(\frac{t -z/c+\tau}{\delta t_{ph}}\right) e^{-i w_0 (t-z/c+\tau)} \hat{a}^\dagger(z) \lvert 0 \rangle_\text{Vacuum} $$
Is there anything else?
 A: In general, any two orthogonal modes of light provide a way of constructing a two-level quantum system that can serve as a qubit. The possibilities are immense! Of course, not every option will be viable in practice. Besides the cases you mention, I know of at least a few others that have been considered and employed in the lab:
1. Dual-rail: A qubit is encoded in the occupation of one single photon in either of two orthogonal modes, that we label A and B. In your notation, the logical qubit states are 
$$|0\rangle=|0\rangle_A|1\rangle_B$$
$$|1\rangle=|1\rangle_A|0\rangle_B$$
See this paper for a good review.
2. Orbital angular momentum: A photons in modes of different optical angular momentum can serve as basis states of a qubit. If $\ell$ is the angular momentum quantum number, we have for example
$$|0\rangle=|1\rangle_{l=0}$$
$$|1\rangle=|1\rangle_{l=1}$$
See this paper for a good review.
3. Occupation number: As stated in the comment above, given one single mode, qubit basis states can be defined in terms of the photon occupation number. The state $|0\rangle$ corresponds to no photons in the mode and $|1\rangle$ to one photon in the mode.
If you are interested in learning about general implementations of quantum information processing with light, there are many other interesting examples, of which I will provide only a few.
There is an entire paradigm for continuous-variable quantum computation with light, as reviewed in this paper. People have demonstrated entanglement between time and energy in photons. Finally, non-orthogonal coherent states have been used in quantum key distribution protocols.
A: Yeah there is a couple of 'em - off the top of my head I can think of:
\begin{align}
|\uparrow\;\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|\downarrow\;\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|g\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|e\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|L\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|R\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|H\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|V\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|0\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|1\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|+\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|-\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}
\end{align}
But these are only the naming conventions in the northern hemisphere. 
In the southern hemisphere they are labeled;
\begin{align}
|\downarrow\;\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|\uparrow\;\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|e\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|g\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|R\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|L\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|V\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|H\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|1\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|0\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|-\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|+\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}
\end{align}
So as you can see you have to be careful about who you are talking to, otherwise you could get your calculations all backwards.
