Polarization vectors in Quantum Electric Field The quantum electric field is written as,
\begin{equation}
\mathbf{E}(\mathbf{r})=i\sum_{\mathbf{k},\lambda}\sqrt{\frac{\hbar \omega}{2 V \epsilon_0}}\left(\mathbf{e}^{(\lambda)}\hat{a}^{(\lambda)}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}} - \mathbf{e}^{(-\lambda)}\hat{a}^{\dagger(\lambda)}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}}\right).
\end{equation}
The $\mathbf{e}^{(\pm\lambda)}$ terms are the polarization vectors.  Do these vectors represent any kind of polarization vector? Or are the only circular polarization vectors?  What if you want to measure something horizontally polarized?  Would you just dot the $E$ field with a vector that yields a polarization vector you need?
 A: So, I actually found this post because I've been trying to answer the same question. Here's what I understand from various sources I've found.
These vectors represent a polarization basis. For example, two orthogonal linear polarizations (which I've seen written before as $e_{\textbf{k}\lambda}$ for $\lambda=1,2$), which I think is the more traditional basis, or an alternate circularly polarized basis (which I've seen written as $e_{\textbf{k}\alpha}$ for $\alpha=1,-1$). These form a basis for the photon polarization, but don't necessarily describe the photon polarization itself. The photon polarization can be expressed in terms of these basis vectors, just like we might express motion in Cartesian or polar coordinates.
These notes here include a pretty good description, just search for "polarization". It's a little technical but it's got a more mathematical description of how the bases work.
This paper is definitely very technical, and I haven't actually read all of it, but the introduction talks some more about how these vectors are used.
I'm not sure how your measurement question fits into this though.
