## Background

First of all, I'll use $\lvert H\rangle$ as a horizontally polarised state and $\lvert V\rangle$ as a vertically polarised state^{1}. There are three modes of light involved in the system: pump (p), taken to be a coherent light source (a laser); as well as signal and idler (s/i), the two generated photons

The Hamiltonian for SPDC is given by $H = \hbar g\left(a^{\dagger}_sa^{\dagger}_ia_p + a^{\dagger}_pa_ia_s\right)$, where g is a coupling constant dependent on the $\chi^{\left(2\right)}$ nonlinearity of the crystal and $a\left(a^{\dagger}\right)$ is the annihilation (creation) operator. That is, there is a possibility of a pump photon getting annihilated and generating two photons^{2} as well as a possibility of the reverse.

The phase matching conditions for frequencies, $\omega_p = \omega_s + \omega_i$ and wave vectors, $\mathbf{k}_p = \mathbf{k}_s + \mathbf{k}_i$ must also be satisfied.

## Type 1 SPDC

This is where the two generated (s and i) photons have parallel polarisations, perpendicular to the polarisation of the pump, which can only be used to perform SPDC when the pump is polarised along the extraordinary axis of the crystal.

This means that defining the extraordinary axis as the vertical (horizontal) direction and inputting coherent light along that axis will generate pairs of photons in the state $\lvert HH\rangle\, \left(\lvert VV\rangle\right)$. This is not of much use, so to generate an entangled pair of photons, two crystals are placed next to each other, with extraordinary axes in orthogonal directions. The coherent source is then input with a polarisation of $45^\circ$ to this, such that if the first crystal has an extraordinary axis along the vertical (horizontal) direction, there is a probability of generating photons in the state $\lvert HH\rangle\, \left(\lvert VV\rangle\right)$ as before from the first crystal, as well as a probability of generating photons in the state $\lvert VV\rangle\, \left(\lvert HH\rangle\right)$ from the second crystal.

However, as the light from the pump is travelling through a material, it will also acquire a phase in the first crystal, such that the final state is $$\lvert\psi\rangle = \frac{1}{\sqrt{2}}\left(\lvert HH\rangle + e^{i\phi}\lvert VV\rangle\right).$$

Due to the phase matching conditions, the emitted photon pairs will be emitted at opposite points on a cone, as shown below in Figure 1.

Figure 1: A laser beam is input into two type 1 SPDC crystals, with orthogonal extraordinary axes. This results in a probability of emitting a pair of entangled photons at opposite points on a cone. Image taken from Wikipedia.

^{1 This can be mapped to qubit states using e.g. $\lvert H\rangle = \lvert 0\rangle$ and $\lvert V\rangle = \lvert 1\rangle$}

^{2 called signal and idler for historical reasons}

References:

Keiichi Edamatsu 2007 Jpn. J. Appl. Phys. 46 7175

Kwiat, P.G., Waks, E., White, A.G., Appelbaum, I. and Eberhard, P.H., 1999. Physical Review A, 60(2) - and the arXiv version