# Numerical simulation for spontaneous parametric down conversion

I want to numerically simulate the SPDC (spontaneous parametric down conversion) process. I have a basic background on nonlinear and quantum optics — I am a graduate student and just took a few classes regarding these subjects. I have no clue where to begin.

So far I have found Gerry & Knight's book, Introductory Quantum Optics, which describes well the phenomenon, but it is unclear to me how to proceed from there. The way I imagine this will go is as follows:

1. I have to solve the wave equation for the electric field in nonlinear medium. Do I have to consider diffraction? If so, I can not simply use beam propagation numerical methods like split step Fourier transformation, right?

2. From there, I have to somehow go to quantum optics and describe the entanglement of the outgoing photons.

I could really use a few good references or pretty much anything. Thanks in advance.

• Which property of SPDC do you want to simulate? Commented Dec 27, 2019 at 10:11
• I want to see the cones and the entangled photons. Commented Dec 27, 2019 at 11:52
• What do you want to see in the entangled photons? Photons aren't something that are generated in a local position in space, and their electric field wave functions make them really not classical. So what exactly do you want to simulate? Commented Dec 27, 2019 at 11:59
• For starter, I would like to see how a beam propagates in a nonlinear medium. Commented Dec 28, 2019 at 1:13

So the most basic way to understand parametric down conversion is that of a two mode squeezed vacuum states. This is typically what you see in introductory textbooks in quantum optics where the resulting state can be written as $$\lvert\psi\rangle = \dfrac{1}{\cosh r } \sum _{n=0} ^{\infty} \left(\tanh r \right)^n \lvert n,n\rangle .$$

This is generated by applying the two mode squeezing unitary to the vacuum state for the two modes. For SPDC we truncate this at the term containing two photons (typically) which gives us

$$\lvert\psi\rangle \sim \lvert 0,0\rangle + p \lvert 1,1\rangle \mathcal+ {O}\left(p^2\right),$$ which is unnormalised with amplitude $$p$$ for pair generation.

Real PDC sources are more complicated than this and contain many different two mode squeezers. These can be different spectro-temporal (longitudinal) modes of the photons emitted, which are described by the joint spectral amplitude (JSA) of the source.

In this case the state is typically described as $$\lvert\psi\rangle \sim \lvert 0,0\rangle + \intop\intop \text{d}\omega_1 \text{d}\omega_2 f\left(\omega_1,\omega_2\right)a^\dagger \left(\omega_1\right) b^\dagger \left(\omega_2\right)\lvert 0,0\rangle,$$

where the two dimensional function $$f\left(\omega_1,\omega_2\right)$$ is the JSA. Taking a Schmidt decomposition (Singular Value Decomposition) of the JSA tells us how many effective two mode squeezers make up the PDC source in the spectral domain. There are a ton of references on this but a good place to start is here. If you Google around you can find lots of PhD thesis on this topic which should be pretty accessible.

This description ignores any spatial modes which may be present. This is the case when we consider PDC in a waveguide or if we collect the two photons into single mode optical fibers (this projects the photons into Gaussian spatial modes which match the fiber mode).

Typically this is the description people talk about since we the photons are almost always collected into single mode fibers.

I see from the comments you want to understand how the cones form, a quick Google shows this paper from the arXiv which describes spatial variation of PDC photons leading to the cones which are well known from BBO PDC sources. There is also this paper which goes a little bit more in depth. Going through these papers and links therein should give you plenty to go on.

If you want to go further you should consider that the two mode squeezing Hamiltonian comes from a three wave mixing interaction in a nonlinear medium. You can find the equations of motion for 3-wave mixing in your favourite nonlinear optics book. The most complete description of PDC then comes from elevating the electric field amplitudes in these equations to operators and solving the equations of motion in a full vectorised 3D model. For 99% of purposes this is overkill.

What type of entanglement are you looking into? Due to the energy and momentum conservation necessary for PDC there are different types of entanglement available.