Student Name: **Andrew Kunkel**

Major: Physics

Minor: Mathematics

Advisor: Cody C. Leary, Second reader: John F. Lindner

In this thesis we model two-photon interference with a nonlinear sample in a Mach Zehnder interferometer to calculate a coincidence signal that can be compared with ongoing experimentation. Specifically, we use entangled biphoton states created by Type I collinear spontaneous parametric down conversion that give valuable symmetries that simplify the formalism. The interferometer constrains the input state to enter the apparatus through one entrance of a beam splitter allowing for a more intuitive model for the sample interaction. The experiment is first modeled using a black box method, with the techniques used then applied to each part of the interferometer one by one. We complete the model with an expression for a particular coincidence signal Rcc that is constructed from three terms, one for the linear interaction contribution to the signal denoted R0 and two for the nonlinear parts R1 and R2. In this analysis, only the R2 contribution was successfully evaluated. We implemented a second order Taylor expansion to approximate the unknown nonlinear susceptibility of the sample χ(3) to evaluate R2. The evaluation of R1 and the quantitative com- parison of both R1 and R2 to ongoing experiment is left to future researchers.

Andrew will be online to field comments on April 16: 10am-noon EDT (Asia: late evening, PST 6am-8am, Africa/Europe: late afternoon).

Very difficult and impressive calculation! Could it be automated using computer algebra?

I am confident certain sections could be easily automated, such as most of the integrals and factoring. However, I am not experienced with handling commutation relations in computer algebra so I am uncertain as to how difficult it is to ensure the computer chooses the right order to commute in certain situations. Assuming that is an easy problem to solve, automating the process would allow us to more quickly evaluate other possible configurations of four-wave mixing that were omitted from this project.

For example, FeynCalc is a Mathematica package for symbolic evaluation of Feynman diagrams and algebraic calculations in quantum field theory and elementary particle physics. Automating you calculations would probably be a yearlong project in itself, but it might be a path forward.

I enjoyed your project.

Congratulations, Andrew, on making a solid contribution to a very difficult problem. Given the promising results, I am very curious how modeling the phase matching function for the sample more carefully would affect the outcome, in addition to alternate interferometer configurations. Mathematica can handle commutation relations, but I personally find it hard to gain initial intuition with these kinds of quantum systems by turning to a computer too early…

Thank you. I would say that a more accurate model for the phase matching function of the sample could leave more characteristics of the sample’s nonlinear susceptibility as parameters in the result. With regards to the interferometer configuration, there could be several different setups using the same devices that would result in a different experiment to model. For example, the biphoton state could be configured to enter the interferometer in each input of the first beam splitter instead of just one. Another configuration could place the time delay along a different path. Each of these examples would require alterations to the model and a different experiment to compare with, but I believe the steps for the calculation would be similar.

Congratulations Andrew!

Congratulations on a job well done!

Andrew, great work. Congratulations on your project.

Congratulations, Andrew! 🙂

Andrew, amazing work! Congratulations!