This example demonstrates how to use the biphoton wavefunctions generated from two independent nonclassical source models, such as Spontaneous Parametric Down-Conversion (SPDC) Photon Source ( example ) or Spontaneous Four-wave Mixing (SWFM) Microring Resonator Photon Source, ( example ) as the input to a circuit in qINTERCONNECT designed to model Hong-Ou-Mandel (HOM) interference. The visibility of the HOM interference is calculated, which is directly related to the indistinguishability between the signal photons from each source.

## Overview

Understand the simulation workflow and key results

A HOM interferometer consists of a single 50:50 beamsplitter with one photon incident upon each input port. If the photons are perfectly indistinguishable the photons will interfere, and both exit from the same output power. However, if there exists information to distinguish between the two photons, such as a timing delay or a difference in frequency profiles, the interference will not be perfect and there will be some chance of detecting one photon at each of the output ports of the beamsplitter, referred to as a coincident detection. In the limit of completely distinguishable photons, it becomes equally probable to measure a coincidence detection as it is to detecting two photons exiting from the same port.

Signal photons from two independent heralded photon sources can be interfered using the setup depicted in the figure above, where MZI filters are used to filter the signal and idler photons from each source. The indistinguishability of the signal photons will be limited by any spectral correlations between the signal and idler photons in each pair, which can be seen from the biphoton wavefunction. If a biphoton wavefunction cannot be factorized as a single product of functions separately describing the signal and idler envelope, the detection of an idler photon will reveal frequency information about the signal photon. This frequency information ultimately limits the indistinguishability of independent signal photons.

The simulations in this example are performed with the INTERCONNECT and qINTERCONNECT solvers. The simulation is set up and run through a Lumerical script file in INTERCONNECT. Next, a python script is run to generate the state that is used as an input to qINTERCONNECT. The qINTERCONNECT simulation is then run through the Python API, and can be either run directly in INTERCONNECT or in an external Python environment. Since qINTERCONNECT requires an INTERCONNECT GUI license to run, at least two GUI licenses are required to run qINTERCONNECT directly from the INTERCONNECT environment. Documentation for the Python API is available here.

### Step 1: Generate biphoton wavefunction

Biphoton wavefunctions for each source are calculated using the parameters determined from [ Spontaneous Four-wave Mixing (SWFM) Microring Resonator Photon Source – Ansys Optics ].

### Step 2: Design MZI Filter

The parameters are determined for an MZI filter that separates photons at the signal and idler frequencies.

### Step 3: Simulate HOM interferometer

The HOM interferometer is simulated in qINTERCONNECT. Coincident counts are measured and used to determine the HOM visibility.

### Step 4: Parameter sweep

A sweep is performed over the coupling value of the directional coupler.

## Run and Results

Instructions for running the model and discussion of key results

### Step 1: Generate biphoton wavefunction

- Open and run the script [[biphoton_wavefunction_step1.py]] in the INTERCONNECT environment.

The biphoton wavefunction is calculated for a pair of signal and idler photons produced by spontaneous four-wave mixing in a microring resonator. For information regarding how these specific parameters are determined see [ Spontaneous Four-wave Mixing (SWFM) Microring Resonator Photon Source – Ansys Optics ]. Additionally, the discretized biphoton wavefunction is also shown below, which spans a set of discrete frequencies and can be expressed using Fock states. The discretized biphoton wavefunction can then be used as an input to circuit in qINTERCONNECT.

### Step 2: Design MZI Filter

- Open the [[mzi_filter.icp]] simulation file using INTERCONNECT.
- Run the [[mzi_filter_step2.lsf]] script file to update the values of the filter.

The script [[ mzi_filter_step2.lsf]] sets the length of the first arm of the MZI filter to:

where the free spectral range is . The filter is centered on the signal and idler frequencies by a small increase in the length of the second arm of the MZI filter, which provides a phase shift. The transmission through the filter is shown below:

### Step 3: Simulate HOM interferometer

- Open and run simulation_step3.py script file in the INTERCONNECT environment.

To decrease the total computation time, the qINTERCONNECT simulation is broken down into three parts. First, the propagation of the signal-idler pairs from each source through the MZI filters is simulated separately, and a density matrix describing the state containing both signal photons is created from the results. Finally, this density matrix is used as the input to 50:50 directional coupler.

Since the frequency range of the signal and idler photons do not overlap, separate simulations are first performed for each component over the signal and idler frequency ranges. Then, a single S-matrix is created from the results of the individual simulations over the signal and idler frequencies.

After running the simulation the probability of coincident detection is determined to be 0.016301. For a perfect 50:50 directional coupler, we can compare this to the analytical result [1]

where and are defined in terms of the biphoton wavefunctions for each source by performing a Schmidt decomposition:

*
*

The above formula gives a coincidence probability of 0.016303, where the discrepancy between the simulated result and the analytical result is due to the imperfectness of the MZI filter. Another useful metric is the HOM visibility:

*
Which describes the decrease in coincidence counts relative to the limit
*
*
describing completely distinguishable photons. Completely distinguishable photons will thus have a visibility of 0 whereas indistinguishable photons will have a visibility of 1. For a perfect 50:50 directional coupler the visibility can be found by
*

We obtain
*
V = 0.96740
*
from both methods.

### Step 4: Parameter Sweep

- Open the waveguide_coupler.icp simulation file using INTERCONNECT.
- Open and run sweep_step4.py script file in the INTERCONNECT environment.

The coupling coefficient for the waveguide coupler is now swept from to in increments of The visibility is plotted below as a function of
*
k
*
:

## Updating the Model With Your Parameters

Instructions for updating the model based on your device parameters

**
Biphoton Wavefunction:
**
The biphoton wavefunction used in the simulation can be changed by importing a ‘biphoton_wavefunction.txt’ file. Instructions on how to simulate the biphoton wavefunction from spontaneous four-wave mixing in a silicon nitride microring resonator can be found here [link].

The number of frequency points used in the simulation can be changed by adjusting
**
frequency_points
**
when generating the biphoton wavefunction.

**
MZI Filter:
**
The MZI filter can be adjusted by specifying the signal and idler frequency. Be sure to update the group index if necessary.

**
Component Values:
**
The MZI filter and directional coupler can be replaced with your own custom components. If performing a sweep over a custom component, ensure that the compound element has a property corresponding to the parameter you are sweeping over in qINTERCONNECT.

## Additional Resources

Additional documentation, examples and training material

###
**
Related Publications
**

- A. M. Brańczyk, “Hong-Ou-Mandel Interference,” Oct. 2017. https://arxiv.org/abs/1711.00080