# Limits on the heralding efficiencies and spectral purities of spectrally-filtered single photons from photon pair sources

## TL;DR

This paper investigates the fundamental trade-offs between spectral purity and heralding efficiency in spectrally-filtered photon pair sources, revealing intrinsic limitations and providing a benchmarking framework.

## Contribution

It demonstrates that filtering both photons reduces heralding efficiency despite improving spectral purity, and introduces a framework for benchmarking these effects.

## Key findings

- Filtering both photons lowers heralding efficiency even with ideal filters.
- Spectral purity can be improved at the expense of heralding efficiency.
- Experimental validation shows the trade-off between purity and efficiency.

## Abstract

Photon pairs produced by parametric down-conversion or four-wave mixing can interfere with each other in multiport interferometers, or carry entanglement between distant nodes for use in entanglement swapping. This requires the photons be spectrally pure to ensure good interference, and have high heralding efficiency to know accurately the number of photons involved and to maintain high rates as the number of photons grows. Spectral filtering is often used to remove noise and define spectral properties. For heralded single photons high purity and heralding efficiency is possible by filtering the heralding arm, but when both photons in typical pair sources are filtered, we show that the heralding efficiency of one or both of the photons is strongly reduced even by ideal spectral filters with 100% transmission in the passband: any improvement in reduced-state spectral purity from filtering comes at the cost of lowered heralding efficiency. We consider the fidelity to a pure, lossless single photon, symmetrize it to include both photons of the pair, and show this quantity is intrinsically limited for sources with spectral correlation. We then provide a framework for this effect for benchmarking common photon pair sources, and present an experiment where we vary the photon filter bandwidths and measure the increase in purity and corresponding reduction in heralding efficiency.