Effect of dark counts on single photon heralding with quasi-number-resolving detection schemes

We consider photon heralding with quasi-number-resolving detection schemes and account for detection efficiencies and dark count probabilities. With a straightforward formalism, we develop closed-form expressions for the heralding probability, photon number distribution of the resulting heralded state, and fidelity of this heralded state to a single photon state. We calculate that, on the basis of optimizing this fidelity, due to the presence of dark counts there is a maximum number of detection modes worth multiplexing for each combination of efficiency and dark count probability.