  What is the most efficient way to generate random numbers
device-independently using a photon pair source based on spontaneous parametric
down conversion (SPDC)? We consider this question by comparing two
implementations of a detection-loophole-free Bell test. In particular, we study
in detail a scenario where a heralded single photon source (HSPS) is used to
herald path-entangled states, i.e. entanglement between two spatial modes
sharing a single photon and where non-locality is revealed using photon
counting preceded by small displacement operations. We start by giving a
theoretical description of such a measurement. We then show how to optimize the
Bell-CHSH violation through a non-perturbative calculation, taking the main
experimental imperfections into account. We finally bound the amount of
randomness that can be extracted and compare it to the one obtained with the
conventional scenario using photon pairs entangled e.g. in polarization and
analyzed through photon counting. While the former requires higher overall
detection efficiencies, it is far more efficient in terms of both the entropy
per experimental run and the rate of random bit generation.
