Introduction to the absolute brightness and number statistics in spontaneous parametric down-conversion

As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi-mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multipair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources.