  Spontaneous parametric down-conversion (SPDC) is the most widely-used method
to generate higher-order Fock states ($n\geq 2$). Yet, a consistent performance
analysis from fundamental principles is missing. Here we address this problem
by introducing a framework for state fidelity and generation probability under
the consideration of losses and multimode emission. With this analysis we show
the fundamental limitations of this process as well as a trade-off between
state fidelity and generation rate intrinsic to the probabilistic nature of the
process. This identifies the parameter space for which SPDC is useful when
generating higher-order Fock states for quantum applications. We experimentally
investigate the strong pump regime of SPDC and demonstrate heralded Fock states
up to $\lvert n\rangle =4$.

Quantum Physics