What is photon blockade? Photon blockade seems to be a well-known phenomena that has been observed in a plethora of devices, and that is often mentionned in physics papers without much explanation. A simple search of "photon blockade" on arXiv.org yields 398 results.
I understand it is based on the Jaynes-Cummings Hamiltonian that features a non-linear eigenspectrum, but I have not been able to find a good reference that explains photon blockade using simple physics and maths.
So, what exactly is photon blockade?
 A: Definition
Photon Blockade is an effect that was brought up in analogy to Coulomb blockade (which refers to the suppression of electric transport due to electronic repulsion). In the photonic case, it means that the interaction of a photon with a nonlinear system prevents the interaction with further photons, yielding a single photon emission, even for incident light with a large stream of photons.
There are two mechanisms on which photon blockade relies: The first one, as you correctly mentioned, uses an anharmonic spectrum to generate photon blockade. This method is known as conventional photon blockade (CPB) [1]. On the other hand, another method was proposed by Liew and Savona [2] and later clarified by Bamba et. al [3], on which photon blockade is achieved by quantum interference. This is known as unconventional photon blockade (UPB).
Mathematical derivation
Although normally the CPB and UPB are examined in different systems, there are some cases on which both mechanisms can be observed for a single system [4]. One of them is in the Jaynes Cummings model, which will be my main example. It turns out that photon blockade arises in a coherently driven Jaynes Cummings model
$$H=\hbar\omega_a\hat{a}^\dagger\hat{a}+\hbar\omega_\sigma\hat{a} \hat{\sigma}^\dagger\hat{\sigma}+\hbar g(\hat{a}^\dagger\hat{\sigma}+\hat{a}\hat{\sigma}^\dagger)+\Omega_a(\hat{a}e^{i\omega_lt}+\hat{a}^\dagger e^{-i\omega_lt})$$
For the moment, consider only the Jaynes Cummings model, given by the following Hamiltonian
$$H=\hbar\omega_a\hat{a}^\dagger\hat{a}+\hbar\omega_\sigma\hat{a} \hat{\sigma}^\dagger\hat{\sigma}+\hbar g(\hat{a}^\dagger\hat{\sigma}+\hat{a}\hat{\sigma}^\dagger)$$ Using the bare state basis $\lbrace |G,n\rangle,|E,n-1\rangle\rbrace$, this system can be written in matrix form as
$$H^{(n)}=
\begin{pmatrix}
  n\omega_a & g\sqrt{n}\\ 
  g\sqrt{n} & \omega_\sigma+(n-1)\omega_a
\end{pmatrix}$$
Where the corresponding eigenvalues are $E_{n}^{(\pm)}=\frac{\omega_\sigma+(2n-1)\omega_a}{2}\pm\frac{1}{2}\sqrt{4ng^2+(\omega_a-\omega_\sigma)^2}$, which are the known energies of the polaritons. Know, to obtain photon blockade from this system, an external laser with frequency $\omega_l$ must be tuned to the one photon polariton. this yields the following property
$$\omega_l=\frac{\omega_\sigma+\omega_a}{2}-\sqrt{4g^2+(\omega_a-\omega_\sigma)^2}$$
After some algebra, and introducing the rotating frame frequencies defined as $\tilde{\omega_c}=\omega_c-\omega_l$, the CPB condition turns to be
$$\boxed{\tilde{\omega}_a\tilde{\omega}_\sigma=g^2}$$
On the other hand, the condition for the UPB can be obtained by quantum interference from two paths that lead to a two photon state. If the probabilities of both paths are equal and opposite, the two photon state will be blockaded and therefore only one photon states can exist on the system. On the Jaynes Cummings models, this paths are $\text{Path 1:}|0,g\rangle\xrightarrow[]{\Omega_a}|1,g\rangle\xrightarrow[]{\Omega_a}|2,g\rangle$ and $\text{Path 2:}|0,g\rangle\xrightarrow[]{\Omega_a}|1,g\rangle\xrightarrow[]{g}|0,e\rangle\xrightarrow[]{\Omega_a}|1,e\rangle\xrightarrow[]{g}|2,g\rangle$. The analytical condition for UPB is a little harder (and longer) to obtain than for the CPB case, but the solution is
$$\boxed{(\tilde{\omega}_\sigma+\tilde{\omega_a})\tilde{\omega}_\sigma+g^2=0}$$
The result for both mechanisms can be visualized for the numerical solution of the second order correlation function. Refer to Fig 7a in [4] for an example.
I hope to have clarified a little bit your view on photon blockade.
References
[1] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble. Photon blockade in
an optical cavity with one trapped atom. Nature, 436(7047):87–90, July 2005.
[2]T. C. H. Liew and V. Savona. Single Photons from Coupled Quantum Modes. Physical Review Letters,
104(18):183601, May 2010
[3]Motoaki Bamba, Atac Imamo ̆glu, Iacopo Carusotto, and Cristiano Ciuti. Origin of strong photon
antibunching in weakly nonlinear photonic molecules. Physical Review A, 83(2):021802, February 2011.
[4]E. Zubizarreta Casalengua, J. C. López Carreño, F. P. Laussy, and E. del Valle. Conventional and uncon-
ventional photon statistics. Laser & Photonics Reviews, 14(6):1900279, June 2020. arXiv: 1901.09030.
