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This page is still largely in progress.
This is our favourite theoretical model, the full-field quantization of zero-dimensional modes: a two-level system, with annihilation operator $\sigma$ that obeys Fermi anticommutation rules, $\{\sigma,\sigma^\dagger\}=\sigma\sigma^\dagger+\sigma^\dagger\sigma=1$, and an harmonic oscillator, that obeys Bose anticommutation rules: $[a,a^\dagger]=aa^\dagger-a^\dagger a=1$. These two modes, with free energy $\hbar\omega_a$ and $\hbar\omega_\sigma$, respectively, are linearly coupled with strength $g$, providing the celebrated Jaynes-Cumming Hamiltonian:
\begin{equation} \tag{1} H_\mathrm{JC}=\hbar\omega_aa^\dagger a+\hbar\omega_\sigma\sigma^\dagger \sigma+\hbar g(a^\dagger\sigma+a\sigma^\dagger)\,. \end{equation}
This is, despite its simple appearance, an exceedingly rich and complex system, proposed by Ed.~Jaynes and his student Fred Cummings to prove that you don't need to fully-quantize light to explain various things deeply rooted into full-field quantization in popular consciousness, such as, more famously, spontaneous emission and the Lamb shift.
People didn't get further interested in Jaynes' original intention (!?) but the model, that is exactly solvable, remained and became the drosophila of quantum optics.
With Liouvillian theory, you can turn it into a richer still system, the dissipative Jaynes-Cummings model. Calling $\gamma_a$, $\gamma_\sigma$ the decay rates of modes $a$ and $\sigma$, respectively, the system is now described by its density matrix $\rho$ according to Liouville-von Neumann equation:
\begin{equation} \tag{2} i\hbar\partial_t\rho=[H_\mathrm{JC},\rho]+\left(\frac{\gamma_a}2\mathcal{L}_a+\frac{\gamma_\sigma}2\mathcal{L}_\sigma\right)\rho \end{equation}
where $\mathcal{L}$ is defined here.