m (→Some papers of us) |
m (→See also) |
||
Line 33: | Line 33: | ||
* [http://en.wikipedia.org/wiki/Jaynes%E2%80%93Cummings_model Wikipedia] | * [http://en.wikipedia.org/wiki/Jaynes%E2%80%93Cummings_model Wikipedia] | ||
* [http://qwiki.stanford.edu/index.php/Jaynes-Cummings_Hamiltonian qwiki] | * [http://qwiki.stanford.edu/index.php/Jaynes-Cummings_Hamiltonian qwiki] | ||
+ | |||
+ | === Some papers === | ||
+ | |||
+ | They are countless and I'll do a selection of favourites one day, but for the time being, let me propose Shore and Knight review.{{cite|shore93a}} | ||
=== Some papers of ours === | === Some papers of ours === |
Contents |
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[1] 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.
They are countless and I'll do a selection of favourites one day, but for the time being, let me propose Shore and Knight review.[2]
There are various, but we recommend particularly Ref.~[3], [4] and [5].