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− | 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. | + | This is, despite its simple appearance, an exceedingly rich and complex system, proposed by Ed.~Jaynes and his student Fred Cummings{{cite|jaynes63a}} 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 <wz tip="which might still be right despite all the evidence and personal opinions from experts, but that is another story">(!?)</wz> but the model, that is exactly solvable, remained and became the drosophila of quantum optics. | People didn't get further interested in Jaynes' original intention <wz tip="which might still be right despite all the evidence and personal opinions from experts, but that is another story">(!?)</wz> but the model, that is exactly solvable, remained and became the drosophila of quantum optics. | ||
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where [[Liouvillian|$\mathcal{L}$ is defined here]]. | where [[Liouvillian|$\mathcal{L}$ is defined here]]. | ||
− | = See also = | + | == See also == |
− | == On this website == | + | === On this website === |
* [[Blog:Notes/Strong-_and_weak-coupling:_a_notion_defined_at_resonance.|Blog post]] arguing that weak- and strong-coupling are concepts defined at resonance only. | * [[Blog:Notes/Strong-_and_weak-coupling:_a_notion_defined_at_resonance.|Blog post]] arguing that weak- and strong-coupling are concepts defined at resonance only. | ||
− | == On the rest of the web == | + | === On the rest of the web === |
* [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}} There is also a [http://iopscience.iop.org/0953-4075/page/Special-issue-on-Jaynes-Cummings-physics Special issue on Jaynes–Cummings physics] in the Journal of Physics B. | ||
+ | |||
+ | === Some papers of ours === | ||
+ | |||
+ | There are various, but we recommend particularly Ref.~{{cite|delvalle09a}}, {{cite|delvalle10d}} and {{cite|delvalle11a}}. | ||
+ | |||
+ | == References == | ||
+ | |||
+ | <references /> | ||
+ | |||
+ | * [https://iopscience.iop.org/book/mono/978-0-7503-3447-1 The Jaynes–Cummings Model and Its Descendants], [[J. Larson]] and [[T. Mavrogordatos]], IOP Series in Quantum Technology, [[2021]]. |
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 is also a Special issue on Jaynes–Cummings physics in the Journal of Physics B.
There are various, but we recommend particularly Ref.~[3], [4] and [5].