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Blog: Notes

This is a collection of technical and research notes. Browse only if curious and/or undemanding.

Strong- and weak-coupling: a notion defined at resonance.

The energies of the dressed states of the dissipative Jaynes-Cummings model (refer to this page for notations) read:

Commutation of Pauli matrices

$\sigma$ being the annihilation operator of a two-level system (one of the Pauli matrices), $\sigma^\dagger$ its conjugate, $\mu,\nu,\eta,\theta$ boolean variables (0 or 1), the following general commutator reads in normal order:

$$[{\sigma}^{\dagger\mu}\sigma^\nu,{\sigma}^{\dagger\eta}\sigma^\theta] = \nu\theta(\eta-\mu)\sigma +\mu\eta(\nu-\theta){\sigma}^\dagger +[(1-\mu)\nu\eta(1-\theta)-\mu(1-\nu)(1-\eta)\theta](1-2{\sigma}^\dagger\sigma).$$

I sometimes need this formula but always have to derive it again, which is very annoying (see this). (Unrelatedly, I also hope it is the last time I typeset something with texvc, as I plan to move to MathJax).

NormalOrder examples

In this note I give some particular cases of the module NormalOrder (see here for this module itself).

NormalOrder module

Something that comes recurrently when you work with quantum fields is, given any operator that consists of products of powers of annihilation $$a$$ and creation $$a^\dagger$$ Bose operators, such as, e.g.,

$a^{\dagger3}a^2a^{\dagger3}a^2a^{\dagger}a$

compute its normal order[1]:

$:a^{\dagger3}a^2a^{\dagger3}a^2a^{\dagger}a:$

Associate module

I am writing a code that computes arbitrary commutation relations. As part of this code is the following module that performs the associative part of the algebra:

Find: