@principia-official
/My notes to understanding Gradient Echo Memory
/Part 1: Foundations
/Two Level Atom With photon Algebra

Two Level Atom With photon Algebra

Contents2 sections
  1. The combined operator
  2. Commutation

The combined operator§

There are four combination worth mentioning on, the rest is trivial / mentioned previously. Which is

a^σ^+a^†σ^+a^σ^−a^†σ^−\hat{a}\hat{\sigma}^+ \quad \hat{a}^\dagger\hat{\sigma}^+ \quad \hat{a}\hat{\sigma}^- \quad \hat{a}^\dagger\hat{\sigma}^-

Since photon operators and atom operators commute, the combination can be seen as a single operator as whole.

Commutation§

[a^σ^+,a^†σ^+]=1(σ+)2=0[\hat{a}\hat{\sigma}^+,\hat{a}^\dagger\hat{\sigma}^+]=1(\sigma^+)^2=0 [a^σ^+,a^σ^−]=(a^)2(σ^e−σ^g)[\hat{a}\hat{\sigma}^+, \hat{a}\hat{\sigma}^-]=(\hat{a})^2(\hat{\sigma}_e-\hat{\sigma}_g) [a^σ^+,a^†σ^−]=a^a^†(σ^e−σ^g)+σ^−σ^+=(n^+1)σ^e−n^σ^g[\hat{a}\hat{\sigma}^+,\hat{a}^\dagger\hat{\sigma}^-]= \hat{a}\hat{a}^\dagger(\hat{\sigma}_e-\hat{\sigma}_g)+ \hat{\sigma}^-\hat{\sigma}^+=(\hat{n}+1)\hat{\sigma}_e-\hat{n}\hat{\sigma}_g [a^†σ^+,a^σ^−]=n^(σ^e−σ^g)−σ^g=n^σ^e−(n^+1)σ^g[\hat{a}^\dagger\hat{\sigma}^+,\hat{a}\hat{\sigma}^-]=\hat{n}(\hat{\sigma}_e-\hat{\sigma}_g)-\hat{\sigma}_g=\hat{n}\hat{\sigma}_e-(\hat{n}+1)\hat{\sigma}_g [a^†σ^+,a^†σ^−]=(a^†)2(σ^e−σ^g)[\hat{a}^\dagger\hat{\sigma}^+,\hat{a}^\dagger\hat{\sigma}^-]=(\hat{a}^\dagger)^2(\hat{\sigma}_e-\hat{\sigma}_g) [a^σ^−,a^†σ^−]=0[\hat{a}\hat{\sigma}^-, \hat{a}^\dagger\hat{\sigma}^-]=0

Discussion

no comments
Commenting as a guest — sign in to comment as yourself.

No comments yet — yours could open the discussion.