@principia-official
/My notes to understanding Gradient Echo Memory
/Part 1: Foundations
/Creation / Annihilation operators

Creation / Annihilation operators

Contents9 sections
  1. Definitions
  2. Properties
  3. Arbitrary states
  4. Commutation
  5. The number operator
  6. Anti-commutation
  7. Modes
  8. General single photon
  9. Other Commutation relation

Definitions§

a^∣n⟩=n∣n−1⟩a^†∣n⟩=n+1∣n+1⟩\hat{a}\ket{n}=\sqrt{n}\ket{n-1}\\ \hat{a}^\dagger\ket{n}=\sqrt{n+1}\ket{n+1}\\

Where, a^\hat{a} is the annihilation operator and it's conjugate is the creation oprator.

Properties§

Arbitrary states§

By direct recursion, we easily find that,

(a^†)n∣0⟩=n!∣n⟩\left(\hat{a}^\dagger\right)^n\ket{0}=\sqrt{n!}\ket{n}

Thus

∣n⟩=1n!(a^†)n∣0⟩\ket{n} = \frac{1}{\sqrt{n!}} \left( \hat{a}^\dagger\right)^n\ket{0}

Commutation§

[a^,a^†]=a^a^†−a^†a^=1[\hat{a},\hat{a}^\dagger]=\hat{a}\hat{a}^\dagger - \hat{a}^\dagger \hat{a} = 1

The number operator§

a^†a^=n\hat{a}^\dagger\hat{a} = n

And by the commutation relation,

a^a^†=n+1\hat{a}\hat{a}^\dagger = n + 1

Anti-commutation§

{a^,a^†}=a^a^†+a^†a^=2n+1\{\hat{a}, \hat{a}^\dagger\}=\hat{a}\hat{a}^\dagger + \hat{a}^\dagger\hat{a} = 2n+1

Modes§

For different modes ( kk or ω\omega ),

[a^k,a^k′†]=δkk′[\hat{a}_k, \hat{a}^\dagger_{k'}]=\delta_{kk'}

General single photon§

∣ψ;f⟩=∑kf(k)a^k†∣0⟩\ket{\psi;f}=\sum_k f(k)\hat{a}^\dagger_k \ket{0}

Other Commutation relation§

[a^†a^,a^†]=a^†[a^,a^†]=a^†[\hat{a}^\dagger \hat{a}, \hat{a}^\dagger]=\hat{a}^\dagger[\hat{a}, \hat{a}^\dagger]=\hat{a}^\dagger [a^†a^,a^]=[a^†,a^]a^=−a^[\hat{a}^\dagger \hat{a}, \hat{a}]=[\hat{a}^\dagger, \hat{a}]\hat{a}=-\hat{a}

Discussion

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

No comments yet — yours could open the discussion.