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@principia-official/My notes to understanding Gradient Echo Memory
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Creation / Annihilation operators

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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}\\a^∣n⟩=n​∣n−1⟩a^†∣n⟩=n+1​∣n+1⟩

Where, a^\hat{a}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}(a^†)n∣0⟩=n!

Thus

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

Commutation

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

The number operator

a^†a^=n\hat{a}^\dagger\hat{a} = na^†a^=n

And by the commutation relation,

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

Modes

For different modes ( kkk or ω\omegaω ),

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

General single photon

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

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