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Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. ˆA=⟨ˆA⟩I+Δˆa
The right hand side is a sum of the "mean" value and the fluctuations about the mean. While I understand that the physical picture is reasonable, is this mathematically correct? If so what are the constraints this imposes? In literature this is designated as a "linearization" process.
My understanding of a linear operator is that it is simply a homomorphism. I have never seen anything done like this and I'm having a hard time finding references which justify this process.
I would be grateful if somebody can point me in the right direction!
Answer
The operation you mentioned Δˆa=ˆA−⟨ˆA⟩ˆI is just shifting of the annihilation operator. Typically people even drop the identity and write ˆanew=ˆaold−α0, where α0 is a complex number. In your question the shift is such that ⟨ˆanew⟩=0, which may be convenient for calculations. In particular when the pump beam has many photons, the quantum part (i.e. related to ˆanew) may be neglected.
Additionally, ˆanew is an annihilation operator with the same anti-commutation relations as ˆaold.
From mathematical point of view it is a perfectly legit operation. Moreover, both operators have the same domain, and spectrum only shifted by α0. To see that domain is the same take any |ψ⟩∈dom(ˆaold). Then denoting |ϕ⟩:=ˆaold|ψ⟩, we check that ˆanew|ψ⟩=|ϕ⟩−α0|ψ⟩ is a well-defined vector.
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