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I want to know under what conditions the operator expectation values of a product of operators can be expressed as a product of their individual expectation values. Specifically, under what conditions does the following relation hold for quantum operators (For my specific purpose, these are bosonic creation/annihilation operators for different modes 1 and 2)? Any reference I can read up would be great too!

$$<\hat{A_1}\hat{A_2}>=<\hat{A_1}><\hat{A_2}>$$

I'm trying to use it in the context of a Bose-Hubbard Hamiltonian with two coupled bosonic modes whose operator expectation values I kinda need to decouple. I've seen it used in other similar circumstances to decouple operator products of Fermionic and Bosonic operator expectations but the exact conditions under which this is valid was not mentioned. I think this is valid when operators commute, but would like to know the exact conditions.

Thanks.