# Independence of Operator expectation values

• A
thariya
Hi!

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.

thephystudent
This decoupling is valid in the classical limit (the operator is its expectation value times unity), doing it means neglecting correlations (entanglement) between the modes. It can be a good approximation if the coupling between the modes is very weak, but then I don't really see the point of studying the BH Hamiltonian, at least not quantum-mechanically.

If you want to capture the most important quantum effects, a Gaussian approximation is a bit better, there you decouple expectation values of a large amount of operators into expectations of two operators trough Wick's theorem.

thariya
thariya
This decoupling is valid in the classical limit (the operator is its expectation value times unity), doing it means neglecting correlations (entanglement) between the modes. It can be a good approximation if the coupling between the modes is very weak, but then I don't really see the point of studying the BH Hamiltonian, at least not quantum-mechanically.

If you want to capture the most important quantum effects, a Gaussian approximation is a bit better, there you decouple expectation values of a large amount of operators into expectations of two operators trough Wick's theorem.

Thanks for this! I think I will go the Gaussian approximation route. Wick's theorem doesn't quite help me since my operators(ladder operators) are actually already normal ordered and their individual operator expectations are not zero. I don't see how I could reduce that using Wick's theorem. Would you know if that is possible?

Thanks.

thephystudent
Thanks for this! I think I will go the Gaussian approximation route. Wick's theorem doesn't quite help me since my operators(ladder operators) are actually already normal ordered and their individual operator expectations are not zero. I don't see how I could reduce that using Wick's theorem. Would you know if that is possible?

Thanks.
You can define a fluctuation operator ##\hat{\delta}## trough ##\hat{A}=\langle \hat{A}\rangle+\hat{\delta}## and substitute. Then, you can separate the mean-field values (already classical numbers) from the fluctuations (for which you can do the Gaussian approximation). And if you want you can do the inverse substitution at the end of the calculation to get rid of fluctuation operators.
I recently published a paper on applying such methods to the dynamics open systems, you may find interesting or some references therein like [61]. If you have further remarks on it, feel free to send them in pm.

thariya
thariya
You can define a fluctuation operator ##\hat{\delta}## trough ##\hat{A}=\langle \hat{A}\rangle+\hat{\delta}## and substitute. Then, you can separate the mean-field values (already classical numbers) from the fluctuations (for which you can do the Gaussian approximation). And if you want you can do the inverse substitution at the end of the calculation to get rid of fluctuation operators.
I recently published a paper on applying such methods to the dynamics open systems, you may find interesting or some references therein like [61]. If you have further remarks on it, feel free to send them in pm.

Thank you for that reference! And thank you for the offer of further discussion! I will take some time to go through your paper and will definitely take you up on that offer.

thephystudent