Commutation of Beam splitter operator with Displacement operator

[deepalakshmi]

TL;DR

Does Beam splitter operator and Displacement operator of coherent state commute with each other?

I have a beam splitter operator (a†)b +(b†)a. Does it commute with exp(αâ†-α*â). Here a and ↠are creation and lowering operator

 

[vanhees71]

Well, just check, whether the BS operator commutes with the argument of the operator exponential!