Undergrad Commutation of Beam splitter operator with Displacement operator

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The discussion focuses on the commutation relationship between a beam splitter operator and a displacement operator. The beam splitter operator is defined as (a†)b + (b†)a, while the displacement operator is expressed as exp(α↠- α*â). Participants suggest checking if the beam splitter operator commutes with the argument of the displacement operator's exponential. This involves evaluating the commutation relations between the creation and lowering operators. Understanding these commutation properties is crucial for applications in quantum optics and related fields.
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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
 
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Well, just check, whether the BS operator commutes with the argument of the operator exponential!
 

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