Beam Splitter Inputs and Commutting Modes

[BeyondBelief96]

Homework Statement

[Not really Homework, but for my research] Background: I am an undergrad doing research in quantum optics, and studying different output states from various inputs into a 50:50 Beam splitter

Question: Do the two annihilation and creation operators of different input ports commute or do they obey the well-known commutation relation $[\hat{a}, \hat{a}^{\dagger}] = 1$

I guess I'm having trouble distinguishing modes and inputs of a beam splitter. Can two separate input ports be of the same mode, or are they different modes? How do the field operators for those two inputs commute with each other?

The reason I need to know this is because I am trying to decompose the Beam Splitter Operator

$\hat{B} = e^{\frac{i\pi}{4}(\hat{a}_0^{\dagger}\hat{a}_1 - \hat{a}_0\hat{a}_1^{\dagger})}$

Using the Baker-Campbell-Hausdorff Relation:

$e^{\hat{A} + \hat{C}} = e^{\hat{A}} e^{\hat{C}}e^{-\frac{1}{2}[\hat{A},\hat{C}]}$

if and only if $[\hat{A},\hat{C}]$ also commutes with $\hat{A}$ and $\hat{C}$

Relevant Equations

$[\hat{a}, \hat{a}^{\dagger}] = 1$

My thinking so far is that if the two different input mode operators of a beam splitter commute, but I can't really give any good reasoning behind it.

I defined $\hat{A} = \frac{i\pi}{4}\hat{a}_0^{\dagger}\hat{a}_1$

and

$\hat{C} = \frac{-i\pi}{4}\hat{a}_0\hat{a}_1^{\dagger}$

and am trying to figure out the commutation relation between these two operators. I am stuck however because I when calculating this I am not sure if I can swap operators around or not of the different modes 0 and 1, where 0 and 1 are two different inputs of the beam splitter.

 

[nrqed]

Homework Statement:: [Not really Homework, but for my research] Background: I am an undergrad doing research in quantum optics, and studying different output states from various inputs into a 50:50 Beam splitter

Question: Do the two annihilation and creation operators of different input ports commute or do they obey the well-known commutation relation $[\hat{a}, \hat{a}^{\dagger}] = 1$

I guess I'm having trouble distinguishing modes and inputs of a beam splitter. Can two separate input ports be of the same mode, or are they different modes? How do the field operators for those two inputs commute with each other?

The reason I need to know this is because I am trying to decompose the Beam Splitter Operator

$\hat{B} = e^{\frac{i\pi}{4}(\hat{a}_0^{\dagger}\hat{a}_1 - \hat{a}_0\hat{a}_1^{\dagger})}$

Using the Baker-Campbell-Hausdorff Relation:

$e^{\hat{A} + \hat{C}} = e^{\hat{A}} e^{\hat{C}}e^{-\frac{1}{2}[\hat{A},\hat{C}]}$

if and only if $[\hat{A},\hat{C}]$ also commutes with $\hat{A}$ and $\hat{C}$
Relevant Equations:: $[\hat{a}, \hat{a}^{\dagger}] = 1$

My thinking so far is that if the two different input mode operators of a beam splitter commute, but I can't really give any good reasoning behind it.

I defined $\hat{A} = \frac{i\pi}{4}\hat{a}_0^{\dagger}\hat{a}_1$

and

$\hat{C} = \frac{-i\pi}{4}\hat{a}_0\hat{a}_1^{\dagger}$

and am trying to figure out the commutation relation between these two operators. I am stuck however because I when calculating this I am not sure if I can swap operators around or not of the different modes 0 and 1, where 0 and 1 are two different inputs of the beam splitter.

You need first to find out what the commutation relations are between the operators with subscript 0 and the operators with subscripts 1. I would expect them to commute but you should check that first.