# Interferometer questions, pure states?

1. Jan 25, 2005

### trosten

The beamsplitters are 50/50 and the setup is symmetric.

initial
|i>
first beamsplitter, where U is unitary and represents the beamsplitter.
U|i> = 1/sqrt(2)( |u> + exp(ix)*|d> )
second beamsplitter, A and B are detectors (upper and lower)
U|d> = 1/sqrt(2)( |A> + exp(ix)*|B> )

U|u> = 1/sqrt(2)( |A> + exp(iy)*|B> )

now what is x and y? I let x = 0 and then assume that <d|u> = 0 and use this to determine y under the assumption that U is unitary this gives y = pi.

U|d> = 1/sqrt(2)( |A> + |B> )
U|u> = 1/sqrt(2)( |A> - |B> )

this gives the state after the second beamsplitter to be
1/sqrt(2)( 1/sqrt(2)( |A> + |B> ) + 1/sqrt(2)( |A> - |B> )) = |A>
and this gives probability of detection in A to be
|<A|A>|^2 = 1
and B
|<B|B>|^2 = 0

1. How come its valid to assume that |u> and |d> are pure states?

2. Will the experimental outcome be different if we put up a wall on the horizontal symmetry line? Or wont that affect the experiment cause all the interference takes place at the second beamsplitter?

3. is possible to view the |u> (and |d>) as being pure states because they are sort of statistically pure as long as the paths are separated well enough? Has this anything to do with the statistical interpretation of QM ?

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Last edited: Jan 25, 2005