From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Quantum analysis of Mach-Zehnder Interferometer
Organization: Whole Earth 'Lectronic Link
Date: Wed, 18 Nov 1992 20:51:40 GMT


Clarification of Ramsay Sarfatti debate in simple case.

#1 Mach-Zehnder Interferometer (MZI) with ordinary unconnected one photon
input.

The MZI consists of two 50-50 non-absorbing beam splitters (\\) and two
nonabsorbing mirrors (\).

---->\\---------\
|a>  |    |t>   |
     |          - phase plate (phi)
     |          |
     \----------\\----- |tr+rt> = |I1> interferogram 1
          |r>   |
                |
                |
                 |tt+rr> = |I2> interferogram 2

Fig.1A MZI (orthogonal ket)

A. The first beam splitter resolves the input photon ket |a> into

|a> -> [|t> + i|r>]/sqrt2

The i is from reflection at the first beam splitter.

<t|r> = 0

<t|a> = 1/sqrt2

<r|a> = i/sqrt2

The effect of the phase plate (all optical paths otherwise equal) is

|t> -> e^iphi |t>

so that

[|t> + |r>]/sqrt2 -> [e^iphi|t> + |r>]/sqrt2

Interferogram 1 is then

[ie^iphi|t> + i|r>]/2 = |I1>

<I1|I2> = 1

The probability of counter 1 to "click" is then

p(1) = |<1|I1>|^2

= |[ie^iphi<1|t> + i<1|r>]/2|^2

= [|<1|t>|^2 + |<1|r>|^2 + 2|<1|t>||<1|r>|cos(phi + arg<1|t> - arg<1|r>)]/4

similarly, interferogram 2 is

[-|r> + e^iphi |t>]/2 = |I2>

<I2|I2> = 1

note that <I2|I1> = 0

and

p(2) = |<2|I2>|^2

= |[e^iphi<2|t> - <2|r>]/2|^2

= [|<2|t>|^2 + |<2|r>|^2 - 2|<2|t>||<2|r>|cos(phi + arg<2|t> - arg<2|r>)]/4


p(1) + p(2) = 1

is satisfied if

<1|t> = <1|r> = <2|t> = <2|r>

|<1|t>| = 1

so that

p(1) = [1 + cos(phi)]/2

p(2) = [1 - cos(phi)]/2

* Note that phi is definite. There is no reason to assume that phi is
random as Ramsay requires. There is no sum over phi required in an
individual experiment in order to conserve probability. This is a
significan difference from the classic 2-slit experiment. The MZ experiment
has a "discrete" screen of only two resolution-limited "points" in contrast
to the large number of resolution-limited points in the 2-slit experiment.

This model in which <t|r> = 0, requires 8 adhoc unobservable real
parameters which must be adjusted to get the right answer.

B. A much simpler model based upon the Feynman histories picture is:

---->\\---------\
|a>  |    +     |
     |          - phase plate (phi)
     |          |
     \----------\\----- |tr+rt> = |1> interferogram 1
          -     |
                |
                |
                 |tt+rr> = |2> interferogram 2

Fig.1B MZI (Feynman histories)

|a> = |1><1|a> + |2><2|a>

<1|a> = [a(+1) + a(-1)] = [ie^iphi + i]/2

<2|a> = [a(+2) + a(-2)] = [e^iphi -  1]/2

p(1) = |<1|a>|^2 = [1 + cos(phi)]/2

p(2) = |<2|a>|^2 = [1 - cos(phi)]/2

*Now we can see more clearly what I really mean when in the photon pair
case I write |a,e,+> parallel to |a,o,+>. That corresponds to the coherent
addition of the indistinguishable amplitudes a(+1) + a(-1) in which photon
a takes path + and it takes path - in both cases detected at counter 1.
I will develop this more for photon pair case in future postings.

It appears that Ramsay and I have been talking at cross purposes since
either picture gives the same answer in this simple case at least.