In interferometry with coherent light, the final output is differenced detectors. That is,
$\left<N\right> = \left<N_1\right> - \left<N_2\right>$
where $N_i$ is the number operator of that mode.
However, in interferometry with NOON states, the Wikipedia article states
consider the observable
A = | N , 0 ⟩ ⟨ 0 , N | + | 0 , N ⟩ ⟨ N , 0 |The expectation value of A {\displaystyle A} A for a system in a NOON state switches between +1 and −1 when the phase changes from 0 to $π/N$.
My confusion is with the expression and physical meaning of the operator A. If the NOON state is
$\left|\psi\right> = \frac{\left|N\right>_a\left|0\right>_b+e^{iN\phi}\left|0\right>_a\left|N\right>_b}{\sqrt{2}}$
as taken in the Wikipedia article. Then, because I'm confused about the notation of $A$, there are two possible expressions of $A$ for me. $A_1$ and $A_2$:
$A_1 = \left|N\right>_a\left|0\right>_b\left<N\right|_a\left<0\right|_b + \left|0\right>_a\left|N\right>_b\left<0\right|_a\left<N\right|_b$,
which I think would be the sum of the number operators on both the output arms of the Interferometer. However, this operators doesn't give the correct result of
$\left<A\right> = \cos 2\phi$
which Wikipedia alludes to in its change of phase from 1 to -1 statement. On the other hand, the expression
$A_2 = \left|N\right>_a\left|0\right>_b\left<0\right|_a\left<N\right|_b + \left|0\right>_a\left|N\right>_b\left<N\right|_a\left<0\right|_b$
does give the correct answer. However, I don't understand the physical realization of this operator. And if this is the measurement being performed, then this paper refers to it as
If one now carries out a simple measurement scheme in the N-photon detecting analyzer
So, what is this measurement being carried out? Which one is the correct representation, $A_1$ or $A_2$, of $A$ as written by Wikipedia? And what is the physical meaning of this operator?
