Light as a Probability Wave — Question 2

Question 2

Fringe visibility reveals path imbalance with a labeling ambiguity

At a lossless mixer, the input amplitudes are $a=\sqrt p$ and $b=\sqrt{1-p}\,e^{i\phi}$, with $0\le p\le1$. One output has amplitude $(a+b)/\sqrt2$. Visibility is $\mathcal V=(P_{\max}-P_{\min})/(P_{\max}+P_{\min})$.

1. Derive the phase-dependent detection probability and visibility.

2. Infer all possible values of p from $\mathcal V=.800$.

Question 2 - Solution

Fringe visibility reveals path imbalance with a labeling ambiguity

(a) Method and calculation

$$P(\phi)=\tfrac12\bigl[1+2\sqrt{p(1-p)}\cos\phi\bigr].$$$$P_{\max,\min}=\tfrac12\pm\sqrt{p(1-p)},\quad\boxed{\mathcal V=2\sqrt{p(1-p)}}.$$

(b) Interpretation and further calculation

$$p(1-p)=.160\quad\Rightarrow\quad p^2-p+.160=0.$$$$\boxed{p=.200\ \text{or}\ .800}.$$

Physical check

Visibility cannot identify which input path carries the larger probability. An additional path-resolved measurement is required to break that symmetry.