Light as a Probability Wave — Question 10

Question 10

Interference phase sensitivity follows from both slope and counting noise

A balanced interferometer detects N independent photons with plus-port probability $p(\phi)=(1+\cos\phi)/2$. The estimated fraction has standard deviation $\sigma_p=\sqrt{p(1-p)/N}$. Use local linear propagation $\sigma_\phi=\sigma_p/|dp/d\phi|$ where the slope is nonzero.

1. Derive the phase uncertainty away from fringe extrema.

2. Find it at $\phi=\pi/2$ for $N=10^4$, and state why direct linear propagation is invalid exactly at $\phi=0$.

Question 10 - Solution

Interference phase sensitivity follows from both slope and counting noise

(a) Method and calculation

$$\left|\frac{dp}{d\phi}\right|=\frac{|\sin\phi|}{2},\quad p(1-p)=\frac{\sin^2\phi}{4}.$$$$\boxed{\sigma_\phi=1/\sqrt N}\qquad(\sin\phi\ne0).$$

(b) Interpretation and further calculation

$$\phi=\pi/2,\ N=10^4:\quad\boxed{\sigma_\phi=.0100\,\mathrm{rad}}.$$$$\phi\simeq0:\quad p=1-\phi^2/4+O(\phi^4),\quad\boxed{p\prime(0)=0}.$$

Physical check

The limiting formula does not license dividing zero by zero at an extremum. A nonlinear likelihood analysis is needed there, and a single fringe also has phase ambiguities.