Light as a Probability Wave — Question 5

Question 5

A random phase mixture differs from a stable dark-port state

A balanced interferometer has $P_+(\phi)=(1+\cos\phi)/2$. In ensemble A every photon has $\phi=\pi$. In ensemble B the phase is independently uniform on $[0,2\pi)$ for each photon. Both ensembles contain exactly N photons.

1. Find the mean plus-port count in each ensemble.

2. For ensemble B, find the count variance and explain why averaging amplitudes would give the wrong rule.

Question 5 - Solution

A random phase mixture differs from a stable dark-port state

(a) Method and calculation

$$\langle P_+\rangle_A=0,\quad\langle P_+\rangle_B=\frac1{2\pi}\int_0^{2\pi}\frac{1+\cos\phi}{2}\,d\phi=\frac12.$$$$\boxed{\langle K_A\rangle=0,\qquad\langle K_B\rangle=N/2}.$$

(b) Interpretation and further calculation

$$K_B\sim\operatorname{Bin}(N,1/2)\quad\Rightarrow\quad\boxed{\operatorname{Var}K_B=N/4}.$$$$\boxed{\langle|c_+|^2\rangle=1/2\ne|\langle c_+\rangle|^2=1/4},\quad c_+=(1+e^{i\phi})/2.$$

Physical check

Different members of a statistical ensemble are averaged as probabilities. Coherent alternatives within one member are combined as amplitudes.