Light as a Probability Wave — Question 3

Question 3

Global phase cancels while a relative phase changes interference

Two input amplitudes are initially $(a,b)=(1,1)/\sqrt2$. Compare transformation G, which multiplies both by $e^{i\alpha}$, with transformation R, which multiplies only b by $e^{i\alpha}$. A mixer has outputs $(a\pm b)/\sqrt2$.

1. Find output probabilities under G for arbitrary $\alpha$.

2. Find output probabilities under R at $\alpha=\pi$ and identify the observable distinction.

Question 3 - Solution

Global phase cancels while a relative phase changes interference

(a) Method and calculation

$$c_+^{(G)}=e^{i\alpha},\qquad c_-^{(G)}=0.$$$$\boxed{P_+^{(G)}=1,\quad P_-^{(G)}=0}\quad\text{for every }\alpha.$$

(b) Interpretation and further calculation

$$c_\pm^{(R)}=\tfrac12(1\pm e^{i\alpha}),\quad P_\pm^{(R)}=\tfrac12(1\pm\cos\alpha).$$$$\alpha=\pi:\quad\boxed{P_+^{(R)}=0,\quad P_-^{(R)}=1}.$$

Physical check

An overall complex phase leaves all absolute squares unchanged. A phase difference between coherent alternatives changes their cross term.