Light as a Probability Wave — Question 4

Question 4

Partial which-path information reduces the interference cross term

A photon has equal path amplitudes and becomes correlated with normalized marker states $|m_a\rangle$ and $|m_b\rangle$. Their overlap is real, $\langle m_a|m_b\rangle=\mu=.600$. At one mixer output the unnormalized marker state is $(|m_a\rangle+e^{i\phi}|m_b\rangle)/2$. Ignore the marker when counting photons.

1. Calculate the output probability from the squared norm.

2. Find its maximum, minimum and visibility; compare with orthogonal markers.

Question 4 - Solution

Partial which-path information reduces the interference cross term

(a) Method and calculation

$$P_+=\tfrac14[2+e^{i\phi}\mu+e^{-i\phi}\mu]=\tfrac12(1+\mu\cos\phi).$$$$\boxed{P_+=\tfrac12+.300\cos\phi},\quad P_-=1-P_+.$$

(b) Interpretation and further calculation

$$\boxed{P_{\max}=.800,\quad P_{\min}=.200,\quad\mathcal V=.600}.$$$$\mu=0:\quad\boxed{P_+=P_-=1/2,\quad\mathcal V=0}.$$

Physical check

The loss of visibility follows from correlations with distinguishable marker states, even when nobody reads the marker.