- #1
boxfullofvacuumtubes
- 20
- 2
Homework Statement
Suppose two polarization-entangled photons A and B in the following Bell state:
\begin{equation}
\Phi=\frac{1}{\sqrt{2}}\bigl(\left|H_{A},H_{B}\right\rangle + \left| V_{A},V_{B}\right\rangle\bigr)
\end{equation}
1. What is the state if the photon A passes through a double-slit?
2. What is the state if the photon A passes through a double-slit and the photon B passes through a linear polarizer oriented at the +45 angle?
2. The attempt at a solution
My attempt to solve (1):
A double-slit in the photon A's path creates a phase shift $$e^{i\Delta\phi}$$ because of unequal paths from each slit to a particular place on a screen. As the photon A can now take a path through one or the other slit, and there is a phase shift between the two,
\begin{equation}
\left|H_{A},H_{B}\right\rangle \longrightarrow \frac{1}{\sqrt{2}}\bigl(\left|H_{A},H_{B}\right\rangle\bigr) + e^{i\Delta\phi}\frac{1}{\sqrt{2}}\bigl(\left|H_{A},H_{B}\right\rangle\bigr)
\end{equation}
Similarly:
\begin{equation}
\left|V_{A},V_{B}\right\rangle \longrightarrow \frac{1}{\sqrt{2}}\bigl(\left|V_{A},V_{B}\right\rangle\bigr) + e^{i\Delta\phi}\frac{1}{\sqrt{2}}\bigl(\left|V_{A},V_{B}\right\rangle\bigr)
\end{equation}
Therefore:
\begin{equation}
\Phi=\frac{1}{2}\bigl(\bigl(1+e^{i\Delta\phi}\bigr)\left|H_{A},H_{B}\right\rangle + \bigl(1+e^{i\Delta\phi}\bigr)\left|V_{A},V_{B}\right\rangle \bigr)
\end{equation}
My attempt to solve (2):
The photon B passing through a 45-degree polarizer has a 50% probability of being absorbed and a 50% probability of converting:
\begin{equation}
H_{B} \longrightarrow \frac{1}{\sqrt{2}}\left|+45_{B}\right\rangle = \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr)
\end{equation}
\begin{equation}
V_{B} \longrightarrow \frac{1}{\sqrt{2}}\left|+45_{B}\right\rangle = \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr)
\end{equation}
Putting (1) and (2) together:
\begin{equation}
\Phi=\frac{1}{2}\bigl(\bigl(1+e^{i\Delta\phi}\bigr)\left|H_{A}\right\rangle \bigotimes \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr) + \bigl(1+e^{i\Delta\phi}\bigr)\left|V_{A}\right\rangle \bigotimes \frac{1}{2}\bigl(\left|H_{B} + V_{B}\right\rangle\bigr) \bigr)
\end{equation}
The rest is easy, just manual work, but I'm wondering if I made any mistakes up to this point. Anyone willing to help?