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Advanced Physics Homework Help
Quantum state of entangled photons
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[QUOTE="boxfullofvacuumtubes, post: 6124574, member: 621453"] [h2]Homework Statement [/h2] 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? [B]2. The attempt at a solution [/B] 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? [/QUOTE]
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Quantum state of entangled photons
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