6) Quantum Optics Problem:
Pairs of photons are captured in separate fibers, with quantum-entangled polarization.
For either photon, we define two circular polarization quantum states. For the first
photon right and left polarization states are referred to as $\left| R \right\rangle_1$, and $\left| L \right\rangle_1$. For the
second photon the states are referred to as $\left| R \right\rangle_2$, and $\left| L \right\rangle_2$.
We also can define linear polarization states for photon 1 called $\left| H \right\rangle_1$, and $\left| V \right\rangle_1$, for
horizontal and vertical polarized light (a photon that was passed through a horizontal
or vertical polarizer).
$\left| R \right\rangle_1 = \frac{1}{\sqrt{2}} (\left| H \right\rangle_1 + i \left| V \right\rangle_1)$
$\left| L \right\rangle_1 = \frac{1}{\sqrt{2}} (\left| H \right\rangle_1 - i \left| V \right\rangle_1)$
and similarly for photon 2.
Remember the rotation operator for rotations around the z axis is:
$Rot(\theta) = e^{-i \frac{S_z \theta}{\hbar}}$
$S_z \left| m \right\rangle = m\hbar \left| m \right\rangle$
and for circular polarization states for E&M vector fields (massless photons) m = ±1.
a. The state $\left| R \right\rangle_2$ is measured in a particular coordinate system. How does the
state look in a new coordinate system, rotated around the z axis by $\theta$. (the R
state is the m = 1 one with z component of angular momentum = $m\hbar$.
b. If the two photons are entangled and in the state
$\frac{1}{\sqrt{2}} (\left| R \right\rangle_1 \left| R \right\rangle_2 + \left| L \right\rangle_1 \left| L \right\rangle_2)$.
Rewrite this state in terms of linear polarization states.
c. The photon from each fiber is passed through two ideal horizontal polarizers.
After many such trials, for what fraction of the trials will have:
i. A photon is detected after both polarizers.
ii. No photon is detected from either polarizer.
iii. A photon emerges from only the polarizer after the 1$^{st}$ fiber.
iv. A photon emerges from only the polarizer after the 2$^{nd}$ fiber.
d. If one of these polarizers rotate by 1 degree, how do the results from part c
change.
