Undergrad Phase shift after double-slit for entangled photons

[boxfullofvacuumtubes]

Hi all,

I'm trying to understand how to describe the quantum state of entangled photons, including their phase, if one of them encounters a double-slit.

Here's a simple example:

Suppose you have 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}

Suppose the photon A passes through a double-slit.

Is my understanding correct that 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, is the following true?

\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}

\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, can the resulting state be described by this?

\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}

Or, is this a wrong way of including the phase shift in the quantum state of entangled photons?

 

[DrChinese]

I have never heard of what you are describing about phase shift (although that shouldn't be taken as particularly meaningful by itself).

Generally, entangled photons do not exhibit interference on a double slit setup. That is because they are not coherent. If they are made coherent, then they will not be entangled. Not sure that affects your description, but somehow I think it would.

 

[boxfullofvacuumtubes]

Generally, entangled photons do not exhibit interference on a double slit setup. That is because they are not coherent. If they are made coherent, then they will not be entangled. Not sure that affects your description, but somehow I think it would.

There have been some interesting experiments that have achieved interference either through a double-slit or through a Mach-Zehnder interferometer while using entangled photons. They typically use a laser pumping a nonlinear crystal that creates a pair of entangled photons by spontaneous parametric down-conversion (SPDC). One or both of these photons then go through an interferometer before detection. So, I guess the coherence length of this setup is sufficient if it's done right.

What I'd like to understand is the mathematical formalism that could describe this type of situations. But I'm not sure if the logic used in my post is all right.

 

[DrChinese]

There have been some interesting experiments that have achieved interference either through a double-slit or through a Mach-Zehnder interferometer while using entangled photons. They typically use a laser pumping a nonlinear crystal that creates a pair of entangled photons by spontaneous parametric down-conversion (SPDC).

If you know of any papers showing SPDC photons and a double slit (with interference), I'd love to see one. Always looking for good new references.

This is what Zeilinger had to say, p. 290, Figure 2: Experiment and the foundations of quantum physics

"FIG. 2. A source emits pairs of particles with total zero momentum.
Particle 1 is either emitted into beams a or a' and
particle 2 into beams b or b' with perfect correlations between
a and b and a' and b', respectively. The beams of particle 1
then pass a double-slit assembly. Because of the perfect correlation
between the two particles, particle 2 can serve to find
out which slit particle 1 passed and therefore no interference
pattern arises."

I have not seen an experimental demonstration of this effect, however.

 

[boxfullofvacuumtubes]

If you know of any papers showing SPDC photons and a double slit (with interference), I'd love to see one. Always looking for good new references.

Here are some papers that observed interference experimentally:

• https://www.univie.ac.at/qfp/publications3/pdffiles/1995-17.pdf
• Double-slit

Of course, interference arises only if there's no information about which path through the interferometer or double-slit the photon took. If two photons are entangled, the second photon may carry such information about the first photon, depending on the experimental setup.

 

[DrChinese]

Here are some papers that observed interference experimentally:

• Double-slit

Of course, interference arises only if there's no information about which path through the interferometer or double-slit the photon took. If two photons are entangled, the second photon may carry such information about the first photon, depending on the experimental setup.

This one I am familiar with. When there is interference here, I believe it is only seen via coincidence counting.