Classical treatment of quantum eraser?

[Aidyan]

The below experimental setup is of the sort one finds frequently discussed in the delayed choice quantum eraser (QE) experiments, such as that of Kim et al. (https://arxiv.org/abs/quant-ph/9903047) I extracted only the essential part I'm wondering about and did not find a satisfying answer in other threads.

To recapitulate, on the left a light source (say a laser), then the usual double slits, in front of each slit there is a polarizer which filters the horizontal component (H) and vertical component (V) for slit S1 and S2 respectively. A Glan-Thompson prism splits the two orthogonally polarized beams and sends the H-pol. beam towards mirror M1 and the V-pol. beam towards mirror M2. These then are reflected or transmitted at beamsplitter BS, which recombines the two beams and is supposed to erase the 'which-way' information. The question is: what will we observe on screen 1 and 2?

My first instinctive answer was that to think of an interference pattern, just as in the two slits experiment, eventually with on one screen displaying a fringe- on the other an anti-fringe interference pattern (note that the two sides at the BS are not symmetric: the beam going towards screen 1 is a superposition of a reflected H-pol. and transmitted V-pol. beam, whereas the opposite is true for the beam going towards screen 2). This is also what the literature on QE seems to suggest (I say 'suggest' because they did not use screens in place, only detectors).

However, later I could not convince myself that this is the case. Because, taking a more classical EM wave theory approach, one must not forget that two mutually orthogonal polarized light beams going through double slits do not produce interference fringes (just google 'interference of polarized light'). This will also lead to a different interference pattern than that we are accustomed with in the conventional Young experiment, and can be captured by modifying the intensity function as follows:

$I\left(\delta \phi,\delta\theta\right)= I_1+I_2+2 \sqrt{I_1 I_2} \, (cos{\delta (\phi))} \cdot c o s{(\delta\theta)}$,

where $I_1$ and $I_2$ are the intensities from slit 1 and 2 respectively, $\delta \phi$ is the usual angular phase path difference of the beams that modulates the interference term and $\theta$ is the relative separation angle between the polarization vectors of the two beams. This latter term is usually omitted in analyzing the standard double slit experiment because if no polarizer is in place $\delta\theta=0$. If however $\delta\theta=\pi/2$ the entire interference term is canceled and only the two intensities $I_1$ and $I_2$ remain, that is, we have no interference fringes. So, should we therefore expect to see only the usual Gaussian probability function on both screens?

On the other side, are we not always told that if the photons 'which-way' information is erased we should expect the interference fringes to reappear?

So, I'm confused. There seems to me a contradiction between the classical and quantum perspective. Where did I get it wrong?

 

[eljose79]

Thank you for bringing up this interesting topic. The delayed choice quantum eraser experiment is indeed a fascinating experiment that challenges our understanding of the nature of quantum mechanics.

To answer your question, we need to first understand the fundamental principle of quantum mechanics, which is the superposition principle. This principle states that a quantum system can exist in multiple states simultaneously until it is observed or measured. In the double-slit experiment, this means that the photon can take multiple paths, going through both slits at the same time, until it is measured at the screen.

In the setup you described, the polarizers and the Glan-Thompson prism act as "which-way" detectors, as they determine the polarization state of the photon. However, the key aspect of the delayed choice quantum eraser experiment is that the measurement of the "which-way" information can be delayed until after the photon has passed through the double slits and reached the screen.

Now, let's consider the two possible scenarios:

1. If we do not measure the "which-way" information (i.e. remove the polarizers and the Glan-Thompson prism), we will see an interference pattern on the screen, as expected in the double-slit experiment.

2. If we measure the "which-way" information after the photon has passed through the double slits (i.e. before it reaches the screen), the interference pattern will disappear. This is because the act of measurement collapses the superposition of states, and the photon can only be in one state at a time.

In the second scenario, if we then use a second beamsplitter to recombine the two beams, the interference pattern will reappear on the screen. This is because the "which-way" information has been erased, and the photon can once again exist in a superposition of states.

Therefore, in the delayed choice quantum eraser experiment, we can see both the interference pattern and the usual Gaussian probability function on the screens, depending on whether we measure the "which-way" information or not.

I hope this helps to clarify your confusion. If you have any further questions, please do not hesitate to ask.