A Heisenberg lens and probabilities

[Heidi]

Hi Pf
in her experiment Birgit Dopfer uses an https://www.researchgate.net/figure/color-online-The-Dopfer-experiment-of-the-Zeilinger-Group-Innsbruck-If-detector-D2-is_fig7_265787833
the distance between the source and the lens is 2f and the detector may be at the distance f or 2f behind the lens.
in classical optics approximation all the light is focused on the image at 2f.
when the detector is a f what is the probability to get one photon?

 

[vanhees71]

The idea is to use the distance between the lens and the screen to decide whether you want to measure through which way the photon came or which momentum it had when coming from the double slit.

If you put the screen at a distance $d_S$ from the lens and the lens from the double slit $d_D$ you must have $d_D, d_S>f$ such that the lens equation $1/d_S+1/d_D=1/f$ is fulfilled. Then the registration on the screen is in one-to-one correspondence through which slit the photon came, and you simply see to sharp spots on the screen when accumulating many photons (it's looking as with classical light), i.e., you see a sharp picture on the screen.

On the other hand, if you make $d_S=f$, then the spot where the photon ends only tells you the direction of the photon's momentum, because then formally $d_D \rightarrow \infty$. So in this case you measure the momentum of the photon coming from the double slit. There's now no way to get the which-way information from this measurement but the Fraunhofer diffraction picture in full contrast when accumulating many photons.

Thus using a lens you can either measure the which-way information of the photon momentum exactly but not both. Of course by choosing $d_S$ and $d_D$ in an arbitrary way, not fulfilling the condition for a sharp image, leads to something in between, i.e., you get neither a completely accurate which-way information nor a completely accurate momentum information and thus some interference but not with full contrast. You get the better contrast of the interference picture the more accurate you measure the photon momenta with worse which-way information and vice versa.

Dopfer's thesis is available from Vienna University (in German)

https://www.univie.ac.at/qfp/publications/thesis/bddiss.pdf

 

[Heidi]

its seems that you use the analogy between the pdc and a mirror.
Can we do without that to explain what happens?
in which page in the thesis?

 

[vanhees71]

What is a "pdc"?

The Heisenberg microscope experiment (also with the entangled photon pairs and delayed choice) is described in detail in Chpt. 4.

 

[Heidi]

it was for parametric down conversion.
I am looking in chapter 4 to the figures 4.3 and 4.4.
Birgit Dopfer writes that fig 4.3 is a first idea for the Heisenberg microscope. there is a source of electrons in front of the slits and behind a source of photons under the lens. so they interact between the slits and the Heisenberg lense.
Do this corresond to the Lilo kristall where the two photons come from?
can we replace like that an electron by a photon?

 

[Heidi]

I have another question:
Is there a projection when the photon is detected in the image plane?
Of course the photon is registered at a given position in the plane but it seems that something else is selected by this partial measurement on the pair of entangled photons. there is also a path information. Does it come from a projection? on which states?

 

[vanhees71]

Let's discuss the two extremes separately.

(a) maximum diffraction-picture contrast

Then you have the setup as in the following figure from Dopfer's PhD thesis:

In the thesis Dopfer explains the result a bit different in an equivalent gedankenexperiment description, but I find it simpler to describe the real experiment.

From the LiIO3 crystal you get parametric-down converted entangled photons. What's used in this experiment is only the entanglement of the photon momenta.

Measured are only such entangled pairs, i.e., the photons at the detectors D1 and D2 get only registered when they arrive in coincidence such that you are sure to always measure only photons belonging to an entangled pairs.

With D1 in the focal plane registering photons at a specific position (i.e., making this detector sufficiently small) you select photons of a specific momentum (direction, the magnitude is anyway fixed from the phase-matching condition of the down-conversion process), because in the focal plane there's a one-to-one map between the point of photon detection and the momentum direction of the photon, i.e., you prepare (almost ideally) a plane wave.

For the so registered plane wave the partner photon registered at D2 went through the double slit and is also described by an (almost ideal) plane wave with the corresponding momentum and thus you get a double-slit interference pattern with full contrast when detecting many photons with D2 as a function of the position of D2. Since you have "prepared" a plane wave there's no way (not even in principle!) to gain which-way information, i.e., through which slit the photon registered by D2 came.

Shifting D1 within the focal plane shifts the double-slit interference pattern accordingly.

(b) Allowing for which-way information

Now D1 is put in the imaging plane of the Heisenberg lense.

Now photons with any momentum direction are registered. What D1 sees is an image of the illuminated LiIO3 crystal or a smaller part of it depending on the size of D1. Since now the photons registered with D2 correspond to a light with photon momenta in any direction this corresponds to illuminating the Double slit with incoherent light, i.e., for a large detector D1 you get the incoherent superposition of the two single slits. If the slits are small enough you get a single-slit interference pattern from both slits and just add the intensities of these two. If D1 is small you just get the single-slit interference pattern. In any case since in principle you can know the precise position from which slit each photon registered by D2 came there is no two-slit interference pattern anymore.

 

[Heidi]

thand you.
As in the figures D1 was on the lens symmetry axis i wondered how we could get path information there. Of course D1 can be elsewhere (as D2 behind the slits) ! it is clear now.

 

[Heidi]

Maybe another question:
if D1 is always in the focal plane on the lens axis, we have a given pattern. Have we another pattern if it is displaced to another place in the focal plane?
I asd this question for this reason: when D1 is in the image plane it may be on the image of one slit or on the other. and in this case there are two patterns associated to each slit.

 

[vanhees71]

Sure, the images (double slit interference pattern or single-slit interference pattern) shift when shifting D1 within the focal or image plane.