One particle and two slits

[Mentz114]

The single photon experiment of Grangier showed that if a phase shift ##\theta## is introduced in one arm of a Mach-Zehnder interferometer then there is interference when the two beams are recombined - even if only one photon is introduced.

If a particle encounters a double slit of suitable geometry then one possibility becomes two which are recombined at some point on the screen. I wondered what the result would be if the phase difference in the MZ could be expressed in terms of the two slit geometry. This is easily done and the result is ##\theta_s=2\pi(L_1-L_2)/\lambda## where ##L_1^2=X^2+(y-d)^2,\ L_2^2=X^2+(y-2d)^2##.

One of the detectors in the MZ setup has probability ##(1+\cos(\theta))/2 =\cos(\theta/2)^2##. Putting ##\theta_s## into this looks like this for X=75, λ=1, d=5

which at least looks plausible. I'm not sure if this is interesting or valid, but it is all quantum.

 

[tech99]

I think it it true that after a period of time, a pattern of dots will build up. I am not sure why your pattern does not taper away from the centre. Each slit has its own radiation pattern.

 

[Derek P]

The single photon experiment of Grangier showed that if a phase shift ##\theta## is introduced in one arm of a Mach-Zehnder interferometer then there is interference when the two beams are recombined - even if only one photon is introduced.

If a particle encounters a double slit of suitable geometry then one possibility becomes two which are recombined at some point on the screen. I wondered what the result would be if the phase difference in the MZ could be expressed in terms of the two slit geometry. This is easily done and the result is ##\theta_s=2\pi(L_1-L_2)/\lambda## where ##L_1^2=X^2+(y-d)^2,\ L_2^2=X^2+(y-2d)^2##.
View attachment 222136

One of the detectors in the MZ setup has probability ##(1+\cos(\theta))/2 =\cos(\theta/2)^2##. Putting ##\theta_s## into this looks like this for X=75, λ=1, d=5
View attachment 222137

which at least looks plausible. I'm not sure if this is interesting or valid, but it is all quantum.

I think you'll find the L2 formula should just be ..+d.., not ...-2d... But yes, the Young's slit pattern shows the same stretching off-centre as your plot. We often use a simple formula for calculating the spacing of interference bands but sometimes we forget that it is an approximation.

The introduction of a phase shift of more than 4πd/λ can't be accommodated in the YS setup for geometrical reasons, and this is reflected in the fact that there is no real solution for y. But the M-Z formula handles it without a hitch.

My maths is generally stochastic so I'll echo your comment: I'm not sure if this is interesting or valid.

 

[Derek P]

I think it it true that after a period of time, a pattern of dots will build up. I am not sure why your pattern does not taper away from the centre. Each slit has its own radiation pattern.

Sure, but the plot is for the interferometer not for a double slit. So there's no geometrical factor to worry about.
Certainly you will get a pattern of dots build up in both cases. A DS creates a pattern of dots with a detector array, the MZI builds up a histogram of detections against theta or y. That's the nice thing about quantum systems. They obey quantum mechanics.

 

[Mentz114]

I think it it true that after a period of time, a pattern of dots will build up. I am not sure why your pattern does not taper away from the centre. Each slit has its own radiation pattern.

There was a mistake in the algebra ( I was tired !)

I think you'll find the L2 formula should just be ..+d.., not ...-2d... But yes, the Young's slit pattern shows the same stretching off-centre as your plot. We often use a simple formula for calculating the spacing of interference bands but sometimes we forget that it is an approximation.

.

Thanks for pointing that out.

Calculating the phase in this way is not quantum, of course so it is not the right pattern. It needs a proper QFT treatment.

Here is a corrected plot.

 

[Derek P]

Calculating the phase in this way is not quantum, of course so it is not the right pattern. It needs a proper QFT treatment.

Are you sure?

 

[Mentz114]

Are you sure?

No. I will have a dabble with this later but I don't think the two slits can be handled by the beam splitter equation. The size of the slits is not a parameter and I'm not sure if changes in the distance between them (d) affects the pattern in the way this is observed in experiments.

I made a picture from a plot. The parameters are X=75, d=3, λ=1.

The MZ adaptation shown shown above does not agree with the result in
Quantum interference with slits by Thomas V Marcella
which is "the familiar intensity distribution for Fraunhofer diffraction as given in Guenther
[8], among others."

Marcella's result has been deprecated by some responsible https://www.researchgate.net/publication/46584717_Quantum_Interference_with_Slits_Revisitedso its status is not certain.

 

[Derek P]

No. I will have a dabble with this later but I don't think the two slits can be handled by the beam splitter equation. The size of the slits is not a parameter and I'm not sure if changes in the distance between them (d) affects the pattern in the way this is observed in experiments.

Yes, I agree with that. The size of the slits just adds a sinc term to the Fourier transform so you could insert it into the interferometer model as a shaped uncertainty in the "introduced" phase shift. The behaviour with simple coherent light would still be the same in both models because the complex amplitudes have the same form as a classical EM wave. Unfortunately, you haven't actually described the experiments you're talking about, but Grainger's name is associated with highly non-classical states and for all I know, the beam splitter may behave differently. If so then maybe the additional phase uncertainty (reflecting the non-zero width) will make a difference. Good luck with analysing that, I wouldn't know where to begin!

 

[Derek P]

The MZ adaptation shown shown above does not agree with the result in
Quantum interference with slits by Thomas V Marcella
which is "the familiar intensity distribution for Fraunhofer diffraction as given in Guenther
[8], among others."
Marcella's result has been deprecated by some responsible https://www.researchgate.net/publication/46584717_Quantum_Interference_with_Slits_Revisitedso its status is not certain.

Well as an irresponsible critic I would agree. Superficially skimming through, it looks like a standard treatment with a few workarounds to avoid explicitly solving wave equations. Lande took the same approach in New Foundations of Quantum Mechanics, 1965, saying something like "the particle interacts with the slits as a whole" implying that the lateral momentum imparted to the particle is a probability distribution whose amplitude is given by the Fourier transform of the slits. Unless there is something deep and mysterious buried in the paper, I cannot agree with their conclusion "It is interesting that for particles scattered from a double slit, the probability amplitude that gives rise to the interference is due to a superposition of delta functions." That is exactly what any integral is and a Fourier transform is no exception. It is true - but not interesting