# Three-slit Interference

(or: How Not to Axiomatically Construct Quantum Mechanics)

Faculty of Mathematics, Physics, and Computation

Schmid College of Science and Technology

## An Initial Dilemma:

Naively, the Born rule seems to imply only pair-wise interference. Initial experiments seemed to confirm this naive suggestion (within experimental noise).

"The consequences of detecting even a small amount of three-way interference (by deviating from the quantum mechanical null prediction) would be tremendous. A modification to Born’s rule that leads to multi-order interference would have repercussions on the allowable dynamics. In particular, if probability must be conserved, then Schrödinger’s equation would likely have to be modified as well." - Sinha (2010)

Data from Science Paper (2010)

Seems to suggest a null result (< 10^-2)

But data is noisy

Not convincing

(Vertical axis renormalized difference parameter between two and three-slit results)

## But wait, there's more!

"It is not widely appreciated that the superposition principle is incorrectly applied in most textbook expositions of interference experiments both in optics and quantum mechanics" - Sinha (2015)

"The non-zeroness ... is essentially due to boundary condition considerations and should affect both classical as well as quantum physics. In existing experimental results in literature ..., the experimental inaccuracies have prevented us from concluding that [third-order interference] is non-zero." - Sinha (2015)

## So what is going on?

• Third-order interference should exist, for both classical field theories and quantum wave equations
• Solving the wave equations numerically (with the correct boundary conditions, red squares below) shows the effect immediately - pairwise interference is only an approximation
• Third-order interference is a near-field effect at the boundary

## An analytic approximation: exotic Feynman paths

Helmholtz Eq. for plane wave:

Point-to-point streamline propagator:

Expand in perturbative series with number of "loops" around slits:

Path-integral propagator representation:

("classical paths" are streamlines of the Poynting vector field in the optical case, and equivalent to the Bohmian trajectories)

(no plasmons - small effect)

(with plasmons - large effect)

(Poynting vector field follows red path above)

(Gaussian focused on left-most slit)

(Measured single-photon CCD camera images of each distribution)

Verified 3-slit interference term

# Looped Paths Matter

(No looped paths - only direct "Born Rule")

(With looped paths - correct boundary conditions)

"When these slits are illuminated with light polarized along the long axis of the slits, surface
plasmon modes are not excited in the structure and the interference pattern shown in Fig. 2a is shown
in the far field.
This pattern is practically identical to that obtained by simply applying the Fourier
transform of the three slits
. However, this simple experiment shows a striking interference structure
when the slits are illuminated with light polarized along the short axis of the slits. In this case, surface
plasmon modes are efficiently excited, leading to an increased probability for looped paths, which in
turn leads to the significantly different interference pattern shown in Fig. 2b.
"

# Take-away Messages?

• Boundary conditions matter
• The Born Rule is not broken, when used correctly
• Axiomatic reformulations must be careful not to discount relevant physics by oversimplifying the problem
• Even knowledgeable researchers make simple mistakes

(Note: heralded single photons!)

#### Three-slit Interference

By Justin Dressel

# Three-slit Interference

A brief summary of the state-of-play for three-slit interference: history and recent results.

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