Wavefunction Coherence Considerations in the Dual Slit Experiment

by | Jun 20, 2025 | Physics, Quantum Mechanics | 0 comments

Thomas Lee Abshier
Author: Thomas Lee Abshier

Wavefunction Coherence Considerations in the Dual Slit Experiment
by Thomas Lee Abshier, ND, Copilot, and Poe Assistant
6/20/2025

Thomas: A photon obviously has a wavelength inferred by E=hf. Wavelength, of course, refers to the longitudinal distance over which the E and B fields act. My question is whether whether we can infer the approximate diameter of action of the EM wave by the outer spacing of the slits in a dual slit experiment. Can we determine the transverse width/extent of a photon by the dimensions of the slits (outer edges and inner barrier) in a dual slit experiment?

Copilot: That’s a deep and fascinating question—and it touches the heart of quantum weirdness.

In short: we can’t directly determine the transverse width of a single photon just by the dimensions of the slits in a double-slit experiment. But the interference pattern and diffraction effects do give us indirect clues about the spatial coherence and wave-like extent of the photon’s probability amplitude.

Here’s why:

🌊 Photons as Quantum Wavepackets
A photon isn’t a tiny billiard ball—it’s better described as a quantum wavepacket with a certain spatial extent. In the double-slit setup:

  • If the slit separation is smaller than the transverse coherence width of the photon’s wavefunction, you get interference.
  • If the slits are too far apart, the photon behaves as if it only goes through one slit—no interference.
    So, the ability to produce an interference pattern tells us that the photon’s wavefunction must be wide enough to “cover” both slits simultaneously.

🧠 But What About the “Size” of a Photon?
This is where it gets tricky. The photon doesn’t have a well-defined size in the classical sense. Its transverse extent depends on:

  • The source (e.g., laser vs. spontaneous emission)
  • The optical setup (collimation, apertures, etc.)
  • The coherence properties of the light
    In practice, the slit width and separation act as a kind of “ruler” for the transverse coherence of the photon field—but not for the photon’s literal size.

🧪 Experimental Insight
In single-photon double-slit experiments, the interference pattern builds up over time, even when photons are sent one at a time. This suggests that each photon interferes with itself, and its wavefunction must span both slits. But the pattern’s fringe spacing is determined by the wavelength, not the photon’s “width.”

So while the slit geometry constrains what kind of interference you can observe, it doesn’t pin down a hard boundary for the photon’s transverse size. It’s more accurate to say the slits probe the coherence and spatial structure of the photon’s wavefunction.

Thomas: The question is thus, what is the coherence and spatial structure of the photon’s wavefunction? For example, what would be the outer/most extreme/largest dimensions of a dual slit (outer extent and inner barrier (and slit width) that produce an interference pattern for a 650 nm, series of single photons, e.g. from a quantum dot?

You (Copilot) said, “In single-photon double-slit experiments, the interference pattern builds up over time, even when photons are sent one at a time. This suggests that each photon interferes with itself, and its wavefunction must span both slits. But the pattern’s fringe spacing is determined by the wavelength, not the photon’s “width. So while the slit geometry constrains what kind of interference you can observe, it doesn’t pin down a hard boundary for the photon’s transverse size. It’s more accurate to say the slits probe the coherence and spatial structure of the photon’s wavefunction.”

Thomas: The question is thus, what is the coherence and spatial structure of the photon’s wavefunction? For example, what would be the outer/most extreme/largest dimensions of a dual slit (outer extent and inner barrier (and slit width) that produce an interference pattern for a 650 nm, series of single photons, e.g. from a quantum dot?

Copilot: That’s a beautifully precise and probing question—and it gets to the heart of how quantum optics bridges theory and experiment.

🌌 Coherence and Spatial Structure of a Single-Photon Wavefunction
A single photon emitted from a quantum dot typically has a wavefunction that is spatially extended, especially in the transverse direction. This wavefunction isn’t a sharply bounded object—it’s a probability amplitude that decays smoothly, often modeled as a Gaussian beam or a wavepacket with finite spatial coherence.
The transverse coherence width (or spatial coherence length) of the photon’s wavefunction determines whether it can “cover” both slits and produce interference. This coherence width depends on:

  • The emission aperture and optical setup (e.g. collimation, lenses)
  • The angular divergence of the photon beam
  • The source size and emission mode of the quantum dot

📏 What Slit Dimensions Still Produce Interference at 650 nm?
Let’s assume you’re using single photons at 650 nm from a well-collimated quantum dot source. Based on experimental data and coherence theory:

  • Slit separation (center-to-center):
    Interference is typically observed for separations up to 10–50 microns, depending on the beam’s transverse coherence. Beyond that, the photon’s wavefunction may not span both slits coherently.
  • Slit width:
    Usually 1–10 microns. Narrower slits increase diffraction and fringe visibility but reduce photon throughput.
  • Inner barrier (between slits):
    Often 1–20 microns. This is essentially the gap that the wavefunction must “bridge” to interfere with itself.
  • Outer extent (total slit array width):
    Could be 20–100 microns or more, but only the coherent overlap region matters for interference.
    So, for a 650 nm photon, a double-slit with 5 μm wide slits, 10 μm separation, and total width ~20 μm would be well within the range to observe interference—assuming good spatial coherence.

🧠 What This Tells Us About the Photon’s Wavefunction
If interference is observed, the photon’s transverse wavefunction must have non-negligible amplitude across both slits. That implies a transverse coherence width at least as large as the slit separation—say, 10–20 microns in this case.
But this doesn’t mean the photon “has a size” of 20 microns. Rather, its probability amplitude is spread over that region, and the interference pattern is a manifestation of that spread.

Thomas: Excellent answer! I think the bottom line is that for a wavelength of 650nm (.65 microns), there is a probability distribution of the photon acting on the space over a region of 20 microns. Is this a reasonable summation of your answer?

Copilot: Yes, that’s a very reasonable summation—with just the right nuance.

You’re capturing the core idea: for 650 nm photons to produce interference in a double-slit setup with, say, 10–20 micron slit separations, their wavefunction must be coherently spread over that 10–20 micron region. So while the photon itself isn’t a “blob” of that size, its probability amplitude—its ability to exhibit wave-like effects—is nonzero across that span.

In other words, the photon “acts on” that region through its quantum wavefunction, not because it occupies it like a rigid object, but because its presence is delocalized over that range. The interference pattern is the physical footprint of that delocalized action.
It’s a beautiful example of how quantum particles refuse to fit neatly into classical molds.

Thomas: I think your qualification of my answer was to emphasize that this 10-20 micron region was not a uniform region of influence, but rather the region over which the probability of the wavefunction acting was strong to some arbitrary cutoff, such as 2 standard deviations of intensity. Was this the point of your emphasis?

CoPilot: Exactly—that’s beautifully put.

The emphasis was to highlight that the wavefunction’s influence isn’t uniform across that 10–20 μm span. Rather, it’s shaped—typically something like a Gaussian distribution—so the probability amplitude peaks near the center and tapers off toward the edges. When we talk about the photon “covering both slits,” we mean that the amplitude is still appreciable at those points—enough to enable interference.
Using something like 2 standard deviations of intensity as a cutoff is a very reasonable way to conceptualize the effective transverse coherence width, even though quantum mechanics doesn’t prescribe a hard boundary. This kind of soft-edged extent is exactly what allows one photon to “feel” both slits, as strange as that sounds.

Would you like to explore how to calculate the transverse coherence width from a source’s spatial or angular properties? That opens the door to designing precision experiments—or engineering quantum light sources. ✨🔬

Thomas: I would you like to explore how to calculate the transverse coherence width from a source’s spatial or angular properties? That opens the door to designing precision experiments—or engineering quantum light sources.

 

8:40 AM Bot image for Assistant Assistant
Absolutely! Let’s explore how to calculate the transverse coherence width of a photon’s wavefunction based on its spatial and angular properties. Understanding the transverse coherence width is essential for designing experiments and optimizing quantum light sources for applications like interferometry, spectroscopy, and quantum communication.

1. Transverse Coherence Width: What Is It?

The transverse coherence width is a measure of the spatial extent over which the photon’s wavefunction maintains its coherence. In simpler terms:

  • It describes the region where the phase relationship of the wavefunction is consistent enough for interference effects to occur.
  • For a single photon, it’s the effective width of the probability amplitude in the transverse plane.

The coherence width depends on:

  1. The angular divergence of the light source.
  2. The spatial size of the emission aperture or source.
  3. The wavelength of the light.

2. Key Parameters and Formulas

a. Angular Divergence (\( \theta \))

The angular divergence of the light beam determines how much the wavefront spreads as it propagates. It is related to the source size and wavelength.
For a Gaussian beam, the angular divergence is given by:
$$ \theta = \frac{\lambda}{\pi w_0} $$
Where:
– \( \lambda \): Wavelength of the light
– \( w_0 \): Beam waist, the narrowest part of the beam (related to the source size)

b. Transverse Coherence Width (\( L_c )\)

The transverse coherence width is inversely related to the angular divergence:
$$ L_c = \frac{\lambda}{\Delta \theta} $$
Where:
– \( \Delta \theta \): Angular spread of the beam (similar to \( \theta \), but measured for the source)
– \( \lambda \): Wavelength of the light
If the source is collimated (low divergence), \( L_c \) becomes larger, meaning the photon’s wavefunction is spread over a broader transverse region.

c. Spatial Coherence and Source Size

For a light source with finite spatial size ( d ), the coherence width is related to the source size and distance ( z ) from the source:
$$ L_c \sim \frac{\lambda z}{d} $$
Where:
– \( d \): Effective size of the light source
– \( z \): Distance from the source to the plane of interest
– \( \lambda \): Wavelength of the photon

This equation shows that smaller sources (or better-collimated beams) produce larger coherence widths.

3. Example Calculation

Let’s calculate the transverse coherence width for a 650 nm photon emitted from a quantum dot source.

Assumptions

– Wavelength: \( \lambda = 650, \text{nm} = 6.5 \times 10^{-7}, \text{m} \)
– Emission aperture size: \( d = 1, \mu\text{m} = 1 \times 10^{-6}, \text{m} \)
– Distance to the double-slit plane: \( z = 1, \text{m} \)

Using the Formula
$$ L_c = \frac{\lambda z}{d} $$
Substituting values:
$$ L_c = \frac{(6.5 \times 10^{-7})(1)}{1 \times 10^{-6}} = 0.65, \text{m} $$

$$ L_c = 0.65, \text{m} $$

This result suggests that the transverse coherence width of the photon’s wavefunction can extend as much as 0.65 meters in this setup, assuming perfect collimation. This is an idealized scenario, and practical factors like beam divergence and imperfect collimation would reduce this value.

4. Practical Implications

a. Double-Slit Interference

For a double-slit experiment:
– The slit separation must be smaller than or comparable to the transverse coherence width \( L_c \) for interference to occur.
– If the slits are too far apart (e.g., greater than \( L_c \)), the photon’s wavefunction cannot coherently span both slits, and interference patterns will disappear.

b. Source Engineering

– To maximize \( L_c \), use a smaller source size (\( d \)) and better collimation optics to reduce the angular divergence (\( \Delta \theta )\).
– Quantum dot sources often emit light with high spatial coherence, making them suitable for interference experiments.

5. Insights from Coherence Theory

a. Gaussian Beam Approximation

In many cases, the photon’s wavefunction can be modeled as a Gaussian beam, with a transverse intensity profile given by:
$$ I(r) = I_0 \exp\left(-\frac{2r^2}{w(z)^2}\right) $$
Where:
– \( r \): Radial distance from the beam center
– \( w(z) \): Beam width at distance ( z ), which expands with distance

b. Effective Coherence Width

The transverse coherence width can be approximated as the region within 1–2 standard deviations of the beam’s intensity distribution. This captures the region where the probability amplitude is significant, even if it technically extends to infinity.

6. Summary

  • The transverse coherence width of a photon’s wavefunction depends on its wavelength, source size, and angular divergence.
  • For a 650 nm photon emitted by a quantum dot, and assuming a small source size and good collimation, the coherence width can span tens of microns to meters.
  • In a double-slit experiment, the slit separation and geometry probe the coherence properties of the photon’s wavefunction, not its “size.”

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