Double Slit

The double-slit experiment distills quantum mechanics. A photon travels from a source to a screen through two narrow slits separated by distance \(d\). Classically you might expect two bright bands — one from each slit. Instead, the intensity oscillates as \[ I(\theta)\propto \cos^2\!\Big(\frac{\pi d}{\lambda}\sin\theta\Big)\, \mathrm{sinc}^2\!\Big(\frac{\pi a}{\lambda}\sin\theta\Big), \] where \(a\) is the slit width. The fringes come from amplitudes adding, not probabilities. Send photons one at a time: each lands as a pointlike click, but the histogram of many clicks draws the same interference pattern.

The crucial lever is which-path information. If the setup allows you (even in principle) to know which slit the photon used, interference fades. If no information exists, the pattern returns. Visibility is a quantitative knob, not a binary switch.

A true single-photon source produces one photon at a time, evidenced by \(g^{(2)}(0)<0.5\) in a Hanbury Brown–Twiss test. In this regime, detections on the screen are governed by the probability density \(P(x)=|\psi_1(x)+\psi_2(x)|^2\). Over minutes or hours the dots accumulate into bright and dark fringes: particles arrive discretely, yet their probabilities interfere.

The fringe spacing on a distant screen is \(\Delta y \approx \lambda L/d\) for screen distance \(L\). Narrow slits (small \(a\)) broaden the envelope via the \(\mathrm{sinc}^2\) term. Finite coherence length, detector blur, and source bandwidth reduce the contrast.

Attach tiny “path tags” to the photon — polarization rotators at each slit, for example. If the tags are perfectly distinguishable, the cross-term \(2\operatorname{Re}(\psi_1^\*\psi_2)\) averages to zero and fringes vanish. Quantitatively, fringe visibility \(V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})\) and path distinguishability \(D\) obey \[ V^2 + D^2 \le 1 . \] You can trade interference for which-path knowledge, but you cannot have both at once.

Decoherence is the environment performing a relentless which-path measurement. Even imperceptible scattering can carry away phase information and reduce \(V\).

In a quantum eraser, you first imprint which-path markers (destroying fringes in the raw sum), then later measure in a basis that erases the path information. Conditioning on the eraser’s outcome recovers complementary fringes in sub-ensembles. Nothing travels back in time: you are sorting data by a later choice that correlates with path phase.

Wheeler’s delayed-choice variant emphasizes that the decision to observe interference or path information can be made after the photon enters the apparatus — the outcomes remain consistent with a single, unitary quantum description until measurement.

Alignment tolerance scales with wavelength. For \(\lambda=650\text{ nm}\), \(d=100\,\mu\text{m}\), and \(L=1\text{ m}\), fringe spacing is ~6.5 mm; slit parallelism must be better than a few arcminutes. Use narrowband sources or filters to boost coherence; suppress ambient light to keep dark counts low; stabilize air currents to prevent phase jitter.

With electrons, neutrons, even large molecules, the same logic holds: de Broglie wavelength replaces \(\lambda\), and environmental coupling sets how fast fringes wash out.