Quantum Eraser

The double-slit shows interference when no which-path record exists. If you imprint path markers — say, orthogonal polarizations at the two slits — the cross term in \(|\psi_1+\psi_2|^2\) averages away and fringes disappear. A quantum eraser measures those markers in a basis that makes the paths indistinguishable again. Interference reappears, not in the raw sum, but in conditioned sub-ensembles sorted by the eraser’s outcome.

Nothing travels back in time. The total (unconditioned) distribution still shows no fringes; only after sorting by the eraser result do complementary patterns emerge. This is careful bookkeeping with entangled degrees of freedom.

Let slit 1 attach \(|H\rangle\) and slit 2 attach \(|V\rangle\). The state at the screen is \[ \lvert\Psi\rangle=\psi_1(x)\lvert H\rangle + \psi_2(x)\lvert V\rangle . \] If you ignore polarization, \(P(x)=|\psi_1|^2+|\psi_2|^2\) — no cross term, no fringes. Insert a polarizer at \(45^\circ\), projecting onto \(|\pm\rangle=\frac{1}{\sqrt2}(|H\rangle\pm |V\rangle)\). Conditioned on the “+” outcome the wave at the screen is \(\propto \psi_1+\psi_2\) (bright fringes); conditioned on “−” it is \(\propto \psi_1-\psi_2\) (dark-shifted fringes). Adding both sub-ensembles cancels the interference, as it must.

You can choose the eraser setting after the photon has passed the slits — even after it was detected at the screen. The statistics remain consistent: the undecorrelated totals show no fringes; only when you later label each detection by the eraser outcome do fringes appear in the labeled piles. The order of spacelike-separated choices does not change local marginals, so there’s no signaling.

Use SPDC to create a signal photon that goes to the screen and an idler that undergoes a choice: detectors \(D_3,D_4\) reveal which-path (no fringes in coincidences); detectors \(D_1,D_2\) mix the paths (eraser), yielding complementary fringes when you build histograms of signal positions conditioned on those idler clicks. Summing all four coincidence sets destroys the interference again.

The signal alone never shows fringes; only coincidence-conditioned subsets reveal interference — the essence of the eraser.

Fringe visibility requires good mode overlap, narrow spectral bandwidth, and stable alignment. Polarization erasers need high extinction-ratio optics. For entangled setups, timing resolution and coincidence windows set the usable signal-to-noise. Remember: any uncontrolled coupling that carries away path info (stray scattering, birefringence drift) acts like a partial which-path tag and reduces visibility.