Q005 · Two fringe patterns inside one record

Which Path & Quantum Eraser

Can sorting the hits by their which-path tags bring the interference fringes back? Yes, if you read the tags in a basis that mixes the two path labels. Fringes then return in each group, and the two sets are offset by half a fringe. Added together they fill each other in and give the same fringe-free total as before. The interference was never in the total; it appears only when the hits are sorted by tag result.

Tag the two arms of a single-photon interferometer with polarization. Turn the analyzer, record all four output counters, and sort the same events into two complementary groups. See what changes within each group and what stays the same in their sum.

Ideal all-record fringe visibility $\text{—}$

Marker-state overlap $\text{—}$

First, make a prediction

Does sorting restore fringes to the whole record?

A single photon passes an interferometer with two arms, two polarization tags and four counters. Compare all events with the two groups sorted from that same record.

Choose a prediction, then acquire and inspect the record.

No events yet. Replay only illustrates the modes and adds no records.

Change one condition at a time

Compare the cases

Arm phase

\phi

0^{\circ}

Tag separation angle

\theta

90^{\circ}

Analyzer basis angle

\beta

0^{\circ}

Drag the orange analyzer axis; the slider gives keyboard access.

Reveal polarization tags

Hide the tags before acquisition, then reveal them later. Only the display changes; no events are resampled or edited. This is delayed review, not a delayed-choice measurement.

Preparation, coherence and detector settings

Remaining path coherence

v

1

Four-port detection efficiency

\eta

1

Trials per phase

N

600

Sampling seed

\mathrm{seed}

1982

The interferometer is balanced in path and polarization, and every detector loses photons at the same rate. Both tags belong to one photon; this is not a delayed-choice experiment with two separated photons. Showing the tags later only delays when you look at a record that already exists. Classical coherent optics can produce the same intensity law.

What the same clicks show

Simulated detector events

Sorting into two groups leaves the total unchanged

Acquire events to begin.

White points: output A among all clicks. Orange and blue: output A conditional on each revealed tag, with pointwise 95% Wilson intervals. Gray: ideal all-click reference. Weight the groups by their sample sizes when recombining.

Detected / emitted

\text{—}

First tag / other tag

\text{—}

Conditional fractions at current phase

\text{—}

Nothing to estimate until events are acquired.

Joint bins recomputed from clicks
Phase (deg)	Emitted / detected	A first / A other / B first / B other

Latest 12 raw trials; CSV contains every trial
Trial	Phase (deg)	Output	Tag

Start with the essentials

Focus question: If you reveal the stored tags later, can that change counts that were already recorded?
One-sentence intuition: Use the same polarization basis at both output ports. A basis that mixes the path tags splits the events into complementary fringes, and adding the raw counts back together returns the unchanged total.

Core mathematical model

State before recombination

|\Psi\rangle=\frac{|A\rangle|m_A\rangle+e^{i\phi}|B\rangle|m_B\rangle}{\sqrt2}

Both arms have equal amplitude. Phases from the source and mirrors are folded into the adjustable arm phase. A balanced, real Hadamard recombiner defines which output is A and which is B.

Symmetric polarization markers

|m_A\rangle=\cos(\theta/2)|H\rangle+\sin(\theta/2)|V\rangle,\quad |m_B\rangle=\cos(\theta/2)|H\rangle-\sin(\theta/2)|V\rangle

The marker angle measures how far apart the two polarizations are, not how a plate is turned in the optical layout. Orthogonal markers remove the overall interference even if the tags are hidden on screen.

Analyzer basis

|b_0\rangle=\cos\beta|H\rangle+\sin\beta|V\rangle,\quad |b_1\rangle=-\sin\beta|H\rangle+\cos\beta|V\rangle

After each spatial output, an ideal polarization rotation and a polarizing splitter separate the two basis states. All four detectors have the same efficiency.

Joint detector probabilities

P(s,q)=\frac{a_q^2+b_q^2+2sv a_qb_q\cos\phi}{4},\quad a_q=\langle b_q|m_A\rangle,\quad b_q=\langle b_q|m_B\rangle

The sign picks output A or B. The remaining path coherence multiplies the off-diagonal elements of the density matrix. Each trial gives at most one click among the four detectors, and a lost photon leaves a blank entry.

The total output probability stays the same

P(A)=\frac{1+v\cos\theta\cos\phi}{2}=\sum_{q=0}^{1}P(q)P(A\mid q)

Changing the analyzer basis changes the two groups but leaves the total output probability unchanged. With orthogonal markers this total is flat.

Adding the actual counts back together

\widehat P(A)=\frac{N_{A0}+N_{A1}}{N_{\mathrm{det}}},\quad N_{\mathrm{det}}=N_{A0}+N_{A1}+N_{B0}+N_{B1}

Each group's fraction is divided by the clicks in that group. Weighting by group size gives back exactly the fraction over all clicks. The Wilson ranges describe finite counts at each point separately; they are not a joint confidence region for a fitted visibility.

Common difficulties

Fringes in one group of events

Typical misconceptionFringes in a selected group prove that all the earlier counts were changed.

Better mental modelBoth tag groups stay in the same joint record. Their opposite fringes cancel when you add them with the correct weights.

What is entangled here?

Typical misconceptionThe diagram sends two photons to observers far apart.

Better mental modelA single photon carries both a path and a polarization degree of freedom, and the four counters read out the two together. This Lab does not simulate a source that sends entangled pairs far apart.

This effect alone does not prove quantum light

Typical misconceptionThis optical intensity effect alone proves nonclassical photon statistics.

Better mental modelClassical coherent polarized light obeys the same interference law. Reading the counts as single-photon events relies on the stated assumption about the source.

Run the experiment

01
Predict what comes back
Choose Erase the distinction, make a prediction and acquire a phase scan.
What to observe: The two tag groups oscillate in opposite directions. Their weighted sum stays flat, apart from count fluctuations.
02
Read or erase the path distinction
Compare Read the path tag with Erase the distinction. Acquire again after each physical change.
What to observe: The analyzer sorts the events into different groups. Hiding a tag that could be read does not make the two polarization states the same.
03
Review the same events later
Hide the tags, acquire, then reveal them. Check the trial count and export the raw CSV.
What to observe: The two groups appear without any new event being created. Looking at the record later is not a delayed-choice experiment with twin photons.
04
Separate marking from dephasing
Try partial marking, then reduce the path coherence.
What to observe: Markers that partly overlap leave some interference in the total. Once path coherence is lost, choosing a different analyzer basis cannot bring it back.

Which Path & Quantum Eraser

Sorting into two groups leaves the total unchanged

What changes when we sort a record?