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Sandbox Physics

Q005 · One record / two conditional fringes

Which Path & Quantum Eraser

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

Interactive modelWhich Path & Quantum Eraser
Ideal all-record fringe visibility—\text{—}
Marker-state overlap—\text{—}
Evidence from the same raw record

SIMULATED DETECTOR EVENTS

Two groups, one unchanged sum

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{—}

No inference until events are acquired.

Joint bins recomputed from clicks
Phase (deg)Emitted / detectedA first / A other / B first / B other
Latest 12 raw trials; CSV contains every trial
TrialPhase (deg)OutputTag

Physics tutorial

What changes when we sort a record?

BackgroundThe two arms carry different polarization states. Recombining their spatial modes does not automatically remove the distinction in polarization. The same photon is finally recorded by one of four output counters.

Why it mattersA conditional fringe can be present while the summed output looks flat. Retaining both tag groups prevents the selected fringe from being mistaken for a change in the whole ensemble.

Start with the essentials

Focus question
Can revealing stored tags later change the output counts already recorded?
One-sentence intuition
Analyze both output ports in the same polarization basis. A tag basis that mixes the path labels gives complementary fringes. Recombining the raw counts reproduces the unchanged marginal.

Core mathematical model

State before recombination

∣Ψ⟩=∣A⟩∣mA⟩+eiϕ∣B⟩∣mB⟩2|\Psi\rangle=\frac{|A\rangle|m_A\rangle+e^{i\phi}|B\rangle|m_B\rangle}{\sqrt2}

Arms have equal amplitudes. Source and mirror phases are absorbed into the controlled arm phase. The balanced real Hadamard recombiner defines the named output ports.

Symmetric polarization markers

∣mA⟩=cos⁡(θ/2)∣H⟩+sin⁡(θ/2)∣V⟩,∣mB⟩=cos⁡(θ/2)∣H⟩−sin⁡(θ/2)∣V⟩|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 is a separation in polarization space, not the mechanical orientation of a plate in the optical plan. Orthogonal markers remove unconditional interference even if the tags are hidden on screen.

Analyzer basis

∣b0⟩=cos⁡β∣H⟩+sin⁡β∣V⟩,∣b1⟩=−sin⁡β∣H⟩+cos⁡β∣V⟩|b_0\rangle=\cos\beta|H\rangle+\sin\beta|V\rangle,\quad |b_1\rangle=-\sin\beta|H\rangle+\cos\beta|V\rangle

An ideal polarization rotation and polarizing splitter after each spatial output separate both basis states. All four detectors share the same efficiency.

Joint detector probabilities

P(s,q)=aq2+bq2+2svaqbqcos⁡ϕ4,aq=⟨bq∣mA⟩,bq=⟨bq∣mB⟩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 selects output A or B. Remaining path coherence multiplies the off-diagonal density-matrix elements. Each trial yields at most one of the four detector clicks; losses give a blank record.

Unchanged summed marginal

P(A)=1+vcos⁡θcos⁡ϕ2=∑q=01P(q)P(A∣q)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 conditional groups while leaving the total output probability invariant. Orthogonal marking makes this marginal flat.

Recombine actual counts

P^(A)=NA0+NA1Ndet,Ndet=NA0+NA1+NB0+NB1\widehat P(A)=\frac{N_{A0}+N_{A1}}{N_{\mathrm{det}}},\quad N_{\mathrm{det}}=N_{A0}+N_{A1}+N_{B0}+N_{B1}

The conditional denominators are the clicks in each tag group. Weighting by those group sizes exactly restores the all-click fraction. Pointwise Wilson ranges describe finite binary counts, not a simultaneous visibility-fit confidence region.

Common difficulties

A fringe in a subset

Typical misconceptionA selected fringe proves all earlier detection counts were changed.

Better mental modelBoth tag groups remain in the same joint record. Their complementary modulation cancels in the correct weighted marginal.

What is entangled here?

Typical misconceptionThe diagram sends two photons to separated observers.

Better mental modelOne photon carries a path degree of freedom and a polarization degree of freedom. The four counters resolve their joint output. This lab does not simulate a remote entangled-pair source.

No unique quantum witness

Typical misconceptionThis optical intensity effect alone proves nonclassical photon statistics.

Better mental modelClassical coherent polarization fields obey the same interference law. The single-photon event interpretation requires the declared source assumption.

Run the experiment

  1. 01

    Predict what is restored

    Choose Erase the distinction, make a prediction and acquire a phase scan.

    What to observe: The two tagged groups oscillate oppositely. Their weighted sum remains flat, within count fluctuations.
  2. 02

    Read or erase the path distinction

    Compare Read the path tag with Erase the distinction; reacquire after each physical change.

    What to observe: The analyzer sorts different groups. Hiding a readable tag does not make the polarization states identical.
  3. 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 partition appears without generating a new event. This delayed review is not a physical delayed-choice twin-photon experiment.
  4. 04

    Separate marking from dephasing

    Try partial marking, then lose path coherence.

    What to observe: Partially overlapping markers leave some total interference. Lost path coherence cannot be repaired just by selecting another analyzer basis.