Q036 · Indistinguishable alternatives

Hong–Ou–Mandel Two-Photon Interference

Bring two single-photon modes to the same beam splitter. Predict what happens when they overlap, change their arrival delay or polarization, then collect individual detector clicks and build the interference dip from those records.

Ideal single-pair split probability $\text{—}$

Input-mode overlap $\text{—}$

FIRST, MAKE A PREDICTION

Does mode overlap reduce the split probability?

Two input modes, one beam splitter, two counters. The colored envelopes describe the inputs; the bars show possible output occupations.

Choose a prediction, then compare the cases and acquire events.

No events acquired. Input replay does not generate detector records.

CHANGE ONE THING

Compare the cases

Arrival delay

\tau

0\,\mathrm{ps}

Polarization mismatch

\theta

0\,\mathrm{deg}

Drag the orange delay handle. Keyboard sliders set the same controls.

Source, detectors and sampling

Spectral standard deviation

\sigma_\nu

0.25\,\mathrm{THz}

Center-frequency mismatch

\Delta\nu

0\,\mathrm{THz}

Beam-splitter transmission

T

0.5

Detection efficiency per photon

\eta

1

Electronics timing standard deviation

\sigma_d

100\,\mathrm{ps}

Coincidence half-window

w

500\,\mathrm{ps}

Background-click chance per gate

p_{\mathrm{bg}}

0

Two-pair admixture

p_2

0

Trials per acquisition / nominal setting

N

800

Acquisition seed

s

1987

Electronic jitter is much slower than the optical packet. The time gate selects recorded coincidences; narrowing it does not improve optical indistinguishability. Two-pair input is a declared four-photon Fock admixture.

Pure Gaussian input modes and a lossless beam splitter. Output bars are ideal photon-number probabilities, not tracked paths. Threshold detection, independent loss, electronic timing, background and a declared two-pair admixture act only in the recorded data. This is a simulation, not a source certification.

Evidence from your detector records

SIMULATED ACQUISITION

Build the dip from clicks

Acquire events to begin.

Gray: ideal single-pair reference before loss or electronic gating. Orange: coincidence count per emitted trial, with 95% Wilson intervals. The references and acquired points need not coincide.

Count-only Gaussian fit

\text{—}

Visibility / baseline from counts

\text{—}

Inference requires a complete scan or all four settings.

Delay bins recomputed from raw clicks
Delay (ps)	Trials	Coincidences	Fraction

Start with the essentials

Focus question: Can a narrow electronic time gate make orthogonally polarized photons interfere?
One-sentence intuition: Change delay and polarization separately. Then compare ideal occupation probabilities with coincidences derived from the recorded click times. Optical overlap and electronic event selection have different roles.

Core mathematical model

Normalized pure spectral modes

f_j(\nu)=(2\pi\sigma_\nu^2)^{-1/4}e^{-(\nu-\nu_j)^2/(4\sigma_\nu^2)}

Both inputs have equal Gaussian intensity standard deviation in THz. Delay is in ps, so their product is dimensionless. The envelope view is a schematic mode display.

Mode overlap

M=|\langle f_1|f_2\rangle|^2\cos^2\theta=e^{-(2\pi\sigma_\nu\tau)^2-\Delta\nu^2/(4\sigma_\nu^2)}\cos^2\theta

Relative delay, spectral detuning and polarization all distinguish the inputs. This formula assumes pure, equal-width modes; mixed spectral states require a density-operator overlap.

Single-pair occupations

P_{11}=T^2+R^2-2TRM,\quad P_{20}=P_{02}=TR(1+M),\quad R=1-T

These are probabilities before detectors and sum to unity. The real unitary splitter convention fixes the relative reflection phase. It does not subtract classical intensities.

Recorded coincidence

C_i=\mathbf1(\text{both click})\,\mathbf1(|t_{A,i}-t_{B,i}|\le w),\quad \widehat p=\frac{\sum_i C_i}{N}

Every emitted trial is retained, including no-click trials. Detection is thresholded; electronics jitter dominates the optical arrival width. Background is an independent Bernoulli click per gate. The table and CSV use exactly this rule.

Conditional count fit

p(\tau)=B[1-Ve^{-(2\pi\widehat\sigma_\nu\tau)^2}]

A grid-assisted count likelihood selects the width after baseline and dip amplitude regression. No model bandwidth enters the fit. The fit is disabled with background or two-pair contamination; weak or boundary fits are unresolved. No fit confidence interval is claimed.

Common difficulties

Bunching is not a hidden path

Typical misconceptionTwo drawn particles must choose the same route before the measurement.

Better mental modelThe scene displays input modes and output occupations. No hidden path is stored as an observed record.

Detector timing is not optical overlap

Typical misconceptionA narrower gate creates indistinguishable photons.

Better mental modelThe gate only filters existing click timestamps. It cannot restore amplitudes removed by orthogonal polarization or spectral mismatch.

A dip alone is not source certification

Typical misconceptionDip depth directly equals purity or timing resolution for any source.

Better mental modelThe model specifies pure modes, independent losses and a limited four-photon admixture. Mixed spectra, correlated SPDC pairs, dead time and device calibration require additional models.

Run the experiment

01
Predict before counting
Choose Arrive together, predict the output, and acquire one trial repeatedly.
What to observe: Only one threshold counter can click in the ideal balanced case. A single click does not measure how many photons arrived at that port.
02
Distinguish the modes
Choose Arrive apart, then Orthogonal polarization. Compare a batch in each case.
What to observe: Coincidences reappear. The inputs need matching polarization as well as temporal and spectral overlap.
03
Build a dip from events
Start over and acquire a delay scan. Inspect the bins and export the raw CSV.
What to observe: Orange points are reconstructed from those events. Wilson ranges describe finite binary counts; the gray line is a separate ideal reference.
04
Separate apparatus effects
Open source and detector settings. Change efficiency, gate width, background or the two-pair fraction and reacquire.
What to observe: Loss and a narrow gate reduce recorded counts. Background and four-photon input can fill the dip. The conditional single-pair inference is disabled when its assumptions fail.