Q036 · Two photons meet at a beam splitter

Hong–Ou–Mandel Two-Photon Interference

When two photons enter a beam splitter from opposite sides, why do the two counters stop clicking together? Two photons in the same pure mode leave a balanced beam splitter together because the alternatives cancel. Identical mixed preparations still contain distinguishable modes, so coincidences remain. Delay and polarization mismatch can also weaken the dip.

Send two single photons into the same beam splitter and predict what happens when they overlap. Change their arrival delay or polarization, then collect detector clicks one by one and build the interference dip from them.

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

Density-operator overlap $\text{—}$

Internal-state purity · each source $\text{—}$

Make a prediction

First, make a prediction

Does mode overlap reduce the split probability?

Two input modes meet at one beam splitter, and two counters record the outputs. The colored envelopes describe the inputs; the detector row below shows the possible output photon numbers.

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

No events acquired yet. Replaying the inputs does not create detector records.

Change one condition at a time

Compare the cases

Arrival delay

\tau

0\,\mathrm{ps}

Polarization mismatch

\theta

0\,\mathrm{deg}

Drag the orange delay handle, or use the sliders, which also work from the keyboard.

Source, detectors and sampling

Joint source phase

\varphi

0\,\mathrm{deg}

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

The electronic timing jitter is much slower than the optical wave packet. The time gate only selects which recorded coincidences are kept; narrowing it does not make the photons more indistinguishable. The joint source phase changes an independent two-mode internal state in both inputs. This factorized model keeps the Gaussian arrival envelope separate. Mixed internal states use single-pair input; increasing the two-pair fraction restores pure internal states.

Model assumptions

Each input has a Gaussian arrival envelope and a separate two-mode internal state. Their independence is assumed. The joint source phase changes internal purity in both independent inputs; the beam splitter is lossless. Ideal output probabilities and synthetic click records appear separately. Mixed internal states use single-pair input; the optional four-photon admixture restores pure internal states. Threshold detectors include independent loss, electronic timing and background clicks. The Lab is a simulation and does not certify a real photon source.

Evidence from your detector records

Simulated acquisition

Build the dip from clicks

Acquire events to begin.

Both in A

0.25

One in each port

0.25

Both in B

0.25

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.

Expand the fit and raw trials to inspect the numerical evidence.

Fit and interpretation

Count-only Gaussian fit

\text{—}

Visibility / baseline from counts

\text{—}

Complete the scan, or acquire all four settings, to see an inference.

Inspect counts and individual trials

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

Physics tutorial

When two alternatives cancel

BackgroundOne photon enters each input port. There are two ways to end up with one photon in each output: both photons reflect, or both pass through. When the splitter is balanced and the input modes are identical, the amplitudes for these two ways cancel.

Why it mattersThe coincidence dip tests how indistinguishable the two input modes are. Detector loss, accidental clicks and extra photon pairs can change the recorded dip, but they do not change the ideal interference law for a single pair.

Start with the essentials

Focus question: Can a narrow electronic time gate make orthogonally polarized photons interfere?
One-sentence intuition: Optical overlap and electronic event selection do different jobs. Change delay and polarization one at a time, then compare the ideal output probabilities with the coincidences counted from the recorded click times.

Core mathematical model

Internal mixture and HOM visibility

M_{\mathrm{int}}=\operatorname{Tr}(\rho^2),\qquad P_{11}(0)=\tfrac12[1-\operatorname{Tr}(\rho^2)]

The two independent inputs have identical internal density operators. At a balanced splitter and matched envelope and polarization, a maximally mixed two-mode input has a coincidence floor of one quarter.

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 Gaussian spectra with the same intensity standard deviation, in THz. Delay is in ps, so their product is dimensionless. The envelope view is only a sketch of the modes.

Mode overlap

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

A relative delay, a frequency offset or a polarization difference each makes the inputs distinguishable. The Gaussian envelopes have equal width. A separate internal two-mode state multiplies this overlap by its density-operator purity; these internal labels factor from the continuous delay.

Output probabilities for one photon pair

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

These are probabilities before detection, and they add up to one. The splitter is described by a real unitary matrix, which fixes the relative phase on reflection. The calculation does not subtract classical intensities.

How a coincidence is counted

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 kept, including trials with no click. Each detector either clicks or stays silent and cannot count photons, and the electronic jitter is much wider than the optical pulse. Background adds an independent chance of a click in each gate. The table and the CSV use exactly this rule.

Fitting the dip width from counts

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

For each width on a grid, the baseline and dip depth are fitted first, and the count likelihood then picks the width. The model bandwidth is not used. The fit is switched off when background or two-pair events are present, and weak fits or fits at the grid edge are reported as unresolved. No confidence interval is given for the fit.

Common difficulties

Bunching does not mean the photons pick a path in advance

Typical misconceptionThe two particles in the picture must pick the same route before they are measured.

Better mental modelThe scene shows input modes and output photon numbers. No hidden path is ever stored as if it had been observed.

A time gate cannot create optical overlap

Typical misconceptionNarrowing the gate makes the photons indistinguishable.

Better mental modelThe gate only filters click times that already exist. It cannot bring back amplitudes removed by orthogonal polarization or a spectral mismatch.

A dip alone cannot certify a source

Typical misconceptionFor any source, the depth of the dip directly gives its purity or its timing resolution.

Better mental modelFor identical independent inputs with all other degrees of freedom matched, the ideal balanced HOM visibility equals internal-state purity. Mismatch, background and extra pairs also lower the contrast. Correlated pair preparation and calibrated device response require a source model.

Run the experiment

01
Predict before counting
Choose Arrive together, predict the output, then acquire one trial at a time, several times.
What to observe: In the ideal balanced case, only one counter clicks. That click cannot tell you how many photons arrived at its port.
02
Make the photons distinguishable
Choose Arrive apart, then Orthogonal polarization, and compare a batch from each case.
What to observe: Coincidences come back. The photons must overlap in time and spectrum, and they must also share the same polarization.
03
Build the dip from clicks
Start over and acquire a delay scan. Inspect the bins and export the raw CSV.
What to observe: The orange points are computed from those clicks. The Wilson ranges show the uncertainty of a finite number of yes-or-no counts, and the gray line is a separate ideal reference.
04
Separate the effects of the apparatus
Open the source and detector settings. Change the efficiency, gate width, background or two-pair fraction, then acquire again.
What to observe: Loss and a narrow gate lower the recorded counts. Background and four-photon input can fill in the dip. When the single-pair assumptions fail, the fit is switched off.

Hong–Ou–Mandel Two-Photon Interference

Build the dip from clicks

When two alternatives cancel

Core mathematical model

Internal mixture and HOM visibility

Normalized pure spectral modes

Mode overlap

Output probabilities for one photon pair

How a coincidence is counted

Fitting the dip width from counts

Common difficulties

Bunching does not mean the photons pick a path in advance

A time gate cannot create optical overlap

A dip alone cannot certify a source

Run the experiment

Predict before counting

Make the photons distinguishable

Build the dip from clicks

Separate the effects of the apparatus

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