Two-Photon Interference and the Hong–Ou–Mandel Effect

Two indistinguishable photons meet at a half-silvered mirror, and something with no classical shadow happens: they refuse to come out separately. This article takes the machinery the chapter has already built — the quantized field with its creation operators $\hat a^\dagger$ , the Fock states that carry exactly one photon, and the bosonic algebra $[\hat a,\hat a^\dagger]=1$ that lets photons share a mode — and turns it on the simplest two-photon experiment imaginable: one photon into each input port of a beam splitter, two detectors watching the outputs. Classically each photon independently transmits or reflects, so a quarter of the time one goes each way and the detectors click together. Quantum mechanically the two ways of producing that coincidence — both photons reflected, both transmitted — are indistinguishable amplitudes, and the unitarity of a lossless beam splitter forces them equal and opposite. They cancel; the coincidence rate drops to zero, and the photons always leave through the same port. This is not one photon interfering with itself, and it has no first-order coherence analog — the individual output intensities show no fringes whatever. It is interference between two-particle probability amplitudes, and it survives even when the photons come from independent sources, provided they are made spectrally, temporally and polarization-identical. The signature is a dip in coincidences versus the delay $\tau$ : zero at perfect overlap, rising to a flat baseline once the photons distinguish. The depth measures indistinguishability, the width the wavepacket duration, and the effect underwrites Bell-state measurement, boson sampling, and every scalable scheme for photonic quantum information.

Key takeaways

One photon enters each port of a balanced 50:50 beam splitter; if the photons are indistinguishable and arrive together, the probability of finding one in each output port drops to zero — they coalesce and exit the same port.
The effect is destructive interference between two two-photon amplitudes, "both reflected" and "both transmitted," opposite in sign because a lossless splitter imposes a $\pi/2$ reflection phase.
One counts coincidences versus relative delay $\tau$ ; the rate traces the Hong–Ou–Mandel dip, zero at $\tau=0$ and rising to a flat baseline for large $\tau$ .
The dip width is the photon coherence time, so HOM measures temporal overlap to femtosecond precision, far below any detector jitter; the depth measures indistinguishability.
Attenuated laser light caps the visibility at $1/2$ ; a dip deeper than half is an unforgeable certificate of single-photon character.
Key idea: for a balanced splitter the coincidence amplitude is a both-transmitted term $t^2=+\tfrac12$ plus a both-reflected term $r^2=(i/\sqrt2)^2=-\tfrac12$ ; equal and opposite, they vanish exactly when the photons are indistinguishable.

The core insight

For a balanced beam splitter the coincidence amplitude is a both-transmitted term ( $t^2=+\tfrac12$ ) plus a both-reflected term ( $r^2=-\tfrac12$ , from two factors of the $i$ reflection phase); they cancel exactly, so coincidences disappear whenever the two photons are truly indistinguishable.

Where this lives today

Two-photon interference is now the acceptance test for quantum-light hardware. Every serious single-photon source — a quantum dot in a micropillar cavity, a trapped ion, a nitrogen-vacancy centre, a heralded down-conversion pair — is benchmarked by interfering two successive emissions and quoting the HOM visibility; the best quantum dots exceed $0.95$ , below which photonic error correction fails. On the same principle, a beam splitter watched for coincidences is a partial Bell-state analyzer: only the antisymmetric singlet $|\psi^-\rangle$ produces a coincidence while the symmetric states bunch, and that projection drives quantum teleportation, entanglement swapping in repeaters, and the fusion gates of photonic computing. The most spectacular scaling is boson sampling. Push many indistinguishable photons through a large multiport interferometer and the output statistics are governed by permanents of the scattering matrix — believed classically intractable — with HOM the elementary two-mode block repeated thousands of times. The Chinese Jiuzhang processors pushed $50$ to $76$ detected photons through hundred-mode interferometers to claim a quantum computational advantage, each event a HOM dip in disguise. What Mandel measured with two photons on a lamp-lit table now certifies the largest photonic quantum machines built.

The physical picture

Forget waves for a moment and count histories. A coincidence — one photon at $c$ , the other at $d$ — arises two ways: both photons transmitted ( $a\!\to\!c$ , $b\!\to\!d$ ), or both reflected ( $a\!\to\!d$ , $b\!\to\!c$ ). If the photons are identical, no measurement tells these apart: the outcome "one in $c$ , one in $d$ " is reached by two indistinguishable paths, and quantum mechanics says we add their amplitudes before squaring. Unitarity forces reflection to lead transmission by a quarter cycle, so reflecting twice stamps the both-reflected amplitude with $i^2=-1$ . For a 50:50 splitter the magnitudes are equal, the signs opposite, and the coincidence amplitude is annihilated.

Figure

A balanced beam splitter takes one photon into port a and one into port b. A coincidence can happen two ways — both photons transmitted, amplitude t² = +1/2, or both reflected, amplitude r² = (i/√2)² = −1/2 — equal and opposite for a 50:50 splitter, so they cancel and the photons always leave together.

A second, equivalent view appeals to nothing but symmetry. Photons are bosons, so the two-photon output state must be symmetric under exchange. For a balanced splitter the only exchange-symmetric combinations downstream are both in $c$ and both in $d$ ; the one-in-each combination is antisymmetric and simply not produced — given the chance, bosons bunch. The delay $\tau$ switches the effect off: if the photons arrive at distinguishably different times, an experimenter could timestamp the two histories apart, and that which-path information is what interference cannot survive. Because the dip tracks the wavepacket overlap, its shape is the Fourier transform of the spectrum: a narrow spectrum means a long coherence time and a broad dip.

Figure

The Hong–Ou–Mandel dip: coincidence probability P_c versus delay τ for single photons at 810 nm through a 3 nm filter (rms bandwidth Δω ≈ 3.7×10¹² rad/s). At τ = 0 the photons are indistinguishable and P_c falls to 0; once τ exceeds the coherence time P_c rises to the classical baseline 1/2. The full width at half depth is about 455 fs.

The core equations

A beam splitter is a linear, energy-conserving four-port, fixed by how it maps the input creation operators onto the output modes — all of HOM follows by pushing a state through this map. Write input modes $a,b$ , output modes $c,d$ , amplitudes $t$ (transmission) and $r$ (reflection).

\hat{a}_c^{\dagger} = t\,\hat{a}_a^{\dagger} + r\,\hat{a}_b^{\dagger}, \qquad \hat{a}_d^{\dagger} = r\,\hat{a}_a^{\dagger} + t\,\hat{a}_b^{\dagger} \qquad (1)

where: $\hat{a}_a^{\dagger},\hat{a}_b^{\dagger}$ — input-mode creation operators; $\hat{a}_c^{\dagger},\hat{a}_d^{\dagger}$ — output-mode creation operators; $t,r$ — complex transmission and reflection amplitudes with $T=|t|^2$ , $R=|r|^2$ . Each output is a coherent superposition of the two inputs; the whole quantum content of the device is the pair $(t,r)$ . Energy conservation makes the transformation unitary, and that pins down the magnitudes and, crucially, the relative phase of $t$ and $r$ — not a convention but the constraint that drives the interference.

|t|^2 + |r|^2 = 1, \qquad t^{*}r + r^{*}t = 0 \qquad (2)

where: the first relation is conservation of probability ( $T+R=1$ ); the second, $\mathrm{Re}(t^{*}r)=0$ , forces $t$ and $r$ to differ in phase by $\pi/2$ . Taking $t$ real and $r=it$ , reflection carries a factor $i$ — a quarter-cycle lead that becomes a sign when it happens twice. The experiment prepares one photon in each input port — in Fock language, one creation operator on each input mode of the two-mode vacuum.

|1_a,1_b\rangle = \hat{a}_a^{\dagger}\hat{a}_b^{\dagger}\,|0\rangle \qquad (3)

where: $|0\rangle$ — two-mode vacuum; $|1_a,1_b\rangle$ — one photon in $a$ , one in $b$ . This definite-number state has no classical analog: exactly the "one per port" condition a laser cannot supply. Allowing the photons to be only partially indistinguishable — overlap $\gamma=\langle\psi_1|\psi_2\rangle$ of their modes — the coincidence probability takes a compact form for any splitter ratio.

P_c = R^2 + T^2 - 2RT\,|\gamma|^2 \qquad (4)

where: $P_c$ — probability of one photon in each output; $R,T$ — reflectivity and transmissivity; $|\gamma|^2$ — mode overlap, the indistinguishability. The first two terms are the classical baseline; the last is the interference, biting only as far as the photons overlap. For $R=T=\tfrac12$ this collapses to $P_c=\tfrac12(1-|\gamma|^2)$ . Scanning the delay $\tau$ turns $|\gamma|^2$ into a function of time set by the spectrum. For identical photons with a Gaussian power spectrum of rms bandwidth $\Delta\omega$ , the overlap is Gaussian in $\tau$ and the rate traces the dip.

P_c(\tau) = \tfrac{1}{2}\left(1 - e^{-\Delta\omega^{2}\,\tau^{2}}\right) \qquad (5)

where: $\tau$ — relative delay; $\Delta\omega$ — rms bandwidth of each photon's power spectrum. (This is the measured spectral rms; defining $\Delta\omega$ from the field amplitude instead adds a factor $\tfrac12$ in the exponent — a convention trap.) The dip runs from $0$ at $\tau=0$ to the baseline $\tfrac12$ , with $1/e$ half-width $1/\Delta\omega$ , of order the coherence time. Finally, the standard figure of merit is the visibility — how far the rate falls from baseline to floor, the number every source datasheet quotes.

V = \frac{P_c(\infty) - P_c(0)}{P_c(\infty)} \qquad (6)

where: $V$ — visibility, $0\le V\le 1$ ; $P_c(0),P_c(\infty)$ — rates at zero and large delay. For a balanced splitter $V=|\gamma|^2$ is the indistinguishability; $V=1$ is a perfect dip, $V=\tfrac12$ the ceiling for classical light.

Derivation AVanishing coincidences for a balanced beam splitter

The goal is to push the one-photon-per-port input state through the beam-splitter transformation and watch the coincidence term destroy itself for a balanced splitter. In the symmetric convention $t=1/\sqrt2$ , $r=i/\sqrt2$ , we invert the mode relations, expand the product, and read off the coefficient of the one-in-each-output term. That coefficient is $t^2+r^2$ , and the whole effect lives in the fact that it is zero.

Step 1. Invert equation (1) so the input operators act on the output modes: $\hat a_a^\dagger = t\,\hat a_c^\dagger + r\,\hat a_d^\dagger$ and $\hat a_b^\dagger = r\,\hat a_c^\dagger + t\,\hat a_d^\dagger$ . Why: the detectors sit in modes $c,d$ , so we express the prepared state in the output operators before it hits the vacuum.

Step 2. Form the product that builds the input state:

\hat a_a^\dagger\hat a_b^\dagger = tr\,\hat a_c^{\dagger 2} + (t^2+r^2)\,\hat a_c^\dagger \hat a_d^\dagger + rt\,\hat a_d^{\dagger 2}.

Why: the cross term $\hat a_c^\dagger\hat a_d^\dagger$ is the coincidence — one photon in $c$ , one in $d$ ; the other terms put both photons in the same port.

Step 3. Insert the balanced amplitudes into the coincidence coefficient:

t^2+r^2 = \left(\tfrac{1}{\sqrt2}\right)^2 + \left(\tfrac{i}{\sqrt2}\right)^2 = \tfrac12 - \tfrac12 = 0.

Why: $t^2=+\tfrac12$ and $r^2=-\tfrac12$ are equal and opposite — the reflection phase $i$ , squared, supplies the minus sign, and the coincidence pathway interferes itself to death.

Step 4. Collect the survivors ( $tr=i/2$ ) and act on vacuum with $\hat a^{\dagger 2}|0\rangle=\sqrt2\,|2\rangle$ :

|1_a,1_b\rangle \;\longrightarrow\; \tfrac{i}{2}\left(\hat a_c^{\dagger 2}+\hat a_d^{\dagger 2}\right)|0\rangle = \tfrac{i}{\sqrt2}\left(|2_c,0_d\rangle + |0_c,2_d\rangle\right).

Why: only same-port terms remain, so the photons always leave together — both to $c$ or both to $d$ — and $P_c=0$ . The state is normalized, $|i/\sqrt2|^2+|i/\sqrt2|^2=1$ .

Go deeper · Derivation B · 1 minGeneral coincidence probability with partial overlap

Derivation BGeneral coincidence probability with partial overlap

The cancellation above assumed a perfect 50:50 splitter and identical photons; real photons overlap only partially and real splitters are never exactly balanced. The goal is the general coincidence probability $P_c=R^2+T^2-2RT|\gamma|^2$ interpolating between the ideal dip and the classical baseline. We label the input modes by states $|\psi_1\rangle,|\psi_2\rangle$ with overlap $\gamma=\langle\psi_1|\psi_2\rangle$ , track the both-transmitted and both-reflected amplitudes, and let the overlap weight their interference.

Step 1. The two coincidence amplitudes: "both transmitted" ( $\propto t^2$ ) leaves each photon in its original mode; "both reflected" ( $\propto r^2$ ) swaps the photons between output ports and modes. Why: these are the only two routes to a coincidence, but they deposit the wavepackets differently — which is what lets distinguishability matter.

Step 2. Square the summed amplitude, keeping the interference cross term:

P_c = |t|^4 + |r|^4 - \left(t^{*2}r^2 + r^{*2}t^2\right)|\gamma|^2.

Why: $|t|^4$ and $|r|^4$ are the incoherent (classical) contributions; the cross term is the interference, weighted by $|\gamma|^2$ because only the overlapping part of the wavepackets interferes.

Step 3. Evaluate the phase factor with $r=it$ ( $t$ real):

t^{*2}r^2 + r^{*2}t^2 = 2\,\mathrm{Re}(t^{*2}r^2) = 2RT.

Why: the $\pi/2$ reflection phase turns the cross term into a clean $-2RT|\gamma|^2$ , giving $P_c=R^2+T^2-2RT|\gamma|^2$ .

Step 4. The limits: distinguishable photons ( $\gamma=0$ ) give $P_c=R^2+T^2$ (baseline $\tfrac12$ when balanced); identical photons ( $\gamma=1$ ) give $P_c=(R-T)^2$ , zero only when $R=T=\tfrac12$ . Why: both indistinguishability and balance are necessary — either failure leaves coincidences behind.

Go deeper · Derivation C · 1 minThe dip as the Fourier transform of the spectrum

Derivation CThe dip as the Fourier transform of the spectrum

The delay $\tau$ enters only through the overlap $\gamma$ , so the dip's shape is entirely a statement about the spectrum. The goal is the Gaussian dip $P_c(\tau)=\tfrac12(1-e^{-\Delta\omega^2\tau^2})$ and the general rule behind it: the dip is the Fourier transform of the power spectrum, so its width is the coherence time. We give one photon amplitude $\phi(\omega)$ , delay the other by $\tau$ (multiplying its amplitude by $e^{i\omega\tau}$ ), then compute the overlap.

Step 1. Write the delayed overlap as a Fourier integral of the power spectrum $S(\omega)=|\phi(\omega)|^2$ :

\gamma(\tau) = \int \mathrm d\omega\; \phi^{*}(\omega)\,\phi(\omega)\,e^{i\omega\tau} = \int \mathrm d\omega\; S(\omega)\,e^{i\omega\tau}.

Why: the overlap of two identical wavepackets displaced by $\tau$ is the characteristic function of their power spectrum — where "how much do the photons overlap" becomes "what is the spectrum's Fourier transform."

Step 2. For a Gaussian power spectrum centred at $\omega_0$ with rms width $\Delta\omega$ , the transform is Gaussian:

\gamma(\tau) = e^{i\omega_0\tau}\,e^{-\Delta\omega^{2}\tau^{2}/2}, \qquad |\gamma(\tau)|^2 = e^{-\Delta\omega^{2}\tau^{2}}.

Why: the modulus removes the carrier phase $e^{i\omega_0\tau}$ , leaving a real Gaussian in $\tau$ of width $1/\Delta\omega$ — a narrow, long-coherence photon gives a broad, easily resolved dip.

Step 3. Substitute into the balanced-splitter result $P_c=\tfrac12(1-|\gamma|^2)$ :

P_c(\tau) = \tfrac12\left(1 - e^{-\Delta\omega^{2}\tau^{2}}\right).

Why: the measured curve — zero at $\tau=0$ , climbing to $\tfrac12$ past the coherence time, its full width at half depth $2\sqrt{\ln 2}/\Delta\omega$ reading the wavepacket duration off the counter.

Limiting cases

The coincidence probability depends on the splitter ratio ( $R=|r|^2$ , $T=|t|^2$ ) and the distinguishability (overlap $|\gamma|$ or delay $\tau$ ). The corners are what you reason with:

Case	Result	Meaning
Balanced splitter, identical photons ( $\tau=0$ )	$P_c=0$	Perfect HOM dip: full coalescence. The ideal result, requiring $R=T=\tfrac12$ and complete indistinguishability.
General ratio, identical photons	$P_c=(T-R)^2=(1-2R)^2$	Coincidences vanish only at $R=\tfrac12$ ; an unbalanced splitter never fully suppresses them, even with perfect photons.
Distinguishable photons (large $\tau$ )	$P_c=R^2+T^2\;\xrightarrow{R=T=1/2}\;\tfrac12$	No interference; photons behave independently and half the trials coincide — the baseline.
Partial indistinguishability, balanced	$P_c=\tfrac12(1-\|\gamma\|^2)$	Dip depth measures the squared overlap; visibility $V=\|\gamma\|^2$ is the standard indistinguishability metric.
Weak coherent (laser) input	$V_{\max}=\tfrac12$	Poissonian numbers let ports hold zero or two photons; the resulting coincidences cannot interfere away, capping visibility at one half.

Common trapHOM is just one photon interfering with itself, and a bright laser would show the same dip

Both halves are false. HOM is two-particle interference between two-photon amplitudes — it produces no first-order fringes, and each output detector alone sees a constant intensity as you scan $\tau$ . Nothing interferes with itself; what interferes are the "both reflected" and "both transmitted" histories of the pair. And a laser cannot reproduce it: the effect demands exactly one photon per port, a number state no classical field supplies. Attenuated coherent pulses have a Poissonian spread that lets ports hold zero or two photons, seeding coincidences no interference can cancel — visibility capped at $\tfrac12$ . A dip deeper than half is thus an unforgeable signature of single-photon physics. A separate trap: the dip depth reports indistinguishability, not alignment, so any spectral, temporal, or polarization mismatch fills it in even with perfect spatial overlap.

Worked examples

1 · How wide is the dip? (the real, concrete one)

Two single photons at $\lambda_0=810\ \mathrm{nm}$ are each passed through a $3.0\ \mathrm{nm}$ (FWHM) interference filter before meeting at a 50:50 splitter. One photon's path is scanned by a translation stage. Find the frequency bandwidth, the dip's full width in delay, and how far the stage must move to trace out that width.

Step 1. Convert to a frequency bandwidth with $\Delta\nu=c\,\Delta\lambda/\lambda_0^2=(2.998\times10^8)(3.0\times10^{-9})/(810\times10^{-9})^2 = \mathbf{1.37\times10^{12}\ Hz}$ , i.e. $1.37\ \mathrm{THz}$ — the FWHM of the measured power spectrum.

Step 2. Get the rms angular bandwidth. For a Gaussian $\mathrm{FWHM}=2\sqrt{2\ln2}\,\sigma$ , so $\sigma_\nu=\Delta\nu/2.355=5.82\times10^{11}\ \mathrm{Hz}$ and $\Delta\omega=2\pi\sigma_\nu=\mathbf{3.66\times10^{12}\ rad/s}$ .

Step 3. Apply equation (5): the dip is $\tfrac12 e^{-\Delta\omega^2\tau^2}$ , with half-depth where $e^{-\Delta\omega^2\tau^2}=\tfrac12$ , i.e. $\tau_{1/2}=\sqrt{\ln2}/\Delta\omega$ . So $\Delta\tau_{\mathrm{FWHM}}=2\sqrt{\ln2}/\Delta\omega = 2(0.833)/(3.66\times10^{12}) = \mathbf{4.55\times10^{-13}\ s\approx 455\ fs}$ .

Step 4. The stage introduces a delay $\tau=\Delta L/c$ , so the dip width in position is $\Delta L=c\,\Delta\tau_{\mathrm{FWHM}}=(2.998\times10^8)(4.55\times10^{-13})=\mathbf{136\ \mu m}$ . The effective time resolution ( $\sim\!455\ \mathrm{fs}$ ) is set by the photon bandwidth, not the detectors — whose jitter is a thousand times coarser. Δν ≈ 1.37 THz, dip FWHM ≈ 455 fs, traced over ≈ 136 μm of stage travel.

2 · Reading indistinguishability off the counts

A balanced HOM setup records $1000$ coincidences in a fixed integration time at large delay, falling to $60$ at zero delay. Find the visibility and the photon indistinguishability, and decide whether the source could be an attenuated laser.

Step 1. Visibility from equation (6): $V=(N_\infty-N_0)/N_\infty=(1000-60)/1000=\mathbf{0.94}$ .

Step 3. Attenuated coherent light caps visibility at $V_{\max}=\tfrac12$ . A measured $V=0.94$ sits far above that, so no weak-laser input can explain the data — it certifies genuine single photons, $94\%$ indistinguishable. V = 0.94, indistinguishability |γ|² = 0.94; well above the 0.5 classical ceiling, so these are true single photons.

3 · Why balance matters as much as purity

A fabrication error leaves a splitter at $R=0.40$ , $T=0.60$ instead of 50:50. If the photons are otherwise perfect ( $|\gamma|=1$ ), what dip depth and visibility result?

Step 1. Coincidence floor from equation (4): $P_c(0)=(T-R)^2=(0.60-0.40)^2=\mathbf{0.04}$ — perfect photons no longer reach zero, because the unequal amplitudes cannot fully cancel. Baseline: $P_c(\infty)=R^2+T^2=0.52$ .

Step 2. Visibility: $V=(0.52-0.04)/0.52=\mathbf{0.92}$ . A $10\%$ imbalance alone knocks $8\%$ off a nominally perfect dip — HOM visibility conflates photon quality with device quality, so characterize the splitter ratio before blaming the source. P_c(0) = 0.04, baseline 0.52, V = 0.92 despite perfect photons.

Physics of the portrait

The Hong–Ou–Mandel effect carries three names, but the deep signature is the third: the capstone of a career spent proving that light has features no classical wave can imitate. Mandel interfered two photons because he had spent two decades teaching the community how to count them.

Origins · 1 minLeonard Mandel and how this was discovered

Leonard Mandel

1927–2001 · Rochester quantum optics — photon statistics, antibunching, and the 1987 two-photon interference experiment

Leonard Mandel was born in Berlin in 1927, emigrated to England as a child, and took his doctorate at Birkbeck College, London, before turning to optical coherence. His long partnership with Emil Wolf at the University of Rochester produced a generation of experiments and the definitive treatise Optical Coherence and Quantum Optics (1995). Through the 1960s he pioneered the photon-counting statistics of light, sharpening the line between classical fields and true quantum states. In 1977, with Kimble and Dagenais, he observed photon antibunching in resonance fluorescence — the first light shown to have a manifestly nonclassical intensity correlation. The 1987 experiment with his students Chung Ki Hong and Zheyu Ou was the natural sequel: having shown that single photons exist, he sent two into a beam splitter and measured a coincidence dip whose subpicosecond width lay far below any detector's timing resolution — the interference itself was the clock. His instinct was always to find the observable that cleanly separates quantum from classical, and the HOM dip is arguably the cleanest in optics. He remained active at Rochester until his death in 2001.

History

1963

— Glauber formulates the quantum theory of optical coherence, giving photon correlations a rigorous language

1970

— Burnham and Weinberg observe correlated photon pairs from parametric down-conversion

1977

— Kimble, Dagenais and Mandel measure photon antibunching, light with no classical analog

1987

— Hong, Ou and Mandel discover the two-photon coincidence dip at a beam splitter

2002

— Santori and co-workers interfere indistinguishable photons emitted by a semiconductor quantum dot

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 6 · Mandel and Wolf, Optical Coherence and Quantum Optics, Ch. 12 and 22 · Gerry and Knight, Introductory Quantum Optics, Ch. 6 · Saleh and Teich, Fundamentals of Photonics, 2nd ed. · Grynberg, Aspect and Fabre, Introduction to Quantum Optics, Ch. 5.

Provenance. The beam-splitter transformation, the unitarity-forced $\pi/2$ phase, and the vanishing-coincidence result follow Loudon, Ch. 6. A verifier should recompute: (1) $t^2+r^2=0$ for $t=1/\sqrt2$ , $r=i/\sqrt2$ , giving $\tfrac{i}{\sqrt2}(|2_c,0_d\rangle+|0_c,2_d\rangle)$ ; (2) $P_c=R^2+T^2-2RT|\gamma|^2$ and its balanced reduction $\tfrac12(1-|\gamma|^2)$ ; (3) the Gaussian dip — with $\Delta\omega$ the rms of the power spectrum $|\gamma(\tau)|^2=e^{-\Delta\omega^2\tau^2}$ , whereas the field-amplitude rms inserts a $\tfrac12$ in the exponent; (4) $\Delta\nu=c\,\Delta\lambda/\lambda_0^2=1.37\ \mathrm{THz}$ , dip FWHM $\approx455\ \mathrm{fs}\approx136\ \mu\mathrm{m}$ , and $V=0.94$ , $0.92$ . The $50\%$ coherent-state ceiling and the zero-delay suppression are from Hong, Ou and Mandel, Phys. Rev. Lett. 59, 2044 (1987).

Prerequisites. Quantization of the Electromagnetic Field · Fock States and the Photon-Number Basis · Coherence Functions and the Degree of Coherence · Vacuum State and Zero-Point Energy. Leads to. Photon Antibunching · The Hanbury Brown–Twiss Experiment · Single-Photon Sources.

Ask the tutor: Ask me why the reflection phase $i$ turns "both reflected" into a minus sign, or to turn your coincidence counts into a photon indistinguishability.

Is the HOM dip the same thing as single-photon interference in a Mach–Zehnder interferometer?

No, and the distinction is the whole point. A Mach–Zehnder fringe is first-order: a single photon's amplitude splits, recombines, and the intensity at one detector rises and falls with phase. HOM is second-order: the individual output intensities stay flat as you scan the delay; what oscillates is the joint probability of both detectors firing. It needs two photons, the interfering objects being the amplitudes "both reflected" and "both transmitted." That is why HOM has no classical-field analog while a Mach–Zehnder fringe does.

Where exactly does the minus sign that cancels the coincidences come from?

From the unitarity of a lossless beam splitter. Energy conservation makes the scattering matrix unitary, and its off-diagonal condition $t^{*}r+r^{*}t=0$ requires $t$ and $r$ to differ in phase by $\pi/2$ . That means $r=it$ : reflection carries a factor $i$ , so "both reflected" picks up $i^2=-1$ relative to "both transmitted." For a balanced splitter the magnitudes are equal, so the amplitudes are $+\tfrac12$ and $-\tfrac12$ and sum to zero. The minus sign is not a modelling choice — it is a consequence of the device not absorbing light.

Do the two photons have to come from the same source?

No — and that is one of the most striking features. HOM works between photons from completely independent sources, even different lasers or distant emitters, provided they are made indistinguishable in every degree of freedom: central frequency, spectral and temporal profile, polarization, spatial mode. Independent-source HOM is exactly what quantum repeaters and networked computers rely on, since the point is to entangle qubits in separate modules. The requirement is indistinguishability, not common origin.

What physically fills in the dip as I increase the delay?

Which-path information — in the time domain. When the photons arrive together the two histories are indistinguishable and interfere destructively. Introduce a delay $\tau$ larger than the coherence time and the photons carry distinct timestamps; an experimenter could tell which history occurred by timing, and that information destroys the interference. Formally $|\gamma(\tau)|^2=e^{-\Delta\omega^2\tau^2}$ shrinks, so $P_c$ climbs toward the baseline $R^2+T^2$ . The dip is a direct picture of complementarity: interference exactly where no timing information exists.

Why can't a strong laser just be attenuated down to fake single photons for HOM?

Because attenuation changes the mean, not the statistics. A coherent state has Poissonian photon numbers, so even at very low intensity each port sometimes holds zero photons and sometimes two, not reliably one. Those extra photons make coincidences with no partner amplitude to interfere away, so the visibility cannot exceed $\tfrac12$ . This is why a measured visibility above $50\%$ is treated as proof of nonclassical, sub-Poissonian light — a boundary no classical field can cross.

How can the dip measure femtosecond delays when the detectors are a thousand times slower?

Because the timing resolution comes from the photons' coherence, not the electronics. The detectors only need to register two clicks in the same coarse window; the depth of the dip encodes the sub-coherence-time overlap. Scan the delay by a fraction of the coherence time — hundreds of femtoseconds here — and the coincidence rate changes measurably, because $|\gamma(\tau)|^2$ is steep there. You read a femtosecond delay off a slow count rate, using interference as an intrinsic clock.